A study on the influence of particle shape and shape approximation on particle mechanics in a rotating drum using the discrete element method

Powder Technology 253 (2014) 256–265

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A study on the influence of particle shape and shape approximation on particle mechanics in a rotating drum using the discrete element method D. Höhner ⁎, S. Wirtz, V. Scherer Department of Energy Plant Technology, Ruhr University Bochum, 44780 Bochum, Germany

a r t i c l e

i n f o

Article history: Received 13 May 2013 Received in revised form 31 October 2013 Accepted 16 November 2013 Available online 23 November 2013 Keywords: DEM Rotating drum Particle geometry

a b s t r a c t In this study experimental and numerical investigations with the discrete element method (DEM) on the mechanical interactions of spheres and polyhedral dices in a rotating drum are conducted. In DEM the dices are approximated by polyhedra and smoothed polyhedra respectively and hence allow examining the influence of sharply-edged and smooth particle geometries on the mechanical behavior. Simulation results are in good general agreement with the experiments and hence demonstrate the adequacy of DEM as well as polyhedral and smoothed polyhedral approximation schemes to simulate non-spherical particle geometries. It was observed that an increase of particle angularity leads to an increase of the dynamic angle of repose. On the other hand, while spheres mix faster than the polyhedral dices, no significant difference in the mixing behaviors of the dices can be observed. © 2013 Elsevier B.V. All rights reserved.

1. Introduction Granular materials can be found in various shapes and quantities in a wide variety of application areas such as the energy sector, the food and pharmaceutical industry as well as geophysics. Knowledge about the mechanical behavior of these materials is of major importance for a proper design of processing units such as silos, rotating drums and firing systems. Rotating drums in particular exhibit a wide range of phenomena such as avalanching, mixing and segregation that involve granular materials and are not fully understood yet. The discrete element method (DEM), as introduced by Cundall and Strack in 1979 [1], has proven to be a capable tool for predicting the dynamics of large particle assemblies. A comprehensive overview about the major applications of the DEM in recent years is given in [2]. Several authors such as [3–6] identified particle shape approximation as one of the key challenges of DEM simulations, due to its significant influence on the mechanical behavior of granular materials. Nevertheless, most simulations conducted with the DEM involve spheres due to their simplicity in terms of contact detection, which results in lowest possible computing times. The major drawback of using spherical particle representation in the DEM is that most industrial granular materials exhibit a significantly different shape. As a result DEM predicts a deviating mechanical behavior on the single grain level as well as in larger particle assemblies, which renders the physical meaning of the obtained simulation results questionable [7]. For this reason several non-spherical particle shape approximation schemes have been proposed in literature. The most commonly used approaches include ellipsoids [5,8,9] and superquadrics [10–12], clustered⁎ Corresponding author. Tel.: +49 234 3223503; fax: +49 234 3214227. E-mail address: [email protected] (D. Höhner). 0032-5910/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.powtec.2013.11.023

spheres approaches [13–15], polyhedra [16–19], sphero-cylinders [20] and discrete function representations [21,12]. Especially polyhedra pose an interesting option due to their flexibility to approximate arbitrarily shaped objects of both symmetrical and asymmetrical shape at various accuracy levels. Moreover it was shown by Höhner et al. [6] that polyhedra allow approximations of sharply-edged as well as smoothed particle shapes, further increasing its versatility. In the following a brief overview over recent experimental and numerical studies on the influence of particle shape on the mechanical behavior of particle assemblies is given. Cleary and Sawley [13] investigated particle shape effects in 2D hopper flow using the discrete element method. In particular they studied the effect of both particle angularity and aspect ratio independently and together. It was shown that increasing angularity increases the flow resistance and reduces the flow rate of the hopper significantly while the flow behavior is only influenced moderately. On the other hand an increase of the particle aspect ratio not only reduces the flow rate in the hopper but also leads to substantial changes in the structure of the particle flow. Cleary [22] conducted 2D DEM simulations of a Couette shear cell with elliptical particles of varying angularity and aspect ratio. Some of the major findings are that both angularity and aspect ratio lead to significant flow changes but a variation of the aspect ratio has the overall stronger effect. With an increase of the non-circularity the shear resistance of the granular material increases. Cleary attributes this to the ability of non-circular particles to interlock due to the presence of sharp corners, flat surfaces and higher numbers of contacts with surrounding particles. A similar effect on the particle–wall interaction could be observed, with non-spherical particles having an increasingly better grip on the walls resulting in a decline of the wall slip. Markauskas et al. [5] investigated the piling of ellipsoidal particles and ellipsoids approximated with a multi-sphere approach in discrete

