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An experimental study of packing of ellipsoids under vibrations
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An experimental study of packing of ellipsoids under vibrations C.X. Li a, R.P. Zou b, D. Pinson c, A.B. Yu b, Z.Y. Zhou b,⁎ a b c
School of Materials Science and Engineering, The University of New South Wales, Sydney, NSW 2052, Australia ARC Research Hub for Computational Particle Technology, Department of Chemical Engineering, Monash University, Clayton, Victoria 3800, Australia Steelmaking Technology and Planning, BlueScope Steel, 202, Port Kembla, NSW 2505, Australia
a r t i c l e
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Article history: Received 6 June 2019 Received in revised form 4 October 2019 Accepted 31 October 2019 Available online xxxx Keywords: Packing density Ellipsoidal particles Vibration frequency Aspect ratio Feeding method
a b s t r a c t Packing of ellipsoidal particles with a range of aspect ratios is experimentally studied under vibration conditions. The effects of operational conditions such as dropping heights, feeding methods, and vibration modes on packing density are investigated systematically. The results indicate that packing density first increases with dropping height, and then tends to be a certain value when dropping height is over 180dv. The relationship between packing density and aspect ratios gives an M-shaped curve, irrespective of operational conditions. This is also consistent with literature observations. The packing density obtained by batch-wised feeding method is higher than that obtained by total feeding method, especially when three-dimensional vibration is applied. The packing density increases with the increase of vibration frequency and then decreases, i.e. there is an optimum frequency to achieve maximum packing density. The optimum frequency varies with vibration dimensions. The local particle orientation order can be found under three-dimensional vibration with proper amplitude and frequency. © 2019 Elsevier B.V. All rights reserved.
1. Introduction Particle packing, as a core research area in particle technology, is quite significant to many industrial processes of handling granular materials. Many experimental and mathematical studies have been conducted to investigate packing systems of spheres [1–18]. However, real particles widely encountered in nature and industries are not spherical, for example, sands with a range of shapes and sizes. Particle shape can affect packing structures and hence transport properties such as permeability related to pore connection, and thermal conductivity related to particle connection in a packing. Therefore, understanding the fundamental of packing of non-spherical particles is high desirable. In the past decades, experimental and numerical investigations regarding the packing of non-spherical particles have been carried out extensively. However, as the contact detection and force calculation in numerical methods among non-spherical particles are quite complex, most investigations mainly focus on the packing of regular shaped particles such as cube [19–22], cylinder [23–27], sphero-cylinder [28], ellipse and ellipsoid [29–35], and other regular shaped particles [36,37]. Among these shaped particles, ellipsoids are typical as they can represent a large range of particle shapes varying from disk-like to cylinderlike. Many kinds of granular materials resemble these shapes [38]. The packing of ellipsoids can show some fascinating phenomena, for example, higher packing density than spheres [29,32,33].
⁎ Corresponding author. E-mail address: [email protected] (Z.Y. Zhou).
Donev et al. [29] investigated the packing of mono-sized ellipsoids by using the so-called Lubachevsky-Stillinger (LS) algorithm, and found that packing density of ellipsoids can reach 0.71 at aspect ratio of 0.67 (close to that of M&M candies) for oblate spheroids and 1.5 for prolate spheroids. It should be noted here that the aspect ratio of a spheroid is defined as the ratio of the principal diameter in the polar direction to the principal diameter in the equational plane [33]. Man et al. [32] also conducted an experimental study on the random packing of ellipsoids, showing a good agreement with numerical results. Donev et al. [31] also investigated the unusually dense crystal packing of ellipsoids by a molecular dynamics technique, and found that the maximal packing density of 0.770732 can be achieved whenever the maximal aspect ratio of the ellipsoids is greater than or equal to 1.732. Zhou et al. [33] used the DEM and investigated the packing of ellipsoids with a range of aspect ratios from 0.1 to 7.0. The packing properties with aspect ratios were discussed in terms of packing density, coordination number and radial distribution function. Local ordered structure was found in the packings of ellipsoids with small or large aspect ratios. Gan et al. [34] studied the packing of fine ellipsoidal particles, and correlations were established to describe the relationship between packing porosity and aspect ratio, particle size or force ratio (the ratio of particle-particle contact force or Van der Waals force to gravity). Vibration can significantly affect packing structures, and hence has been applied in the studies of packing of spherical [9,10,14–18] and non-spherical particles [20,22,25–27,35]. The feeding methods also play an important role in particle packing. For example, the effects of total feeding and batch-wised feeding methods on packing density of spheres under vibration conditions have been investigated [16–18].
https://doi.org/10.1016/j.powtec.2019.10.115 0032-5910/© 2019 Elsevier B.V. All rights reserved.
