prediction of impeller torque in high shear powder mixers

Chemical Engineering Science 56 (2001) 4457–4471www.elsevier.com/locate/ces

Prediction of impeller torque in high shear powder mixers

P. C. Knighta, J. P. K. Sevillea;b; ∗, A. B. Wellma, T. InstonecaSchool of Chemical Engineering, University of Birmingham, Birmingham B15 2TT, UK

bIRC in Materials for High Performance Applications, University of Birmingham, Birmingham B15 2TT, UKcUnilever, Quarry Road East, Bebington, Wirral L63 3JW, UK

Received 28 February 2000; received in revised form 16 March 2001; accepted 26 March 2001

Abstract

An investigation was made of the factors that determine the impeller torque of vertical axis high speed mixers containing granularsolids of low cohesion, the experimental material being sand. The diameter of the mixer bowls, which were constructed of stainlesssteel, ranged from 0.13 to 0:30 m. Disc impellers with both smooth and grooved surfaces were used. Two and three blade 7atimpellers were used with heights in the range 3–12 mm and bevel angles ranging from 11

◦to 90

◦. The study was supported by

the application of positron emission particle tracking (PEPT) to investigate the 7ow of the material in the mixer. A dimensionalanalysis was made of the data. The e:ects of the mass of powder, M , and the bowl radius, R, could be satisfactorily representedby the dimensionless torque group, T=MgR. In the case of disc impellers, the dimensionless torque was independent of impellerrotational speed. For the blade impellers, the dimensionless torque was found to be a function of the impeller Froude number anda dimensionless blade height. A powder mechanics analysis was made of the 7ow of material in the mixer ;tted with both discand blade impellers. The 7ow of the powder was modelled as ‘rigid’ body rotation and both frictional and inertial interactionswith the impeller were accounted for. The analysis provides a ;rst order representation of the e:ects of scale, mass ;ll, impellerrotational speed, blade height and blade bevel angle on the torque. The assumptions made in the model are critically discussed. ?2001 Elsevier Science Ltd. All rights reserved.

Keywords: Particle technology; Mixing; PEPT

1. Introduction

This paper is concerned with the torque required to turnthe impeller in a powder mixer. Although this is arguablya basic requirement for mixer design, little guidance onit can be found in the literature. Mixer designs can bedivided into those in which the axis of rotation is verticaland those in which it is horizontal. The research reportedhere was concerned with the ;rst of these types of mixer,which comprises a bowl and a centrally mounted impellerrotating about a vertical axis. The tip speed of the impellerin these mixers is typically of the order of 10 m=s, andfor this reason they are sometimes known as high speedor high shear mixers. The speed of rotation employeddepends on both the size of the mixer and the application.For example, in the case of a small-scale unit with a bowl

∗ Corresponding author. School of Chemical Engineering, Uni-versity of Birmingham, Birmingham B15 2TT, UK. Tel.:+44-121-414-5322; fax: +44-121-414-5377.E-mail address: [email protected] (J. P. K. Seville).

diameter of 0:25 m, the rotation speed typically rangesfrom 100 to 1000 (or more) revolutions per minute (rpm).With a large-scale unit of bowl diameter 2 m, the rotationspeed typically ranges from 20 to 100 rpm. A major useof these mixers is in the agglomeration of powders by ad-dition of liquid binders, for example in the manufactureof granular products in the ;ne chemical, pharmaceuti-cal, foodstu:, household product, metallurgical and mi-crobiological product industries. This type of equipmentis also used intimately to mix cohesive powders and todistribute highly viscous liquids and pastes into powders.The same class of mixer is employed in the compoundingof polymers, by mixing solid pellets and utilising highmixing speeds and consequently high power input to meltand mix the polymer. The mixers give good containmentof dusts and aerosols and are more easily cleaned thanthe horizontal axis types of mixer. A survey of centrifu-gal mixers has been reported by Le Lan (Le Lan, 1978).A number of factors are important in the design of

these mixers. Besides aspects regarding mechanical

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4458 P. C. Knight et al. / Chemical Engineering Science 56 (2001) 4457–4471

reliability, a factor of major importance is the design ofthe impeller blade. There are considerable di:erences inthe designs of impeller blades used by di:erent manufac-turers. In agglomeration processes, the swept volume ofthe blade and the design of the tip are known to be im-portant (SchGfer, 1996). Control of granule size, shapeand size distribution in agglomeration is problematicaland size distributions are often bi-modal in character(Knight, Instone, Pearson, & Hounslow, 1998). There istherefore a de;nite need to improve the performance forthese applications. Fundamental progress has been inhib-ited by a lack of understanding of how the motion of thepowder within the mixer is induced by the impeller. Thismotion has not been characterised experimentally. It isalso the case that no models have been published thatpredict the powder motion from the mixer design andoperating parameters. Consequently, little fundamentalprogress has been made on the design of impeller bladesto give improved performance.In this work, we have addressed these issues by in-

vestigating, both experimentally and theoretically, howthe torque required to turn the impeller varies with im-peller blade design, rotational speed, ;ll and bowl size.The torque (or equivalently the power consumption) isused to monitor and control agglomeration processes (e.g.Leuenberger, Luy, & Studer, 1990) and it is thereforeimportant to understand the factors that determine it. Itis also sensitive to changes in operating conditions andcan therefore be used as a suitable parameter to probemixer design. Moreover, it can be measured rapidly andwith reasonable precision. Recently, positron emissionparticle tracking (PEPT) has been applied to characterisepowder motion in a number of types of mixer (Forster,Seville, Parker, & Ding, 2000). PEPT results for the mo-tion of powder in vertical axis mixers were utilised in thisinvestigation.There have been a number of studies of powder motion

in horizontal ribbon blade and ploughshare type mix-ers (Rumpf & MJuller, 1962; MJuller, 1982; Broadbent,Bridgwater, Parker, Keningley, & Knight 1993; Bridg-water, Broadbent, & Parker, 1993). There have also beenexperimental and theoretical investigations of powdermotion in vertical axis ribbon mixers (Reichert, Vock,Kolk, & Sinn, 1978; Derrenbecher, Faulhaber, Kurz, &Sartor, 1984; Cooker & Nedderman, 1987a; Cooker &Nedderman, 1987b) and, recently, of the vortex motion inenclosed screw conveyors (Roberts, 1999). High speed,vertical axis mixers have received little theoretical atten-tion. A study was reported of a vertical axis device formeasurement of dynamic friction (Rademacher, 1978)and a torque balance model was proposed for rotary mo-tion in a spouted bed with tangential gas inlets (Evans,Seville, & Clift, 1989). As far as we are aware, there areno published studies speci;cally on powder motion invertical axis mixers, or on the prediction of the impellertorque. There have been a number of studies of the force

required to move a single blade linearly through powdercontained within a trough (Bagster & Bridgwater, 1967;Bagster, 1969; Bagster & Bridgwater, 1970; MJuller,1982). The measurements by Bagster were made at ve-locities up to about 0:25 m s−1 and were thus concernedwith low velocity, frictional, behaviour. The measure-ments reported by MJuller were made at velocities up toabout 3 m s−1 and thus encompassed inertial as well asfrictional forces. It has been proposed (MJuller, 1982;Satoh & Terashita, 1983) that when inertial forces areimportant, the force to move a blade can be obtainedby addition of two terms: a frictional term which is in-dependent of velocity and an inertial term which varieswith the square of the velocity. Below, we build on theseideas to develop a model for the motion of powder andprediction of torque in vertical axis high-speed mixers.

