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AMERICAN JOURNAL OF ADVANCED SCIENTIFIC RESEARCHFarzin Salehpour-Oskouyi and Amin Hadidi, Vol. 1, Issue. 2, pp. 44-51, 2012

Development of a Numerical Model of the Granular MaterialFlow under Vibration based on Discrete Element Method (DEM)

Farzin Salehpour-Oskouyi1*, Amin Hadidi

1

1 Department of Mechanical Engineering, Ahar Branch, Islamic Azad University, Ahar, Iran*Corresponding Author Email Address: [email protected]

Various methods have been used to study theAbstract—In recent years, one of important aspects of

the granular material physician industries is controlling wayof the granular flow. One of the best ways to do this is use ofmechanical vibrations to guide the flow of particles indesirable way. This paper theoretically and experimentallyinvestigates the effect of horizontal sinusoidal vibration onthe particles flow. Certainly, the base vibration causes thespherical particles collide with each other and with the wall.These collisions cause their moving and displacement whichis interpreted as flow. In this study, a new numerical modelof spherical particles movement and the exerted forces fromthem to each other and also on the wall due to the vibrationis proposed. Discrete Element Method (DEM) is used here.Finally, the presented numerical model is compared withexperimental model and a good agreement between them isnoticed.

Index Terms—granular materials, mechanical vibrations,discrete element method, vibration acceleration.

I. INTRODUCTIONThe dynamic analysis of granular particles which

present as discrete and segregated in a system andhave the force and energy interactions with each othercan be observed in various fields such as chemistry,agriculture, pharmaceutical, powder metallurgy, soilmechanics, casting, cement manufacturing, civil, andetc. In all these fields, the interaction between particlesand the effect of internal and external parameters onthe behavior of these particles have been studied. Oneof the important aspects related to the subject,considered today in most industries, is the control wayof granular particles flow. One of the best ways to dothis is use of mechanical vibrations to guide the flow ofparticles in a desirable way. For example, for thedischarge of particles from tanks and silos, compressionof set of particles, separation and equalizing of system,the best solution is use of external vibrations on theconsidered set.

Manuscript Received Aug. 18 2012 ; Revised Aug. 28 ; Accepted Aug.29

dynamics of granular materials each has its uniqueadvantages and disadvantages. In this paper, DiscreteElement Method (DEM) is used which is proposed byCundall in 1979 for the first time [1]. This method isbased on the modeling of each particle separately in away that it moves and by calculating all the appliedforces and moments the motion of the particle ispredicted in the next time step. In this method, wehave nothing to do with the whole set and we study thedynamics of individual particles.

The point that distinct the different problemsrelated to the vibration of granular materials isboundary conditions and type of external appliedexcitation on the system and also other assumptionsthat vary regarding the aims of researchers and theirneeds. For example, it may be assumed as a two-dimensional problem [2] which in this case relativelysimple assumptions are applied on the system andforce equations take a simple form. While the three-dimensional case, is more complicated [3] and [4]. Also,the assumptions related to the used force model canhave different conditions which according to theresearchers’ objectives the number of forces may beeliminated [5-6]. Various methods are used tonumerically solve the force equations depending on therequired accuracy in the problem [6]. On the otherhand, some researchers just address the quality aspectof problem [7] whereas others analyzed the problemanalytically [8-10]. The application of it in variousindustries has caused scientists to study various aspectsof the subject. The dominant view in any studydepending on the aim it seeks has its own uniquecharacteristics.

II. DISCRETE ELEMENT METHOD

Discrete element method is a novel method insolving of granular materials problem. Recently

44AMERICAN JOURNAL OF ADVANCED SCIENTIFIC RESEARCHFarzin Salehpour-Oskouyi and Amin Hadidi, Vol. 1, Issue. 2, pp. 44-51, 2012

researchers have utilized this method to study andanalyze the various granular issues [11]. In this method,the problem is analyzed at consecutive time steps. Thewhole of a collision process can be in several successivetime steps. Since, the onset of ball-ball or ball-wallcontact until the interference and indentation of themwith each other and then onset of separation stage untiltheir complete separation from each other occur inseveral steps (Figure 1). For better understanding of thiscase we can say that from a general view it seems thatthe balls begin to move inside the chamber andsometimes collisions take place. They interfere in eachother and then separation onsets until they arecompletely separated and each under its own velocityand gravity force continue moving along new paths.

Figure 1. Simulation of collision [4].