Please cite this article as: C.X. Li, R.P. Zou, D. Pinson, et al., An experimental study of packing of ellipsoids under vibrations, Powder Technol., https://doi.org/10.1016/j.powtec.2019.10.115
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The results show that the packing densities of spheres are strongly affected by feeding methods and vibration conditions. The ordered structure of spherical particle packing can be obtained by using batch-wised feeding method under three-dimensional vibration conditions with proper amplitude and frequency. In this work, the effects of different variables under vibration conditions on the packing of ellipsoids are investigated experimentally. The packing of mono-sized ellipsoids with a range of aspect ratios, and effects of dropping height, packing methods (total feeding and batch-wised feeding), and vibration modes (one-, two-, and threedimensional vibrations) are examined in detail. 2. Experimental method and conditions The physical experiments are carried out by a 3D vibration device as shown in Fig. 1. This setup can vibrate independently in three directions with different amplitudes and frequencies. The vibrations in the three directions are driven by three motors whose amplitudes and frequencies can be controlled independently by cams and transducers. The cam used is the eccentric wheel, which is a circular disk and solidly fixed to a rotating axle with its centre offset from the axle. In this work, we employed the same amplitude (A) and frequency (ω) in the three vibration directions. The phase difference is zero in the three components and the movement is in a straight line. Under this condition, it is considered that the phase angle of vibration has no effect. Note that in this work, 1D vibration means the motion of container in the vertical direction, 2D vibration means the motion in the horizontal direction, and 3D vibration presents the motion of container in both vertical and horizontal directions. The packing density is affected by many factors, for example, the container size. The packing density may vary with container sizes due to the wall effect, but does not change much if the size ratio of container (inner diameter) to particle diameter is large enough. To eliminate the wall effect, the method used by many investigators on the basis of the extrapolation of packing densities at different sized containers is employed, as done for sphere packings [16–18]. In this work, five different sized (81.18 mm, 109.97 mm, 154.90 mm, 190.96 mm and 242.10 mm in inner diameter) cylindrical containers made of PMMA are used in physical experiments, and shown in Fig. 2. The particles used in the physical experiments are ellipsoids with a range of aspect ratios from 0.1 to 10, indicating from very platy to elongated particles. The
ellipsoids are also made of PMMA materials The volume equivalent diameter is 5 mm. Each type of particles is made with different colours for visualisation as shown in Fig. 3. The experimental conditions are briefly given here. The effects of dropping height and feeding methods on packing density of ellipsoids are investigated first without vibrations. Note that two feeding methods (so called total feeding method and batch-wised feeding method) are used in this work as for spherical particle packing [16–18]. In the process of total feeding, all particles are fed into the container along the side wall of the container which is tipped to an angle before vibration, while the batch-wise feeding method means that particles are fed into the container layer by layer at a certain time interval. In the process of batch-wised feeding, the container is fixed vertically on the platform. Then, vibration is applied to the process of particle packing. The feeding methods and vibration modes (1D, 2D and 3D vibrations) are varied to study their effects on packing density. The effect of vibration frequency is also investigated under different vibration modes. 3. Results and discussion 3.1. Effect of dropping height without vibration The effect of dropping height on packing density is investigated first by using the so-called poured packing method without any vibrations, and the results are shown in Fig. 4. It can be observed that the packing density for all cases increases with the dropping height and tends to be a certain value when dropping height is over 180dv (dv is the volume equivalent diameter of ellipsoids). It should be noted that dropping height has a more obvious effect on the packing density with aspect ratios of 0.5 and 2.0 than those with aspect ratios of 0.25 and 3.5. Generally, increasing dropping height means that more energy is supplied to particles, enabling particles to remove local arching and rearrange themselves to form a denser packing. However, with the deviation of particle shape from spheres, it will result in more restriction of particle movement, lowers the probability of rearrangement, and then leads to lower increase in packing density. Fig. 5 shows the relationship between packing density and aspect ratio at different dropping heights. Clearly, the packing density increases with the increase of dropping height for the packing of all types of ellipsoids. Packing density does not vary much when dropping height is over 180dv. Much more significant effects of dropping height
Fig. 1. Schematic view of the vibration setup.
Please cite this article as: C.X. Li, R.P. Zou, D. Pinson, et al., An experimental study of packing of ellipsoids under vibrations, Powder Technol., https://doi.org/10.1016/j.powtec.2019.10.115
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Fig. 2. (a) Five different-sized cylindrical containers used in physical experiments; and (b) Schematic figures for batch-wised feeding method (left) and total feeding method (right).