2. Experimental

A mixer was designed and constructed in which boththe bowls and impellers could be changed (Wellm, 1997).Four bowls, having diameters of 0.13, 0.16, 0.21 and0:3 m, were employed. They were cylindrical in form,with 7at bases and vertical side walls (Fig. 1). The bowlwas mounted on bearings, which supported it and allowedit to rotate freely in a horizontal plane. Rotation was pre-vented by a load arm ;xed to the bowl and which restedagainst a load cell mounted on the frame of the appara-tus; the torque was computed from the measured load.The drive shaft projected into the centre of the bowl andwas mounted in bearings. Impeller blades of various de-signs, and also disc impellers, could be inter-changeablyattached to the drive shaft. Impeller blades and discs weremanufactured to ;t the bowls with a tip clearance in therange 0.5–1 mm. The clearance between the underside ofthe blades and the base of the bowls was approximately1 mm. The shaft was driven by a 7:5 kW three phase DCdrive electric motor, whose speed could be altered con-tinuously in the range 30–1100 rpm. The speed could becontrolled to within ±5 rpm. Signals for the torque andthe speed were sampled every 0.5–1 Hz by a data log-ger. For each impeller speed, the torque was measuredfor 30 s and an average value calculated. Torque valueswere subject to errors caused by variation in friction inthe drive shaft bearings. The friction changed with timeof operation, making it diMcult to make a precise esti-mate of the magnitude of the error. Individual values oftorque are estimated to be accurate to ±10%. However,low values of torque are subject to greater uncertainty.Both two blade and three blade impellers were used.

All the blades were made to a constant width in the hori-zontal plane of 50 mm. Three depths of blade were used:3, 10 and 12 mm. Blades with both rectangular (90◦) andbevelled (11◦ and 17◦) cross-section were used (bevelledside up). The front 1 mm of the bevelled blades was left

P. C. Knight et al. / Chemical Engineering Science 56 (2001) 4457–4471 4459

Fig. 1. Details of the mixer. (a) General construction, (b) Cross-section of bowl, showing co-ordinates, the actual powder pro;le and the modelpro;le and the toroidal motion of the powder, (c) Section of bowl and impeller blade in the z– surface, viewed in a radial direction. Themotion of the impeller blade is shown relative to that of the powder. A 90◦ bevel angle blade is shown.

vertical, so that the blades were not sharp. Two types ofdisc impeller were used: one had a smooth surface andthe other had a surface with grooves, formed by machin-ing, 1 mm deep at a separation of 2 mm, in a spiral patternemanating from the centre of the disc. The blades andbowls were ;nished with a machined, but not a polished,surface.A limited number of measurements of the motion of

the powder within the mixer were made using the positronemission particle tracking (PEPT) technique (Wellm,1997). Measurements were made with the 0:21 m bowl,;tted with smooth and corrugated disc blades and three-blade impellers of rectangular or bevelled cross-section.Sand of size range 180–250 �m was used as the ex-perimental material and glass spheres of diameter 1.5and 2:0 mm were used as tracers. Measurements car-ried out since this investigation with a new and moresensitive camera, that allows the use of smaller tracers,have shown that coarse size tracers are not subject tosigni;cant segregation. The output of the technique wasa matrix of the co-ordinates of the tracer particle as afunction of time. The data were processed to give thefractional occupancy and mean velocity of the traceras a function of position within the bowl. Additionally,Fourier analysis was employed to determine the mostfrequently occurring angular velocity.Most torque measurements were made with a sand of

size range 180–250 �m and bulk density approximately1200 kg m−3. To investigate the sensitivity of the results

Fig. 2. Dependence of impeller torque on rotational speed for asmooth disc impeller and di:erent mass ;lls. Bowl diameter 0:21 m,180–250 �m sand.

to the particle size range of the material, measurementswere made also with sands of size range 90–180 and500–710 �m.

3. Results

3.1. Disc impellers

Fig. 2 shows torque data for 180–250 �m sand in the0:21 m bowl ;tted with the disc impeller having a smooth


Fig. 3. Comparison of 2 and 3 blade impellers. Dependence of impellertorque on rotational speed for 10 mm high, 90◦ bevel angle impellerblades. Bowl diameter 0:21 m, mass ;ll 3 kg, 180–250 �m sand.

surface. For a given mass ;ll, the torque values were in-dependent of impeller speed within the range investigated(100–1100 rpm). Within experimental uncertainties, themagnitude of the torque increased approximately linearlywith mass ;ll. Torque data obtained with 0.13 and 0:16 mbowls were also independent of impeller rotational speedand were proportional to mass ;ll.Results (not shown) were also obtained with the disc

impeller having a grooved surface. The torque values alsovaried in proportion to the powder mass and were about20% larger in magnitude than those obtained with thesmooth disc.The powder motion could be seen visually to have a pri-

mary rotational motion in the circumferential direction. Asecondary, toroidal, motion could also be seen, illustratedin Fig. 1b. PEPTmeasurements were carried out using thesmooth disc, at disc rotational speeds of 200 and 400 rpmand with a ;ll of 4 kg of the sand. The angular velocityof the circumferential motion of the tracer particle wasobtained by Fourier analysis. The same frequency dis-tribution was obtained for both disc speeds and showeda single, sharp, peak. This is evidence that the mass ofmaterial rotated as ‘rigid’ body at a constant angular ve-locity that was independent of disc speed in the range200–400 rpm. The frequency of the peak was 0:85 s−1,equivalent to a rotational speed of approximately 70 rpm.That the motion of the powder was not signi;cantly af-fected by changes in impeller speed is consistent witha predominantly frictional interaction between disc andpowder.