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element simulations. It was found that the piling behavior of the multisphere particles converges to the behavior exhibited by the ellipsoids with an increasing number of sub-spheres. They concluded that the multi-sphere approach is capable of approximating the mechanics of ellipsoidal particles with a reasonable number of sub-spheres. Hilton et al. [23] carried out simulations on the pneumatic conveying of particles in a rectangular duct. The DEM simulations were coupled to the gas flow using a finite-difference code and showed that particle shape is critical to the transition between different flow modes. While particles with a high sphericity transition to slug flow at high gas flow rates, pronounced non-spherical particles transition to dilute flow. They explained this behavior with the smaller void fraction in the beds of non-spherical particles. Mack et al. [24] conducted experiments and simulations with 322 spheres as well as a mixture of various polyhedral dices in hoppers of three different angles α. In 3D DEM simulations the dices were approximated as smoothed polyhedra, i.e. with rounded edges and vertices. The obtained experimental results showed that polyhedra flow is slightly faster than the flow of spheres and it was assumed that the flat surfaces enable them to slide past each other more easily. Moreover, for a hopper cone angle of α = 30°, polyhedra showed a greater tendency for arching (based on two experiments performed per particle geometry and α). For a flat bottom hopper they additionally observed a higher tendency for polyhedral particles to pile up in the corner of the hopper. A reasonable agreement between experiments and simulations in terms of static packing, flow behavior and hopper discharge rates was accomplished. Höhner et al. [6] also conducted experiments and simulations with spheres and polyhedral dices with varying sphericity in a hopper. In the DEM simulations the dices were represented as sharply-edged polyhedra and as smoothed polyhedra. With the exception of hexahedral particles they obtained a good general agreement between experiments and simulations with regards to discharge rate, flow profile and residual particle fraction in the hopper after discharge. In general they found that angular particles exhibit a lower discharge rate and a higher tendency to form pile-ups at the bottom walls of the hopper than spheres of similar volume and material. A comparison between both particle shape approximation schemes showed that sharply-edged particles lead to a higher increase of the microstructural strength of the particle assembly, which further enhances the effect of increasing particle angularity. Although many studies on particle mechanics in rotating drums can be found in literature [25–29], the studies mentioned are mostly restricted to spherical particles. Kodam et al. [30] conducted experiments and DEM simulations of 250 acrylic glass cylinders in a rotating drum. They approximated the test particles as true cylinders in their DEM simulations and used a contact detection algorithm customized for cylindrical objects. A comparison of the mean dynamic angle of repose showed that DEM predictions lie within the uncertainty interval of the experimental results. Wachs et al. [19] conducted DEM simulations of spheres and three different polyhedral particles in a rotating drum and examined the internal flow structure and the evolution of the dynamic angle of repose as a function of the rotation rate. They found that angular particles lead to an increase of the dynamic angle of repose as well as a loss of the flatness of the particle bed's free surface. In particular they found that the angle of repose increases with decreasing particle sphericity if the volumetric filling ratio of the rotating drum remains constant. Additionally they observed the occurrence of a slumping regime for non-spherical particles at low rotation rates where this could not be observed for spheres. The angular particle shape and the flat surfaces prevent particles from rolling down the free surface of the particle bed and promote avalanching, which results in an intermittent flow behavior. In this study experiments and DEM simulations with non-spherical particles of varying sphericity in a rotating drum were conducted in order to evaluate the influence of particle shape and approximation accuracy on the mechanical behavior. For this purpose a laboratory scale rotating drum was constructed and differently shaped test particles

