Chemical Engineering Science 60 (2005) 4083–4091

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Longitudinal and transverse mixing in rotary kilns:A discrete elementmethod approach

G.J. Finniea, N.P. Kruytb,∗, M. Yec, C. Zeilstrac, J.A.M. Kuipersc

aDepartment of Chemical Engineering, Soshanguve Campus, Tshwane University of Technology, Private Bag X07, Pretoria North, Republic of South AfricabDepartment of Mechanical Engineering, University of Twente, P.O. Box 217, 7500 AE Enschede, The NetherlandscDepartment of Chemical Engineering, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands

Received 13 September 2004; received in revised form 20 October 2004; accepted 4 December 2004Available online 7 April 2005

Abstract

Longitudinal and transverse mixing in horizontal rotary kilns has been studied, using a three-dimensional Discrete Element Methodapproach. The focus is on the effect of the main operating conditions, i.e., the filling degree and the rotational speed of the rotary kiln,on quantitative measures of longitudinal and transverse mixing. Discrete Element Method simulations have been performed for variousvalues of the main operating conditions. Flow regimes have been determined from visualisations of the simulations. It is shown that themixing in the longitudinal direction can be described by a diffusion equation. The dependence of the diffusion coefficient on the operatingconditions has been determined from the results of the simulations. It is found that the diffusion coefficient increases linearly with rotationalspeed, while the influence of the filling degree is relatively small. The mixing in the transverse plane is quantified by an entropy-likemixing index. The results of the simulations show that this mixing index varies exponentially with the number of revolutions of the rotarykiln. The dependence on the operating conditions of the transverse mixing speed, as determined from the exponential behaviour of themixing index, has been determined. This mixing speed decreases with increasing rotational speed and with increasing filling degree. Thetransverse mixing speed that has been determined from the results of additional two-dimensional simulations is generally larger than thatobserved in the corresponding three-dimensional simulations. 2005 Elsevier Ltd. All rights reserved.

Keywords:Rotary kiln; Granular material; Mixing; Discrete Element Method; Diffusion; Simulation

1. Introduction

Rotary kilns are widely used in chemical and mechan-ical engineering for processing of granular materials forvarious purposes, such as for mixing, drying and coat-ing of materials, for calcining of limestone, for reduc-tion of iron ore and for waste incineration. Rotary kilnsare able to handle varying feedstocks, in particular with

∗ Corresponding author. Tel.: +31 53 489 2528; fax: +31 53 489 3695.E-mail addresses:finniegj@tut.ac.za(G.J. Finnie),

n.p.kruyt@utwente.nl(N.P. Kruyt), m.yemao@tnw.utwente.nl(M. Ye),c.zeilstra@tnw.utwente.nl(C. Zeilstra),j.a.m.kuipers@utwente.nl(J.A.M. Kuipers).

0009-2509/$ - see front matter 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.ces.2004.12.048

wide particle-size distributions and significant differences inphysical properties.

The operating conditions of a rotary kiln (as sketched inFig. 1) involve rotational speed around its axis, radiusR,lengthL in longitudinal direction, inclination with respectto the horizontal direction and filling degreeJ (i.e., the frac-tion of the volume of the rotary kiln that is filled with gran-ular material, including voids between particles), as well asparameters related to the feedstock.

In the sequel the focus is on horizontal rotary kilns, i.e., = 0. Inclined rotary kilns were studied bySudah et al.(2002).

Generally, the main operating conditions of rotary kilnsare expressed in terms of non-dimensional numbers bythe filling degreeJ and the Froude numberFr, which is

4084 G.J. Finnie et al. / Chemical Engineering Science 60 (2005) 4083–4091

Fig. 1. Sketch of the geometry of the horizontal rotary kiln.

defined by

Fr = 2R

g, (1)

whereg is the acceleration due to gravity.Two different types of mixing can be distinguished in ro-

tary kilns. The first type occurs in the plane transverse tothe axis of rotation, while the second type occurs in longi-tudinal direction. The first type of mixing is vigorous, whilethe second is much slower.

