Chemical Engineering Science 65 (2010) 5687–5698

Contents lists available at ScienceDirect

Chemical Engineering Science

0009-25

doi:10.1

� Corr

Univers

E-m

journal homepage: www.elsevier.com/locate/ces

Continuous blending of cohesive granular material

Avik Sarkar a, Carl Wassgren a,b,�

a School of Mechanical Engineering, Purdue University, USAb Department of Industrial and Physical Pharmacy (by courtesy), Purdue University, USA

a r t i c l e i n f o

Article history:

Received 14 August 2009

Received in revised form

7 April 2010

Accepted 13 April 2010Available online 6 May 2010

Keywords:

Cohesion

Discrete element method

Dispersion

Granular materials

Mixing

Particulate processes

09/$ - see front matter & 2010 Elsevier Ltd. A

016/j.ces.2010.04.011

esponding author at: School of Mechanical E

ity, USA.

ail address: wassgren@purdue.edu (C. Wassgr

a b s t r a c t

Results are presented from discrete element method (DEM) computer simulations of cohesive particles

in a periodic slice of a continuous blender. The influence of inter-particle cohesion at various impeller

speeds and fill levels is reported. Although increasing cohesion does not significantly change axial flow

rates, mixing rates in the transverse plane and axial direction are affected. Mixing is generally enhanced

for slightly cohesive materials, but decreases for larger cohesion, similar to trends observed in tumbling

batch mixers. Changes in fill level are also shown to affect axial transport rates and mixing. These

results suggest that the controllable operating parameters, such as feed rate and impeller speed, may be

adjusted for cohesive powder formulations to obtain optimal mixing performance.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Continuous blending is becoming an increasingly commonprocess in industries that handle particulate materials such asfood products, pharmaceuticals, and chemicals (Pernenkil andCooney, 2006). Continuous blending offers several advantagesover more traditional batch blending including larger through-puts, more straightforward scaling, and less operator interaction.Unfortunately, knowledge accumulated from batch blendingexperience does not always translate to continuous blending.For example, the residence time in a batch blender is independentof impeller speed while in a continuous blender the two areclosely connected. Therefore, it is worthwhile to investigate theparameters governing a continuous powder mixing process.

This paper is part of an ongoing effort to study bladedcontinuous mixers (Portillo et al., 2008; Sarkar and Wassgren,2009). In particular, this paper is closely related to the workof Sarkar and Wassgren (2009) in which continuous blending offree-flowing granular material was simulated for a range ofoperating conditions. The difference between this previous andthe current work is that the current work includes the effects ofparticle cohesion, which has not been systematically studied forcontinuous blenders. The effects of cohesion on flow rate andmixing over a range of impeller rotation speeds and fill levels arereported.

ll rights reserved.

ngineering, Purdue

en).

2. Background

Blending of non-cohesive particulate materials in batchblenders has been widely studied by a number of researchers,both experimentally (e.g., Bagster and Bridgwater, 1967, 1970;Bridgwater et al., 1968; Bridgwater, 1976) and computationally(e.g., Moakher et al., 2000; Zhou et al., 2004; Sudah et al., 2005;Bertrand et al., 2005). These studies have provided considerableinsight into the physics of blending and segregation processes,such as how the initial loading of materials can result insignificantly different blending rates (Sudah et al., 2005), largernumbers of shorter mixing blades are more effective for mixingthan fewer, but larger blades (Malhotra and Mujumdar, 1990;Malhotra et al., 1988, 1990; Laurent and Bridgwater, 2002b, c),and that, for certain ranges of speed, fill level has a moresignificant influence on mixing than rotation speed (Laurent et al.,2000; Laurent and Bridgwater, 2002a).

Studies investigating the influence of cohesion on blending inbatch blenders are more recent. McCarthy (2003) and Chaudhuriet al. (2006) both found that mixing rates marginally improve forsmall values of particle cohesion, but decrease as cohesionincreases further. Though neither paper identifies a mechanismfor the improved mixing rate, both indicate that their resultsare consistent with the experimental findings of Shinbrot et al.(1999). Shinbrot et al. (1999) demonstrated the emergence ofspontaneous chaotic flow patterns for a fine, cohesive powder,which enhances mixing as compared to free flowing, coarsematerial.

There have been only a few studies concerning blending incontinuous blenders. Reviews concerning continuous mixing can

Fig. 1. Schematic showing a periodic section of a continuous mixer.

A. Sarkar, C. Wassgren / Chemical Engineering Science 65 (2010) 5687–56985688

be found in Fan and Chen (1990) and Pernenkil and Cooney(2006). Fan and Chen (1990) report two continuous mixer types, arotating drum and a continuous ribbon blender. Harwood et al.(1975) compared the performance of seven continuous mixersblending a dry sand–sugar mixture. Mixers that performed bestrelied on creating a high shear gradient between two augers orsurfaces. Rotating drums with an inlet and an outlet have alsobeen used as continuous mixers. Abouzeid et al. (1974) studiedthe effects of drum rotation rate, inclination, feed rate, andparticle size on residence time mean and variance. The variance ofthe residence time can be directly correlated to the axialdispersion of the powder particles. Dimensionless varianceincreased with increasing rotation speed and inclination, anddecreasing feed rate. Sherritt et al. (2003) proposed an axialdispersion model which included the drum rotation speed, filllevel, and drum and particle diameter. This model was applied toboth batch and continuous mixers, since axial dispersion, which iscaused by random collisions, is purely diffusive and is unaffectedby a bulk axial transport rate.