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were selected. The dynamic angle of repose and the mixing behavior were chosen as parameters characterizing the mechanical interaction within the rotating drum. Moreover the effect of shape approximation is analyzed by applying two different methods of approximating nonspherical particle shapes in the DEM. 2. Experiments 2.1. Test particles For all experiments spheres as well as three different polyhedral dices were used (see Fig. 1). The polyhedral geometries are icosahedra, dodecahedra and hexahedra and exhibit a decreasing sphericity, i.e. increasing deviation from the spherical shape, in that order. For this reason they can be interpreted as sphere-approximations at different levels of accuracy. The dices are made from acrylic glass (PMMA) and were dyed with car finish in two different colors (green, red) in order to allow studying the mixing behavior within the rotating drum. Note that the spheres consist of polyoxymethylene (POM), which has similar mechanical properties as PMMA, and were dyed in blue and red for reasons of availability. Tables 1–3 summarize all relevant measurements, contact parameters and material properties of the test particles. A detailed description on how the particle properties were determined can be found in [6]. Note that we considered rolling friction in the DEM simulations only for spherical particles following Wang et al. [31] and Wachs et al. [19]. 2.2. Test facility For the experimental investigations a rotating drum as shown in Fig. 2 was constructed. The rotating drum consists of acrylic glass (PMMA) and has an inner diameter of 290 mm (outer diameter: 300 mm) as well as a length of 500 mm. At the top a transparent PMMA cover plate was attached, allowing a view into the rotating drum during the test runs. Additionally, in the bottom wall a lockable opening was incorporated, which allowed filling particles into the rotating drum without previous dismantling. The drum is seated on 4 length-adjustable staffs, which are attached to a table plate at an angle of 45°. At the top of the staffs plastics rolls are fixated, thus allowing a rotational movement of the incumbent drum. The rotating drum is driven by an electric motor, which is placed in a holding fixture underneath the table plate and can be vertically adjusted by a thread bar. A tooth belt is utilized to apply the torsional moment as well as rotational velocity from the rotating shaft onto the drum. A frequency converter regulates the driving speed of the motor and thus allows setting of the drum's rotational velocity. For additional inspection reflector stripes are fixed along the circumference of the drum, thus permitting to measure the rotational velocity of the drum with a laser control counter. Table 4 summarizes the relevant boundary conditions for the conducted experimental and numerical investigations. For investigating the mixing properties the particles were filled into the drum in two equally sized layers of red and green (blue) particles respectively as depicted in Fig. 3. Note that the number of particles differs between particle shapes as well as between experiments and DEM simulations (see Table 6).

Fig. 1. Test particles from left to right: sphere, icosahedron, dodecahedron, hexahedron.

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Table 1 Particle properties. Particle type Icosahedron Dodecahedron Octahedron Hexahedron Tetrahedron Sphere

Table 3 Modulus of shear G and Poisson's ratio ν of the investigated materials. Diameter [mm]

Edge length [mm]

Mass [g]

Volume [10−6 m3]

Density [kg/m3]

23.8 22.2 23.3 24.8 – 20.0

12.9 8.6 17.8 16.1 21.2 –

6.5 5.6 4.0 5.5 2.3 5.7

4.6 4.3 2.7 3.8 1.5 4.2

1420 1283 1469 1455 1522 1357

The discrete element method fully resolves the movement and interaction of all particles in the investigated system. By calculating the acting forces for all particle–particle and particle–wall contacts, the translational and rotational velocities as well as the positions and orientations of all particles can be determined based on Newton's and Euler's equations of motion. Since a detailed determination of elastic and plastic deformations of the contact area is computationally too expensive for systems with large numbers of particles, several simplified models exist to approximate and evaluate particle contacts. The soft-sphere approach allows particles to overlap and computes the resulting contact forces as a function of the overlap ξ as well as the relative velocity vrel of the contact particles. A detailed overview of force schemes is given by Di Maio and Di Renzo [32,33] as well as Kruggel-Emden et al. [34,35]. The most frequently employed normal force laws in DEM simulations are viscoelastic spring dashpot models [36]. The normal force Fn consists of two parts: while the elastic component (Fn,el) models elastic repulsion, the dissipative component (Fn,diss) accounts for the dissipation of energy during the collision process. n