Depending on the non-dimensional operating conditions,several flow regimes may be observed in the plane trans-verse to the axis of rotation, namely slipping, slumping,rolling, cascading, cataracting and centrifuging (Henein etal., 1983a,b; Mellmann, 2001). In industrial operations therolling regime is mostly encountered, since it promotes goodmixing of particles. This flow regime is characterised by twodistinct regions: (i) a relatively thin active layer is formed atthe top in which the material flows along the inclined slopeand (ii) a thick passive region in which the material rotatesas a rigid body with the angular velocity of the rotary kiln(see for example,Boateng and Barr, 1997). Scaling rela-tionships for rotary kilns in the rolling and slumping regimehave been studied byDing et al. (2001).

The present study is a basic computational study of lon-gitudinal and transverse mixing in horizontal rotary kilns,focusing on the effect of the main operating conditions onquantitative measures of longitudinal and transverse mix-ing. A recent overview of computational methods for study-ing mixing of granular materials is given byMcCarthy etal. (2000). The method used here to study the longitudinaland transverse mixing processes is that of three-dimensionalDiscrete Element Method (DEM) simulations, as proposedby Cundall and Strack (1979a,b), for a number of non-dimensional operating conditions. The particle-size distribu-tion employed in the simulations is fairly narrow, so no sig-

nificant segregation is expected.A recent overview of mixingand segregation of granular materials is given byOttino andKhakhar (2000). A computational DEM study of mixing ofbinary particle mixtures (including density differences be-tween the two sizes) is reported byDury and Ristow (1999).At present, gas flows in the rotary kilns are not consideredhere.

Separate descriptions are formulated for the longitudinaland transverse mixing. The results of the DEM simulationsare analysed to determine quantitatively the influence of non-dimensional operating conditions of the rotary kiln on the pa-rameters describing the longitudinal and transverse mixing.

The outline of this study is as follows. Firstly, the DEMsimulations are described in Section 2. This is followed inSection 3 by the formulation of measures to quantify longi-tudinal and transverse mixing. The results of the simulationsare analysed in Section 4. Finally, findings from this studyare discussed.

2. Discrete element method simulations

The computational method employed here is the three-dimensional DEM (also called Discrete Particle Method), asproposed byCundall and Strack (1979a,b). In this methodthe motion of a large number of particles is determined nu-merically by integrating the equations of motion, using asuitable model, the so-called contact constitutive relation,for the particle–particle interactions and the particle–wallinteractions. Here, equations of motions are solved for par-ticle translations as well as for particle rotations, contrary towhat was done bySchutyser et al. (2001). Particle shapesare restricted to spheres (with varying radii) for reasons ofcomputational efficiency.

The contact constitutive relation employed here is basi-cally that ofCundall and Strack (1979a,b). It involves (lin-ear) elastic behaviour, Coulomb frictional behaviour andviscous contact damping to model dissipative collisions be-tween particles.

A linear spring is acting in the direction normal to thecontact and a tangential spring is acting in the directionof the tangential component of the relative displacement atthe contact. The respective spring constants are denoted bykn andkt . Coulomb friction is modelled by an interparticlefriction coefficient, such that at each contactc we have‖f c

t ‖|f cn | wheref c

n is the (scalar) normal component ofthe contact force vectorf c, i.e., f c

n = f c · n c with n c theunit normal vector at the contactc, ‖f c

t ‖ is the magnitude ofthe tangential component of the force, i.e.,f c

t = f c − f cn n c

and |f cn | is the absolute value of the normal component of

the contact force. Viscous contact damping is implementedin terms of a coefficient of restitutioncr (see for example,Hoomans, 2000). A similar contact constitutive relation isused for the interactions between particles and cylinder wallsand end walls of the rotary kiln, but with a different valuefor the friction coefficient. Unlike in the simulations by

G.J. Finnie et al. / Chemical Engineering Science 60 (2005) 4083–4091 4085

Yang et al. (2003), no rolling resistance was included here inthe contact constitutive relation, considering the problem ofestablishing a manner for determining the material parame-ters involved in the rolling resistance model.

