Ž .Powder Technology 109 2000 49–57www.elsevier.comrlocaterpowtec

Numerical simulation of particulate flow with liquid bridge betweenž /particles simulation of centrifugal tumbling granulator

Yoshitsugu Muguruma a,), Toshitsugu Tanaka b, Yutaka Tsuji b

a Manufacturing DiÕision, Department of Engineering, Shionogi and Co., 2-1-3 Kuise Terajima, Amagasaki, Hyougo 660-0813, Japanb Department of Mechanical Engineering, Osaka UniÕersity, 2-1 Ymada-oka, Suita, Osaka 565, Japan

Abstract

It is well known that a small amount of liquid has significant effects on the motion of particles. A numerical simulation of mono-sizedspherical particle flows with a small amount of water in a centrifugal tumbling granulator was made by using discrete element methodŽ .DEM . A model for the inter-particle force due to a liquid bridge was applied to the present work because granulating processes areusually performed by adding binding liquid. In the present simulation, this behavior was predicted when bridges are formed within thelayer. Once the locations of bridge were determined, the water was assumed to distribute uniformly among all locations. In addition to thesimulation, measurements were made to validate the simulation results. Particle velocities on the surface of a moving particle bed was

Ž .measured with a particle tracking velocimetry PTV technique. The calculated variation of particle velocities against the water contentagreed well with the experiments. q 2000 Elsevier Science S.A. All rights reserved.

Keywords: Centrifugal tumbling granulator; Particle velocity; Liquid bridge; DEM

1. Introduction

Ž .The discrete element method DEM simulation hasbecome a popular method of predicting particulate flows.Most previous works have dealt with comparatively simplecases where various complicated factors inherent to fineparticles are neglected. The adhesion force due to liquidbridges between particles is one of those factors which haslong been attracting attention in the field of powder tech-nology. Theories of the liquid bridge between two particles

w xhave been developed by several researchers 1,2 . Thearrangements of every particle are known in the DEMcalculation. Thus, it is not difficult to combine the theoreti-cal models of a liquid bridge with DEM calculations. The

w xpresent authors 11 attempted to develop a method consid-ering the effects of the liquid bridge in the DEM simula-tion. Results of a rigorous theory of the adhesion forcewere used in the formulation of contact forces betweenparticles, where the adhesion force acting on each particlepair was taken into account.

) Corresponding author. Tel.: q81-6-6401-4406; fax: q81-6-6401-8535; e-mail: yoshitsugu.muguruma@shionogi.co.jp

w x w xThornton et al. 3 and Lian et al. 4 also carried out theDEM simulation of agglomerate collisions taking accountof the liquid bridge. In their study, a specific volume ofliquid was assigned to all particle–particle gaps where theliquid bridges are able to form. In an actual granulator,however, the given condition is the total amount of wateradded to the particle layer. In the present work, we assumethat the water is distributed uniformly in all particle–par-ticle gaps where the liquid bridges are able to form. Toaccomplish this, we also developed the algorithm to treatthe liquid bridge based on the volume of water added inthe system.

The method was applied to the flow in a centrifugaltumbling granulator. The granulating process is made byadding a binding liquid. It is well known that even a smallamount of liquid has a significant effect on the particulateflow. The present numerical simulations revealed that theparticle motion was retarded by the liquid as was expected.Since it is difficult to measure the particle velocities insidethe layer, we tried to measure the particle velocities on thesurface of a moving particle bed for quantitative compari-son between simulations and experiments. The experimen-

Žtal method was based on a PTV particle tracking ve-.locimetry technique. It was confirmed that calculated

particle velocities compared well with measurements.

0032-5910r00r$ - see front matter q 2000 Elsevier Science S.A. All rights reserved.Ž .PII: S0032-5910 99 00226-0

( )Y. Muguruma et al.rPowder Technology 109 2000 49–5750

2. Equipment

Fig. 1 shows a schematic diagram of a granulator towhich the numerical simulation was applied. The granula-tor consists of a rotor, 360 mm in diameter and a stator,376 mm in diameter. The standard content of particles inthe present granulator is 3 kg. The granulator of this size isused for small-scale experiments or production in pharma-ceutical companies. Particles form an annular layer andshow a spiral motion as shown in Fig. 1. In practice, thegranulation is progressed by adding the binder liquid pro-vided by a spray gun, and the binder liquid is evaporatedby dry air supplied through the slit between the rotor andthe stator. However, for simplicity, the dry air and thespraying were not considered in this work. Instead ofspraying, a controlled amount of water was added to thefield at the beginning of experiments. Mono-sized spheri-cal grass beads were used as the test particles.