on packing density can be found when particle shape deviates slightly from spheres, particularly for elongated particles. Note that the relationship between packing density and aspect ratios gives an M-shaped curve, irrespective of dropping height. M-shaped relationship is consistent with the observations in the literature [29,33,34], illustrating that ellipsoidal particles can pack more densely than spherical particles in a certain range of aspect ratios close to 1.0. The packing of ellipsoids becomes looser than spheres when particle aspect ratios are far from 1.0. This is because when particles become flat or elongated, the restrictions among particles increase, which limit particle movements. Hence, it is more difficult for particles to arrange themselves into more stable positions, and thus a looser packing is formed. This is a consequence of the increase in excluded volume effects for particle aspect ratios far from 1.0 [34]. 3.2. Effect of feeding methods The effect of feeding methods, including total feeding method and batch-wised feeding method, on packing density without vibration is studied first. These two feeding methods can be linked to the deposition intensity which means the amount of particles fed into the container within a certain time. The total feeding method is related to high deposition intensity while batch-wised means low deposition intensity. High deposition intensity indicates that more particles move simultaneously, that is, more restrictions among particles, which will result in the
increase of the probability of forming arch and bridge structures in a packing, and the decrease of packing density accordingly. Low deposition intensity or batch-wised feeding, however, means particles are fed into the container layer by layer. In that case, the restrictions on particle movements decrease greatly. More importantly, the latter particles can also easily arrange themselves into more stable positions without so many restrictions in total feeding method. As shown in Fig. 6, the packing density obtained by batch-wised packing method is higher than that by total feeding method in most cases. However, it is not the case for ellipsoids when aspect ratios are 5 and 10 in the present work. Note that generally, pouring particles at a certain height allows particles to have more potential energy, which then translates into kinetic energy for particles to remove local arching and rearrange themselves in forming a denser packing. However, for the process of total feeding method, the cylindrical container is tipped to an angle, and particles are fed into the container along the wall, then the cylinder is slowly rotated about its axis and gradually returned to the vertical position. In this case, the energy for particles is lower than that obtained by batchwised packing method, which leads to a lower packing density. For the packing formed, ellipsoids show a certain preferred orientation, indicating partially ordered structures. For example, as observed in Fig. 7(a), particles tend to align parallel near the wall by using total feeding method. Particles can form the so-called nematic phase in which particles have no positional order, but they tend to align to have directional or orientation order with their long axes. This local ordered
Please cite this article as: C.X. Li, R.P. Zou, D. Pinson, et al., An experimental study of packing of ellipsoids under vibrations, Powder Technol., https://doi.org/10.1016/j.powtec.2019.10.115
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Fig. 3. Ellipsoids used in physical experiments. From left to right, the aspect ratios are: (top) 0.1, 0.25, 0.5, 0.75, 1; (bottom) 2, 3.5, 5, 10.
packing results in an increase in the packing density. For the packing formed by using batch-wised feeding method, less local ordered structures are found as shown in Fig. 7 (b). The combination of these two effects results in that the packing density obtained by batch-wised feeding method is lower than that by total feeding method for the cases of aspect ratios larger than 4. The effects of feeding methods under vibration conditions are then studied, and the results are compared with those using total feeding method without vibration. Fig. 8 shows the results under different vibration modes, i.e., one-dimensional (1D), two-dimensional (2D), and three-dimensional (3D) vibrations, respectively. Clearly, there is a large increase in the packing density when vibration is applied. By vibration, more energy is supplied to particles, which can overcome the mutual locking among particles. Large pores can be effectively eliminated to allow particles to find the most stable positions for a denser packing. Note that the packing density obtained by using batch-wised feeding method is higher than that by total feeding method, irrespective of vibration modes. As discussed above, more arch and bridge structures may be formed in the packing by using the total feeding method, resulting in the decrease of packing density. However, these arch and bridge structures can partly be eliminated by feeding particles into container layer by layer, which increases the packing density. Therefore, higher packing density can be obtained by combining the effects of feeding method and vibration. It should be noted that the difference in packing density obtained by total feeding method and batch-wised feeding method is not obvious
under 1D vibration condition as shown in Fig. 8 (a). However, this is not the case under 2D and 3D vibration conditions shown in Fig. 8 (b) and (c), respectively. This indicates the effects of vibration modes on packing density, which will be discussed in the next section.
3.3. Effect of vibration conditions As discussed above, the vibration has a strong effect on packing density, and different vibration modes have different effect. Fig. 9 (a) and (b) re-plot the results shown in Fig. 8 to demonstrate the effect of vibration mode. It can be seen that the effects of vibration modes on packing density are quite different. Higher packing density can be obtained by 3D vibration than 1D and 2D vibrations. This is mainly because that 1D vibration can only provide energy for particles to move in the vertical direction, and movements in the horizontal directions are restricted. Similarly, 2D vibration can only make particles move in the horizontal directions, but not in the vertical direction. This restriction, however, can be overcome by using 3D vibration. In addition to the effect of vibration modes, Figs. 9 (a) and (b) also show that the variation of packing density obtained by total feeding method is not as obvious as that by batch-wised feeding method, particularly for particle shape being too flat and too elongated. This is because particles have more chances to move and rearrange themselves to form a much more stable position when these particles are fed into container layer by layer, especially when vibration is applied. The vibration frequency can have a significant effect on packing density as demonstrated in the packing of spherical [16–18] and non-
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Please cite this article as: C.X. Li, R.P. Zou, D. Pinson, et al., An experimental study of packing of ellipsoids under vibrations, Powder Technol., https://doi.org/10.1016/j.powtec.2019.10.115
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spherical particle [22]. In this work, the effect of vibration frequency on packing density is carried out at two amplitudes A = 0.5 mm and 1 mm, respectively. Fig. 10 shows the relationship between packing density and frequency for ellipsoidal particles with aspect ratios of 0.1, 0.5, 3.5 and 10 by using total feeding method. The curves have similar trends, i.e. packing density increases first with frequency to a maximum value and then decreases. There is an optimum value for frequency to achieve the maximum packing density. This is similar with the effect of vibration frequency on the packing density of spherical particles [16–18] and other regular shaped particles [22]. In fact, increasing frequency means that more energy is added to the packing particles, which not only accelerates the rearrangement of particles during densification, but also eliminates the bridge or arch structures formed in the initial packing. However, if frequency is too high, the resulted high vibration intensity will over-excite particles, which has negative effects on the formation of a dense structure. The figure also shows that the peak position of each curve remains unchanged, indicating that the maximum packing density can be obtained when frequency remains the optimum value, irrespective of particle shape. Fig. 11 shows the relationship between packing density and frequency under different vibration modes. It can be seen that the peak
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Fig. 6. Effects of feeding methods on packing density for packings of mono-sized ellipsoids.