Laurent and Bridgwater (2000) used positron emission particletracking (PEPT) to study a bladed mixer with an inlet and outlet,thereby producing a continuous system with superimposed axialtransport. They observed that the weirs supporting the bladespartitioned the mixer into three compartments, and causedmaterial holdup in each compartment. Two circulation loopswere also identified in each compartment which hampered axialmixing. Portillo et al. (2008) experimentally characterized theperformance of a horizontal axis, bladed continuous mixer. Theyreported longer residence times, which also corresponded tobetter blend homogeneity, for an upward inclination of the mixeraxis and for smaller impeller rotation speeds. Portillo et al. (2008)also reported that increasing the number of blades from 29 to 34slightly improves mixture homogeneity. Sarkar and Wassgren(2009) performed discrete element method (DEM) parametricstudies using non-cohesive particles in a periodic slice of acontinuous mixer. The fill level and impeller rotation speed weresimultaneously varied to cover a wide range of values. Mixing wasfound to be fastest at small fills and large impeller speeds wherediffusive mixing produced during fluidization was the primarymechanism. At larger fills, a smaller impeller speed was found togive better mixing. It was also shown that decreasing the bladespacing, corresponding to increasing the number of blades,improves the mixing rate, consistent with the experimentalresults of Portillo et al. (2008).

Since the blending dynamics of batch blenders and continuousblenders may be different, due in particular to the fact that thereis axial transport and a limited residence time of material incontinuous blenders, it is worthwhile to examine how cohesivityinfluences particle mixing in continuous blenders. Unfortunately,there have been no studies to date on this topic. This work usesDEM simulations of a periodic ‘‘slice’’ of a horizontal, bladed,continuous blender to examine the flow rate and axial andtransverse mixing as a function of particle cohesion, impellerrotation speed, and fill level.

3. Model

A periodic section of a continuous blender has been modeledusing DEM and is shown schematically in Fig. 1. Material exitingthe periodic section at one end re-enters the section at the other.The periodic section contains two mutually orthogonal bladestages in order to simulate the geometry found in a GEA BuckSystems experimental blender. Each stage has two blades inclined451 to the mixer axis. This periodic section simulates an axialregion near the middle of the continuous blender. Entrance and

exit effects are not captured using this periodic slice. More detailsregarding the geometry and dimensions of the model can befound in Table 1 and also in Sarkar and Wassgren (2009).

A number of cohesion models have been reported in theliterature for use in DEM simulations, including those implement-ing liquid bridges (Lian et al., 1993; Muguruma et al., 2000), JKRsurface cohesion (Johnson et al., 1971; Mishra et al., 2002),constant cohesion (Iordanoff et al., 2005; Chaudhuri et al., 2006),and cohesion as a function of contact area (or contact length in thetwo-dimensional simulations of Matuttis and Schinner, 2001). Inthis work, the cohesion model proposed in Luding (2005) isimplemented, which is a combination of the hysteretic linearspring normal force model of Walton and Braun (1986) and theanalytical elasto-plastic/van der Waals cohesive interactionmodel of Tomas (2004). The cohesive force acts only after contactis made, and only during unloading. The loading, unloading, andcohesive forces are continuous, linear functions of the contactoverlap, with the interaction force equaling zero at separation(refer to Fig. 2). Three parameters define the model: a loadingstiffness, kL, an unloading stiffness, kU, and a cohesive stiffness, kC.When two particles first come into contact, they proceed alongthe loading path OA until a maximum overlap dmax is reached. Theslope of OA is given by the loading stiffness kL. Subsequently, ifthe particles move apart, they follow the unloading path AB (slopekU) and BO (slope �kC). The largest cohesive force experiencedduring the lifetime of the contact is at point B, with the magnitudegiven by |kC|dmin. If a contact is reloaded before completelyseparating, the force follows the unloading path (along BA inFig. 2a) till the maximum contact overlap is reached. Subsequentloading beyond the preexisting maximum overlap proceeds alongthe regular loading path with a linear contact stiffness of kL. TheLuding model was chosen for the current studies since it isphysically based and straightforward to implement.

A simple sliding friction tangential force model is used in all ofthe simulations. As with the other cohesive force modelsmentioned previously, cohesion is assumed to have no influenceon tangential force interactions.

Table 1Simulation parameters and variables.

Parameter/variable Symbol Value (s)

Dimensional constant parameters

Average particle diameter dp 4 mm

Particle density r 1000 kg m�3

Gravitational acceleration g 9.81 m s�2

Simulation time step Dt 2.99�10�5 s

Dimensionless constant parameters

Particle size dispersity f 0.2

Drum to particle diameter ratio Ddrum/dp 37.5

Shaft to particle diameter ratio Dshaft/dp 5.0

Periodic length to particle diameter ratio Lperiodic/dp 30.0

Blade length to particle diameter ratio Lblade/dp 17.5

Blade angle yblade 451

Scaled particle–particle unloading stiffness k�U,pp ¼ kU,pp=ðrd2pgÞ 2000

Particle–particle friction coefficient mpp 0.3

Particle–particle normal restitution coefficient eN,p 0.75

Scaled particle–wall unloading stiffness k�U,pw ¼ kU,pw=ðrd2pgÞ 2000

Scaled particle–wall cohesive stiffness k�C,pw ¼ kC,pw=ðrd2pgÞ 0 (non-cohesive)

Particle–wall friction coefficient mpw 0.3

Particle–wall normal restitution coefficient eN,w 0.75

Dimensionless variables

Scaled particle–particle cohesive stiffness k�C,pp ¼ kC,pp=ðrd2pgÞ {0, 500, 1000, 1500, 2000, 2500, 3000}

Froude number Fr¼o2Ddrum/(2g) {0.21, 0.84, 1.89, 3.35}

Bulk fill volume fraction n {25%, 40%, 55%, 70%}

Fig. 2. The inter-particle normal force model. (a) Normal force vs. displacement curve for the Luding (2005) model and (b) schematic showing of the spring stiffness

modeled in a contact. Additional detail concerning the force model may be found in Luding (2005).