3. Methods

3.1. Model of particle–water system

The particle–water systems show several patterns de-pending on the amount of water. In the case of a smallvolume fraction of water, the liquid bridges formed in thegap of particles are isolated from each other as shown inFig. 2. As the amount of water increases, the bridgesbecome combined. A further increase of water leads to theformation of a slurry. Granulating processes that produceagglomerate particles from fine particles are generallyoperated under the condition of a small amount of binder

w xliquid 5 . Thus, the effect of water in the present

Fig. 1. Schematic diagram of the granulator.

Fig. 2. Model of contact with liquid bridge.

particle–water system is modeled based on the liquidbridge between two particles as shown in Fig. 2.

The liquid bridges cause various effects on the particles,such as the adhesion force caused by the capillary forces,the shearing force caused by the viscosity of the liquid andrelative displacement of the particles. In the centrifugaltumbling granulator, particles are assumed to have a smallrelative displacement. Hence, the lubrication forces areneglected for simplicity. That is, only capillary attractiveforces between the particles are considered.

3.2. Equation of particle motion

The equations of translational and rotational particlemotion are given by

Ý fj i jr s qg 1Ž .¨i mi

Ý Tj i jv s 2Ž .˙ i Ii

where r is the position vector of the particle i, m is thei i

mass of the particle i, f are the forces acting betweeni j

particles i and j, g is the gravitational acceleration vector,v is the angular velocity vector of the particle i, T is thei i j

torque caused by the contact forces, and I is the momenti

of inertia of the particle i. The dot in the above equationsdenotes time derivative; the summation Ý is carried outj

among particles and walls contacted with particle i.

3.3. Inter-particle forces

Fig. 2 shows the forces acting on the particle i. Both ofthe contact forces f and the adhesion force of a liquidC i j

bridge f are taken into account for contacting particles.L i j

If a liquid bridge is formed between separated particles,only the adhesion force f is considered.L i j

The contact force f between particles is given byC i jw xthe model 6 in Fig. 3, which was first proposed by

w xCundall and Strack 7 . This model expresses the contact

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Fig. 3. Model of contact force.

forces using the same concepts as springs, dash-pots andfriction sliders. Concerning the adhesion force of liquidbridge f , its formulation is described in Section 4.L i j

4. Modeling of liquid bridge

4.1. Adhesion force of liquid bridge

When the liquid bridge is formed between two particles,the shape of the liquid bridge and the capillary force are

w xcalculated from the Laplace’s equation 8 . The Laplace’sequation, however, cannot be solved analytically. Before

w xabove exact study, Fisher 1 had supposed that the shapesof liquid bridge approximate to an arc of a circle as shownin Fig. 4. This approximation gives the adhesion force f L

as

1 12f sp r g y q2p r g 3Ž .L 2 2ž /r r1 2

where g is a surface tension of liquid, and r ,r are the1 2

radii of curvatures of the liquid bridge surface. The firstterm in the right hand side arises from the reduced Laplacepressure. The second term is due to the surface tension. Adimensionless force F is given byL

1 R R qRŽ .2 1 2F s f r2p r g s 4Ž . Ž .L L 0 2 R1

where r is the particle radius, R sr rr , and R s0 1 1 0 2

r rr .2 0

4.2. Geometry of liquid bridge

Available data for the calculation of f or F , are theL L

volume of a liquid bridge between particles and the lengthof the gap between particles, h. The length h is easilyobtained from the position of particles. The volume of aliquid bridge is obtained from the volume of total liquidadded in the system by using an algorithm that will bementioned later. Let us define the non-dimensional volumeof a liquid bridge F as follows.

Volume of a liquid bridgeFs 5Ž .