3.2. Rectangular, 90◦ impellers

Fig. 3 shows a comparison of the torque values fortwo- and three-blade rectangular 90◦ impellers in a 0:21 mbowl for a mass ;ll of 3 kg of 180–250 �m sand. Torquevalues increased monotonically with increasing impellerspeed and, at speeds above 100 rpm, exceeded the values

Fig. 4. Dependence of impeller torque on rotational speed for a 10 mmhigh, 90◦ bevel angle, three-blade impeller for di:erent mass ;lls.Bowl diameter 0:13 m, 180–250 �m sand.


with the disc impeller with the same mass ;ll. The mag-nitude of the torque with the three-blade impeller waslarger than that with the two-blade impeller. The di:er-ence, however, was only about 25% instead of 50%, asmight have been expected had the torque been propor-tional to the number of blades. The data below refer tothree blade impellers.Figs. 4–6 show the variation of torque with rotational

speed as a function of mass ;ll for bowls of 0.13, 0.21and 0:30 m diameter, respectively, ;tted with 10 mm, 90◦,three-blade impellers. The capacities of the largest andsmallest bowls di:er by more than an order of magni-tude. In all the results, the torque increased signi;cantlywith the mass ;ll. With the 0:13 m bowl, the dependenceof torque on impeller speed was, within experimental er-rors, linear (Fig. 4). With the 0.21 and 0:30 m bowls(Figs. 5 and 6), the dependence of torque on impellerspeed displayed s-shaped character. This is more markedat high ;ll levels. The reasons for these di:erences arediscussed further below.



Fig. 7. Dependence of impeller torque on rotational speed for sands ofdi:erent size fraction, with a 10 mm high, 90◦ bevel angle, three-bladeimpeller. Bowl diameter 0:21 m, mass ;ll 4 kg, (•) 500–710 �m, (?)180–250 �m, (�) 90–180 �m.

The data presented above are for 180–250 �m sand.Fig. 7 shows a comparison of torque values obtained with90–180 and 500–710 �m sands. The data were obtainedwith a mass ;ll of 4 kg in a bowl of diameter 0:21 m ;t-ted with a 10 mm deep, 90◦, three-blade impeller. Thedata for the 500–710 �m sand were rather inconsistentand gave some high torque values at low impeller speeds.Noticeable comminution of this size fraction of sand wasobserved. The particle size is of the same order of mag-nitude as the gap between the underside of the impellerand the base of the bowl and also the gap between thetip of the impeller and the wall of the bowl. Trappingof particles between the blade and the bowl is thereforeto be expected, giving spurious results. Aside from theseobservations, the insensitivity of the torque to the parti-cle size of the sands suggests that mechanical 7uidisatione:ects were not signi;cant.The PEPT technique was applied to the motion of 4 kg

of the sand in the 0:21 m bowl ;tted with a 10 mm, 90◦,

Fig. 8. Dependence of impeller torque on rotational speed forthree-blade impellers of di:erent heights and bevel angles and fora smooth disc impeller. Bowl diameter 0:16 m, mass ;ll 3 kg,180–250 �m sand.

three-blade impeller, at a rotational speed of 400 rpm.The results con;rmed qualitatively the observation madevisually of a primary rotational motion in a circumferen-tial direction and also of a secondary, toroidal, motion.Fourier analyses were carried out to obtain the frequencydistribution of the angular velocity of the circumferen-tial motion of the tracer particle. In contrast to the sharppeak obtained with the disc impeller, the distribution ob-tained was broad, in the range 0.2–2 s−1, correspondingto rotational speeds in the range 30–300 rpm.

3.3. The e8ect of impeller height and bevel

Fig. 8 shows the e:ect on torque of using impellerblades of di:erent thickness and shape, for a 0:16 m bowland a mass ;ll of 3 kg of 180–250 �m sand. Data pointsfor a smooth disc are also included for comparison pur-poses. The magnitude of the torque was signi;cantlylarger with a 12 mm blade than with a 3 mm blade, butthe increase in torque was less than would have been ex-pected from a linear dependence on blade thickness.With bevelled impeller blades, the magnitude of the

torque was markedly lower than that with a 90◦ blade ofsimilar thickness. The shape of the curves is also verydi:erent. With the 11◦ blade, the torque increased to amaximum at an impeller speed of about 500 rpm and, atimpeller speeds above 700 rpm, the torque fell to a magni-tude similar to that obtained with a smooth disc impeller.With the 17◦ blade, the torque also reached a maximum atabout 500 rpm but declined in magnitude only marginallyat higher speeds. Such behaviour at ;rst appears diMcultto account for, but can be given satisfactory theoreticalinterpretation, as discussed later.


Fig. 9. Dependence of dimensionless torque on impeller rotationalspeed for a smooth disc impeller and di:erent mass ;lls. Bowl diam-eter 0:21 m, 180–250 �m sand. Line is ;tted from powder mechanicstheory with � = 0:75.

4. Dimensional analysis

4.1. Disc impellers

The observed dependence of the disc torque on themass ;ll of material suggests that the torque, T , can beexpressed in the dimensionless form T=MgR, where M isthe powder mass and R is the radius of the mixer bowl.This group is a form of Newton number. Fig. 9 showsthe data from Fig. 2, for a smooth disc in a 0:21 m bowl,expressed in this way, from which it can be seen thatthis is a satisfactory representation. The magnitude of thedimensionless torque was found to be 0.58. In the caseof the grooved disc (results not shown), the magnitudeof the dimensionless torque was found to be 0.68.

4.2. Three-blade, 90◦, impellers

The approximately linear dependence of the impellertorque on the mass ;ll of material, observed with bowls ofdi:ering diameters (Figs. 4–6), again suggests the use ofthe group T=MgR to represent the dimensionless torque.With blade impellers, however, there is also a pronounceddependence of torque on the rotational speed of the im-peller and the torque can be expected to be a functionalso of the impeller height. These types of mixer inducecentripetal forces within the material, which vary as !2R.Thus, for a given impeller rotation speed and for geomet-rically similar impellers, the magnitude of the centripetalforce increases with the radius of the bowl. We thereforesuggest representing the rotational speed of the impellerby a Froude number, FrI=!2

I R=g. However, because thetorque shows a dependence on impeller rotational speedthat is more nearly linear than quadratic, it is appropri-ate to plot the dimensionless torque as a function of thesquare root of FrI .Figs. 10–12 show data, respectively, for 0.13, 0.21 and

0:30 m diameter bowls, ;tted with 90◦ impeller blades.

Fig. 10. Dependence of dimensionless torque on impeller rotationalspeed for a 10 mm high, 90◦ bevel angle, three-blade impeller. Bowldiameter 0:13 m, 180–250 �m sand. Curve is ;tted from powdermechanics theory with �=0:75; �=45◦; �=1=2, F1 =1:2; F2 =3:5.

Fig. 11. Dependence of dimensionless torque on impeller rotationalspeed for 3 and 10 mm high, 90◦ bevel angle, three-blade impellers.The notation of the symbols is the same as that in Fig. 5. Bowldiameter 0:21 m, 180–250 �m sand. Curves are ;tted from powdermechanics theory with �=0:75; �=45◦; �=1=2, F1 =1:2; F2 =3:5.