n

n

n

n

n

ð1Þ

According to their dependence on the overlap, these force models can be classified to be either linear or non-linear. The linear viscoelastic spring dashpot model has the advantage of having a simple analytic solution, which guarantees a simple relation between its macroscopic collision properties, i.e. coefficient of restitution en and collision time tcoll, and its model parameters, i.e. the stiffness of a normal linear spring kn and the constant of a velocity-proportional dashpot γn [37]. 0 e ¼ exp@− n

t coll

n

n

n

γ π 2  meff

k γ − meff 2  meff

n

!2 !−12

n

k γ ¼π − meff 2  meff

!2 !−12 1 A

PVC-U

Aluminum

POM

1700 0.40

80 0.38

26,000 0.35

1050 0.35

m m

3. DEM modeling

n

PMMA

Here meff ¼ mi iþmj j is the effective mass of the contact particles i and j.

All experimental runs were recorded with a video camera (IDSImaging UI-1245-LE-C-HQ) at 25 frames per second, which was placed in front of the cover plate of the drum and allowed computer-based post-processing of the experimental results. In order to provide a better contrast between the particles and the surrounding in the video recordings, the drum was shaded using black cardboard on the outside.

F ¼ F el þ F diss ¼ k  ξ þ γ  vrel

G [MPa] ν [−]

Linear force models are also frequently applied for the tangential force component [32,6]. The earliest approach was proposed by Walton [38], which has found wide application in DEM simulations. In their model, the tangential contact force is calculated based on a linear elastic spring, unless the related Coulomb force is exceeded.       t t  t n t F ¼ −min k  ξ ; μ  F   sign ξ

The friction coefficient μ is a parameter which can be determined experimentally. On the other hand the spring stiffness of the tangential contact direction kt can be derived from the mechanical properties of the contact particles and the collision time. t



k ¼ κ  meff 

π

2

t coll

1−νi 1−ν j þ Gi Gj κ¼ 1−0:5  νi 1−0:5  ν j þ Gi Gj

 n ω Mrolling ¼ μ r   F   rel jωrel j

PVC-U-plate

Aluminum-plate

POM-plate

0.49/0.22/− 0.48/0.11/31.9

0.88/0.27/− 0.88/0.13/29.5

– 0.87/0.12/4.2

ð7Þ

where μr is the rolling friction coefficient and ωrel is the relative rotational velocity of the contact particles.

ð3Þ

0.55/0.39/− 0.88/0.27/27.1

ð6Þ

Both the modulus of shear G and Poisson's ratio ν are material properties of the contact particles i and j. In this study linear force models have been applied for both the normal and tangential direction in all DEM simulations. Additionally, for spherical particles only, a rolling torque resistance Mrolling is added to the total torque acting on the particle. Mrolling is modeled following Beer and Johnson [39] as

ð2Þ

PMMA-plate

ð5Þ

Here κ is the ratio of tangential and normal stiffness and can be determined as

Table 2 Coefficient of restitution [−]/friction coefficient [−]/rolling friction coefficient [10−5 m] determined for PMMA- and POM-spheres on PMMA-, PVC-U-, aluminum- and POMplates.

PMMA-sphere POM-sphere

ð4Þ

Fig. 2. Rotating drum used for the experimental investigations presented here.

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Table 4 Boundary conditions of experiments and DEM simulations. Inner drum diameter

mm

290

Drum length Rotational velocity Filling degree

mm rad/s vol.%

500 0.78 30

3.1. Polyhedra Polyhedra can be described within the DEM by a set of vertices and connectivity lists which connect the vertices to form edges and triangular surface elements. The overall contact forces and moments acting on the polyhedron can be determined as the averaged value of the contact forces and moments acting at each contact point. A thorough definition of the contact area for polyhedral contacts in the DEM can be found in [40]. num ! 1 X ! F ¼ F num k¼1 k

ð8Þ

num ! 1 X ! M M¼ num k¼1 k

ð9Þ Fig. 4. CP between two polyhedra.