Volume of a particle

To calculate the adhesion force, the relation between theknown data F ,h and the curvature R , R is needed.1 2

Assume that the shape of a liquid bridge approximatesto an arc of circle and the contact angle of liquid–particleis zero as shown in Fig. 4. These assumptions are reason-able for the condition of a water–grass beads system.

Integrating the contour curve of the liquid bridge fromxs0 to xsx and then doubling the value corresponding0

to the half side, F is found to be given by

23 X0 2 2(Fs2 R qR y R yx d xH ž /2 1 2½ 4 0

3 X0 2y 1y xy1yH d xŽ .Ž .H 54 0

3 12 2 3s R qR qR X y XŽ .� 41 2 1 0 02 3

2 2 2(y R qR R yx X yR R qR uŽ . Ž .1 2 1 0 0 1 1 2

32y 1qH X yH Hq2 XŽ . Ž .0 02

31 H3 2y X q qH 6Ž .03 3

where Hshrr , and X sx rr . The dimensions of0 0 0 0

liquid bridge are normalized by particle radius r . Refer to0

Fig. 4 for the meaning of other variables.There are following geometrical relations among the

dimensions of liquid bridge, R , R , X , c , u and H.1 2 0

R 1qHŽ .1X s 7Ž .0 R q11

pus yc 8Ž .

2

1qHycos cR s 9Ž .1 cos c

1qH sin cy1 qcos cŽ . Ž .R s 10Ž .2 cos c

Fig. 4. Schematic diagram of liquid bridge.

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The following equation is derived by substituting Eqs.Ž . Ž . Ž .7 and 8 for Eq. 6 :

32Fs R 1qH y R qRŽ . Ž .1 1 2½2

=

2p H Hq3Ž .yc y 11Ž .ž / 52 3

Ž . Ž .Solving Eqs. 9 – 11 simultaneously, the curvature ofŽ .liquid bridge R , R are given by F and H. Eq. 4 gives1 2

the adhesion force of a liquid bridge F from R , R .L 1 2

4.3. Restriction on liquid bridge formation

Liquid bridges are formed not only between contactingparticles, but also between separating particles with a smallgap. It is necessary to give the limit length of the gap inwhich the liquid bridge is formed.

The liquid bridge can be sustained when the capillaryw xpressure is larger than zero. Kousaka et al. 9 derived the

limit length of gap H by solving the Laplace’s equa-max

tions under the condition of the capillary pressure equal tozero. H of neighboring particles is given bymax

1q 1yR( 2H sR ln q 1yR y1 12Ž .(max 2 2ž /R( 2

Finally, H is given by the volume of liquid bridge FmaxŽ . Ž .solving Eqs. 9 – 12 simultaneously.

4.4. Algorithms

We assumed that the water added to the field is dis-tributed uniformly among all gaps in which the formation

Fig. 5. Flow chart of calculation.

Table 1Conditions of simulation

y3 w xParticle diameter, 2 r 2.0=10 m03 3w xParticle density 2.52=10 kgrm

5 w xYoung’s modulus of particle 10 PaCoefficient of restitution 0.9Coefficient of friction 0.3Limits of periodic boundary, u p r32b

Number of particle, N 8880pw xRotational speed of rotor 300 rpm

2w xSurface tension of liquid, g 0.0725 JrmŽ .Water content, f 0.000 Dry , 0.007, 0.018,

w x0.052, 0.089, 0.125 y

of a liquid bridge is possible. This assumption is reason-able for the condition of the present particle–water system.This is because an uneven distribution of water will rapidlybecome uniform with the mixing effects of moving parti-cles. In addition to the above assumption, we have ne-glected the volume of a liquid film formed on the surface

Fig. 6. Snapshots of particle flow.

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Ž .Fig. 7. Velocity profiles of cross-section of particle layer effect of water content .

( )Y. Muguruma et al.rPowder Technology 109 2000 49–5754

of particles. This makes the calculated volume of a liquidbridge somewhat larger than the actual volume. Then, thenon-dimensional volume of a liquid bridge F is given by

N fpFs 13Ž .