Fig. 12. Dependence of dimensionless torque on impeller rotationalspeed for a 10 mm high, 90◦ bevel angle, three-blade impeller. Thenotation of the symbols is the same as that in Fig. 6. Bowl diameter0:30 m, 180–250 �m sand. Curve is ;tted from powder mechanicstheory with � = 0:75, �= 45◦, �= 1=2, F1 = 1:2, F2 = 3:5.


The curves shown in the Figures are theoretical, fromthe powder mechanics analysis described below. Fig. 11includes data for a 3 mm high impeller blade, as well asthat for a 10 mm high impeller blade. The increase indegree of s-shaped character of the curves with increasein bowl diameter, noted above, can be accounted for bythe rise in the maximum value of the impeller Froudenumber with increase in bowl diameter. Up to a valueof Fr1=2I of 9, an approximately linear dependence wasobserved with the three sizes of bowl.Data for impellers with geometrically similar dimen-

sions should, according to these arguments, fall on acommon curve when plotted in the form of dimension-less torque against the square root of the impeller Froudenumber. The present data cannot be directly tested inthis way because the impellers were not geometricallysimilar—neither the height nor the width of the impellerswas scaled in proportion to the bowl radius. The heightof the impeller blade can be seen to have a signi;cant ef-fect on the magnitude of the torque. Hence it is necessaryto consider further the e:ect of blade height. Any e:ectof blade width will not be accounted for.Even a very thin and narrow impeller blade (i.e. a very

thin wire) would require an appreciable force to moveit through granular powder. This is because particles incontact with, and close to, the blade are displaced byit. The thickness of the shear zone near to an immersedblade or wire has apparently not been considered in thepublished literature. A thickness of the shear zone of ;veparticle diameters at a wall in slow moving powder hasbeen reported (Nedderman & Laohakul, 1980). We willassume an e:ective shear zone thickness of ;ve particlediameters above and below the blade. The e:ective bladeheight, he: is then given by

he: = h+ 10d; (1)

where h is the height of the impeller blade and d is themean particle diameter. In the case of 180–250 �m par-ticles, he: was taken to be equal to (h+ 2) mm.The data shown in Figs. 10–12 suggest that, for a

;rst-order analysis, the dimensionless torque, T̂ , can berelated to the impeller Froude number by an expressionof the form

T̂ = T̂ 0 + kFr0:5I where k = �(he:R

)a; (2)

which is linear in impeller speed. The ;rst term on theright hand side of Eq. (2) is the extrapolated interceptvalue of dimensionless torque at an impeller speed ofzero. The gradient, k, was determined for each set of datafor the four sizes of bowl by least squares analysis. Theexponent � was determined from a double logarithmicplot of k against (he: =R), shown in Fig. 13, and wasapproximately 0.5.

Fig. 13. Double logarithmic plot of the parameter k in Eq. (2)versus the dimensionless e:ective impeller height. Data for bowls ofdiameters from 0.13 to 0:30 m, 180–250 �m sand.

Fig. 14. Dependence of dimensionless torque on impeller rotationalspeed for a 10 mm high, 11◦ bevel angle, two-blade impeller. Bowldiameter 0:16 m, 180–250 �m sand. Continuous curve is ;tted fromtheory with � = 0:75, �= 11◦, �= 1=2, F1 = 1:2, F2 = 3:5.

4.3. Three-blade, 11◦ and 17◦ impellers

The data for the bevelled three-blade impellers cannotbe represented by Eq. (2) because the increase in torquewith rotational speed is not monotonic (see Fig. 8). How-ever, use of the dimensionless torque provides a reason-ably satisfactory reduction of the data, as illustrated inFig. 14, which shows data for a bowl of diameter 0:16 m;tted with a three-blade, 11◦ impeller. A theoretical in-terpretation of the dependence of dimensionless torqueon impeller speed is described below.

5. Powder mechanics analysis

5.1. Aims and assumptions

The following simple powder mechanics analysis aimsto describe the e:ects of scale, mass ;ll and design of theimpeller blade on the torque. Expressions are derived for(i) the torque exerted by the rotating mass of powder onthe bowl and (ii) the torque required to rotate the impeller


within the powder. At steady-state, these two torque val-ues must be equal (Newton’s ;rst law). The two equa-tions contain two unknowns: the torque and the rotationalspeed of the mass of powder and can thus be solved. Thepowder is modelled as a cohesionless Coulomb material,with stress-independent coeMcients of internal and wallfriction. The powder is taken not to be aerated and thebulk density of the material is taken to be constant. Airdrag forces at the surface of the powder are neglected.As stated earlier, the disc and blade impellers produce

movement of powder in both circumferential and radialdirections, giving a toroidal motion. Both of these move-ments produce a stress normal to the wall and thus ow-ing to wall friction contribute to the torque generated bythe moving mass of powder. However, for the analysis,the contribution to the wall stress arising from the radialmotion is not included. The powder is taken to rotate cir-cumferentially at a constant angular velocity !P (as forforced vortex 7ow). The volume occupied by the impellerblades is neglected.The toroidal motion a:ects the shape of the cross-

sectional pro;le in the r–z plane. At the side wall, theradial movement is converted to movement in a verticaldirection up the wall. At the top surface, powder slidestowards the centre, where it falls and is again swept radi-ally towards the side wall by the motion of the impeller.The motion produces the cross-sectional pro;le shownschematically in Fig. 1b. For the purposes of the model,the observed shape is simpli;ed and is represented by arectangular cross-sectional pro;le of uniform height H ,inner radius r0 and outer radius R.

For the analysis of the torque required to rotate adisc impeller, the powder is considered to rotate as a‘rigid’ body at a constant angular velocity. The interac-tion between the disc impeller and the powder is takento be solely frictional in nature. For the analysis of thetorque required to rotate a blade impeller, the materialis no longer assumed to be rigid. As the impeller movesthrough it at a relative angular velocity (!I − !P), it isdisplaced upwards as shown in Fig. 1c. Both Coulombicand inertial contributions to the force required to producethis upward displacement are considered. For simplicityof expression, the radius of the disc or the impeller istaken to be equal to the radius of the bowl.