The philosophy of this approach is clear. Balls exertforces to each other during contact. These forces aredifferent but the simplest of them is elasticity force(spring force) which is an attempt to create an opposingforce in direction of balls separation. This forcegradually (during successive time steps) becomes sogreat that eventually causes speed variation. In fact, itcreates a conductive acceleration that after a while itwill stop during balls concurrent, and then onset toseparate from each other. Even during leaving until thetwo objects interfere, the force is still existed andconstantly increases the balls speed. This will continueuntil the balls are completely separated. When no othercontact is existed between the balls, the other force(due to collisions) will not exist. In the DEM, it is notimportant that a particular ball in a specific moment isin interaction with several balls or the wall; but what isimportant is effect of these factors on it. Thesurroundings of a particular ball interacting with it(other balls, obstacles, walls) each, independentlyleaves its force effects on the ball. In this method, theeffect of each ball is independently investigated freefrom of effects of other balls on the central ball. Then,by formation of free diagram of the ball, the ball will be

studied in an isolated environment. The applied totalforces and moments from the surrounding environmentcreate a resultant force and torque that having thesetwo the new position of the ball at the next time step isobtained and the information are updated. This newobtained information is initial conditions of the nexttime step.

First, the collision must be identified beforecalculation of collision force. It should be determinedthat each ball interacts by what factors (ball-ball andball-wall collision). Therefore, collisions can beidentified at the beginning of each time step. Thecreated forces resulted from collisions are computedand then the balls information and whole of system areupdated using the mentioned equations and finally atime step is moved forward.

A. Collision Detection AlgorithmLarge scale simulation of granular materials always

includes many collisions between discrete particles. Theimportance of these collisions and the distinguishingway of them are more when we know that at the nextstage of simulation, including force modeling andanalysis of dynamic particles, detection of collisions andcalculation of forces and moments form the main partof the investigation. Thus, attempt to find suitable andoptimal ways in detection of collisions betweenparticles will greatly help to simulation progress. In thismethod, the whole of system is modeled using a cubicgrid as shown in Figure 2. The grid is consisted of smallcubes which size of each one is equal to ball diameter.Therefore, the center of ball is in the each house, thehouse will belong to it.

Figure 2. The system grid [11].

After meshing the system, a specific number isallocated for each house. On the other hand, every ballhas a unique number. Therefore, a matrix is defined inwhich it determines that which ball lays in which


FnE = Knδ n

house.Hence, positioning of balls is simply done. In thisstudy, this matrix is used as an addressing matrix. In thealgorithm used in this study, instead of individuallystudying of balls, all houses of grid system areinvestigated. So, it is checked that whether a ball is inan assumed house or not. In the absence of ball wemove to the next house. This continues until a house isfound with a ball inside. In this case, 26 houses aroundthe mentioned house are studied as shown in Figure 3.

Figure 4. Modeling of forces [3].

Forces resulted from the collision in the model areconsidered including elastic force, friction force anddamping force. The final force acting on each particle isconsisted of the sum of above forces and externalforces applied on the particle. The collision force invertical direction is consisted of followingcomponents[3],

Figure 3. Available housed around the central ball. Fn = FnE + FnD + FC (1)

B. Collision ModelingWe are faced with two general cases in modeling of

collision between two balls. In the first model called“solid ball” model, balls are assumed to be completelysolid and impervious. In this case, there is no need for

WhereFNE is the normal elastic force, FC is the

cohesion force and FND is the normal damping force.

The collision force in the tangential direction is limited

by Coulomb friction force [3],accurate studying of the ball-ball collision. On the otherhand, systems with high particles can be analyzed usingthis model because of less volume of calculations.

Ft = FFbgs + FtD

Ft = FFags

(2)

(3)

However, this model cannot study simultaneouscollision of a ball with more than a ball. In the secondmodel called “soft ball” model, balls are assumed to becompletely soft and permeable. One of the modelingmethods related to this case is the spring–dampermodel. Different views about the dynamic modeling ofcollision using spring-damper were expressed in theliterature. The two main approaches are widely used inthe DEM. In the first approach, which is simpler than theother approach, the linear model of spring-damper-slider (Figure 4) is used to model interactions betweenthe particles. In the second approach details ofcollisions between particles are more focused. In thisapproach, the Hertz-Mindlin model which is a non-linearmodel is used for collision modeling [11]. The firstapproach is the appropriate approach for problems

where FFbgs is the elastic component of the friction force

prior to gross sliding, FtD is the tangential damping

force, and FFags is the friction force at and after gross

sliding.