Detecting and quantifying a contact between polyhedral bodies can be complicated and many approaches have been described in literature [16–19]. One of the most effective methods is the Common Plane algorithm, which was first introduced by Cundall [16]. The algorithm substitutes the complicated contact detection between two polyhedra by two less complicated contact detection tests between each polyhedron with a plane (see Fig. 4). The crux of the algorithm is the correct positioning of the Common Plane (CP), which has to meet the following conditions: • The centroids of the polyhedra are located on opposite sides of the CP. • d1 and d2 are the distances of the closest vertices of polyhedron 1 and 2 to the CP. d1 and d2 have equal absolute values and opposite signs. • The total gap, which is defined as d2 − d1 is a maximum, such that every rotation of the CP would result in a smaller total gap. In the author's discrete element code the correct position of the CP is determined with a non-iterative procedure based on the Fast Common Plane algorithm [17]. This algorithm identifies that there is only a limited number of possible orientations for the CP, which considerably speeds up its computation.

Fraige et al. [18] and extended for 3D-simulations by Wang et al. [31]. The geometry of smoothed polyhedra and their handling in a discrete element code differ significantly from their mathematical counterparts described in Section 3.1. Fig. 5 shows a 2D-sketch of a mathematical and a smoothed cube. A polyhedron can be understood as the skeleton structure of the smoothed polyhedron. In each vertex of the polyhedron a sphere with radius rSM is inscribed, hence eliminating sharp corners. The same radius is attributed to the faces (i.e. surface triangle segments) and edges of the polyhedron resulting in rounded edges. Depending on the ratio of the selected rSM to the particle size, the shape of the generated smoothed polyhedra may vary from very roundish to almost sharply-edged as regular polyhedra. As described in [41] contact detection between two smoothed polyhedra requires multiple contact checks, which represent simple geometric problems such as calculating the distance between a point and a triangle, a point and a line, two points and two edges. In the authors' implementation particle contact is structured as follows:

3.2. Smoothed polyhedra

1. Reduce computing effort by identifying vertices and edges of particles 1 and 2 which are not relevant for the contact. Only contact-relevant vertices and edges are considered in the following steps.

Another method of incorporating complex shaped particles into the DEM is the usage of polyhedra with rounded edges and vertices (in the following denoted as smoothed polyhedra) as initially introduced by

2. Check for contact between each vertex of particle 1 and every surface triangle of particle 2 and vice versa. 3. Check for contact between each edge of particles 1 and 2.

Fig. 3. Starting position of the particle bed in the rotating drum (here: icosahedra).

Fig. 5. 2D-sketch of a cube (left) and a cube represented as smoothed polyhedra (right).

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and μ of the PMMA-dices and POM-spheres as outlined in Section 2.1 were used. Furthermore setting up DEM simulations requires additional specification of the contact parameters kn, γn and tcoll, which depend on particle properties, such as the particle diameter and volume, and material properties, such as Young's modulus E. In general defining these parameters is complicated, since a consideration of the actual value of E results in small collision times and hence very small DEM time-steps, which would render the computing effort unaffordable. Instead simulations often rely on best practice using the following assumptions [4,19]:

Fig. 6. Determination of the dynamic angle of repose for experimental (left) and numerical (right) results.

4. Calculate contact forces and moments at all determined contact points and accumulate them according to Eqs. (8) and (9).

• The coefficient of restitution en and the friction coefficient μ are the two contact parameters with the most significant influence on particle mechanics. • The collision time is not a primary feature of the system investigated. • The sole purpose of the stiffness coefficient kn is to control the maximum overlap ξmax between contact particles. Using these assumptions, specification of the contact parameters can be accomplished as follows:

3.3. Shape approximation of the test particles 1. Determine en and μ. In this study the three types of dices were approximated by polyhedra and smoothed polyhedra, respectively. Thus it was possible to compare two shape approximation schemes that deliver sharply-edged as well as smooth particle approximations. The smoothed polyhedra were generated such that their diameters are equal to the diameters measured for the respective dice-type. Assuming that rSM = 1 mm, the smoothed polyhedra render good approximations of the real dice-shapes. For obvious geometric reasons, matching diameters of the polyhedra to the real dices would imply an overall smaller particle volume. Since we consider the particle volume as the main property that has to be kept constant in order to make experiments and simulations comparable, we set the particle volume of the polyhedra to the measured volume of the corresponding dices for all shape approximations, which implied slightly larger diameters. 3.4. Simulation parameters In all simulations the experimentally determined values for the volume, density and material properties as well as the contact properties en