A. Sarkar, C. Wassgren / Chemical Engineering Science 65 (2010) 5687–5698 5689

Energy is dissipated in the model through non-elastic normalimpacts and tangential friction. For non-cohesive particles, thecoefficient of normal restitution eN is related to the particleloading (kL) and unloading (kU) stiffness as,

eN ¼

ffiffiffiffiffiffikL

kU

s: ð1Þ

For the present work, a value of 0.75 has been chosen for allnormal contact pairs, based on Muller et al. (2008). Muller et al.(2008) report the ranges of normal restitution coefficients ofgranules for different materials at different impact speeds. Thebaseline force model parameters used in the simulation, as well asthe other simulation parameters are listed in Table 1.

Parametric studies are performed in which the impellerrotational speed, cohesive contact stiffness, i.e. contact strength,and fill volume are varied. Each of these parameters is expressedin dimensionless form. The impeller rotational speed, o, is

expressed in terms of a Froude number, Fr,

Fr¼o2Ddrum

2gð2Þ

where Ddrum is the containing drum diameter and g is theacceleration due to gravity. The Froude number is a ratio of thecharacteristic centripetal acceleration acting on a particle due toimpeller rotation, to the acceleration due to gravity.

The cohesive contact stiffness, kC, is expressed in dimension-less form as,

k�C ¼kC

rd2pg

, ð3Þ

where r is the particle density and dp is the particle diameter. Thedimensionless cohesive stiffness is a ratio of a characteristiccohesive strength acting on a particle, albeit based on an overlapequivalent to a particle diameter, to a particle’s weight.

Table 2Granular bond number values Bog using Eq. (3), kU,p

*¼2000, and eN,p¼0.75.

kC,pp* Bog

0 0.00

500 0.16

1000 0.26

1500 0.33

2000 0.39

2500 0.43

3000 0.47

Fig. 3. Instantaneous velocities and solid fractions at different cross sections along

the mixer length for (a) Fr¼0.21, kC,p*¼0, (b) Fr¼0.21, kC,p

*¼3000, (c) Fr¼3.35,

kC,p*¼0, (d) Fr¼3.35, kC,p

*¼3000, and (e) locations of cross sectional regions (1)–(4).

Arrows represent the local velocity vectors and the shading represents the solid

fraction. Vertical and horizontal lines in regions (2) and (4), respectively, represent

the instantaneous orientations of the blades.

A. Sarkar, C. Wassgren / Chemical Engineering Science 65 (2010) 5687–56985690

A commonly used dimensionless parameter used in cohesiveparticle studies is the granular Bond number Bog, which is definedas the ratio of a characteristic cohesive force to a characteristicgravitational force. The dimensionless cohesive stiffness, kC

*,defined in Eq. (3) is, in fact, a form of a granular Bond number;however, it is based on a contact overlap equivalent to a particlediameter. Hence, a more easily interpreted Bond number is givenin addition to the dimensionless cohesive stiffness. For the Ludingcohesion model a maximum cohesive force cannot be readilydefined for an impact since the maximum contact overlap is afunction of the impact speed. The cohesive force for the moresimple case of a single particle resting on a surface under theaction of gravity can be used to determine the characteristiccohesive force. Therefore, a more easily interpreted granular Bondnumber may be defined as

Bog ¼jFN,minj

mg¼jkC jdmin

kLdmax¼jkC j

kL

kU�kL

kUþjkC j, ð4Þ

where |FN,min| is the magnitude of the largest cohesive forcedeveloped during this contact (point B in Fig. 2a) correspondingto a maximum loading force FN,max¼mg (point A in Fig. 2a). Table 2lists the corresponding Bog values for the values of the stiffnessesused in the simulations (reported in Table 1). These values may beused to approximately compare the cohesion level in the currentsimulations with those in other computational and experimentalworks. However, a consistent definition of Bond number must beused, where the cohesive force is a result of the particle’s self-weight. For example, for the current case, a Bond number of 0.47corresponds to a surface energy value of 16.4�10�3 J/m2 in the JKRcohesion model (Johnson et al., 1971) for the particle propertiespresented in Table 1. At the limit |kC|-N, the maximum Bondnumber Bog,max using the current definition is given by

Bog,max ¼kU�kL

kL¼

1

ðeN,ppÞ2�1, ð5Þ

where eN,p is the particle–particle coefficient of normal restitutionfor the non-cohesive case. Near this limit the time-step requiredto resolve a cohesive impact is infinitesimally small, which cannotbe computationally implemented. For the presently used value ofeN,p¼0.75, the limiting Bond number using the current definitionis Bog,max¼0.78.

