Powder Technology 261 (2014) 154–160

Contents lists available at ScienceDirect

Powder Technology

j ourna l homepage: www.e lsev ie r .com/ locate /powtec

Simulation of vibrated bulk density of anode-grade coke particles usingdiscrete element method

Behzad Majidi a,b, Kamran Azari a,b, Houshang Alamdari a,b,⁎, Mario Fafard b, Donald Ziegler c

a Department of Mining, Metallurgical and Materials Engineering, Laval University, Canadab NSERC/Alcoa Industrial Research Chair MACE3 and Aluminum Research Center – REGAL, Laval University, Canadac Alcoa Primary Metals, Alcoa Technical Center, 100 Technical Drive, Alcoa Center, PA, 15069-0001, USA

⁎ Corresponding author at: Pavillon Pouliot, office 1745Université Laval, Québec, QC, G1V0A6, Canada.

E-mail address: houshang.alamdari@gmn.ulaval.ca (H

http://dx.doi.org/10.1016/j.powtec.2014.04.0290032-5910/© 2014 Elsevier B.V. All rights reserved.

a b s t r a c t

a r t i c l e i n f o

Article history:Received 5 July 2013Received in revised form 26 March 2014Accepted 5 April 2014Available online 14 April 2014

Keywords:Calcined cokeDiscrete Element MethodVibrated Bulk Density

Packing density is an important quality parameter of calcined cokes used in aluminum industry to produce car-bon anodes. Vibrated bulk density (VBD) test is awell establishedmethod tomeasure the packing density of cokesamples. In the present work, Discrete Element Method (DEM) is coupled with a three-dimensional imagingtechnique to investigate the possibility of using DEM to simulate the packing behavior of calcined cokes. As themethod verified, effects of shape, friction coefficient, size and size distribution of the particles on the VBD ofcokes are also investigated. DEM simulations, in accordance with the experiments, show that vibrated bulk den-sity of coke samples decreases as the content of coarse particles in the mixture increases. Moreover, it is shownthat friction coefficient has a negative effect on the VBD value and this effect is more pronounced for the sampleswith lower sphericity. High friction coefficient restricts the movement and rearrangements of the particles andthus vibrational forces cannot effectively rearrange the particles and fill the porosities. Lower sphericity of cokesamples not only induces a higher initial porosity level in the samples but it also increases the chance of formationof locks and particle bridges which result in lower VBD. Results also show that sphericity and friction coefficienthave a synergic effect on the VBD of coke so that, for the samples with lower average sphericity the rate ofdecrease in VBD is higher with increasing the friction coefficient.

© 2014 Elsevier B.V. All rights reserved.

1. Introduction

Calcined coke is an important raw material for production ofconsumable carbon anodes in aluminum industry. Anode is made bymixing coke and pitch to prepare anode paste, which is then compactedand baked. Physical and chemical properties of raw materials must betaken into account to determine the paste formulation,which in turn af-fects the quality of the baked anode. Due to the frequent changes ofsources of rawmaterials, the processing parameters should be adjustedaccordingly in order to keep the quality consistency of the baked an-odes. The adjustment of process parameters requires understandingthe effect of each raw material properties on the anode quality.

Modeling approaches could be useful to simulate the process and todetermine the right process parameters for a given set of rawmaterials.Choosing an appropriate model, which depends on the nature of thepaste and its constitutive laws, is however crucial to obtain reliablesimulation data.

Green anode is made either by pressing or by vibro-compacting theanode paste, which consists of two principal phases; a binder matrix

F, 1065 avenue de la Médecine

. Alamdari).

(pitch + fine) and coke aggregates. Bulk density of calcined coke isused as an important gage in determination of pitch demand in anodeproduction process [1–3].

Several works have been published in the last two decades dealingwith the effects of different parameters on packing density of granularmedia such as [4–7].White andWalton [8] studied the effects of particleshape on packing density of different packing systems using both theo-retical and experimental methods. H.J.H. Brouwers [9] investigated theparticle size distribution and porosity fractions in randomely packedbeds. He addressed the parameters of analytical expressions fordetermining void fraction in packing of unimodal and bimodal spheres.Effects of particle shape on angle of repose of heaps have also beenreported in the literature [10].