2. Define ξmax/r, with r being the particle radius. 3. Estimate the maximum pre-collisional relative velocity vrel,max. 4. Derive kn, γn and tcoll from en and ξmax using Eqs. (2) and (3). Cleary et al. [4] observed that ξmax needs to be in the range of 0.1 to 0.5% of the particle radius to ensure that the particle flow dynamics are independent of the stiffness coefficient kn. Performing step 2 and 3, one has to keep in mind that small ξmax as well as high vrel,max result in small DEM time-steps. In order to keep the computing effort on a reasonable level, we set ξmax to 4% of the particle radius and estimated that vrel,max = 1 m/s in the DEM simulations presented here. As shown in [6], choosing smaller ξmax does not automatically improve the simulation results and thus does not justify the associated increase of the computing effort. The contact parameters used in the DEM simulations were equal for both polyhedral and smoothed polyhedral approximations of the respective dice-types and can be found in [6]. Note that contacts in DEM are integrated in time with more than a single time-step, so that the DEM time-step Δt has to be smaller than tcoll. As in [42] we set Δt = tcoll/25 in our simulations.

Fig. 7. Dynamic angle of repose. Error bars depict the standard deviation of the measured values.

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4. Results and discussion 4.1. Dynamic angle of repose For all experiments and DEM simulations conducted the dynamic angle of repose in the rotating drum was determined by using a specifically designed image processing routine. For this purpose the video recordings, starting from the point in time at which a stationary particle movement was established, were disassembled into individual images. For every single image the surface of the particle bed was scanned by detecting pixels of predefined values within the RGB color space. Fig. 6 illustrates the result of the procedure with the pixels detected on the surface of the particle bed being highlighted: Based on the pixel positions the angle of repose of the particle bed could be determined by using the method of least squares for linear data fitting. A subsequent averaging over all single images rendered the dynamic angle of repose in the rotating drum. Fig. 7 shows the results for experiments and DEM simulations for the four investigated particle shapes: The results indicate that the dynamic angle of repose increases with increasing angularity of the particle shape. While icosahedra and dodecahedra exhibit very similar results, spheres and hexahedra differ significantly in negative and positive direction respectively. DEM is in very good agreement with the experimental results for both shape approximation schemes. Table 5 summarizes the results: As Table 5 indicates, apart from the average value for the dynamic angle of repose, also the standard deviation increases with increasing particle angularity. It was already observed by Höhner et al. [6] that the angular dices increase the strength and shear resistance of the particle assembly, which leads to a relatively inhomogeneous particle movement in the rotating drum and hence higher standard deviations. Interlocking hinders particles to roll freely down the surface of the particle bed and promotes the formation of pile ups, thus, increasing the angle of repose. A zoomed-in snapshot of the hexahedral particle bed illustrates the formation of inner structures which cannot be observed in this form for spherical particles (see Fig. 8). While DEM shows very good agreement with tendencies observed in the experiments, it also predicts slightly greater angles of repose, with spheres being the exception. Considering the uncertainty interval of the results, those deviations are deemed acceptable. Moreover polyhedra tend to form greater angles of repose than their smoothed polyhedral counterparts, indicating that sharply-edged particle shapes increase the interlocking of particles even further and lead to the formation of pile-ups and subsequent avalanches.