The dimensionless bulk fill fraction has also been varied in thisstudy. Assuming a maximum random packing solid fraction of0.64 for spheres, the bulk fill fraction n is defined as

n¼p6 d3

pNp

0:64 � p4 ðD2drum�D2

shaftÞLperiodic

, ð6Þ

where Np is the number of particles in the simulation, Dshaft is theinternal shaft diameter, and Lperiodic is the mixer periodic length. Ina full continuous blender, the bulk fill is dependent on the inletfeed rate, impeller rotation speed, and outlet weir geometry.Currently simulated values of n listed in Table 1 cover a widerange of fills expected within a typical mixer.

4. Results

4.1. Effect of rotation rate and cohesion

Simulations were performed for seven values of dimensionlesscohesive stiffness and four values of Froude number. Allmeasurements were averaged over at least six impeller rotationsduring steady-state operation. The baseline case of 40% bulk fillvolume level contained 15,912 particles.

4.1.1. Flow and transit time

Transverse flow patterns for the non-cohesive (kC,p*¼0) and the

largest cohesion case simulated (kC,p*¼3000) are presented in

Fig. 3 for two Froude numbers (Fr¼0.21 and Fr¼3.35). At thesmaller value of Froude number, nearly all of the particles resideat the bottom of the bed, agitated only during intermittent bladepasses. Particles roll down the surface of the bed, both over andunder the central shaft. Similar flow patterns were observed fornon-cohesive particles in Sarkar and Wassgren (2009), and havealso been reported by Malhotra et al. (1990) and Laurent andBridgwater (2002a). Videos of the simulation do not reveal anysignificant agglomeration at the bed surface, but the angle ofrepose qualitatively appears to be larger (Fig. 3b) than thecorresponding non-cohesive case (Fig. 3a). It is difficult to

A. Sarkar, C. Wassgren / Chemical Engineering Science 65 (2010) 5687–5698 5691

quantify this measurement since the free surface is not welldefined due to some fluidization. For the larger Froude numbercase, the flow patterns are very similar to the corresponding non-cohesive cases in Sarkar and Wassgren (2009). Particles arefluidized by the fast moving impellers in the free space availableabove the bed. The differences between the free flowing andcohesive cases are subtle. The number of particles flowing overthe shaft slightly increases at larger particle cohesion values.However, the overall flow patterns remain nearly unchanged withvarying cohesion, particularly for the larger impeller speeds.

The flow rate through the mixer may be expressed in terms ofthe average axial velocity. The average axial velocity is defined as,

vaxial ¼1

Np

XNp

i ¼ 1

vaxial,i, ð7Þ

where the average is taken over all particles in the periodicsection. Fig. 4a plots the average axial velocities scaled by theimpeller tip speed (oDdrum/2) for different Froude numbers anddimensionless cohesive stiffnesses. Little variation is seen in thescaled axial velocities for larger impeller speeds. At smaller

Fig. 4. Average scaled axial velocity, (a) variation with scaled cohesive stiffness,

(b) variation with Froude number. Scatter bars in (a) represent one standard

deviation of the average scaled velocity measured over multiple impeller

rotations.

impeller speeds, the flow rate increases slightly as particlesbecome more cohesive. The formation of tensile force chainsresults in more coordinated axial particle movement, and thuslarger axial flow rates than for uncoordinated, weakly cohesiveparticles. This effect is not observed at larger Froude numberssince particle momentum differences at larger impeller speedscan easily overcome the cohesive spring forces. This effect isdiscussed in greater detail in the following paragraph. Fig. 4b re-plots the same data to illustrate trends with increasing Froudenumber. Scatter bars have been omitted in this figure for clarity.The scaled axial velocity increases with increasing Froudenumber, with a nearly linear dependence for less cohesivecontacts and larger Froude numbers. The weakly cohesive lineardependence is consistent with the non-cohesive cases examinedin Sarkar and Wassgren (2009), despite differences in particledensity and particle–wall friction.

The axial component of the granular temperature has beenused to determine if increased cohesion does indeed result inmore coordinated axial particle movement. A binning proceduresimilar to Sarkar and Wassgren (2009) has been used to computethe axial component of granular temperature and the dispersionindices (Sections 4.1.2 and 4.2.2). Uniformly spaced, 3D, Cartesiangrid points are generated within the simulation domain. A bin isdefined as a cubical region of side 3dp with the grid point ofinterest at the center. Granular temperature or dispersion at a gridpoint is obtained by averaging over all particles contained withinthe corresponding bin. Parameters associated with the binningprocedure are provided in Table 3. The equation used to define theaxial component of the granular temperature for a group of Nbin

particles contained within a bin is given by,

TZ,bin ¼/v02zSbin ¼1

Nbin

XNbin

i ¼ 1

fðvZ,i�vZ Þ2g ¼

1

Nbin

XNbin

i ¼ 1

v2Z,i�

1

Nbin

XNbin

i ¼ 1

vZ,i

!2

,

ð8Þ

where vZ is the average velocity for the group of Nbin particleswithin the bin. The overall axial component of the granulartemperature of the system is taken as the number weightedaverage of the axial granular temperature, given by,

TZ,overall ¼/v02ZS¼

Pbins

/v02ZSbinNbinPbins

Nbin:ð9Þ

Dispersion indices quantifying mixing are calculated in ananalogous manner (Eqs. (10)–(12)). Fig. 5 plots the Z-componentof the granular temperature, scaled by the square of blade tipspeed. For all impeller speeds, an increased strength in the tensileforce chains results in a decrease in the axial component of thegranular temperature, which in turn indicates a greater degree ofcoordinated axial particle movement. The decrease in granulartemperature is most noticeable for the smallest Froude numbersince a small impeller speed is insufficient to break up cohesivecontacts formed at larger values of particle cohesion. Thereduction in the scaled axial granular temperature is smaller forthe larger Froude number cases, likely due to the fact that atlarger impeller speeds particles have sufficient momentum toovercome the cohesive bonds regardless of their strength.