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Fig. 7. Snapshot of local ordered packing of ellipsoids with aspect ratio = 10 near the wall with total feeding method (a) and batch-wised feeding (b).
position shifts towards to the left from 1D to 3D vibrations. This is because two and three electric motors provide energy for particles under two- and three-dimensional vibration conditions, respectively. According to the definition of vibration intensity Γ = Aω2/g (A – amplitude; ω – frequency, and g – gravity acceleration), the vibration intensity only depends on the frequency when amplitude is fixed. In this case, the value
Please cite this article as: C.X. Li, R.P. Zou, D. Pinson, et al., An experimental study of packing of ellipsoids under vibrations, Powder Technol., https://doi.org/10.1016/j.powtec.2019.10.115
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4. Conclusions In this work, the packing of mono-sized ellipsoids was investigated by physical experiments. The effects of dropping height, feeding methods, and vibration modes on packing density are investigated systematically. The following conclusions can be drawn: • Packing density increases first with dropping height and will tend to be a certain value when dropping height is over 180dv. The M-type relationship of packing density with aspect ratio is not affected much by the dropping height. • The packing density obtained by using batch-wised feeding method is generally higher than that obtained by using total feeding method, especially when three-dimensional vibration applied. • Vibration mode has a strong effect on packing density. This is mainly because particles can move and rearrange themselves into much more stable positions in three directions under 3D vibration conditions. • The packing density increases with the increase of vibration frequency, and then decreases, i.e. there is an optimum frequency to achieve maximum packing density. The optimum frequency changes with vibration modes from 1D to 2D and 3D.
of so-called optimum frequency for reaching the maximum packing density should decrease under 3D vibration to avoid high vibration intensity which will over-excite particles and destroy formed dense structure.
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3.4. Particle orientation
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Particle orientation is one important packing property for nonspherical particles. Much studies has been carried out to investigate the orientation in the packing of ellipsoids (for example, [33–35]). The results showed that oblate ellipsoids prefer facing upwards or downwards while prolate ellipsoids tend to orient horizontally. Fig. 12 shows some pictures taken in physical experiments after vibration. Clearly, oblate particles prefer facing upwards or downwards, and the prolate particles tend to lie flat in the plane normal to the vertical direction. Although the angles between semi-major axis and vertical direction can be obtained by a medical magnetic resonance imaging device [31], this measurement work was not undertaken in this study. Therefore, these pictures are shown here to demonstrate the local particle orientation order in the packing of ellipsoidal particles. It needs further detailed investigation, particularly with the application of numerical method of DEM.
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Frequency, Rad/s Fig. 11. Effects of vibration frequency on packing density of ellipsoids with aspect ratio = 0.5 under 1D, 2D, and 3D vibration conditions, where amplitude A = 1.0 mm.
Please cite this article as: C.X. Li, R.P. Zou, D. Pinson, et al., An experimental study of packing of ellipsoids under vibrations, Powder Technol., https://doi.org/10.1016/j.powtec.2019.10.115
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Fig. 12. Snapshots of packing of ellipsoidal particles after three-dimensional vibration, the aspect ratios from left to right are 0.25, 0.5, 2.0 and 3.5, respectively.
Declaration Competing Interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements The authors are grateful to Australian Research Council (ARC) and BlueScope Steel Research for the financial support of this work, and the NCI National Facility for the support in computation. References [1] J.D. Bernal, J. Mason, Packing of spheres: co-ordination of randomly packed spheres, Nature. 188 (1960) 910–911. [2] G.D. Scott, Packing of equal spheres, Nature. 188 (1960) 908–909. [3] J.B. Knight, C.G. Fandrich, C.N. Lau, H.M. Jaeger, R. Nagel, Density relaxation in a vibrated granular material, Phys. Rev. E 51 (1995) 3957–3963. [4] T.G. Owe Berg, R.L. McDonald, B.J. Trainor, The packing of spheres, Powder Technol. 