Discrete Element Method, introduced for the first time by Cundalland Strack [11] in 1979, is nowused to simulate thebehavior of granularmaterials in industrial applications specially where the dynamics andflow of a particulate material are of interest. In DEM simulations, rigiddiscrete elements, which are spheres in 3D and discs in 2D models,are used to model the granular material. The contact law between theelements defines the mechanical behavior of the bulk material. UsingDEM, El Shourbagy et al. [12] showed that the angle of repose of drygranular materials depends on the particles shape and Columb frictioncoefficient.

155B. Majidi et al. / Powder Technology 261 (2014) 154–160

This work aims at exploring the possibility of using DEM to simulatethe packing behavior of coke particles through vibrated bulk density(VBD) test. As the model verified by the experiments, effects of frictioncoefficient and sphericity of coke particles on VBD value is also investi-gated. VBD test is routinely used to measure the packing density of cal-cined coke and is considered as one of the important anode-grade cokespecifications. The simulation is performed using PFC3D software. Athree-dimensional imaging technique is used to capture the real shapeof coke particles. Effects of particles shape and friction coefficient onthe VBD of coke samples are therefore investigated by means of DEMmodeling with PFC3D.

2. The numerical model

2.1. Principles of DEM

A three dimensional DEM model is composed of a combination ofdiscrete spheres andwalls. At the beginning, the position of all elementsand walls are known so that the active contacts are easily determined.Then, according to themechanical behavior of thematerial an appropri-ate force-displacement law is applied to each contact and the contactforces are calculated. Law of motion, Newton’s second law, is thenused to update the position and velocity of each ball.

One common contact model, which is widely used in DEM simula-tions, is linear contact model. This model is simply defined by assigningnormal and shear stiffness values to the contacting elements (see Fig. 1).Normal and shear stiffness values of a contact are expressed as;

Kn ¼ KnA:K

nB

KnA þ Kn

B

Ks ¼ KsA:K

sB

KsA þ Ks

B

whrere Kn and Ks stand for the normal and shear stiffness of theelements (A and B in Fig. 1).

Having the shear and normal stiffness, one can obtain the forcepropagating at the contact point, if the balls are overlapping. Contactforces can be calculated according to the extent of the overlap (fornormal contact force) and tangent movement (for shear contact force);

Fn ¼ Kn:Un

Fs ¼ −Ks:δUs

Fig. 1. Contact of two elements.

2.2. Movement of non-spherical particles

In PFC any kind of irregular shape particles can be generated as aclump composed of several touching or overlapping balls. Contactforce calculations for balls within a clump is skipped during calculationcycle and only the contact force of a clump with neighboring clump/balls or walls are considered [13]. The basic mass properties of aclump are the total mass (mCl), center of mass (xiCl), and moments andproducts of inertia (Iii and Iij). These properties can be mathematicallyexpressed by the following equations [13];

mCl ¼XNb

n¼1mn½ � ð1Þ

xCli ¼ 1mCl

XNb

n¼1mn½ �x n½ �

i ð2Þ

Iii ¼XNb

n¼1mn½ � x n½ �

j −xClj� �

x n½ �j −xClj

� �þ 25m n½ �r n½ �r n½ �

� �ð3Þ

Iij ¼XNb

n¼1mn½ � x n½ �

i −xCli� �

x n½ �j −xClj

� �n o; j≠i ð4Þ

mCl, Nb are the mass of the clump and number of the balls in the clump,respectively. x[n], r[n], andm[n] are center ofmass, radius and themass ofthe nth ball, respectively.

Moments and products of inertia are calculated with respect to acoordination system, which is attached to the center of mass of theclump and is aligned with the global axis system.

Translational motion of clumps is expressed as;

Fi ¼ m∂2xi∂t2

−gi

!

where Fi is the resultant force,m is the total mass of the clump, ∂2xi∂t2 is the

acceleration vector and gi is the gravity acceleration vector.The resultant force, which is the sum of all externally applied forces

acting on the clump, can be expressed as;

Fi ¼ eFi þXNb

n¼1eF n½ �i þ

XNc

c¼1F n;c½ �i Þ

�ð5Þ

where eFi is the externally applied force on the clump, eF n½ �i is the external-

ly applied force acting on ball (n), and Fi[n,c]

is the force acting on ball (n)at contact (c).