The normal elastic force FnE represents the repulsiveforce between any two particles. It is calculated using asimple Hooke’s linear spring relationship,

(4)

where Kn is the stiffness of the spring in the normaldirection, and δn is the displacement between particlesi and j. If i’th particle collide with j’th particle, assumingri as center of i’th ball, rj as center of j’th ball, Ri asradius of i’th ball and Rj as radius of j’th ball, theamount of can be calculated as [9],

dealing with overall movement of collection. But, thesecond approach is the appropriate approach for

r rδn = Ri + Rj − ri + rj (5)

problems dealing with accurate amounts of forces andstresses between particles. In this study, the firstapproach is used.

The normal damping force FND is modeled as a dashpotthat dissipates a proportion of the relative kineticenergy. It is calculated using a relationship [3],


dv

π + (ln(ε ))2

mij = (10)

Rij = (9)

dδ tCn = 2γ mijKn

ij3(1 −ν )

FnD =

Cn

d δ n dt

(6) where δR is a constant. Langston et al. (1995) showedhow coupling of the tangential displacement to the

where Cn is the normal damping coefficient. In order toobtain Kn and Cn following empirical relationships areused [9],

normal displacement, can be calculated as a function ofPoisson’s ratio and the coefficient of friction of theparticles.

Kn =

R0.5E

2 (7) The tangential damping force FtD is modelled as adashpot that dissipates energy from the tangential

(8)

in which, Eis elasticity modulus, and ν is Poisson’s ratio.

motion. It is calculated using a relationship,

FtD = Ctdt

(16)

Rij and mij are effective radius and mass of two ballsduring collision, respectively and are calculated as [9],

RiR jRi + R jmimjmi + mj

γ is the coefficient of critical damping coefficient and

where Ct is the tangential damping coefficient.Ct iscalculated using following equation [9],

Ct = 2γ mijKt (17)

Note that the calculation of this force is omitted if grosssliding occurs because Kt is zero in this region.The vibration force is introduced to the model byoscillating the vessel. Currently this is limited to the z-

is calculated as [9],

ln(ε)γ = −

2(11)

direction only. Hence the particles move as a result ofcontact between the moving wall and the particles.Hence for particle i, the vibration force FV is calculatedas,

where, ε is elasticity modulus and is defined as theratio of the post- to pre-collisional tangentialcomponent of the relative velocity.The variation of the friction force prior to gross slidingFFbgs is calculated using a simple Hooke’s linear springrelationship,FFbgs = Ktδt (12)

Where Kt is the tangential stiffness coefficient, and

δt is the total tangential displacement between the

surfaces since their initial contact. Kt is calculated byequaling the collision time in vertical and tangentialdirections as following [11],

2Kt = Kn (13)

7As mentioned earlier, the total tangential force is

limited by the Coulomb frictional limit. If exceeds thensliding occurs and does not increase. In this case thefriction force after gross sliding FFags is calculated as,FFags = µFn (14)where µ is the coefficient of friction, and FNE is thenormal elastic force which is calculated by the followingrelationship [3],

Fv = Fny + Fty (18)The vessel movement is defined relative to the vesselbase, which at any time is defined as,t < tstart →Ybase = 0t > tstart →Ybase = A.sin(f.(t −tstart ))where A is the amplitude, f is the frequency, t is thetime, and tstart is the time vibration starts.

C. Application of Euler-Newton law and numericalanalysis

Particle i motion is calculated by Newton’s equationof motion as follows.Translational motion:r r r rmi = FG + ∑ Fn +∑ Ft (19)dt

where m is the mass of particle i, FG is the gravitationalforce, ∑ Fn is the sum of the normal forces (particle

particle and particle–wall) acting on the particle, ∑ Ft is

sum of the tangential forces (particle–particle andparticle–wall) acting on the particle and v is the linearvelocity of the particle. The acceleration of the particleis computed from the net force, which is thenintegrated for velocity and displacement as shown

δtMax = δRδn (15) below.