Table 3Binning parameters used for the dispersion measurements.

Parameter Value

Number of bins in the X, Y, and Z directions 15, 15, 12

Bin spacing in the X, Y, and Z directions 2.46dp, 2.46dp, 2.45dp

Bin dimensions (cubical) 3dp

Fig. 5. Scaled axial (Z) component of granular temperature. Scatter bars represent

the scaled standard deviation of the data obtained over multiple impeller

rotations.

A. Sarkar, C. Wassgren / Chemical Engineering Science 65 (2010) 5687–56985692

The mean residence time of particles through a blender isdirectly related to the mixer length and throughput while the widthof the residence time distribution may be used to estimate axialdispersion (Levenspiel, 1999). Portillo et al. (2008) measured theresidence time of a tracer substance in their full length continuousmixer experiments and found that larger mean residence timescorrelated with improved mixing. They hypothesize that for largerresidence times, the mixing blades make a larger number of passesthrough the bed and thus improve mixing. Since the current worksimulates a periodic section, distributions of particle ‘‘transit time’’through the periodic section, rather than the residence time in theblender, are examined. The transit time ttr for a particle is defined asthe time taken for a particle to travel one periodic length (Lperiodic) inthe axial direction (Fig. 6a). Particles may cross a periodic boundarywhile traversing this length. Figs. 6b–e plot the transit time pro-bability distributions in dimensionless form for all of the particles inthe periodic section. The abscissa is scaled by the impeller rotationperiod 2p/o, thereby presenting the transit time in terms of numberof impeller rotations. A set of transit time probability distributiondata is obtained by following the trajectories of all particles startingfrom a given initial configuration of particle positions. The transittime distributions presented in Fig. 6 were obtained by averagingmore than a hundred sets of distributions, each corresponding to adifferent initial configuration of particle positions. The ordinate isscaled by the shaft angular velocity o, since it is the blade rotationthat drives the flow. In these scaled axes, both the mean and widthof the distribution are found to decrease with increasing Froudenumber, consistent with the data in Figs. 4 and 5. A shorter meantransit time distribution corresponds to a higher average axialvelocity, which is seen for the larger Froude number cases in Fig. 4.A narrower transit time distribution indicates more coherent axialmovement of the particles, reflected in the decrease in the axialcomponent of granular temperature in Fig. 5. Note that most of theparticles traverse the periodic length in under two impellerrotations. The transit time distributions show a weak dependenceon cohesion, especially for larger Froude numbers. As discussedpreviously (refer to Figs. 4 and 5), the influence of cohesion is lesssignificant at larger Froude numbers since larger impeller speedsprevent agglomeration and the bed behaves almost like a freeflowing material for all values of cohesion. Differences caused bycohesion are prominent only at smaller Froude numbers. The transit

time distributions are narrower for the more cohesive particles atsmaller Froude numbers, indicating a more coordinated axialparticle movement, consistent with the smaller axial granulartemperatures in Fig. 5.

4.1.2. Mixing

Following Martin et al. (2007) and Sarkar and Wassgren(2009), dispersion is used to characterize mixing. All results arereported for the particle dispersion produced over one completerotation of the impellers. Dispersions in the transverse plane andthe axial direction are reported separately. Axial dispersion is animportant quantity in and of itself since it is directly related tomaterial residence time (or transit time in the present work).Moreover, axial dispersion helps to remove temporal inhomo-geneities in the input stream; however, too large of an axialdispersion may not be desirable since the powder leaving themixer would have widely ranging residence times and be subjectto different amounts of work by the blender paddles. Dispersionat a point inside the blender is taken as the dispersion of particleslying within a cubical bin of side 3dp with the grid point ofinterest as the center of the bin. Mathematically, the dimension-less transverse and axial dispersions for a bin are defined as

MXY,bin ¼1

dp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

Nbin

XNbin

i ¼ 1

fðDxi�DxÞ2þðDyi�DyÞ2g

vuut , ð10Þ

MZ,bin ¼1

dp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

Nbin

XNbin

i ¼ 1

ðDzi�DzÞ2

vuut , ð11Þ

where MXY,bin and MZ,bin are the transverse and axial dimension-less dispersion indices for a bin, and Nbin is the number ofparticles in the bin. The parameters Dxi, Dyi, and Dzi are thedisplacements of the ith particle in the bin, and Dx, Dy, and Dz arethe average displacements of all the particles in the bin. Theoverall transverse dispersion in the mixer, MXY,overall, is taken tobe a number weighted average of the local dispersions

MXY,overall ¼

Pbins

MXY,binNbinPbins

Nbin:ð12Þ

The binning procedure is identical to that considered in Sarkarand Wassgren (2009) and bin dimensions are provided in Table 3.The overall axial dispersion MZ,overall is defined an analogousmanner. It should be noted that the dispersion indices depend onthe bin size and number of bins, which relate to the scale ofscrutiny and number of samples, respectively. For the currentstudy, the dispersion indices have been used as a comparativemeasure at a constant bin size and number of bins, with eachcubical bin having an edge length 3dp, which is a reasonable binsize to investigate mixing at a particle length scale.