The resultant moment about the center of mass of the clump iscalculated by

Mi ¼ eMi þXNb

n¼1

� eM n½ �i þ ϵijk x n½ �

j −xClj� �

F n½ �k þ

XNc

c¼1ϵijk x c½ �

j −x n½ �j

� �F n;c½ �k

�ð6Þ

inwhich eMi is the externally appliedmoment acting on the clump, eM n½ �i is

the externally applied moment acting on ball (n), Fk[n]

is the resultantforce acting on the centroid of ball (n), and Fk

[n,c]is the force acting on

ball (n) at contact (c).Rotational motion of a clump is given by the vector equation of

Mi ¼ H i, where Hi is the time rate of change of the angular momen-tum of the clump.Hi can be written as

Hi ¼ ωiIii−ω jIij þ∈ijkω j ωkIkk−ωlIklð Þ; j≠i; l≠kð Þ ð7Þ

where I is the moment of inertia, and ω and ω are angular velocityand angular acceleration about the principal axes respectively.

156 B. Majidi et al. / Powder Technology 261 (2014) 154–160

The equations of motions are discretized using a centered finitedifference scheme and then are integrated.

3. Materials and methods

Calcined cokes in this study were sampled from the sources whichare currently used for making prebaked anodes in aluminum industry.Vibrated bulk densities of the samples with different size distributionswere obtained experimentally and numerically. VBD test setup containsthree parts. Part 1 is a vibrating funnel, which transfers the powder tothe main container. Part 2 of the setup is a graduated cylinder of 250ml size with the inside diameter of 37 mm made of glass. The thirdpart is the vibrator whichmust be capable of vibrating a 215 g graduat-ed cylinder containing 100 g coke sample at a frequency of 60 Hz andamplitude of 0.20 to 0.22 mm. A quantity of 100.00 ± 0.1 g of coke ischarged in the funnel and is vibrated for 5 minutes at 60 Hz. Then, bymeasuring the height of the particle column the volume of the sampleand so the VBD value can be calculated.

Coke samples within the size ranges of -6 + 14 and -4 + 6 meshwere selected for the study. Coke particles have irregular shapes. Mor-phological studies on the samples were performed using image analysismethod by optical microscope powered by Clemex software. Shape pa-rameters such as sphericity and also size distribution of the particles ateach size range were obtained and were used to model the samples.Table 1 shows the size distribution of the particles within the two sizefractions used in this work.

Particles shape is considered as one of the most important featuresof particulate assemblies [14,15]. There are different methods to assessand describe the shape of particles, including parameters such as sphe-ricity, aspect ratio, elongation ratio, shape factor, and convexity ratio[14,16]. However, sphericity is considered as a single parameter,which can define the shape of the particle to a very good extent witha simple method. Thus, in the present work, sphericity is used as theshape factor to characterize the particles in different size ranges.

Wadell [17] introduced the term sphericity in 1935. Sphericity ingeneral states how close is the particle geometry to a perfect sphere. Itis worthy to note that this term is also used for 2D projections of theparticles and for that case the term of circularity matches better. Anumber of definitions of circularity and sphericity have been proposedin the literature [18]. Themethod proposed byWadell [17,19] is widelyused in the literature andwas also used in this study as the definition ofsphericity. According to Wadell sphericity is defined as:

Ψ ¼ SAes

SArp

Table 1Size distribution of the coke particles within two size fractions of -4 + 6 mesh and-6 + 14 mesh.

-4 + 6 mesh fraction -6 + 14 mesh fraction

Diameter interval (mm) Percent (%) Diameter interval (mm) Percent (%)

2.7-2.9 0.83 1.0-1.1 0.512.9-3.1 0 1.1-1.3 03.1-3.3 1.65 1.3-1.4 1.023.3-3.5 5.79 1.4-1.6 2.553.5-3.8 9.09 1.6-1.7 4.593.8-4.1 15.70 1.7-1.9 10.714.1-4.3 15.70 1.9-2.0 15.844.3-4.7 11.57 2.0-2.2 14.804.7-5.0 13.22 2.2-2.5 10.715.0-5.3 9.92 2.5-2.7 8.675.3-5.7 6.61 2.7-2.9 12.245.7-6.1 2.48 2.9-3.2 6.636.1-6.5 3.31 3.2-3.5 6.126.5-7.0 2.48 3.5-3.8 2.557.0-7.5 1.65 3.8-4.2 3.06sum 100 sum 100

where SArp is the surface area of the real particle and SAes is the surfacearea of the equivalent sphere with the same volume of the real particle.