The radiir of the spherical particles follow a (truncated)lognormal particle-size distribution withrmax/rmin = 1.4.The ratio of the standard deviation over the mean of theparticle-size distribution equals 0.04. Hence, the particle-size distribution is fairly narrow, so no significant size segre-gation is expected. Furthermore, all particles have the samedensity, so no density segregation will occur.

The elastic parameterkn has been selected such thatthe “overlap” between particles is small, yet the time stepin the (explicit) numerical integration of the equations ofmotion does not become exceedingly small. The stiffnessratio kt/kn =0.29. A fairly high interparticle friction coeffi-cient is 0.5 is selected, since real particles are not perfectlyspherical (as is assumed here). Non-spherical particles havea higher effective (continuum) friction, in comparison tospherical particles with the same interparticle friction co-efficient, due to suppression of rolling (Rothenburg andBathurst, 1991). A high friction coefficient of 1.5 is usedfor the interactions between particles and cylinder wall,corresponding to a roughened wall. The friction coefficientof the end walls of the rotary kiln is 0.01. The selectedrestitution coefficientcr is 0.71.

Note that the value of the (microscopic) interparticle fric-tion coefficient and the so-called (macroscopic) internal fric-tion coefficient of the granular material as a continuum (re-lated to the static angle of repose of the granular material)are generally not equal. In fact, there is no direct relation be-tween these two friction coefficients (Skinner, 1969). Sim-ilarly, the value of the particle–wall friction coefficient isgenerally not equal to the wall friction coefficient of thegranular material as a continuum.

The geometry of the rotary kiln (see alsoFig. 1) is suchthat its aspect ratio,L/R, equals 2.0. The ratio of the kilnradiusR over the average particle radiusr is R/r = 40.0.

Several assemblies were simulated, with the number ofparticles in the simulations,Np, depending on the fillingdegreeJ of the kiln: for J = 20%, Np = 10, 667, for J =30%, Np = 16, 000 and forJ = 40%, Np = 21, 333. Thesenumbers of particles were determined from considerationsof acceptable computer simulation times.

The time step in the explicit numerical integration of theequations of motion is such that it is a factor 30 smaller thanthe maximum time step allowed for stability of the numericalmethod.

Five values for the rotational speed of the kiln,, havebeen considered. These correspond to Froude numbers, see(1), of Fr = 0.25, 0.56, 1.00, 1.56, 2.25. In total, 15 simu-lations have been performed for the five values of FroudenumberFr, and for each Froude number with the three fill-ing degreesJ = 20%, 30%, 40%. For each simulation thesimulated time corresponds to 40 revolutions of the rotarykiln.

3. Description of longitudinal and transverse mixing

Two different types of mixing occur in rotary kilns. Thefirst type is mixing in the transverse plane. This type ofmixing is vigorous and quickly leads to mixed-out states.The second, much slower type is mixing in the longitudinaldirection. Separate descriptions are formulated for these twotypes of mixing.

3.1. Longitudinal mixing

In horizontal rotary kilns there is no net transport in thelongitudinal direction. Motion of the particles in longitudi-nal direction is the result of interparticle collisions inducedby the rotation of the rotary kiln. This motion is therefore ex-pected to be completely random. Consequently, the mixingprocess is considered as analogous to diffusion, and henceFick’s law should apply. This leads to the one-dimensionaldiffusion equation (Hogg et al., 1966)

t

c(x, t) = x

[D∗ c(x, t)

x

](2)

for practically identical particles, wherec(x, t) is the rela-tive concentration of the mixture after timet at positionxin longitudinal direction andD∗ (with units m2 s−1) is thediffusion coefficient that is assumed to be spatially uniformin the sequel.