Fig. 7 plots the overall transverse dispersion in the mixer forvarying cohesion. For all but the Fr¼3.35 case, transverse mixingper rotation increases slightly with a small increase in cohesionand reaches a maximum value in the range k*

C,pp¼500–1000,corresponding to Bog¼0.16–0.26. Thereafter, the mixing ratedecreases with a further increase in cohesion. Similar behaviorhas been reported in batch rotating drums by McCarthy (2003)and Chaudhuri et al. (2006). McCarthy’s (2003) simulationsconsidered cohesion due to liquid bridges and found that thebest mixing occurred at Bog¼1.0. Although a different cohesiveforce model (constant cohesive force; Iordanoff et al., 2005) wasused, Chaudhuri et al. (2006) found that mixing was best forparticle interactions with Bog¼0.1. Shinbrot et al. (1999) reportedthat for fine, cohesive particles in a rotating drum, cohesioncaused stick–slip powder movement which resulted in chaotic

Fig. 6. (a) Transit time ttr is defined as the time it takes for a particle to traverse the length Lperiodic. Scaled transit time probability distributions plotted against scaled transit

time for varying cohesion, (b) Fr¼0.21, (c) Fr¼0.84, (d) Fr¼1.89, and (e) Fr¼3.35.

A. Sarkar, C. Wassgren / Chemical Engineering Science 65 (2010) 5687–5698 5693

mixing patterns. Such behavior was not clearly observed in thecurrent simulations. The cohesive interaction between particles isexpected to hinder particle diffusion and promote agglomerateformation, both of which decrease mixing. It is hypothesized thatmoderate levels of cohesion at small Froude numbers improvesconvective mixing due to more coordinated particle movement,

while at the same time not being sufficiently large to adverselyaffect diffusive mixing. Larger values of cohesion maysignificantly hinder diffusive mixing by forming agglomeratesand strong cohesive bonds.

Mixing in the transverse plane also generally increases withFroude number, except for dimensionless cohesive stiffnesses less

Fig. 7. Transverse dispersion in the mixer as a function of dimensionless cohesive

strength for varying Froude numbers. Scatter bars represent one standard

deviation of the transverse dispersions produced over multiple impeller rotations.

Fig. 8. Axial dispersion in the mixer plotted as a function of dimensionless

cohesive stiffness for varying Froude number. Scatter bars represent one standard

deviation of the axial dispersions produced over multiple impeller rotations.

Fig. 9. Scaled overall dispersion over the average transit time versus scaled

cohesive stiffness for (a) transverse plane dispersion and (b) axial dispersion.

Scatter bars in (a) represent one standard deviation of the dispersion data over

multiple measurement intervals.

A. Sarkar, C. Wassgren / Chemical Engineering Science 65 (2010) 5687–56985694

than approximately 1500. For cohesive stiffnesses smaller thanthis value, transverse mixing decreases from Fr¼0.21 to Fr¼0.84,but then increases for larger values of Froude number. Similarbehavior was observed for the free flowing case in Sarkar andWassgren (2009). At larger Froude numbers, particle fluidizationbecomes more apparent with a corresponding increase in mixing.The reason for the dip in transverse mixing at an intermediatevalue of Froude number for less cohesive material remainsunclear.

Fig. 8 plots the overall axial dispersion in the continuousmixer. Axial dispersion decreases monotonically with increasingcohesion for all Froude numbers, consistent with the idea thatincreasing cohesion, especially at smaller Froude numbers,increases the coordination of axial particle movement (Fig. 5).Larger Froude numbers improve axial mixing, consistent withFigs. 5–7.

The dispersion trends suggest that for a given set of operatingconditions, the best mixing will be achieved for non-cohesive tomildly cohesive material. In addition, larger impeller Froudenumbers can significantly enhance mixing as long as care istaken to ensure that the powder experiences a sufficient numberof blade passes. Both the dispersion data and transit timedistributions should be considered simultaneously when selectingthe operating conditions. Keeping this is mind, the overalldispersions defined by Eqs. (10)–(12) are plotted in Fig. 9 overthe average transit time rather than over an impeller rotation. Theaverage transit time for a given cohesion and impeller speed isobtained by taking the arithmetic mean of the correspondingtransit time distribution presented in Fig. 6. The dispersionproduced over the average transit time measures the mixingachieved as the powder traverses a length Lperiodic along the mixer.Fig. 9a and b plot the scaled transverse (MXY,transit) and axial(MZ,transit) dispersions, respectively. The standard deviations of theaxial dispersion data, which have been omitted for clarity, arelarger than that for the transverse component since the allowableaxial particle displacements are not bounded by the drum

A. Sarkar, C. Wassgren / Chemical Engineering Science 65 (2010) 5687–5698 5695

diameter. For both the transverse and axial dispersioncomponents, the mixing trends for a given Froude number withrespect to cohesion remain essentially the same as those in Figs. 7and 8 (dispersion per impeller rotation), but the dependence onFroude number is different. The best mixing is obtained at thesmallest Froude number for free flowing and small cohesivitymaterial for both components of dispersion. At small speeds, theresidence time is large owing to the slow axial flow rate. Thematerial experiences a larger number of blade passes leading tobetter homogeneity. In contrast, though the mixing achieved perimpeller rotation is greater for the larger Froude numbers, a muchsmaller residence time leads to the free flowing material passingthrough the blender without experiencing many blade passes. Atthe largest cohesion values, an intermediate value of Fr¼0.84 isfound to give the best transverse mixing, suggesting that greaterimpeller agitation is necessary to achieve mixing, but one that isnot so large that the number of blade passes that the materialexperiences is small. The axial dispersion values for the mostcohesive case are found to be numerically close to each other,with the Fr¼3.35 case being slightly larger than the others.