5.2. Analysis of the torque exerted by the powder onthe bowl

Cylindrical co-ordinates are used, with the verticalco-ordinate, z, measured downwards from the top surfaceof the powder (Fig. 1). The torque, TB;s, exerted by therotating powder on the side wall of the bowl is given by

TB;s = �w∫ H

02�R2 dz: (3)

The torque, TB;b, exerted by the rotating powder on thebase of the bowl (which is obviously absent with discimpellers) is given by

TB;b =∫ R

r0�w2�r2

(1− nw

2�R

)dr: (4)

The term in parentheses is an approximate correction forthe fractional area of the bowl that is occupied by the im-peller blades and where friction cannot act. For simplic-ity, the correction is based on the fractional length of thebowl perimeter occupied by the impeller blades. This isgiven by (nw=2�R), where n is the number of blades ofwidth w. An exact (and more complex) correction factorfor blades of constant width has been derived elsewhere(Wellm, 1997). However, the torque on the base is usu-ally much smaller in magnitude than that on the side walland hence the use of an approximate correction factor in-troduces negligible error in prediction. In Eqs. (3) and(4), �w is the shear stress at the side wall or the base of thebowl, which is related to the normal stress, N , througha bowl wall friction coeMcient, �w;B, by

�w = �w;B N : (5)

The problem is thus to compute the normal stress at theside wall and base of the bowl. An approximate form ofEuler’s equation for the radial direction applied to ‘rigid’body rotation of the powder is as follows:

@ rr@r

= #!2Pr; (6)

where r is the radius and # is the bulk density of the7owing powder. Similarly, in the vertical direction, anapproximate form for ‘rigid’ body rotation is

@ zz@z

= #g: (7)

At low rotational velocities, the e:ects of gravity willbe larger than those from the centrifugal motion and theFroude number for the rotation of the powder (de;nedby FrP =!2

P R=g, in which !P is the angular velocity ofthe powder) will be small. In this case an analysis resem-bling the Janssen analysis (see, for example, Nedderman,1992) can be made with principal stresses oriented ap-proximately vertical and horizontal, and with the powderin the active state. The centripetal term may be taken toaugment the component of the horizontal stress arisingfrom the gravitational force. Analyses of this type weremade by Cooker and Nedderman (Cooker & Nedderman,1987a) and recently by Roberts (Roberts, 1999). How-ever, the present analysis is concerned with more rapidpowder rotation in which FrP has an order of magnitudeof unity and in which the powder has a cross-sectionalpro;le like that shown in Fig. 1b. Eq. (6) may then beintegrated subject to the boundary condition that at the


inner radius of the powder, r0, the radial stress is zero.It follows that the normal stress, N;s, at the side wall(r = R) is given by

N;s = 12#!

2P(R

2 − r20): (8)

Substitution of Eqs. (5) and (8) in Eq. (3) and integrationgives an expression for the torque acting on the side wallof the bowl:

TB;s = ��w;B#!2P(R

2 − r20)R2H: (9)

Now the mass of the powder is given by

M = #�(R2 − r20)H: (10)

Hence the torque on the side wall, produced by centrifugalforce, can be expressed as

TB;s =MR�w;B:!2PR: (11)

This can be written in non-dimensional form

T̂ B; s =TB;sMgR

= �w;B!2pRg

: (12)

As noted already, the last group is the Froude number,FrP, for the rotation of the powder.The analysis thus far applies to both disc and blade

impellers. However for blade impellers we also need toconsider the torque on the base of the bowl. Integrationof Eq. (7) gives for the normal stress on the base of thebowl:

N;b = #gH (13)

Substitution of Eqs. (5) and (13) into Eq. (4) gives onintegration

TB;b =23��w;B#gH (R3 − r30)

(1− nw

2�R

): (14)

Using Eq. (10), Eq. (14) can be written in non-dimensionalform

T̂ B;b =TB;bMgR

=23�w;B

R3 − r30R(R2 − r20)

(1− nw

2�R

): (15)

Evaluating the second term on the right-hand side ofEq. (15) for the cases of r0 =0, r0 =1=2R, r0 =3=4R, thefollowing values are found:

R3 − r30R(R2 − r20)

= 1;76; 1:32: (16)

In normal operation, with a typical ;ll, r0 is observed to beapproximately 1=2R. It is evident that if the value of 7=6is adopted for the magnitude of this term, correspondingto r0 = 1=2R, the maximum error in the torque is likelyto be less than 15%, which is acceptable in this simplemodel. The dimensionless torque on the base of the bowlthen becomes

T̂ B;b =TB;bMgR

=79�w;B

(1− nw

2�R

): (17)

5.3. Analysis of the torque exerted on the powder by adisc impeller

The dimensionless torque required to rotate a disc im-peller is the same as that for friction on the base of thebowl with an impeller blade mixer, but without the cor-rection term for the area covered by the blades. It is givenfrom Eq. (17) by

T̂ D =79�w;D; (18)

where �w;D is the friction coeMcient for the powdercontacting the disc. A torque balance may now be made,assuming that the disc is rotating fast enough for slip tooccur between the disc and the powder (otherwise thepowder will rotate at the same speed as the disc, asdiscussed later):

T̂ D = T̂ B; s: (19)

After substitution using Eqs. (12) and (18) and rearrang-ing, the Froude number, FrP, for the powder rotation isobtained:

FrP =!2PRg

=79�w;D�w;B

: (20)

If the coeMcients of friction are taken to be the same forthe disc and the wall, as is reasonable in the present workwith the smooth disc, since the materials of constructionand ;nishes were the same, the powder Froude numberis found to be independent of the interface friction coef-;cient and to have a value of 7

9 . The torque can be cal-culated from Eq. (18) and requires a value for the wallfriction coeMcient.

5.4. Analysis of the torque exerted on the powder by ablade impeller

One of the principal di:erences between the case ofthe disc impeller and the blade impeller is that in the lat-ter case, as the blade moves through the powder, the bedis deformed as shown schematically in Fig. 1c. The forceto move the blade is composed of two terms, assumed tobe independent and additive: a Coulombic term, whichis independent of speed, and an inertial, speed-dependentterm, to account for acceleration imparted to the powderby the movement of the impeller blade. The Coulombicterm accounts for the frictional forces in the displacementof powder over the blade. It is a function of both the in-ternal friction coeMcient of the powder and the frictioncoeMcient of the powder against the blade surface. Ap-plication of Coulomb’s method of wedges with surcharge(Nedderman, 1992) suggests the use of the following ex-pression for the force, dFI;f to move an immersed bladeof length dr:

dFI;f = K2#gHhe: dr: (21)


The parameter K2 is a proportionality coeMcient for pas-sive failure and he: is the e:ective height of the impellerblade (Eq. (1)). Hence the torque to rotate the impellerconsisting of n blades is given by

TI;f = nK2#gHhe:∫ R

r0r dr: (22)

Hence

TI;f = 12nK2#gHhe: (R2 − r20): (23)

Substitution using Eq. (10) and rearranging into anon-dimensional form gives

T̂ I;f =TI;fMgR

= F1nhe:R; (24)

where F1 is a proportionality coeMcient to be determinedby experiment.Consider next the inertial term. As the blade moves

through the powder at a relative velocity (!I − !P)r,powder is de7ected through an angle �, creating a reac-tion force (Fig. 1c). Because of the frictional nature ofthe material, a large proportion of the total mass of pow-der is de7ected upwards by the impeller blade. The massof powder de7ected per unit length of blade is thus ap-proximately proportional to the depth of the powder. Notethat this is di:erent from the behaviour of a 7uid. In thecase of a 7uid the mass de7ected is normally taken to beproportional to the height of the blade. This is becausethe 7uid is taken to be frictionless and continuity is sat-is;ed by the 7uid accelerating over the blade. In the caseof 90◦ impeller blades, it is reasonable to take � = 45◦

for continuously sheared powder, taking the directions ofprincipal stress and strain to be coincident and in the direction. In the case of bevelled blades of angle less than45◦, the angle � can be equated with the blade angle.The force to move a blade of length dr is given by the

rate of change of momentum, dFI; i, in the r surface:

dFI; i = E#H (!I −!P)2r2(1− cos �) dr: (25)