47

& &

& &

&t

&t

&&

& & &t

&t

&t&t

TABLE I. REQUIRED PARAMETERS VALUES IN SIMULATION

dω


Rotational motion:r rIi = ∑M (20)dt r

Where Ii is the moment of inertia, ω is the angularvelocity, M is the moment, which is calculated as,r r rM = R × Ft (21)Where R is the radial vector from the particle centre tothe point of contact. With the assumption thatacceleration is constant over the time step, the velocityvector of each particle is then determined from,r +∆t = r + r&+∆t∆t

θt+∆t = θt +θ&

+∆t∆t

(22)

(23)

Where r is the position vector, rt is the linear velocity

vector, r&+∆t is the linear acceleration vector, θt is the

angular velocity vector and θ&

+∆t is the angular

acceleration vector.The positions and orientations at the end of the nexttime step are then determined using an explicitnumerical integration, where the positions at areobtained directly from the acceleration at t. Applyingthe Euler method, the position is determined asfollows:

Figure 5. The simulated model.

Now, it is time to excite the system. It can be done byapplying a harmonic fluctuating motion on thecontainer (Figure 6). Now the final model is created insoftware environment and is ready for analysis. The

rt+∆t = rt + 0.5(rt + rt+∆t )∆t

θt+∆t = θt + 0.5(θt +θt+∆t )∆t

(24)

(25)

analysis is begun by choosing a suitable time step.

The time step is a constant value that is chosen toensure the stability and accuracy of the numericalsimulation, particularly the integration.After performing the above steps once it comes to atime step to move forward. Data obtained in thespecified time interval is the initial conditions of thenext step. Table 1 shows the model parameters usedhere.

III. COMPUTER SIMULATION

In order to simulate the system, firstly differentparts of the system including container and particlesare created. In the next step, boundary conditions areapplied on the system. Then, container and particlesmaterial and other specifications are determined.

Figure 6. Final model under vibration.

IV. EXPERIMENTAL PROCEDUREAccording to Figure 7, the used equipments include

Shaker, data acquisition system, accelerometer, andrelated softwares. The excitation conditions of shakerare determined through Pulse Labshop software andare transferred to amplifier of shaker by dataacquisition system. After, amplifying the signal inamplifier, the signal is transferred to the core of shakerand its plane onsets to oscillate. Different types of

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Parameter symbol ValueStiffness coefficient (PP) Kn 10000 N/mStiffness coefficient (PW) Kn 8000 N/m

Tangential stiffness coefficient (PP) Kt 2850 N/mTangential stiffness coefficient (PW) Kt 2300 N/m

Friction coefficient (PP) µ 0.01Displacement ratio (PP) δ 0.036

Normal damping coefficient (PP) Cn 0.3 N.s/mTangential damping coefficient (PP) Ct 0.03 N.s/m

Time step dt 0.001


vibration and different oscillation conditions aredetermined by software. The validation of proposedmodel is studied using experiments.

particles is studied. The magnitude of frequency is seton 10Hz. Increasing the amplitude from Γ = 0 to Γ = 2the following positions are obtained.

Figure 7. Laboratory conditions.

V. SURVEY RESULTSIn this section we examine the effects of vibration

parameters on how the particles flow. The studiedparameters are frequency and amplitude of vibration.As described in previous sections, a dimensionless

Figure 9. Final form of mixture for a:Γ=0, b:Γ=1,c:Γ=1.5,d:Γ=2.

As seen in Figure 9, slope of the line increases gradually.Now with imposing the conditions of problem on thetheoretical model, we review the results of it. Initial

number Γ defined as Γ =A(2πf )2

gwhere A is amplitude conditions, boundary conditions and excitation

conditions are applied on the mixture exactly as theof vibration, f is excitation frequency and g is gravity. Byusing this parameter, the effect of frequency andamplitude of vibration can easily be studied separately.System studied in this section is as figure 8.

experimental conditions applied. Following results(figure 10) are obtained after 10 seconds.

Figure 8. Initial condition.

One way to study the particles flow is to check the finalform of mixture after specified time. In this section,according to this subject, the effect of describedparameters on the final form of mixture is studied.

A. Effect of Changes in the Amplitude of Vibrationin a Constant Frequency on the Final form of Mixture

In this section, assuming a constant frequency ofvibration and increasing the magnitude of Γ, the flow of

Figure 10. Final form of the theoretical model in 10Hz fora:Γ=1, b:Γ=1.5,c: Γ=2,d: Γ=2.5, e: Γ=3.

Comparing theoretical samples with the experimentalones, it is observed that the numerical model which ispresented for a mixture of particles under vibrationgives a satisfactory result. Change of slope of mixturesurface is shown in Figure 11.


Applying the same conditions on the theoreticalmodel, final form of mixtures after 10 seconds are obtainsas figure 13.

Figure 11. Slope of mixture surface as a result of increasing thevibration acceleration in the frequency of 10Hz.