3 (1969/70) 183–188. [5] D.L. Blair, N.W. Mueggenburg, A.H. Marshall, H.M. Jaeger, S.R. Nagel, Force distributions in three-dimensional granular assemblies: effects of packing order and interparticle friction, Phys. Rev. E 63 (2001) 413041–413048. [6] K.E. Daniels, R.P. Behringer, Hysteresis and competition between disorder and crystallization in sheared and vibrated granular flow, Phys. Rev. Lett. 94 (2005) 1–4. [7] Y. Nahmad-Molinari, J.C. Ruiz-Suárez, Epitaxial growth of granular single crystals, Phys. Rev. Lett. 89 (2002) 1–4. [8] D.A. Weitz, Packing in the spheres, Science 303 (2004) 968–969. [9] R. Amirifar, K.J. Dong, Q.H. Zeng, X.Z. An, Self-assembly of granular spheres under one-dimensional vibration, Soft Matter 14 (2018) 9856–9869. [10] X.Z. An, R.Y. Yang, K.J. Dong, A.B. Yu, DEM study of crystallization of monosized spheres under mechanical vibrations, Comput. Phys. Commun. 182 (2011) 1989–1994. [11] G.D. Scott, Radial distribution of the random close packing of equal spheres, Nature. 194 (1962) 956–957. [12] J.L. Finney, Random packings and the structure of simple liquids. I. the geometry of random close packing, Proc. Roy. Soc. Lond. A. 319 (1970) 479–493. [13] J.L. Finney, Random packings and the structure of simple liquids. II. The molecular geometry of simple liquids, Proc. Roy. Soc. Lond. A. 319 (1970) 495–507. [14] X.Z. An, R.Y. Yang, K.J. Dong, R.P. Zou, A.B. Yu, Micromechanical simulation and analysis of one-dimensional vibratory sphere packing, Phys. Rev. Lett. 95 (2005) 1–4. [15] A.B. Yu, X.Z. An, R.P. Zou, R.Y. Yang, K. Kendall, Self-assembly of particles for densest packing by mechanical vibration, Phys. Rev. Lett. 97 (2006) 1–4. [16] X.Z. An, C.X. Li, R.Y. Yang, R.P. Zou, A.B. Yu, Experimental study of the packing of mono-sized spheres subjected to one-dimensional vibration, Powder Technol. 196 (2009) 50–55. [17] X.Z. An, C.X. Li, Experiments on densifying packing of equal spheres by twodimensional vibration, Particuology. 11 (2013) 689–694.
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Powder Technology journal homepage: www.elsevier.com/locate/powtec
An experimental study of packing of ellipsoids under vibrations C.X. Li a, R.P. Zou b, D. Pinson c, A.B. Yu b, Z.Y. Zhou b,⁎ a b c
School of Materials Science and Engineering, The University of New South Wales, Sydney, NSW 2052, Australia ARC Research Hub for Computational Particle Technology, Department of Chemical Engineering, Monash University, Clayton, Victoria 3800, Australia Steelmaking Technology and Planning, BlueScope Steel, 202, Port Kembla, NSW 2505, Australia
a r t i c l e
i n f o
Article history: Received 6 June 2019 Received in revised form 4 October 2019 Accepted 31 October 2019 Available online xxxx Keywords: Packing density Ellipsoidal particles Vibration frequency Aspect ratio Feeding method
a b s t r a c t Packing of ellipsoidal particles with a range of aspect ratios is experimentally studied under vibration conditions. The effects of operational conditions such as dropping heights, feeding methods, and vibration modes on packing density are investigated systematically. The results indicate that packing density first increases with dropping height, and then tends to be a certain value when dropping height is over 180dv. The relationship between packing density and aspect ratios gives an M-shaped curve, irrespective of operational conditions. This is also consistent with literature observations. The packing density obtained by batch-wised feeding method is higher than that obtained by total feeding method, especially when three-dimensional vibration is applied. The packing density increases with the increase of vibration frequency and then decreases, i.e. there is an optimum frequency to achieve maximum packing density. The optimum frequency varies with vibration dimensions. The local particle orientation order can be found under three-dimensional vibration with proper amplitude and frequency. © 2019 Elsevier B.V. All rights reserved.
1. Introduction Particle packing, as a core research area in particle technology, is quite significant to many industrial processes of handling granular materials. Many experimental and mathematical studies have been conducted to investigate packing systems of spheres [1–18]. However, real particles widely encountered in nature and industries are not spherical, for example, sands with a range of shapes and sizes. Particle shape can affect packing structures and hence transport properties such as permeability related to pore connection, and thermal conductivity related to particle connection in a packing. Therefore, understanding the fundamental of packing of non-spherical particles is high desirable. In the past decades, experimental and numerical investigations regarding the packing of non-spherical particles have been carried out extensively. However, as the contact detection and force calculation in numerical methods among non-spherical particles are quite complex, most investigations mainly focus on the packing of regular shaped particles such as cube [19–22], cylinder [23–27], sphero-cylinder [28], ellipse and ellipsoid [29–35], and other regular shaped particles [36,37]. Among these shaped particles, ellipsoids are typical as they can represent a large range of particle shapes varying from disk-like to cylinderlike. Many kinds of granular materials resemble these shapes [38]. The packing of ellipsoids can show some fascinating phenomena, for example, higher packing density than spheres [29,32,33].
⁎ Corresponding author. E-mail address: [email protected] (Z.Y. Zhou).