Three-dimensional digitized and meshed shapes of coke particleswere obtained by 3D scanning at Cogency Co, in South Africa. AutomaticSphere-clump Generator (ASG) software, developed by Cogency Co.,was then used to generate the 3D shapes using spherical elements for60 coke particles. Fig. 2 shows three examples of 3-D shapes of particlesgenerated by spheres. However, the number of particles used to modeleach fraction was 60. By mixing 60 different particles for each fractionand resizing some particles (if required), size distribution and averagesphericity of the numerical models defined to match the real cokesamples (as shown in Table 1 for size distribution).

Twomono-size range coke samples (S1 and S6) and fourmixed-sizesamples (S2, S3, S4 and S5), as given in Table 2, were examined for theVBD experimentally and numerically.

It is noteworthy that the internal porosity of coke samples is an im-portant quality factor of anode-grade calcined cokes. Apparent densityof coke samples of two size ranges of -4 + 6 and -6 + 14 mesh wasfirst measured and then the obtained values were used to create thenumerical models. Apparent density for the size ranges of -4 + 6 and-6+14meshwere 1.377 g/cm3 and1.532 g/cm3 respectively. Coke par-ticles are presented by rigid super-particles (clumps) in this work. Lin-ear contact model (which was introduced earlier) was applied for allclumps in the model. Normal and shear stiffness of the balls making

Fig. 2. Coke particles modeling by overlapping spheres for DEM simulations (gray: 3Dshape of particles obtained by scanning, green: equivalent clump generated by usingspherical elements).

Table 2Particle size distribution of the coke samples; weight percentage of each size range hasbeen given.

Samples Mesh Size range

-4 + 6 -6 + 14

S1 - 100%S2 30% 70%S3 50% 50%S4 70% 30%S5 80% 20%S6 100% -

157B. Majidi et al. / Powder Technology 261 (2014) 154–160

the clumpswere considered as 1e4 Nm-1. This valuewas set by calibrat-ing the system to have the minimum overlap between the contactingclumps while giving the lowest compromise for the time-step value.VBD of the samples were first obtained experimentally. Then, three-dimensional discrete element method was used to simulate the VBDtests using Particle Flow Code (PFC3D) software developed by ItascaConsulting Group Inc. Experimental and simulation results were thencompared. In the second step, effects of cokeparticles shape (sphericity)and inter-particle friction coefficient on the VBD of coke samples wereinvestigated using 3D-DEM simulations.

Fig. 3. Estimation of friction coefficient between the plate and coke particles; a) experi-ment, b) simulation.

4. Results and discussion

4.1. Friction coefficient estimation

Friction coefficient between the particles is an important parameter,which affects the particle packing and should be measured prior to VBDsimulation. It has been shown that the effect of internal friction coeffi-cient on macroscopic properties of granular materials can be exposedby angle of repose test [20,21]. Angle of repose, was therefore used toestimate the friction coefficient between the particles. In this test, theparticles are charged from a funnel on a horizontal plate and then theinternal angle (θ) between the surface of the pile and the horizontalplate is measured [22].

The angle of repose is affected by internal friction coefficient and thefriction coefficient between particles and the horizontal plate. Thus, thefriction coefficient between the plate and the particleswas first estimat-ed. Coke particles were placed on a horizontal plate, which was tiltedslowly. The inclination angle at which the particles start to slip was re-corded. This process simulated in the same way in PFC3D. The frictioncoefficient was adjusted in a way that the particles slip at the sameangle as recorded experimentally (Fig. 3). Using thismethod the frictioncoefficient between particles and the plate estimated as 0.45.

As shown in Fig. 4, 10 g of coke particles were allowed to fall downthrough a funnel on the horizontal plate and the angle of repose (θ)was measured. Ten tests were conducted and the average of the valuesrecorded at each testwas considered as the angle of repose of coke sam-ples. A mean value of 31 degrees with standard deviation of 1.03 wasobtained for the angle of repose of the coke particles used in thisstudy. The obtained value was the same for two size fractions in thisstudy. Angle of repose test was then simulated by a three dimensionalDEM model using PFC3D software. Funnel with the same geometryand scales was created in the model (as shown in Fig. 4-b) and 10 g ofcoke particles with the real size and shape parameters were allowedto fall down from the funnel on the plate. Again, the internal friction be-tween particles was adjusted until the angle of repose matched withthat obtained experimentally. Internal friction coefficient of 0.27 result-ed in the best match between the experimental and simulation results.

The coefficient of friction between coke particles was thus estimatedto be μ=0.27 andwas used in simulations of vibrated bulk density testsas a property of material.