Fig. 8. Zoomed-in snapshot of the hexahedral particle bed. Dotted lines highlight the piling up of the dices.

angle of repose, the video recordings of experiments and simulations were disassembled into single images and pixels belonging to both particle types (green/blue and red) were detected based on their color value. For all single images a 10 × 10 grid was applied (see Fig. 9) and the ratio of green/blue to red pixels was determined for every cell. By calculating the deviation of the pixel ratio for every cell to the ratio of the single image, a standard deviation could be determined, which is used as a measure for the mixing state of the particle bed in the single image. The standard deviations are normalized with the starting situation at t = 0 s (see Fig. 3), so that 1 can be interpreted as state of ideal segregation, and 0 as ideal mixing respectively. Fig. 10 summarizes the experimental and numerical results: In the experiments spheres mix faster than the three types of dices. On the other hand no significant difference in the mixing behavior between the dices can be observed. Obviously a switch from spherical to non-spherical particles influences the mixing considerably, while a further decrease of the particle sphericity has only small effects. While the results obtained with the DEM, for both methods of shape approximation, show a good agreement with these tendencies, the deviating mixing behavior of spheres is less pronounced. Fig. 11 shows the results for the mixing behavior separately for each particle shape. The experimental results are depicted as zone of

4.2. Mixing For all experiments and DEM simulations conducted the mixing behavior within the rotating drum was monitored with an in-house image processing software. Similar to the evaluation process of the dynamic

Table 5 Dynamic angle of repose.

Sphere (experiment) Sphere (DEM) Icosahedron (experiment) Icosahedron (DEM polyhedra) Icosahedron (DEM smoothed polyhedra) Dodecahedron (experiment) Dodecahedron (DEM polyhedra) Dodecahedron (DEM smoothed polyhedra) Hexahedra (experiment) Hexahedra (DEM polyhedra) Hexahedra (DEM smoothed polyhedra)

Average value [°]

Standard deviation [°]

27.19 25.41 36.52 38.12 37.79 37.30 39.83 37.96 42.06 44.49 44.58

1.54 0.68 2.04 1.07 0.94 2.11 1.35 1.07 2.92 2.31 1.79

Fig. 9. Evaluation grid used for determining the state of mixing in the rotating drum in every individual image.

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Fig. 10. Mixing behavior within the rotating drum.

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Fig. 11. Mixing behavior within the rotating drum for each particle shape.

fluctuation as could be derived from the three test runs conducted for each particle shape. Fig. 11 indicates that the DEM generally predicts a faster and better mixing of both particle types. A possible explanation is the difference between the video qualities and properties of experiments and simulations (see Fig. 6). In the pixel-based evaluation method used here, varying lighting conditions and contrasts in the video recordings have a strong influence on the number of detected pixels and thus the mixing state of the particle bed. For this reason certain deviations between experimental and numerical results are inherent to the selected evaluation procedure. Fig. 12, which shows the particle bed at t = 40 s after starting the rotation of the drum, confirms the explanation. The better mixing state in the DEM simulations, as implied by Fig. 11, cannot be confirmed by a mere visual assessment. The difference between both utilized shape approximation schemes in the DEM simulations are marginal and may be attributed to fluctuations, since only one simulation per particle shape and approximation scheme was conducted. Nevertheless a small tendency for smoothed polyhedra to mix faster and better than regular polyhedra can be observed in Fig. 11. The increased strength and shear resistance of the particle bed due to sharply-edged particle shapes, as outlined in Section 4.1, is a likely explanation for this behavior. 4.3. Static packing Another possible reason for deviating mixing behavior between experiments and simulations, as depicted in Fig. 11, can be found in the

starting positions of the particle bed. The numbers of particles in the rotating drum (see Table 6) as well as the initial packing structures are not exactly the same in both experiment and simulation, which can be partly attributed to the different filling processes used. While this has no significant effect on the dynamic angle of repose, which is not measured until the particle bed has reached a steady state in the drum, a strong influence on the mixing state is coherent. The required numbers of particles for the desired filling fraction were determined experimentally by pouring them, handful by handful, from the top into the vertically positioned drum up to a predefined height of 150 mm. In the DEM simulations this was accomplished by positioning the particles with random starting velocities and orientations above the vertically oriented drum, with a subsequent settling until rest under gravity. Hence the dynamic variables such as dropping height and dropping rate were not exactly the same in both filling processes, which may lead to differences of the particle packing as was observed by Zhang et al. [43] for spheres. Nevertheless, the DEM results, with hexahedra being the exception, are in good agreement with the experimental values. It is worth noticing in this context that the smoothed polyhedral approximations of the dices exhibit smaller deviations from the experimental results, indicating that the static packing structure is better predicted by this particle shape approximation scheme. Additionally the filling process before every test run was different between experiment and simulation. In order to have two layers of differently colored particles in the drum, the previously determined numbers of particles were poured handful by handful into the already horizontally positioned and fully assembled drum through a lockable opening at the