By performing the transformationn = (t)/(2), thisequation can be expressed in terms of the (non-dimensional)number of revolutions,n, of the rotary kiln instead of timet. The resulting equation becomes

n

c(x, n) = x

[D

c(x, n)

x

], (3)

where the units ofD are m2. The diffusion coefficientsD∗andD are related by

D = 2

D∗. (4)

With initial conditionsc(x, 0) = 0 for −L/2< x < 0 andc(x, 0)=1 for 0x < L/2 (corresponding to “blue” particleson the left and “red” particles to the right of the interfaceat x = 0; see also the sketch of the initial configuration forthe longitudinal mixing on the left-hand side ofFig. 2) andboundary conditionsc(x, n)/x|x=±L/2=0 (correspondingto zero flux at the boundaries), the solution of the diffusionequation (3) is (Hogg et al., 1966)

c(x, n) = 1

2+ 2

∞∑j=1

1

2j − 1exp

[−(2j − 1)22Dn

L2

]

× sin(2j − 1)x

L. (5)

4086 G.J. Finnie et al. / Chemical Engineering Science 60 (2005) 4083–4091

“blue” “red” “gray” “black”

Longitudinal Transverse

Fig. 2. Sketch of initial configurations for longitudinal mixing (left) andtransverse mixing (right).

The (time-dependent) non-dimensional mean position ofthe “red” particles is defined by

〈c〉(n) = 1

L2

∫ L/2

−L/2xc(x, n) dx. (6)

Using (5), it follows (Kohring, 1995) that〈c〉(n) is given by

〈c〉(n) = 8

3

∞∑j=1

(−1)j+1

(2j − 1)3 exp

[−(2j − 1)22Dn

L2

]. (7)

After some algebra it follows that for small values of theFourier number(Dn)/L2, this expression can be approxi-mated by

〈c〉(n)1

4− 2

Dn

L2 forDn

L2 1. (8)

Thus, for small Fourier number, the relationship betweenmean position of the “red” particles and (non-dimensional)time is linear.

It is expected that the diffusion coefficientD depends onthe process parameters andJ , the gravitational accelera-tion g, as well as on the geometry of the rotary kiln, i.e., itsradiusR and lengthL. Furthermore, it assumed that of thematerial properties only the average particle sizer, density and frictional material parameters (collectively denotedby here) are important. This means that the influence ofelastic and damping parameters of the granular material isexcluded. Thus we obtain the following functional relation-ship:

D = F(, J, g, R, L, r,,). (9)

From dimensional analysis we find the functional relation-ships forD andD∗

D

R2 = G

(Fr, J,,

R

r,L

R

),

D∗

R2 = G∗(Fr, J,,

R

r,L

R

). (10)

An analogous dimensional analysis for the motion in thetransverse plane was performed byDing et al. (2001).

3.2. Transverse mixing

In analogy to the case of longitudinal mixing, as describedin Section 3.1,Heydenrych (2001)solved a simplifiedconvection-diffusion equation for the mixing in the trans-verse plane. This complex approach was not pursued here,considering the spatial variations in velocity that have to betaken into account in the convective part of the convection-diffusion equation for the concentrationc(x, n) and thepossible spatial variations of the diffusion coefficientD inthe transverse plane.

Instead, transverse mixing is quantified here using an“entropy”-like quantity, as proposed bySchutyser et al.(2001, 2002). In this approach a grid with dimensions(I, J )

is defined that overlaps the region of interest. For each gridcell (i, j), an “entropy”Si,j is calculated as

Si,j = −kV i,j [pi,j ln pi,j + qi,j ln qi,j ], (11)

wherek is a constant,Vi,j is the volume occupied by theparticles in cell(i, j), pi,j is the concentration (fraction) of“black” particles, whileqi,j = 1 − pi,j is the concentration(fraction) of “grey” particles. By definition,pi,j is the ra-tio of the total volume in cell(i, j) occupied by “black”particles over the total volume in cell(i, j) occupied by“black” and “grey” particles. Hence, the voids between par-ticles do not affect the concentrationspi,j andqi,j . Note thatmany particles, with differing longitudinal position, may bemapped into the same cell in the transverse plane. Initially,the colours of the particles are set to “grey” when they arelocated in the left-hand side (when viewed along the kilnaxis) of the rotary kiln, and to “black” when they are lo-cated in the right-hand side; see also the sketch of the initialconfiguration for the transverse mixing on the right-handside ofFig. 2. Note that this “entropy”Si,j is not related tothe thermodynamical entropy of the granular material. Suchinformation-theoretical entropies are used in many applica-tions (Katz, 1967; Kapur and Kesavan, 1992).