4.2. Experimental and simulated vibrated bulk density of coke particles

Numerical coke particles (clumps), as shown in Fig. 2, are composedof an average of 30 spherical elements per particle. The average weightof the particles within the size range of -6+ 14mesh is 9.3 mg. Thus, tohave a sample of 100 g, around 11000 particles comprising at least330000 spherical elements are required. On the other hand the time-step of finite difference discretization of the equations for this systemwas around 1e-6 seconds. Taking into account the number of elementsand the finite difference discretization of 1e-6 seconds, the simulationof the VBD process of 100 g coke requires a very long calculation time.

In order to have a reasonable calculation time, the mass of the sam-ples decreased from 100 g to 10 g. The effect of mass reduction on VBDvalues needs however to be elucidated first. Therefore, VBD tests withboth standard 100 g and also reduced mass of 10 g were conducted on6 samples with different size ranges described in Table 2. It was ob-served that for the studied size ranges given in Table 2, the obtainedvalues of VBD for 10 g samples were 1.0 to 7.9% less than the ones ofstandard 100 g samples. Vibrated bulk density of the numerical samplesis measured in the same way of experiments. It means the height of theparticles column is measured and the occupied volume is obtained asthe diameter of the container is known.

Experimental data of VBD (obtained by the standard 100 g samplesand also those obtained from themodifiedmethod) have been comparedto the simulation results in Fig. 5. As it can be seen, mono-sized sample of-6 + 14mesh (S1) has a higher VBD compared to mono-sized sample of-4+6mesh (0.835 compared to 0.715 g/cm3, respectively). The same re-sults on the effect of increasing the size of particles on the VBD of cokesamples have been already observed by two-dimensional DEMsimulations [23].

Fig. 4. Angle of repose test; experiment (a) and simulation (b).

158 B. Majidi et al. / Powder Technology 261 (2014) 154–160

For themixtures of coarse and fine particles, S2 to S5, as the contentof coarse particles increases the VBD value decreases. It should be notedthat the obtained results on effects of coarse particles addition on VBDvalue must be interpreted carefully and it cannot be generalized to allparticulate systems. If all the particles have the same apparent density,size ratio of large to fine particles is the factor determining the trend.It has been reported that the particles size ratio of at least 7 is requiredfor optimal packing of binary systems [24]. However, large particleshere (-4 + 6 mesh) are between 3.36 and 4.76 mm and fines here are

Fig. 5. Experimental and simulation results of VBD tests for 10 g sam

within the 1.41-3.36 mm range. It means in the particles mixtures inthis experiment, the ratio of the largest particle to the smallest particleis only 3.38. Thus, the small particles are not fine enough to fill thegap between the large particles. Furthermore, larger particles havehigher internal porosities which results in lower apparent density andso lower VBD. Container wall also has an effect on the packing density.It is believed that the ratio of the container size to the particle size isalso an important parameter in packing density of granular materials[25]. The container wall induces a local low density region nearbywhich reduces the total packing density of the material. This effect ismore pronounced when the particle size increases. Thus, it can be saidthat the observed trend for VBD of themixed samples is due to simulta-neous effects of particles size ratio, large particles apparent density andalso container wall.

In VBD test the volume of the particle assembly ismeasured by read-ing the occupied volume in a container with the minimum divisions of1 mm3. As a result, for the mixtures used in this study, this measure-ment is not enough precise to show the variations in the VBD valuefor different samples. As it can be seen in Fig. 5, the experimental values(blue columns) of VBD for samples S1, S2, S3 and S4 are the same and itonly drops for S5 and S6. However, the model is capable to expose thesmall variations in VBD values.

Fig. 6 compares two samples (S2 and S5) having the same weight of10 g. S5 contains 80wt.% large particles compared to S2with 30 wt.%. Itcan be clearly seen that S5 occupies a higher volume which meanslower vibrated bulk density.

4.3. Effects of friction and sphericity

Effects of friction coefficient and sphericity of particles on the VBD ofcoke samples were also investigated in a separate set of simulations. Allparameters of themodels are the same as used in previous VBD simula-tions, but friction coefficient and the average sphericity of the particlesin the samples were changed to study their effects. The individual parti-cles can have different values of sphericity. For example, it is clear thatthe particle in the middle of Fig. 2 is more elongated than the othertwo particles and thus it has a lower sphericity. Therefore, the averagesphericity of the numerical sample can be reduced by adding moreparticles of this type.