Ns

where N is the number of particles, f is the volumep

fraction of the total liquid to the total volume of particles,and N is the number of gaps smaller than H .s max

Fig. 5 shows the flow chart of the present numericalŽ . Ž .simulation. In the steps c and f , the simultaneous

equations were solved numerically by using the Newton’sŽ . Ž .method. In this algorithm, the steps a and f affect each

other because H gives F , while F gives H . Calcu-max max

lation was carried out, however, sequentially as Fig. 5 withthe initial value H s0. H and F were convergedmax max

with progress of time step.

5. Results of calculation

The conditions of the simulation are shown in Table 1.The present simulation dealt with mono-sized sphere parti-cles. Actual physical properties of grass beads were usedexcept the Young’s modulus. For the calculation to be fastusing longer time steps, a smaller Young’s modulus than

w xthe actual one was used 6 . Periodic boundaries wereapplied for the region of u ;yu as shown in Fig. 1 inb b

order to reduce the number of particles treated. The num-ber of particles 8880 is equivalent to 3 kg of the wholecontent. The effects of water on the particulate flow werestudied by changing the amount of water.

An example of a calculated particle flow is shown inFig. 6 which were made by combining results correspond-ing to the periodic boundaries. The snapshots shown in thefigure are illustrated as a cross-sectional view. For making

Ž .the initial condition as shown in Fig. 6 a , particles wereplaced randomly at a distance over the granulator, and thenparticles were allowed to fall due to gravity and move untilthey were coming to rest. Initially, three different colorsare given to the particles according to the distance from

Ž . Ž .the axis as shown in Fig. 6 a . Fig. 6 b shows a steadystate of the flow. Particles form an annulus layer which isthe same behavior as observed in actual granulator. The

Ž .result of Fig. 6 b was obtained at the time when thecalculation proceeded by 1.5 s and so particles are notfully mixed. However, as the time goes further, mixingwill be more completed.

In order to observe the particulate flow, we focused onthe particle velocity profile in the cross-section of the

Ž . Ž .particle layer. Fig. 7 a – f shows the contours of thecircumferential components of particle velocity and vec-tors of the components normal to the circumferential one.fs0.000 means the dry condition in which the effect ofliquid bridge is not considered. These results were ob-

tained by local and time averaging. The time averagingwas done over 2.0–3.0 s after confirming a steady state ofthe flow.

First, let us see the global nature of the flow. It is clearfrom Fig. 7 that shear from the rotor to the stator isobserved in the circumferential flow. At the same time,circulating flow occurs in the cross-sectional plane. Inother words, a spiral motion of particles is produced. Thisresult is the same as observed in actual granulator.

Next, we focus on the effect of water on particleŽ . Ž .motion. It is shown from a comparison of Fig. 7 a and b

that the circulating flow is clearly reduced by the water.The more the water, the smaller the particle velocity as

Ž . Ž .shown in Fig. 7 b – f . As for circumferential components,there is little change in the velocity profiles due to thewater.

It is clear from the present simulation that the particu-late flow is retarded by the water. The effect of water ismore remarkable for the circulating flow.

6. Experiment

The particle velocity on the surface of the particle layerw xwas measured 10 by using a PTV technique. The appara-

tus used for the measurement is shown schematically inFig. 8. The particle flow was recorded by using a high-speed VTR system with speed of 250 images per second.Many images were extracted from VTR and those imageswere transmitted to PC consecutively. Particle positions

Žwere measured from each image. Brown grass beads 1.923 3.mm, 2.52=10 kgrm were used as test particles. As

tracer particles, a small amount of white Alumina beadsŽ 3 3.1.90 mm, 3.98=10 kgrm were supplied. To distin-guish the tracer particles from test particles, a polarized-

w xfiltering technique 10 was employed. Fig. 9 shows anexample of image obtained.

The particle velocity consists of three components, butin this experiment, only two components, Õ and Õ , wereu r

measured. Õ is the circumferential component and Õ isu r

Fig. 8. Schematic diagram of apparatus for particle velocity measurement.