4.2. Effect of fill volume fraction

The bulk fill volume fraction in a continuous mixer is acomplex function of the inlet feed rate, impeller rotation rate, andthe geometry of the outlet weir. In the periodic sectionsimulations performed here, the fill fraction has been artificiallyprescribed by specifying the number of simulated particles. Allresults in this section are reported for an intermediate Froudenumber value of Fr¼0.84 (100 rpm). The number of simulatedparticles ranged from 9944 for 25% bulk volume fill to 27845 for70% fill.

4.2.1. Flow and transit time

The average scaled axial velocity is plotted in Fig. 10 as afunction of the bulk fill volume fraction for two values ofcohesion. In both cases, the average axial velocity reaches amaximum near 40% bulk fill volume fraction. This result is similarto that reported for non-cohesive particles in Sarkar and

Fig. 10. Average scaled particle axial velocity scaled by impeller tip speed for

varying bulk fill volume fraction. Scatter bars represent one standard deviation of

the data obtained over multiple impeller rotations.

Wassgren (2009), where the peak value is in the range between45% and 50% bulk fill fraction. At smaller fill fractions, the bladesspend a short fraction of the rotation time period immersed in thebed resulting in a small flow rate. At larger fills, the increasedblade thrust (due to a larger time spent in the bed) results in anincreased average particle axial velocity. However, increasing thefill fraction also results in an increased wall resistance, which actsto decrease the bed’s axial speed. At the largest fill fraction, thelack of available free volume also hinders particle mobility in boththe axial and transverse directions. This competition of effectsresults in the observed maximum. Increasing particle cohesionincreases the average scaled velocity, but only slightly. As waspreviously discussed in Section 4.1.1, increased cohesion resultsin more coordinated axial particle movement and thus a slightlylarger flow rate.

The scaled transit time distributions for the varying fillfractions are plotted in Fig. 11. The transit time distributions arebroader for smaller fill fractions as compared to those for largerfills. This trend suggests that the axial dispersion is greater atsmaller fills (Levenspiel, 1999), and is indeed found to be the case(discussed further in Section 4.2.2). As was demonstrated in Fig. 5of Sarkar and Wassgren (2009), temporal fluctuations in localaxial velocities are larger at smaller bulk fills. With increasing fillfraction, both spatial and temporal variations in axial flowdecrease as both blade stages remain immersed longer in thebed, thus continuously propelling the bed forward. A bed that hassmaller axial flow rate spatial variations will result in a narrowertransit time distribution. The transit time distributions for the

Fig. 11. Scaled transit time distributions for varying bulk fill volume fraction,

(a) kC,p*¼500 and (b) kC,p

*¼2500. The baseline value of Fr¼0.84 (100 rpm) has been

used in all cases.

A. Sarkar, C. Wassgren / Chemical Engineering Science 65 (2010) 5687–56985696

larger cohesion cases in Fig. 11b are narrower than thecorresponding cases for smaller cohesion (Fig. 11a). Increasingcohesion results in a more coherent axial flow leading to anarrower transit time distribution.

4.2.2. Mixing

For both cohesion values, transverse mixing per rotationincreases with increasing fill up to a fill fraction of 55%, anddecreases for larger fill fractions (Fig. 12). A similar trend wasobserved in Sarkar and Wassgren (2009). The transverse mixingper axial periodic length shows identical behavior. At the 100 rpmimpeller speed, increasing fill improves mixing by establishingcirculation loops in which particles are moved up by the bladesand then cascade down the free surface under the agitator shaft(refer to Fig. 3). As the bulk fill fraction increases, a secondary loopbegins to emerge with particles passing over the shaft. For evenlarger fills where the central shaft is fully immersed in the bed,

Fig. 12. Dimensionless overall transverse dispersion over one impeller rotation as

a function of bulk fill volume fraction. Scatter bars represent one standard

deviation of the data obtained over multiple impeller rotations.

Fig. 13. Dimensionless axial dispersion over one impeller rotation as a function of

bulk fill volume fraction. Scatter bars represent one standard deviation of the data

obtained over multiple impeller rotations.

nearly all particles flow over the shaft. At the largest fills, particlemobility is hampered by the limited free volume available, and adecrease in transverse mixing is observed. Larger cohesion valuesalso hinder particle mobility resulting in a further decrease intransverse mixing.

Fig. 13 plots the dimensionless axial dispersion per rotation asa function of bulk fill volume fraction for two values of cohesion.An increase in either fill fraction or cohesion is detrimental toaxial mixing (axial dispersion per axial periodic length showsidentical behavior). This result is consistent with the transit timedistributions (Fig. 11), which are narrower for both increasing filland increasing cohesion. The same observation has been reportedin Sarkar and Wassgren (2009) for free-flowing particles.

5. Conclusions

Results from DEM simulations of a periodic section of acontinuous mixer section have been presented. The first set ofsimulations investigates the influence of varying cohesion andimpeller speed on flow and mixing. The second set studies theinfluence of the mixer bulk fill fraction for varying levels of cohesion.