As it can be seen in Fig. 7, increasing the inter-particle frictioncoefficient decreases the packing density in all cases. For example, foran equal sphericity of 0.7, vibrated bulk density drops from 0.737 to0.605 g/cm3 by increasing friction coefficient from 0.2 to 0.45. Similar

ples. The points present the values for standard 100 g samples.

Fig. 6. 3D simulations of VBD test; a) sample S5 with 80% of coarse particles; b) sample S2 with 30% of coarse particles.

159B. Majidi et al. / Powder Technology 261 (2014) 154–160

results on the effect of friction coefficient on packingdensity of sphericalparticles have been reported previously [26].

Increasing the friction coefficient enhances the chance of inter-particle lock and bridge formation by restricting the particlesmovements and rearrangements. High friction coefficient also causesdeveloping high frictional forces between particles, which can over-come the vibration forces imposed by container vibration. Therefore,vibration cannot effectively rearrange the particle packing and breakthe bridges.

It can be seen in Fig. 7 that shape of particles has a clear effect onVBDof coke samples. Higher sphericity results in better packing and sohigher VBD. It is also worthy to note that lower sphericity can amplifythe effect of friction coefficient on the VBD. This effect has been clearlyshown in Fig. 8 in which the percentage of the decrease in VBD valueby raising the friction coefficient from 0.2 to 0.45 has been plotted for3 values of sphericity. For example, increasing the friction coefficientfrom 0.27 to 0.35 causes 2.7% and 3.9% of drop in VBD of cokes withthe average sphericity of 0.8 and 0.78, respectively. However, thiseffect is more pronounced for the coke with an average sphericity of0.7 for which the decrease in the VBD value is 8.9%. Coke particleshave a range of completely irregular shapes (some examples havebeen shown in Fig. 2). There are more elongated particles in the sample

Fig. 7.Effects of friction coefficient and sphericity on vibratedbulkdensity of coke samples.

with sphericity of 0.7 compared to the one with sphericity of 0.78. Thiscauses an increased chance of formation of bridges. Higher frictioncoefficient makes the bridges solid enough so that vibration cannotbreak them. Therefore, porosity level does not drop by vibration andas a result VBD of the coke sample drops when sphericity is low andfriction coefficient is high.

5. Conclusions

Discrete element methoe (DEM) was used to simulate the vibratedbulk density of calcined coke samples. A three dimensional imagingtechnique was used to capture the real shape of coke particles andthen the particles were modeled by overlapping spheres in ParticleFlow Code 3D. Coefficient of inter-particle friction was experimentallyestimated as μ = 0.27 by performing the angle of repose test on thecoke samples. Angel of repose test was then used to callibrate thefriction coefficient in DEMmodel.

Results of the used numerial model are in accordance with the ex-perimental data of VBD tests. Since the objective of this work was toinvesigate the application of DEMmodeling combined with 3D imaging

Fig. 8. Rate of decrease of VBD by friction coefficient for particles with different sphericity.

160 B. Majidi et al. / Powder Technology 261 (2014) 154–160

to simulated the packing behavior of calcined cokes, it can be concludedthat the used approach is capable to predict the expected results for theparticle packing. Thus, themodel can be used to invesigate the effects ofdifferent parametes such as shape, friction coefficient and apparentdensity on final packing density/VBD of particulate systems.

The model was then used to investigate the effect of sphericity andinter-particle friction coeffcient on VBD values. Samples with higherfriction coefficient and lower average sphericity had the lowest VBDamong diffeent samples. It was shown that sphericity and frictioncoefficient have a synergic effect and the rate of decrease in VBD withincreasing the friction coefficient is higher for the samples with loweraverage sphericity.

Acknowledgements

The authors gratefully acknowledge the financial support provid-ed by Alcoa Inc., theNatural Sciences and Engineering Research Councilof Canada and Centre Québécois de Recherche et de Développement del'Aluminium. A part of the research presented in this article was fi-nanced by the Fonds de recherche du Québec - Nature et Technologiesby the intermediary of the Aluminium Research Centre – REGAL.

References

[1] M. Paz, J.R. Boero, F. Milani, Coke density and anode quality”, Light Met. 1993 (1993)549–553.

[2] W.K. Fischer, R.C. Perruchoud, Bench scale evaluation of the mechanical and chem-ical behavior of coke in anode manufacturing”, Anodes for Aluminum Industry, R&DCarbon Ltd., Sierre, Switzerland, 1995, pp. 93–101.