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Fig. 12. Particle bed after t = 40 s in the rotating drum. For the dices: experiment (left), polyhedra (middle), smoothed polyhedra (right).

bottom wall. In contrast, in the DEM simulations particles were generated with initial random velocities and orientations within the drum and subsequently let fall to rest under gravity. The differences between both filling processes are likely to have an effect on the initial packing structures and hence the mixing behaviors illustrated in Figs. 10 and 11. 5. Summary In this study the mechanical interactions of spherical and nonspherical particles in a rotating drum were investigated experimentally and numerically. For this purpose three different types of dices, which correspond in shape to the platonic solids icosahedron, dodecahedron and hexahedron, were selected and approximated as sharply-edged polyhedra and smoothed polyhedra in DEM simulations. The three

Table 6 Number of particles required for a filling fraction of 30% in the drum. Bracketed values show the differences to the corresponding experimental values.

Spheres Icosahedra Dodecahedra Hexahedra

Experiment

DEM (polyhedra)

DEM (smoothed polyhedra)

1448 1301 1370 1678

1413 (−2.42%) 1271 (−2.31%) 1308 (−4.53%) 1551 (−7.57%)

1304 (+0.23%) 1349 (−1.53%) 1566 (−6.67%)

types of dices exhibit a varying sphericity and can be considered as sphere-approximations at different accuracy levels, hence allowing an evaluation of the parameter “shape approximation accuracy” on particle mechanics. In the experiments the dynamic angle of repose increased with decreasing particle sphericity. The increasing angularity of the particle shapes increased the formation of pile-ups as well as the interlocking of particles while limiting their free movement, which resulted in greater angles of repose as well as a more inhomogeneous movement of the particle assembly. The experiments showed that while spheres mix faster and better than the dices, there are no significant differences in the mixing behavior of the three dice-types. While a switch from spherical to non-spherical particle geometries had a significant influence on the mixing behavior, a further decrease of the sphericity had only marginal effects. The DEM simulations with spheres as well as the polyhedral and smoothed polyhedral dice-approximations showed good agreement with the experimental results. The dynamic angle of repose for all particle shapes was well within the uncertainty interval set by the experiments. Moreover the influence of varying particle sphericity was predicted correctly by the DEM. On the other hand the numerical mixing results deviated from the experimental results, exhibiting an overall faster and more complete mixing of the binary particle system. The filling process of the rotating drum, significantly impacting the

D. Höhner et al. / Powder Technology 253 (2014) 256–265

initial structure of the particle bed, as well as the evaluation method of the mixing process were identified as sources for the observed deviations. Comparing both particle shape approximation schemes showed that polyhedra tend to form greater angles of repose than their smoothed polyhedral counterparts, indicating that sharply-edged particle shapes further increase the interlocking of particles and the formation of pileups. This observation was confirmed by smoothed polyhedra mixing faster, albeit only marginally, than polyhedra. The results obtained in this study have underlined the adequacy of the DEM to simulate the mechanics of particle assemblies consisting of non-spherical particles. Moreover the influence of approximation accuracy as well as the selection of an appropriate method of shape approximation on the mechanical particle behavior was demonstrated numerically and experimentally. Future work will extend the investigation of the influence of particle geometry, method of shape approximation and approximation accuracy on particle mechanics to complex industrial applications with irregularly shaped granular materials. Additionally these parameters will be evaluated on different scales, i.e. different ratios of particle size to primary system dimensions. These studies will help clarify in which size ranges particle geometry becomes a relevant factor for the mechanical behavior of granular materials.

Acknowledgments The current study was funded by the German Science Foundation (DFG) within the project SCHE 322/6-2. The authors would like to acknowledge the generous support.

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