For varying cohesion and Froude number, changes in flow ratesare subtle. Differences in axial flow are more significant at smallerimpeller speeds where coordinated particle movement is ob-served due to cohesion. At larger speeds, larger shear ratesovercome particle bonding so that cohesive particles flow in amanner similar to non-cohesive material.

Transverse and axial dispersions are found to be dependent onboth cohesion and Froude number. Mixing per rotation improvesat larger Froude numbers, aided by an increase in impelleragitation and fluidization. Optimal transverse mixing is observedat a moderate cohesion value, similar to what has been previouslyobserved in batch blenders. Mixing per unit axial length for freeflowing powders is found to be best at low impeller speeds, as alarger number of blade passes are encountered during thematerial’s residence time in the blender. For more cohesiveparticles, an intermediate value of impeller speed is found to givethe best mixing over a fixed axial length, as both the dispersionproduced per pass and the number of blade passes felt by thepowder need to be sufficiently large.

Variations in fill fraction produce trends similar to thoseobserved in the non-cohesive work of Sarkar and Wassgren(2009). The flow and mixing trends seen with increasing fill areidentical for both the smaller and larger cohesion cases con-sidered presently. A maximum value in the degree of transversemixing per rotation occurs at a fill fraction of approximately 55%,whereas the largest axial mixing per rotation is observed at thelowest simulated bulk fill volume fraction of 25%.

The average transit time in the blender is a strong function ofthe Froude number, with larger Froude numbers resulting insmaller average transit times. Although larger Froude numbersproduce better mixing, the resulting smaller residence times maylead to the mixture components passing through the blenderwithout being well homogenized. Operating a continuous mixerat a higher speed would require a longer mixer axial length toensure sufficient agitation of the material by the blades. Portilloet al. (2008) showed that increasing the backward tilt of themixer increases residence time and also results in improvedmixing. Increasing cohesion narrows the transit time distribution,which in turn decreases axial mixing. Although varying fillfraction does not significantly affect the average transit time,the width of the distribution is affected. Smaller fills have a widertransit time distribution than larger fill fractions. The cause isattributed to the decline in unsteady axial flow behavior as fillfraction increases.

A. Sarkar, C. Wassgren / Chemical Engineering Science 65 (2010) 5687–5698 5697

Nomenclature

Bog granular Bond numberBog,max maximum granular Bond number for the Luding modelDdrum mixer drum diameter, [L]Dshaft central shaft diameter, [L]dp particle diameter, [L]Fr Froude number given by o2Ddrum/(2g)FN,max normal contact force in the Luding (2005) force model

[MLT�2]FN,max, FN,min maximum and minimum normal force in the Luding

force model, [MLT�2]f(ttr) transit time probability distribution, [T�1]g acceleration due to gravity, [LT�2]kU, kL, kC unloading, loading, and cohesive stiffness for the Luding

normal force model, [MT�2]kC

* dimensionless cohesive stiffness for the Luding normalforce model

kU,pp* , kC,pp

* dimensionless unloading and cohesive particle–parti-cle stiffness for a particle–particle contact, madedimensionless by rdp

2g

kU,pw* , kC,pw

* dimensionless unloading and cohesive stiffness for aparticle–wall contact, made dimensionless by rdp

2g

Lblade impeller blade length, [L]Lperiodic mixer periodic length, [i]MXY,bin, MZ,bin local radial-plane (XY) and axial (Z) dispersion

indices in a (cubical) bin over one shaft rotationMXY,overall, MZ,overall overall radial-plane (XY) and axial (Z) disper-

sion indices for the mixer over one shaft rotationMXY,transit, MZ,transit overall radial-plane (XY) and axial (Z) disper-

sion indices for the mixer over the average transit timem mass of a particle given by 1=6prd3

p , [M]Nbin number of particles in a (cubical) binNp total number of particles in a simulationTZ,bin axial component of the granular for a group of particles

in a bin, [L2T�2]TZ,overall overall axial component of granular temperature taken

over all bins, [L2T�2]ttr particle transit time, [T]Dt simulation time step, [T]vaxial,i axial velocity for particle i, [LT�1]vaxial average axial velocity averaged over all particles, [LT�1]v0Z deviation of particle axial (Z) velocity from the mean

axial velocity, [LT�1]vZ,i axial (Z) velocity of particle i, [LT�1]vZ average axial (Z) velocity for a group of particles, [LT�1]wblade impeller blade width, [L]Dxi, Dyi, Dzi displacement of particle i over one impeller rotation,

[L]Dx, Dy, Dz average displacement of all particles in a bin over one

shaft rotation, [L]d contact overlap in the Luding normal force model, [L]dmax, dmin, dres maximum, minimum, and residual overlap in the

Luding force model, [L]eN,pp particle–particle normal restitution coefficienteN,pw particle–wall normal restitution coefficientyblade impeller blade inclination, degreesmpp particle–particle friction coefficientmpw particle–wall friction coefficientn bulk fill fractionr particle density, [ML�3]f particle size dispersity; all particle radii fall in the

interval [dp(1�f),dp(1+f)] with uniform probabilitydistribution

o rotation speed of the impeller blades, [T�1]

Acknowledgements

The authors are grateful to the National Science Foundation Engi-neering Research Center for Structured Organic Particulate Systems(NSF ERC-SOPS, EEC-0540855) for financial support. The authors alsothank members of the Particulate Systems Laboratory (PSL) at PurdueUniversity for their constructive comments.

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