An eMciency factor, E, has been included, which is dis-cussed below. The torque to rotate the impeller with nblades is given by

TI; i = nE#H (!I −!P)2(1− cos �)∫ R

r0r3 dr; (26)

Hence

TI; i = 14nE#H (!I −!P)2(1− cos �)(R4 − r40): (27)

After rearrangement and substitution using Eq. (10), thefollowing non-dimensional equation is obtained:

T̂ I; i =TI; iMgR

=nE4�

(!I −!P)2Rg

(1− cos �)R2 + r20R2 : (28)

The second term in Eq. (28) is the Froude number for therelative motion of the impeller and the powder, Fr−1=2

I−P .The last term in R and r0 takes the values 1.0, 1.25, 1.56for the cases of r0=0, r0=1=2R, r0=3=4R. For simplicityof analysis, it is convenient to take the magnitude of thelast term to be constant at its value when r0=1=2R. Smallvalues of r0 occur when the powder rotational speed islow and hence when the magnitude of the inertial termis small compared with that of the frictional term. Inthis case the error in the total torque is small. At highpowder rotational speeds r0 is greater than 1=2R and thesimpli;cation causes signi;cant underestimation of thetorque, e.g. by about 25%.An eMciency factor, E, is required to account for two

e:ects. Firstly, the de7ection of the powder, and hencethe torque, is dependent to some extent on the height ofthe blade in relation to the size of the bowl. This e:ect canbe incorporated by taking the eMciency to depend on thedimensionless term (he: =R)�. Secondly, the blade movesthrough powder which has been disturbed and lifted bythe blade “in front”. We propose that, over the rotationalspeed range of the data, the eMciency factor should varyinversely with the frequency of the disturbance, i.e. in-versely with both the number of blades and the rotationalspeed of the impeller relative to the powder. Since bothgravity and centrifugal forces are involved, the e:ect ofrelative rotational speed on eMciency can be taken tobe proportional to the inverse square root of the Froudenumber for the relative motion of the impeller and thepowder, Fr−1=2

I−P . Eq. (28) then becomes

T̂ I; i =TI; iMgR

= F2(!I −!P)R1=2

g1=2

(he:R

)�(1− cos �);

(29)

where F2 is a dimensionless proportionality coeMcient tobe determined by experiment.A torque balance may now be made between the torque

generated by the impeller and the torque applied to thebowl under steady-state conditions:

T̂ I;f + T̂ I; i = T̂ B;b + T̂ B; s: (30)

Substituting for the dimensionless torques in Eq. (30),using Eqs. (12), (17), (24), and (29), gives a quadraticequation from which the powder angular velocity, !P,can be determined as a function of the impeller angularvelocity, !I :

!P =12D

(−B+√B2 + 4D(!IB− (C − A)); (31)

where

A= F1nhe:R; B= F2

(Rg

)1=2 (he:R

)�(1− cos �);

C =79�w

(1− nw

2�R

); D= �w

Rg:


At high impeller speeds, the powder rotational speed isgiven approximately by

!P =(BD

)1=2 √!I (32)

and thus theoretically varies with the square root of theimpeller speed.The impeller torque may be determined by substitution

for !P into the expression on the left hand of Eq. (30).It should be noted that the above torque balance requiresthe material to slip at both the impeller and wall surfaces.At low impeller speeds, the above torque balance maynot be satis;ed for one of two reasons:

1. The powder may remain stationary. In this case thetorque applied by the impeller is less than the torquerequired to make the powder slip at the wall. Thiscan occur with an impeller blade of low height andsmall angle �. The impeller torque as a functionof impeller rotational speed can be evaluated fromthe left hand side of Eq. (30) (substituting usingEqs. (24) and (29)), with !p set equal to zero.

2. The powder may rotate at the same speed as the im-peller. In this case the torque applied by the impelleris greater than the torque required to make the powderslip at the wall. This can occur with an impeller bladeof large height and large angle �. In this case, the im-peller torque is numerically equal to the bowl torqueand should be evaluated from the right hand side ofEq. (30) with !P set equal to the impeller speed.

However, as the impeller speed is increased, the powderwill start to move (;rst case) or the impeller will start tomove through the powder (second case) and the torquebalance of Eq. (30) is satis;ed.For bevelled blades, some additional factors must be

taken into account. Firstly, there are two contributions tothe inertial term: that from the front edge of the bladewhere powder is de7ected through an angle, �, of 45◦

and that from the surface of the blade where powder isde7ected through an angle, �, equal to the blade angle.The contributions depend on the magnitude of the term(1− cos �). For example with values of �=45◦ and 17◦,respectively, the term (1−cos �) takes values of 0.29 and0.044. Consequently at low speeds, the inertial contribu-tion to the torque comes predominantly from the ;rst ofthese contributions.A second factor with bevelled blades is that powder is

lifted by the passage of each blade and then falls. At lowspeeds, the powder falls to the base of the bowl. How-ever, with increase in speed, the powder falls through adistance which is less than the blade height. The powderthen interacts only with the blade surface. Thus at a crit-ical speed, there is a sharp decrease in the magnitude ofthe torque. The critical speed, !I;c, is related to the fall

distance, f, by the approximate expression

!I;c ≈ 2�n

(g2f

)1=2

(33)

At impeller rotational speeds larger than the critical value,the torque should be evaluated from a torque balance,using Eqs. (30) and (31), with � set equal to the bevelangle. However, with blades having a small bevel angle,of for example 11◦, the torque is produced mainly bypowder friction on the top surface of the blade. Since theweight of the powder is carried by the impeller blade,the torque in this case is best evaluated from Eq. (18),derived for a disc impeller.