Donev et al. [29] investigated the packing of mono-sized ellipsoids by using the so-called Lubachevsky-Stillinger (LS) algorithm, and found that packing density of ellipsoids can reach 0.71 at aspect ratio of 0.67 (close to that of M&M candies) for oblate spheroids and 1.5 for prolate spheroids. It should be noted here that the aspect ratio of a spheroid is defined as the ratio of the principal diameter in the polar direction to the principal diameter in the equational plane [33]. Man et al. [32] also conducted an experimental study on the random packing of ellipsoids, showing a good agreement with numerical results. Donev et al. [31] also investigated the unusually dense crystal packing of ellipsoids by a molecular dynamics technique, and found that the maximal packing density of 0.770732 can be achieved whenever the maximal aspect ratio of the ellipsoids is greater than or equal to 1.732. Zhou et al. [33] used the DEM and investigated the packing of ellipsoids with a range of aspect ratios from 0.1 to 7.0. The packing properties with aspect ratios were discussed in terms of packing density, coordination number and radial distribution function. Local ordered structure was found in the packings of ellipsoids with small or large aspect ratios. Gan et al. [34] studied the packing of fine ellipsoidal particles, and correlations were established to describe the relationship between packing porosity and aspect ratio, particle size or force ratio (the ratio of particle-particle contact force or Van der Waals force to gravity). Vibration can significantly affect packing structures, and hence has been applied in the studies of packing of spherical [9,10,14–18] and non-spherical particles [20,22,25–27,35]. The feeding methods also play an important role in particle packing. For example, the effects of total feeding and batch-wised feeding methods on packing density of spheres under vibration conditions have been investigated [16–18].
https://doi.org/10.1016/j.powtec.2019.10.115 0032-5910/© 2019 Elsevier B.V. All rights reserved.
Please cite this article as: C.X. Li, R.P. Zou, D. Pinson, et al., An experimental study of packing of ellipsoids under vibrations, Powder Technol., https://doi.org/10.1016/j.powtec.2019.10.115
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The results show that the packing densities of spheres are strongly affected by feeding methods and vibration conditions. The ordered structure of spherical particle packing can be obtained by using batch-wised feeding method under three-dimensional vibration conditions with proper amplitude and frequency. In this work, the effects of different variables under vibration conditions on the packing of ellipsoids are investigated experimentally. The packing of mono-sized ellipsoids with a range of aspect ratios, and effects of dropping height, packing methods (total feeding and batch-wised feeding), and vibration modes (one-, two-, and threedimensional vibrations) are examined in detail. 2. Experimental method and conditions The physical experiments are carried out by a 3D vibration device as shown in Fig. 1. This setup can vibrate independently in three directions with different amplitudes and frequencies. The vibrations in the three directions are driven by three motors whose amplitudes and frequencies can be controlled independently by cams and transducers. The cam used is the eccentric wheel, which is a circular disk and solidly fixed to a rotating axle with its centre offset from the axle. In this work, we employed the same amplitude (A) and frequency (ω) in the three vibration directions. The phase difference is zero in the three components and the movement is in a straight line. Under this condition, it is considered that the phase angle of vibration has no effect. Note that in this work, 1D vibration means the motion of container in the vertical direction, 2D vibration means the motion in the horizontal direction, and 3D vibration presents the motion of container in both vertical and horizontal directions. The packing density is affected by many factors, for example, the container size. The packing density may vary with container sizes due to the wall effect, but does not change much if the size ratio of container (inner diameter) to particle diameter is large enough. To eliminate the wall effect, the method used by many investigators on the basis of the extrapolation of packing densities at different sized containers is employed, as done for sphere packings [16–18]. In this work, five different sized (81.18 mm, 109.97 mm, 154.90 mm, 190.96 mm and 242.10 mm in inner diameter) cylindrical containers made of PMMA are used in physical experiments, and shown in Fig. 2. The particles used in the physical experiments are ellipsoids with a range of aspect ratios from 0.1 to 10, indicating from very platy to elongated particles. The
ellipsoids are also made of PMMA materials The volume equivalent diameter is 5 mm. Each type of particles is made with different colours for visualisation as shown in Fig. 3. The experimental conditions are briefly given here. The effects of dropping height and feeding methods on packing density of ellipsoids are investigated first without vibrations. Note that two feeding methods (so called total feeding method and batch-wised feeding method) are used in this work as for spherical particle packing [16–18]. In the process of total feeding, all particles are fed into the container along the side wall of the container which is tipped to an angle before vibration, while the batch-wise feeding method means that particles are fed into the container layer by layer at a certain time interval. In the process of batch-wised feeding, the container is fixed vertically on the platform. Then, vibration is applied to the process of particle packing. The feeding methods and vibration modes (1D, 2D and 3D vibrations) are varied to study their effects on packing density. The effect of vibration frequency is also investigated under different vibration modes. 3. Results and discussion 3.1. Effect of dropping height without vibration The effect of dropping height on packing density is investigated first by using the so-called poured packing method without any vibrations, and the results are shown in Fig. 4. It can be observed that the packing density for all cases increases with the dropping height and tends to be a certain value when dropping height is over 180dv (dv is the volume equivalent diameter of ellipsoids). It should be noted that dropping height has a more obvious effect on the packing density with aspect ratios of 0.5 and 2.0 than those with aspect ratios of 0.25 and 3.5. Generally, increasing dropping height means that more energy is supplied to particles, enabling particles to remove local arching and rearrange themselves to form a denser packing. However, with the deviation of particle shape from spheres, it will result in more restriction of particle movement, lowers the probability of rearrangement, and then leads to lower increase in packing density. Fig. 5 shows the relationship between packing density and aspect ratio at different dropping heights. Clearly, the packing density increases with the increase of dropping height for the packing of all types of ellipsoids. Packing density does not vary much when dropping height is over 180dv. Much more significant effects of dropping height
Fig. 1. Schematic view of the vibration setup.