[3] A.L. Proulx, Optimum Binder Content for prebaked Anodes”, Light Met. 1993 (1993)657–661.

[4] F. Lange, H. Mörtel, V. Rudert, Dense packing of cement pastes and resulting conse-quences on mortar properties”, Cem. Concr. Res. 27 (10) (1997) 1481–1488.

[5] Gary E. Mueller, Radial void fraction distributions in randomly packed fixed beds ofuniformly sized spheres in cylindrical containers, Powder Technol. 72 (3) (1992)269–275.

[6] A.J. Sederman, P. Alexander, L.F. Gladden, Structure of packed beds probed byMagnetic Resonance Imaging, Powder Technol. 117 (3) (2001) 255–269.

[7] Zhongcheng Wang, Artin Afacan, K. Nandakumar, Karl T. Chuang, Porosity distribu-tion in random packed columns by gamma ray tomography, Chem. Eng. Process. 40(3) (2001) 209–219.

[8] H.E. White, S.F. Walton, Particle packing and particle shape, J. Am. Ceram. Soc. 20(1937) 155–166.

[9] H.J.H. Brouwers, Particle-size distribution and packing fraction of geometric randompackings, Phys. Rev. E 74 (2006) 031309–031322.

[10] Y.C. Zhou, B.H. Xu, A.B. Yu, Numerical investigation of the angle of repose ofmonosized spheres, Phys. Rev. E 64 (2001) 021301–021309.

[11] P.A. Cundall, O.D.L. Strack, A discrete numerical model for granular assemblies,Geotechnique 29 (1979) 47–65.

[12] El Shourbagy, A.M. Salah, Hans-Georg Matuttis, Dependence of the angle of reposeof heaps on the particle shape, Math. Asp. Pattern Formation Complex Fluids 1413(2005) 75–83.

[13] Particle Flow Code in 3-Dimensions (PFC3D)Manual, Version 4.00, Itasca ConsultingGroup, Minneapolis, MN, 2004.

[14] F. Podczeck, A shape factor to assess the shape of particles using image analysis,Powder Technol. 93 (1997) 47–53.

[15] M.A. Taylor, Quantitative measures for shape and size of particles, Powder Technol.124 (2002) 94–100.

[16] C.F. Mora, A.K.H. Kwan, Sphericity, shape factor, and convexity measurement ofcoarse aggregate for concrete using digital image processing, Cem. Concr. Res. 30(2000) 351–358.

[17] H. Wadell, Volume, shape and roundness of quartz particles, J. Geol. 43 (1935)250–280.

[18] I. Cavarretta, C. O’Sullivan, M.R. Coop, Applying 2D shape analysis techniques togranular materials with 3D particle geometries, AIP Conf. Proc. 1145, Allahabad,India, January 5-8 2009, pp. 833–836.

[19] H. Wadell, Volume, shape, and roundness of rock particles, J. Geol. 40 (1932)443–451.

[20] L.R. Carr, Evaluating flow properties of solids, Chem. Eng. 72 (2) (1965) 163–168.[21] Y.C. Zhou, B.H. Xu, A.B. Yu, P. Zulli, Experimental and numerical study of the angle of

repose of coarse spheres, Powder Technol. 125 (2002) 45–54.[22] K.E. Ileleji, B. Zhou, The angle of repose of bulk corn stover particles, Powder

Technol. 187 (2008) 110–118.[23] B. Majidi, K. Azari, H. Alamdari, M. Fafard, D. Ziegler, Discrete Element Method Ap-

plied to the Vibration Process of Coke Particles”, Light. Met. 2012 (2012) 1273–1277.[24] R.M. German, Particle Size Distribution as a Predictor of Suspension Flow Behavior,

in: J.P. Bennett, J.D. Smith (Eds.), Fundamentals of Refractory Technology”, TheAmerican Ceramic Society, 735 Ceramic Place, Westerville, Ohio 43081, 2006.

[25] R.M. German, S.J. Park, Mathematical relations in particulate materials processing:ceramics, powder metals, cermets, carbides, hard materials and minerals”, JohnWiley and Sons, Inc., New York, NY, 2008. 52.

[26] L.E. Silbert, D. Ertaş, G.S. Grest, T.C. Halsey, D. Levine, Geometry of frictionless andfrictional sphere packings, Phys. Rev. E 65 (031304) (2002) 1–6.