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Fig. 9. Example of image of visual field with tracer particle.

the radial one, respectively. These particle velocities wereobtained from a comparison of particle positions measured

Ž . Ž .from consecutive two images. Fig. 10 a , b shows exam-ples of measured particle velocities which are plottedagainst the non-dimensional distance rrR, where r is the

Fig. 10. Experimental data of particle velocity profile.

particle position from the center of the rotor and R is thestator radius. Open symbols are raw data. Solid symbolsare those obtained by locally averaging the raw data withrespect to each rrR. From here on, a discussion of theresults will be based on solid symbols.

A prescribed amount of the water was poured onto theflowing particle layer by using a beaker. After a while,when particles and water were mixed sufficiently, VTRrecording was performed. The volume fraction of the waterf was obtained as follows. About 150 g of grass beadswere sampled and weighed after VTR recording. Sampleswere dried at 1208C in an oven. After that, the sampleswere cooled in a desiccator and were weighed again. Thedifference in weight before and after drying gives thequantity of water.

7. Comparison of experimental and calculated results

Ž . Ž .Fig. 11 a , b shows the comparison between calcula-tion and measurement about the particle velocity profile.The calculated results are shown by lines and experimentalones by points. These lines were obtained by averaging

Fig. 11. Particle velocity profile on the surface of particle layer.

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data over 2.0–3.0 s when the particulate flow becamestationary.

The circumferential components of particle velocity areŽ .shown in Fig. 11 a , where it is found that experimental

and calculated results show a good agreement. The veloci-ties decrease along rrR in common. Observing the figurein detail, the measured velocities are slightly reduced bywater. On the other hand, the calculated velocities of wetparticles lie around the result of the dry condition. Areduction in velocity due to the water is scarcely found inthe calculated results. If we see the results in figures morein detail, the calculation gives qualitatively the same ten-dency as the experiment even concerning minor points.With a small amount of water, the change in velocityagainst rrR is monotonous. However, with a furtherincrease of water, the change is not monotonous, butshowed a small peak near the stator. Both experiment andsimulation show this tendency.

The radial components of particle velocity are shown inŽ .Fig. 11 b . In the dry condition, the results of measurement

and calculation agree well. In the wet condition, the calcu-lated results are smaller than the measured values. Con-cerning the shape of velocity profile, however, the resultsof calculation and measurement are close, showing convexshape with a peak around rrRs0.8. Further, both resultsshow that the more the water, the smaller the velocity.

To focus on the variation of velocity profiles against theamount of water, average height of velocities in Fig. 11 are

Fig. 12. Average particle velocity on the surface of particle layer.

Ž . Ž .plotted against the amount of water in Fig. 12 a , b . As isclear from Fig. 12, the calculated results and the experi-mental results agree well concerning the variation againstthe water fraction, though there is some difference quanti-tatively.

8. Conclusion

Numerical simulations of a three-dimensional particu-late flow with a small amount of water were performed byusing the DEM taking account of the adhesion force due towater. The adhesion force acting on two particles is formu-lated by using a theory of the liquid bridge.

The present method was applied to the flow in acentrifugal tumbling granulator. The simulation indicatedthat the adhesion force due to liquid bridges largely affectsparticulate flow, particularly the circulating flow in thecross-sectional plane being retarded.

Also, the surface particle velocities were measured byusing a high-speed VTR. Concerning the velocity profiles,the results of calculation and measurement showed similartrends although quantitatively, some difference was ob-served. Further, the tendency of a velocity variation againstthe water fraction agreed well between both experimentaland calculated results.

9. Nomenclature

f Forces acting between particles i and ji j

f Contact force between particles i and jC i j

f Adhesion force of liquid bridge betweenL i j

particles i and jF Dimensionless adhesion forceL

g Gravitational acceleration vectorh, H Length of the gap between particlesI Moment of inertia of the particle ii

m Mass of the particle ii

N Number of particlesp

N Number of gaps smaller than Hs max

r Position vector of the particle ii

r Particle radius0

r ,r , R , R Radii of curvatures of the liquid bridge1 2 1 2

surfaceT Torque caused by the contact forcesi j

x , X x coordinate of the position where a0 0

liquid bridge contact with a particleg Surface tension of liquidF Non-dimensional volume of liquid bridgef Volume fraction of the total liquid to the

total volume of particlesc ,u Angles of triangle shown in Fig. 4v Angular velocity vector of the particle ii j

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