Please cite this article as: C.X. Li, R.P. Zou, D. Pinson, et al., An experimental study of packing of ellipsoids under vibrations, Powder Technol., https://doi.org/10.1016/j.powtec.2019.10.115
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Fig. 2. (a) Five different-sized cylindrical containers used in physical experiments; and (b) Schematic figures for batch-wised feeding method (left) and total feeding method (right).
on packing density can be found when particle shape deviates slightly from spheres, particularly for elongated particles. Note that the relationship between packing density and aspect ratios gives an M-shaped curve, irrespective of dropping height. M-shaped relationship is consistent with the observations in the literature [29,33,34], illustrating that ellipsoidal particles can pack more densely than spherical particles in a certain range of aspect ratios close to 1.0. The packing of ellipsoids becomes looser than spheres when particle aspect ratios are far from 1.0. This is because when particles become flat or elongated, the restrictions among particles increase, which limit particle movements. Hence, it is more difficult for particles to arrange themselves into more stable positions, and thus a looser packing is formed. This is a consequence of the increase in excluded volume effects for particle aspect ratios far from 1.0 [34]. 3.2. Effect of feeding methods The effect of feeding methods, including total feeding method and batch-wised feeding method, on packing density without vibration is studied first. These two feeding methods can be linked to the deposition intensity which means the amount of particles fed into the container within a certain time. The total feeding method is related to high deposition intensity while batch-wised means low deposition intensity. High deposition intensity indicates that more particles move simultaneously, that is, more restrictions among particles, which will result in the
increase of the probability of forming arch and bridge structures in a packing, and the decrease of packing density accordingly. Low deposition intensity or batch-wised feeding, however, means particles are fed into the container layer by layer. In that case, the restrictions on particle movements decrease greatly. More importantly, the latter particles can also easily arrange themselves into more stable positions without so many restrictions in total feeding method. As shown in Fig. 6, the packing density obtained by batch-wised packing method is higher than that by total feeding method in most cases. However, it is not the case for ellipsoids when aspect ratios are 5 and 10 in the present work. Note that generally, pouring particles at a certain height allows particles to have more potential energy, which then translates into kinetic energy for particles to remove local arching and rearrange themselves in forming a denser packing. However, for the process of total feeding method, the cylindrical container is tipped to an angle, and particles are fed into the container along the wall, then the cylinder is slowly rotated about its axis and gradually returned to the vertical position. In this case, the energy for particles is lower than that obtained by batchwised packing method, which leads to a lower packing density. For the packing formed, ellipsoids show a certain preferred orientation, indicating partially ordered structures. For example, as observed in Fig. 7(a), particles tend to align parallel near the wall by using total feeding method. Particles can form the so-called nematic phase in which particles have no positional order, but they tend to align to have directional or orientation order with their long axes. This local ordered
Please cite this article as: C.X. Li, R.P. Zou, D. Pinson, et al., An experimental study of packing of ellipsoids under vibrations, Powder Technol., https://doi.org/10.1016/j.powtec.2019.10.115
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Fig. 3. Ellipsoids used in physical experiments. From left to right, the aspect ratios are: (top) 0.1, 0.25, 0.5, 0.75, 1; (bottom) 2, 3.5, 5, 10.
packing results in an increase in the packing density. For the packing formed by using batch-wised feeding method, less local ordered structures are found as shown in Fig. 7 (b). The combination of these two effects results in that the packing density obtained by batch-wised feeding method is lower than that by total feeding method for the cases of aspect ratios larger than 4. The effects of feeding methods under vibration conditions are then studied, and the results are compared with those using total feeding method without vibration. Fig. 8 shows the results under different vibration modes, i.e., one-dimensional (1D), two-dimensional (2D), and three-dimensional (3D) vibrations, respectively. Clearly, there is a large increase in the packing density when vibration is applied. By vibration, more energy is supplied to particles, which can overcome the mutual locking among particles. Large pores can be effectively eliminated to allow particles to find the most stable positions for a denser packing. Note that the packing density obtained by using batch-wised feeding method is higher than that by total feeding method, irrespective of vibration modes. As discussed above, more arch and bridge structures may be formed in the packing by using the total feeding method, resulting in the decrease of packing density. However, these arch and bridge structures can partly be eliminated by feeding particles into container layer by layer, which increases the packing density. Therefore, higher packing density can be obtained by combining the effects of feeding method and vibration. It should be noted that the difference in packing density obtained by total feeding method and batch-wised feeding method is not obvious
under 1D vibration condition as shown in Fig. 8 (a). However, this is not the case under 2D and 3D vibration conditions shown in Fig. 8 (b) and (c), respectively. This indicates the effects of vibration modes on packing density, which will be discussed in the next section.