B. The Effect of Vibration FrequencyChanges in aFixed Vibration Accelerationon the Final Form ofMixture

In this section, assuming a constant vibrationacceleration and changing the magnitude of frequency,the flow of particles is studied. The following results(figure 12) are obtained in Γ=2 after 10 seconds.

Figure 13. Final form of theoretical model for a: f=10Hz, b: f=15Hz, c:f=20Hz.

Comparing the two models, it can be inferred thatincreasing the frequency (vibration accelerationremains constant) will cause the slope of mixturesurface to decline. So with increasing frequency, theparticles and their collisions with each other is lesssevere. This is evident in both theory and empiricalmodel. Effect of frequency on the slope of mixturesurface is shown in Figure 14.

Figure 12. Final form of experimental model for a: f=10Hz, b: f=15Hz,c: f=20Hz.

Figure 14. Change of slope as a result of increasing frequency.

VI. CONCLUSIONIn this paper, a comprehensive model of interactions

resulted from collision of particles has been presented.Then, the post-collision motion of particles was studiedusing DEM. A system of spherical balls under vibrationwas modeled and motion of them at different


conditions was investigated. According to the results ofexperiments and theoretical model analysis, it wasfound that the proposed model is a complete modeland is near to actual case. Furthermore, various systemswith different initial conditions can be simulated usingthis model.

REFERENCES[1] Cundall,P.A., “Formulation of a three dimensional distinct

element model”, International Journal of Rock Mechanicsand Mining Sciences & Geomechanics, Vol. 25 (3), pp.107-116, 1988.

[2] Cleary, P.W., “DEM simulation of industrial particle flows:case studies of dragline excavators, mixing in tumblersand centrifugal mills”,Powder Technology, Vol. 109, pp.83-104, 2000.

[3] Asmar, B. N.,Langston, P. A.,Matchett, A. J.,Walters, J.K.,“Validation tests on a Distinct Element Model ofVibrating Cohesive Particle System”, Computer andChemical Engineering, Vol. 26, pp.785-802, 2002.

[4] Aste, T., Di Matteo, T., Tordesillas, A., “Granular andComplex materials”, World Scientific Lecture Notes inComplex Systems, Vol. 8, 2007.

Farzin Salehpour-Oskouyireceived his B.Sc. degree inMechanical Engineering from Tabriz University in 2008and his M.S. from Tabriz University in 2011. Mr.Salehpour-Oskouyi is currently working on his Ph.D.thesis at Department of Mechanical Engineering, inSahand University of Technology. His researchinterests include the Reliability analysis and systemmaintenance. He is a lecturer of Islamic AzadUniversity.

Amin Hadidireceived his B.Sc. degree in MechanicalEngineering from Tabriz University in 2009 and hisM.S. from TarbiatModares University in 2011. Mr.Hadidi is currently working on his Ph.D. thesis atDepartment of Mechanical Engineering, inTarbiatModares University. His research interestsinclude the numerical and experimental study ofmagnetohydrodynamics. He is a lecturer of IslamicAzad University.

[5] Balevicevius, R., Kacianauskas, R., “DEMMAT code fornumerical simulation of multi-particle systemsdynamics”, Information technology and control, Vol. 34,2005.

[6] Renzo, A., Di Maio, F.P., “An improved integral non-linearmodel for the contact of particles in distinct elementsimulations”, Chemical engineering science, Vol. 60, pp.1303-1312, 2005.

[7] Raihane, A.,Bonnefoy, O.,Gelet, J. L.,Chaix, J.M.,“Experimental study of a 3D dry granular mediumsubmitted to horizontal shaking”, Powder technology,Vol.190, pp. 252-257, 2009.

[8] André, D., Iordanoff, I., Charles, J., Néauport, J., “Discreteelement method to simulate continuous material byusing the cohesive beam model” , Computer Methods inApplied Mechanics and Engineering, Vol. 213, pp. 113–125, 2012.

[9] Jin, F., Zhang, C., Hu, W., Wang, J., “3D mode discreteelement method: Elastic model”, Vol. 48 (1), pp. 59–66,2011.

[10] Hastie, D.B.,Wypych, P.W.,“Experimental validation ofparticle flow through conveyor transfer hoods viacontinuum and discrete element methods”, Vol. 42 (4),pp. 383–394, 2010.

[11] Munjiza, A., The combined finite-discrete elementmethod, John Wiley & Sons Ltd, pp. 332-333, 2004.

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