A “mixing index” M(n) is now defined by

M(n) =∑i,j

Si,j (n)

S∞, (12)

where the “entropy” of the perfectly mixed state,S∞, isgiven byS∞ = −kV [p0 ln p0 + q0 ln q0], V is the total vol-ume occupied by the particles,p0 is the concentration (frac-tion) of “black” particles in the system andq0=1−p0 is theconcentration (fraction) of “grey” particles in the system.In the initial state the mixing indexM(0) = 0, while in theperfectly mixed state that is attained asn → ∞, the mixingindex equalsM(n → ∞)=1. Note that the constantk in Eq.(11) cancels in expression (12) for the mixing indexM(n).

In the determination of the concentrationspi,j , it wastaken into account that particles may occupy multiple cells.Thus, the fractional volumes of particles inside the cellswere used to determinepi,j andqi,j , instead of basing theirdetermination solely on the coordinates of particle centres.

G.J. Finnie et al. / Chemical Engineering Science 60 (2005) 4083–4091 4087

Fig. 3. Employed fractional volumes in the determination of the cellconcentrationspi,j and qi,j .

This procedure is shown schematically inFig. 3 for disks.The (approximate) computation of the fractional volumes isstraightforward (Hoomans, 2000), provided that the lengthof the cells is larger than the maximum particle diameter.This procedure reduces the dependence of the computed“entropy” on the grid sizes (especially when the numberof particles in each cell is fairly small, such as in two-dimensional simulations) that was reported bySchutyseret al. (2001).

4. Results

The results of the three-dimensional DEM simulationshave been analysed with respect to occurring flow regimes,longitudinal mixing and transverse mixing. Additional two-dimensional DEM simulations have been performed in orderto compare the transverse mixing in two-dimensional andthree-dimensional simulations.

4.1. Flow regimes

Depending on the non-dimensional process parameters,such as Froude numberFr and filling degreeJ, various flowregimes may be distinguished: sliding, surging, slumping,rolling, cascading, cataracting and centrifuging.An overviewof the flow regimes, together with transition criteria betweenthe flow regimes, is given byMellmann (2001).

Table 1Flow regimes in three-dimensional simulations

Fr = 0.25 Fr = 0.56 Fr = 1.00 Fr = 1.56 Fr = 2.25

J = 20% Rolling Rolling Rolling/cataracting Cataracting CataractingJ = 30% Rolling Rolling Rolling/cataracting Cataracting CataractingJ = 40% Rolling Rolling Rolling/cataracting Cataracting Cataracting

Kiln speed, Ω⋅R Kiln speed, Ω⋅R

Fig. 4. Velocity of particles in a transverse cross-section. Left: rollingregime forJ =40% andFr=0.25; right: cataracting regime forJ =40%andFr = 2.25. Note that the magnitude of the velocity vectors has beenscaled for visual clarity. For comparison, the kiln speed,R, has alsobeen plotted at the top of the figures.

Based on analyses of the particle-velocity profiles andon visualisations of the results of the simulations, the flowregimes were identified, seeTable 1. Examples of the ob-served velocity profiles are shown inFig. 4 for the rollingregime (left) and for the cataracting regime (right).

A direct quantitative comparison of these results with thetransition criteria given byMellmann (2001)is not possible,since these criteria involve the continuum internal frictionand wall friction coefficients. As noted in Section 2, thesecontinuum friction coefficients are not (necessarily) equalto the particle–particle and the particle–wall friction coeffi-cients, respectively.

Note that the transition from the cataracting regime to thecentrifuging regime does not have to correspond toFr=1, asis discussed byMellmann (2001). The centrifuging regimehas not been clearly observed in the simulations. This isattributed to the low effective friction between the granularmaterial and the wall of the rotary kiln that is caused bythe rolling tendency of spheres and disks (see for example,Rothenburg and Bathurst, 1991).