5.5. Application and evaluation of the powdermechanics analyses

Consider ;rst application to disc impellers. For thebowl of diameter 0:21 m, with the smooth disc, the theo-retical powder rotation speed is calculated from Eq. (20)to be approximately 80 rpm. In comparison, the powderrotational speed determined from PEPT measurementswith this mixer at disc impeller rotational speeds of 200or 400 rpm was found to be approximately 70 rpm. Thepowder mechanics treatment, although simple, may there-fore give useful results.To calculate the torque, a value for the wall friction

coeMcient is required. However, a value appropriate tothe situation (7owing powder, low stress magnitude) isnot available. The friction coeMcient evaluated from theexperimentally determined value (0.58) of dimensionlesstorque, is 0.75. The magnitude of the friction coeMcientis not implausible, but is somewhat higher than values(e.g. 0.45–0.7) commonly obtained for interfacial fric-tion. The value will be utilised in the application of theoryto blade impellers.Consider next application to 90◦ impeller blades. In

Figs. 10–12, the theoretically predicted torque values areshown as continuous lines. The parameters used were:�w = 0:75, � = 45◦, � = 1=2 (based on the dimensionalanalysis described above) and F1 and F2 set, somewhatarbitrarily to 1.2 and 3.5, respectively. The theory givesan adequate ;rst-order representation of the data, but thetheoretical curves do not display the s-shaped characterobserved experimentally with the larger diameter bowls.Also the experimental dimensionless torque values arenot independent of ;ll but tend to decrease in magnitudewith increase in ;ll. This is actually as expected fromEq. (28) because at a given powder rotational speed, thevalue of r0 will decrease with increase in ;ll. For sim-plicity of analysis, r0 was set equal to 1=2R, thus leadingto underestimation of the dimensionless torque at low ;lllevels.In the model, the distribution of powder circumferential

velocities observed by the PEPT technique is representedby an average velocity. The theoretically calculated


Fig. 15. Theoretical powder rotational speed as a function of impellerspeed for 90◦ bevel angle, three-blade impellers of di:erent heights.Bowl diameter 0:21 m, with � = 0:75, � = 45◦, � = 1=2, F1 = 1:2,F2 = 3:5.

rotational speed of the powder, as a function of impellerspeed and blade height, is shown in Fig. 15 for the caseof a bowl of diameter 0:21 m. The e:ect of blade heightis particularly noticeable at low speeds. With a 3 mmhigh 90◦ impeller, the powder is predicted to remain sta-tionary at impeller speeds less than ∼ 110 rpm. Visualobservations con;rmed that the top surface of the powderremained stationary at low impeller rotational speeds.Although the predicted data should be regarded as

semi-quantitative in nature, there is no doubt that the av-erage powder rotational speed is substantially lower thanthe impeller speed. With 90◦ blade impellers, the powderrotational speed can be 25% of the impeller speed, whenoperating with relatively low impeller speeds. At highimpeller speeds, this percentage is much lower, around15%. With impeller blades of low height and, particu-larly those with low bevel angles, this percentage can belower still.Consider next application to bevelled blades. At low

impeller speeds the torque is produced from impact withthe front edge of the blade. The powder is lifted by thepassage of each blade and then falls. At, and above, acritical speed the fall of the powder is such that it nolonger interacts with the front edge, but with the bev-elled surface. Consequently, at this critical speed, thereis predicted to be a sharp decrease in the magnitude ofthe torque. In Fig. 14, a comparison is made between thetheoretical dimensionless torque, shown as a continuousline, and experimental data for an 11◦ bevel blade. Thetheoretical data show a step change in torque at the criti-cal speed, whereas the actual transition is smooth. Agree-ment with experiment is semi-quantitative in nature.In the case of the 17◦ impeller blade, theoretical values

of torque can be calculated for a 0:16 m diameter bowland a 3 kg ;ll and compared with the appropriate data inFig. 8. As with the 11◦ impeller blade, the actual tran-sition at the critical speed is less sharp than predicted.

The theoretical value of torque for a ;ll of 3 kg at an im-peller speed of 1100 rpm, evaluated using the values ofF1 and F2 used to ;t the 90◦ blade data, is 2:4 Nm. Thisis in reasonable agreement with the experimental valueof 2:5 Nm, taken from Fig. 8.

6. Discussion

6.1. Equipment variables

Perhaps the most signi;cant result from the present in-vestigation is that, for a disc impeller, the torque is pro-portional to the mass ;ll and, in the case of a blade im-peller, the torque decreases somewhat with increase inmass ;ll. Our experience in the use of vertical axis mixersfor agglomeration is that the kinetics of size enlargementare fairly insensitive to ;ll, which is consistent with thepresent observations, since the energy dissipation per unitvolume is then constant. Both McTaggart et al. (McTag-gart, Ganley, Sickmueller, &Walker, 1984) and Schaeferet al. (Schaefer, Taagegaard, Thomsen, & Gjelstrup Kris-tensen, 1993) reported that, at a given granulation time,the mean granule size decreased somewhat with increasein ;ll. Schaefer et al. also reported that the speci;c powerconsumption (power consumption divided by mass ;ll)decreased somewhat with increase in mass ;ll, in qualita-tive agreement with present results. The e:ect of mass ;llon size enlargement was attributed partly to the change inspeci;c energy consumption and partly to other factors.A second signi;cant observation concerning blade im-

pellers is that the height of the blade is less importantthan would be expected from 7uid mechanics concepts.There appear to be two reasons for this. Firstly, the massof powder de7ected by the blade is not proportional tothe blade height as it would be with a 7uid of low vis-cosity, but is instead approximately proportional to themass ;ll. This di:erence is attributable to the frictional,Coulombic behaviour of powders. Secondly, the e:ectiveblade height is larger than the actual height, because ofthe e:ects of the appreciable size of the particles relativeto the height of the blade. This has the e:ect of reducingthe sensitivity of the torque to blade height. The seconde:ect will obviously be more pronounced in small scaleequipment.The bevel angle, �, of the blade is an important vari-

able and theory suggests that, for values of � less than45◦, the inertial contribution to the torque should varyas (1− cos �). The bevel angles of blades in commercialequipment typically vary in the range 20–45◦. Such dif-ferences in blade angle can be expected to produce largedi:erences in the impeller torque and hence the mixingand granulation performance of equipment is also ex-pected to show considerable variation.With blades of small height or with low bevel angles,

the frictional component makes a signi;cant (and in some


cases a dominant) contribution to the total torque. Thismeans that in practice, with some designs of mixer, thetorque is not very sensitive to impeller rotational speed.With knife type blades having a small bevel angle of,for example, 11◦, the inertial contribution to the torqueis small provided the powder contacts only the blade sur-face. Under these conditions, the interaction is almost en-tirely frictional in nature and the impeller performs muchlike a disc impeller. However, at low impeller speeds,the powder impacts on the front of the blade, causing de-7ection through a larger angle and giving a larger torquethan that at high speeds. The e:ects of changing the bevelangle can therefore be complex.It has been reported (MJuller, 1982) that the force to

move a single blade linearly in a trough of powder varieswith the square of the velocity. The present results dif-fered from the single blade results in that, for blades witha bevel angle of 90◦, the torque displayed a variation withimpeller rotational speed that was approximately linear.A second observation was that, although three blade im-pellers gave torque values that were larger than those oftwo blade impellers of the same height and width, the val-ues were not 50% larger. It seems probable, as assumedabove, that the e:ects of the impeller speed and the num-ber of blades on the torque are related and arise from thefact that the blades move through disturbed powder. Con-sequently, data obtained for the movement of an isolatedblade in a trough may not be applied directly to interpretthe performance of a high speed mixer.