Shinbrot et al. (1999)experimentally observed chaotictransverse mixing for very fine particles. Since the particlesconsidered here are fairly large, this chaotic mixing was notobserved here.

4.2. Longitudinal mixing

In the initial configuration atn=0, all particles on the leftwere “blue” and all particles to the right of the interface at

4088 G.J. Finnie et al. / Chemical Engineering Science 60 (2005) 4083–4091

0 5 10 15 20 25 30 35 400.22

0.225

0.23

0.235

0.24

0.245

0.25

n

<c

>(n

)

Fig. 5. Evolution of mean position of the “red” particles with time(expressed in terms of the number of revolutions,n). Results forFr=1.00andJ = 30%.

x=0 were “red” (see also the left-hand side ofFig. 2). Sincethe evolution of all particle positions is determined in theDEM simulations, concentration profiles can be determinedby dividing the longitudinal extent of the rotary kiln intoslices and computing, for each time instant, the volume frac-tion of “red” particles in each slice. Thus, time-dependent(with time expressed in terms of the number of revolutions,n) concentrations profilesc(x, n) were obtained, as well astime-dependent mean positions〈c〉(n) of the “red” particles.

The diffusion coefficientsD were determined from thesimulations by considering the evolution of the mean po-sition 〈c〉(n) with number of revolutionsn. According toEq. (8),〈c〉(n) depends linearly on Fourier number(Dn)/L2

for small (Dn)/L2. The results of the simulations confirmthis linear relationship between〈c〉(n) andn, seeFig. 5.

According to the theory presented in Section 3.1 for thelongitudinal mixing, the concentration profiles are given byEq. (5). A typical comparison of the actual and the theo-retical concentration profiles is given inFig. 6. This Figureshows that the actual concentration profiles, as determinedfrom the simulations, closely correspond to the theoreticalconcentration profiles that follow from the solution of thediffusion equation (3). This confirms the correctness of de-scribing the longitudinal mixing process by the diffusionequation (3). Note that all theoretical concentration profilesshown inFig. 6 (for a single operating condition) employthe same value for the diffusion coefficient.

The dependence of the diffusion coefficientD, as deter-mined from the results of the simulations using Eq. (8), onthe non-dimensional operating parametersFr andJ is shownin Fig. 7. Note that the diffusion coefficient is presented ina non-dimensional form (compare Eq. (10)).

Fig. 7 shows that the influence ofFr (i.e., the rotationalspeed) on the diffusion coefficientD is small. This meansthat the diffusion coefficientD∗ will be proportional to, as

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x/L

c(x,

n)

n=40 revs

n=20 revsn=10 revs

n=30 revs

Fig. 6. Concentration profiles for various time instants (expressed in termsof the number of revolutions); results forFr = 1.00 andJ = 30%. Themarkers give the actual concentration as determined from the simulations,while the solid lines indicate the theoretical concentration profiles, seeEq. (5), with a (single) diffusion coefficientD.

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

-3

Fr

D /R

2

J=20%J=30%J=40%

Fig. 7. Dependence of non-dimensional diffusion coefficient on FroudenumberFr and filling degreeJ.

follows from Eq. (4). Hence there is no diffusion when therotary kiln does not rotate, = 0, as there is no equivalentof thermal movement of the particles in this case.

For filling degreeJ =20% andFr1.5, the diffusion co-efficientsD are higher than forJ =30% and 40%, while forFr1.0 they are practically the same. The diffusion coeffi-cients forJ = 30% and 40% are practically identical.

Results for the dependence of the diffusion coefficientD∗ on operating conditions were given byDury and Ristow(1999)(using DEM simulations withJ55% and a binarymixture of grain sizes),Rutgers (1965)andRao et al. (1991)(based on experiments) andSheritt et al. (2003)(based onexperiments and an analysis of data from the literature).

G.J. Finnie et al. / Chemical Engineering Science 60 (2005) 4083–4091 4089

0 5 10 15 20 25 30 35 40

n

M(n

)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 8. Evolution of mixing indexM(n). The markers denote the mixingindex as determined from the simulations, while the solid line correspondsto the exponential form (13) with a fitted parameterS. Results forFr=1.00andJ = 30%.

Caution must be exercised in quantitative comparison ofthese results, since they have been obtained for differentmaterials.

The reported dependence of diffusion coefficientD∗ onrotational speed is contradictory:Dury and Ristow (1999)give D∗ ∝ 2.0, Rao et al. (1991)find D∗ ∝ 1.0, whileRutgers (1965)andSheritt et al. (2003)obtainD∗ ∝ 0.5

andD∗ ∝ 0.4, respectively. The dependence observed here,D∗ ∝ 1.0, is identical to that ofRao et al. (1991).

Rutgers (1965), Rao et al. (1991)andSheritt et al. (2003)find thatD∗ generally decreases with increasing filling de-greeJ , as is also noted here. In particular,Rutgers (1965)andSheritt et al. (2003)give D∗ ∝ J−0.5 in the rolling andcascading flow regime. This dependence is stronger thanwhat is observed here.

4.3. Transverse mixing

The evolution with number of revolutionsn of the in-dex M(n) for mixing in the transverse plane, as defined inEq. (12), is illustrated inFig. 8. This evolution can be accu-rately described by an exponential form

M(n)1 − exp(−Sn), (13)

where the parameterSis a measure of the speed of the mixingprocess in the transverse plane (largeScorresponds to quickmixing). Note that this transverse mixing is expressed interms of the number of revolutionsn of the rotary kiln, notin time t. The parameterSwas determined by fitting the datafrom the simulations with the exponential form (13).

The dependence of the parameterS on the non-dimensional operating parameters Froude numberFr andfilling degreeJ is shown inFig. 9. This transverse mix-ing speedS (in terms of number of revolutions!) decreases

0 0.5 1 1.5 2 2.5Fr

S

3D simulationsJ=20%J=30%J=40%

0

0.1

0.2

0.3

0.4

0.5

0.6

Fig. 9. Dependence of transverse mixing speedS, see Eq. (13), onoperating parametersFr andJ in the three-dimensional DEM simulations.

with increasing rotational speed. This means that slow ro-tation, in comparison to faster rotation, enhances mixingin the transverse plane atidentical number of revolutions(not identical time). Similarly, low filling degreesJ alsoenhance transverse mixing.

4.4. Two-dimensional versus three-dimensionalsimulations: transverse mixing

Additional two-dimensional DEM simulations, with farfewer particles but identical particle-size distribution andparticle properties as described in Section 2, have been per-formed in which particle motions were restricted to a two-dimensional transverse plane. In these two-dimensional sim-ulations the filling degree is defined as the fraction of thearea of the transverse plane that is filled with granular ma-terial (including voids between particles).

The resulting flow regimes and transverse mixing speedsin these two-dimensional simulations have been studied,analogous to those for the three-dimensional case (see Sec-tions 3.2, 4.1 and 4.3). This additional investigation has beenperformed to determine the effect of the dimensionality ofthe simulations (two-dimensional versus three-dimensional)on the flow regimes and the mixing speed. Note that the ac-curacy of the results in the two-dimensional case is smallerthan in the three-dimensional case, since the number of parti-cles in the simulation is fairly small (Np500 forJ =40%).

The observed flow regimes in the two-dimensional simu-lations, seeTable 2, differ from those observed in the three-dimensional simulations, seeTable 1. In the two-dimensionalsimulations the transitions, from rolling to cataracting andfrom cataracting to centrifuging, seem to occur at lower val-ues of Froude numberFr than in the corresponding three-dimensional simulations.

4090 G.J. Finnie et al. / Chemical Engineering Science 60 (2005) 4083–4091

Table 2Flow regimes in two-dimensional simulations

Fr = 0.25 Fr = 0.56 Fr = 1.00 Fr = 1.56 Fr = 2.25

J = 20% Rolling Rolling Rolling/cataracting Cataracting CataractingJ = 30% Rolling Rolling Rolling/cataracting Cataracting CataractingJ = 40% Rolling Rolling Cataracting Cataracting/centrifuging Centrifuging

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

Fr

S

2D simulationsJ=20%J=30%J=40%

Fig. 10. Dependence of transverse mixing speedS, see Eq. (13), onoperating parametersFr andJ in the two-dimensional DEM simulations.