6.2. Powder mechanics analysis

The main problem with the powder mechanics analysisis to make suitable simplifying approximations about thestresses within the mass of powder in order to ;nd the nor-mal stresses at the walls. The model proposed treated themass as a ’rigid’ body rotating at a constant angular ve-locity. The approximations made concerning the stressesare similar to those made in the Janssen analysis. In thiscase, however, it is not clear whether the approximationslead to over- or under-estimation of the normal stresses.An additional factor, unaccounted for in the model, is thatthe impeller imparts motion in the radial as well as in thecircumferential direction. Material 7owing in the radialdirection impacts with the side wall, giving an additional,inertial, component of stress. For these reasons, the wallfriction coeMcient is e:ectively a variable parameter, foruse in the model. The need for a friction coeMcient ofsomewhat larger magnitude than that expected suggeststhat the model gives an underestimation of the normalstresses.From the PEPT results, with a disc impeller, the pow-

der was found to rotate circumferentially at an angularvelocity which did not vary with height. It may gener-ally be the case that, at low powder Froude numbers, the

circumferential velocity of the powder is approximatelyconstant. However, with blade impellers at high pow-der Froude numbers, the PEPT results showed that thiswas not the case. Because of the e:ect of wall friction,the circumferential velocity can be expected to decreasewith height above the impeller, i.e. the assumption of‘rigid’ body rotation is, at best, a ;rst order approxima-tion. Consequently, in the case of mixers ;tted with im-peller blades having large bevel angles, which producelarge powder Froude numbers, the model over-estimatesthe magnitude of the torque at high impeller speeds. Themodel cannot therefore predict the s-shaped character ofthe torque–impeller speed plots observed at high impellerspeeds with the larger diameter bowls.

7. Conclusions

An investigation wasmade of the factors that determinethe impeller torque of vertical axis high speed mixerscontaining granular solids. Both disc and blade impellerswere investigated and bowl sizes ranging from 0.13 to0:30 m were used, corresponding to an order of magnitudechange in bowl capacity. The study was supported by theboth visual observation and the application of positronemission particle tracking (PEPT) to investigate the 7owof the material in the mixer.In the case of disc impellers, the torque was found

to be proportional to the mass ;ll and independent ofdisc speed. PEPT studies showed that the powder rotatedas a ‘rigid’ body at a constant angular velocity. For a0:21 m diameter bowl at disc rotational speeds of 200 and400 rpm, the rotational speed of the powder was found ineach case to be approximately 70 rpm. The data were rep-resented by a dimensionless torque (T=MgR), which wasindependent of disc speed in the range 100–1100 rpm.

A powder mechanics analysis was made of the bowl;tted with a disc impeller, assuming ‘rigid’ body rotationof the material in the bowl. Expressions were derivedfor the dimensionless torque and the speed of rotationof the mass. The expression for the speed of rotationof the mass contained no ;tted parameters. The valueobtained (80 rpm) was in acceptable agreement with theexperimental value.In the case of blade impellers, the torque was found

to vary considerably with both the height and bevel an-gle of the blades. Knife blades, with a small bevel angle,gave signi;cantly smaller torque values than 90◦ blades.In all cases, the torque was found to be approximatelyproportional to the mass ;ll and data were again analysedby the use of the dimensionless torque, T=MgR. With 90◦

blades, to a ;rst order approximation, the torque variedlinearly with impeller speed. The data for dimensionlesstorque were correlated against the square root of the im-peller Froude number. An e:ective blade height was in-troduced to account for the e:ective shear zone thickness


above and below the blade. The dimensionless torque wasfound to vary as the square root of the dimensionless ef-fective blade height, he: =R. PEPT studies of the 7ow witha 10 mm high, 90◦ blade impeller showed that the powderrotated with a range of circumferential velocities.A powder mechanics analysis was made of the bowl

;tted with a blade impeller, again modelling the 7ow ofpowder in the bowl as ‘rigid’ body rotation at a constantangular velocity. Both frictional and inertial interactionswith the blade were accounted for. The analysis providesa ;rst order representation of the e:ects of scale, mass ;ll,impeller rotational speed, blade height and blade bevelangle on the torque. It contains two dimensionless ;ttedparameters. A critical discussion is presented of the as-sumptions made in the model. The rotational speed ofthe powder was obtained from the analysis and found tobe signi;cantly lower than the impeller rotational speed.With a blade impeller of height typically used in practicalmixing and granulation equipment at typical impeller ro-tational speeds, the average rotational speed of the pow-der was calculated to be in the region 10–30% of theimpeller rotational speed.

Notation

A parameter used in Eq. (31)B parameter used in Eq. (31)C parameter used in Eq. (31)d particle mean diameterD diameter of bowlD parameter used in Eq. (31)E eMciency factor in Eq. (25)f fall distance in Eq. (33)F1 proportionality coeMcient in Eq. (24)F2 proportionality coeMcient in Eq. (29)FI;f friction force per unit length of impeller bladeFr Froude numberFrI Froude number for impellerFrI−P Froude number for impeller relative to powderFrP Froude number for powderg gravitational accelerationh height of impeller bladehe: e:ective height of impeller bladeH height of powderk parameter in Eq. (2)K2 coeMcient in Eq. (21)M mass of powdern number of blades per impellerr radial co-ordinater0 inner radius of powderR radius of bowlT torqueT̂ dimensionless torque (T=MgR)TB;b torque on the base of the bowlTB;s torque on bowl side wall

TI;f torque on impeller from the frictional compo-nent

TI; i torque on impeller from the inertial componentTD torque to rotate the disc impellerT̂ 0 extrapolated dimensionless torque in Eq. (2)w width of bladez vertical co-ordinate

Greek letters

� exponent in Eq. (2)� parameter in Eq. (2)� angle of de7ection of powder over the blade angular co-ordinate�w;B wall friction coeMcient for the bowl surface�w;D wall friction coeMcient for the impeller disc sur-

face# bulk density of powder compressive stress N normal stress at the bowl surface N;b normal stress at the bowl base N;s normal stress at the bowl side wall�w wall shear stress! angular speed!P angular speed of powder!I angular speed of impeller!I;c critical angular speed of impeller, Eq. (33)!I−P angular speed of impeller relative to powder

Acknowledgements

The research was sponsored by Unilever PLC and weacknowledge the ;nancial support from Unilever whichenabled ABW to study for a Ph.D. degree. We would alsolike particularly to acknowledge the contributions madeby Prof. R Clift at the start of the work. The equipmentused was constructed by the technical sta: of the Schoolof Chemical Engineering, whose workmanship is grate-fully acknowledged.

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