In comparison to the transverse mixing speedS in three-dimensional DEM simulations (seeFig. 9), the mixing speedS in two-dimensional simulations is higher (seeFig. 10), al-though the trends are the same: decrease of mixing speedwith increasing rotational speed and increasing filling de-gree.

In the two-dimensional case particles have fewer neigh-bours (in comparison to the three-dimensional case) andhence are less constrained by other particles. This increasedmobility is seen as the cause of the changes in the transi-tions between the flow regimes and of the increase in thetransverse mixing speedS.

5. Discussion

A computational study, using three-dimensional DEMsimulations, has been performed of mixing in horizontalrotary kilns. Separate descriptions have been formulatedfor mixing in longitudinal and transverse directions. Thetransverse mixing is vigorous, while the longitudinal mix-ing is slow. DEM simulations have been performed forvarious values of the primary operating conditions, i.e., therotational speed and the filling degree.

The results of the simulations show that longitudinal mix-ing can be described by a one-dimensional diffusion equa-

tion. The dependence of the corresponding diffusion co-efficient on the operating conditions has been determined.The diffusion coefficientD∗ is proportional to the rotationalspeed, while its dependence on filling degreeJ is weak.

Mixing in the transverse plane has been characterised byan “entropy”-like mixing indexM(n). The results of thesimulations show that the evolution with number of revo-lutions of this mixing index is exponential. From this ex-ponential evolution, a transverse mixing speedS has beendefined. The dependence of this mixing speed on the oper-ating conditions has been determined from the simulations.It has been shown that, at identical number of revolutions,the speed of mixing in the transverse plane decreases withincreasing rotational speed and filling degreeJ.

A comparison of the transverse mixing speedS intwo-dimensional and three-dimensional DEM simulationsshowed that this mixing speed is higher in the two-dimensional case. Furthermore, transitions between flowregimes occur at different values of Froude numberFr. Thus,caution should be exercised when using two-dimensionalsimulations to quantity (three-dimensional) transversemixing.

The current investigation focused on the effect of the mainoperating conditions on the mixing behaviour, while keepingthe particle properties constant. The result of the dimensionalanalysis for the diffusion coefficient, see Eq. (10), suggeststhat the non-dimensional parametersR/r and may alsobe of importance. Evidence for the influence of the inter-particle friction coefficient is presented byDury and Ristow(1999), while evidence for the influence of the parameterR/r is given byRao et al. (1991). Further research on theinfluence of such parameters is recommended. A study ofthe effect ofR/r will make it necessary to perform simu-lations with a larger number of particles. If this is done, adetailed study of the velocity profile in the active layer inthe rolling regime can also be performed (seeBoateng andBarr, 1997). The present number of particles in the simula-tions is not sufficient for this purpose.

It is recommended to extend the DEM simulations by alsoconsidering non-spherical particles, since rotary kilns gen-erally operate with feedstock consisting of angular particles.Furthermore, with non-spherical particles the rolling mech-anism exhibited by spherical particles is suppressed. DEMsimulations with non-spherical particles would lead to anincreased effective friction coefficient (at an identical inter-particle friction). A major disadvantage of DEM simulationswith non-spherical particles is the increased computational

G.J. Finnie et al. / Chemical Engineering Science 60 (2005) 4083–4091 4091

complexity. A promising approach to this problem has re-cently been reported byKuhn (2003), who uses approxi-mate axisymmetric ellipsoids, the so-called “ovoids”. Thecomputational complexity of a DEM simulation employingthese ovoidal particles is about twice that of a similar DEMsimulation using spherical particles.

Future work will also deal with inclined rotary kilns, withmodels for gas flow through rotary kilns and with experi-mental studies of longitudinal and transverse mixing in ro-tary kilns.

Acknowledgements

The authors acknowledge financial support of the firstauthor, during his stay at the University of Twente, byIMPACT, the research institute of the University of Twenteon Mechanics, Processes and Control.

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