3.3. Effect of vibration conditions As discussed above, the vibration has a strong effect on packing density, and different vibration modes have different effect. Fig. 9 (a) and (b) re-plot the results shown in Fig. 8 to demonstrate the effect of vibration mode. It can be seen that the effects of vibration modes on packing density are quite different. Higher packing density can be obtained by 3D vibration than 1D and 2D vibrations. This is mainly because that 1D vibration can only provide energy for particles to move in the vertical direction, and movements in the horizontal directions are restricted. Similarly, 2D vibration can only make particles move in the horizontal directions, but not in the vertical direction. This restriction, however, can be overcome by using 3D vibration. In addition to the effect of vibration modes, Figs. 9 (a) and (b) also show that the variation of packing density obtained by total feeding method is not as obvious as that by batch-wised feeding method, particularly for particle shape being too flat and too elongated. This is because particles have more chances to move and rearrange themselves to form a much more stable position when these particles are fed into container layer by layer, especially when vibration is applied. The vibration frequency can have a significant effect on packing density as demonstrated in the packing of spherical [16–18] and non-
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Please cite this article as: C.X. Li, R.P. Zou, D. Pinson, et al., An experimental study of packing of ellipsoids under vibrations, Powder Technol., https://doi.org/10.1016/j.powtec.2019.10.115
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spherical particle [22]. In this work, the effect of vibration frequency on packing density is carried out at two amplitudes A = 0.5 mm and 1 mm, respectively. Fig. 10 shows the relationship between packing density and frequency for ellipsoidal particles with aspect ratios of 0.1, 0.5, 3.5 and 10 by using total feeding method. The curves have similar trends, i.e. packing density increases first with frequency to a maximum value and then decreases. There is an optimum value for frequency to achieve the maximum packing density. This is similar with the effect of vibration frequency on the packing density of spherical particles [16–18] and other regular shaped particles [22]. In fact, increasing frequency means that more energy is added to the packing particles, which not only accelerates the rearrangement of particles during densification, but also eliminates the bridge or arch structures formed in the initial packing. However, if frequency is too high, the resulted high vibration intensity will over-excite particles, which has negative effects on the formation of a dense structure. The figure also shows that the peak position of each curve remains unchanged, indicating that the maximum packing density can be obtained when frequency remains the optimum value, irrespective of particle shape. Fig. 11 shows the relationship between packing density and frequency under different vibration modes. It can be seen that the peak
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Fig. 7. Snapshot of local ordered packing of ellipsoids with aspect ratio = 10 near the wall with total feeding method (a) and batch-wised feeding (b).
position shifts towards to the left from 1D to 3D vibrations. This is because two and three electric motors provide energy for particles under two- and three-dimensional vibration conditions, respectively. According to the definition of vibration intensity Γ = Aω2/g (A – amplitude; ω – frequency, and g – gravity acceleration), the vibration intensity only depends on the frequency when amplitude is fixed. In this case, the value
Please cite this article as: C.X. Li, R.P. Zou, D. Pinson, et al., An experimental study of packing of ellipsoids under vibrations, Powder Technol., https://doi.org/10.1016/j.powtec.2019.10.115
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4. Conclusions In this work, the packing of mono-sized ellipsoids was investigated by physical experiments. The effects of dropping height, feeding methods, and vibration modes on packing density are investigated systematically. The following conclusions can be drawn: • Packing density increases first with dropping height and will tend to be a certain value when dropping height is over 180dv. The M-type relationship of packing density with aspect ratio is not affected much by the dropping height. • The packing density obtained by using batch-wised feeding method is generally higher than that obtained by using total feeding method, especially when three-dimensional vibration applied. • Vibration mode has a strong effect on packing density. This is mainly because particles can move and rearrange themselves into much more stable positions in three directions under 3D vibration conditions. • The packing density increases with the increase of vibration frequency, and then decreases, i.e. there is an optimum frequency to achieve maximum packing density. The optimum frequency changes with vibration modes from 1D to 2D and 3D.
of so-called optimum frequency for reaching the maximum packing density should decrease under 3D vibration to avoid high vibration intensity which will over-excite particles and destroy formed dense structure.
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3.4. Particle orientation
Packing density
Particle orientation is one important packing property for nonspherical particles. Much studies has been carried out to investigate the orientation in the packing of ellipsoids (for example, [33–35]). The results showed that oblate ellipsoids prefer facing upwards or downwards while prolate ellipsoids tend to orient horizontally. Fig. 12 shows some pictures taken in physical experiments after vibration. Clearly, oblate particles prefer facing upwards or downwards, and the prolate particles tend to lie flat in the plane normal to the vertical direction. Although the angles between semi-major axis and vertical direction can be obtained by a medical magnetic resonance imaging device [31], this measurement work was not undertaken in this study. Therefore, these pictures are shown here to demonstrate the local particle orientation order in the packing of ellipsoidal particles. It needs further detailed investigation, particularly with the application of numerical method of DEM.
0.70 0.68 0.66 0.64 0.62 0.60 0
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Frequency, Rad/s Fig. 11. Effects of vibration frequency on packing density of ellipsoids with aspect ratio = 0.5 under 1D, 2D, and 3D vibration conditions, where amplitude A = 1.0 mm.
Please cite this article as: C.X. Li, R.P. Zou, D. Pinson, et al., An experimental study of packing of ellipsoids under vibrations, Powder Technol., https://doi.org/10.1016/j.powtec.2019.10.115
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Fig. 12. Snapshots of packing of ellipsoidal particles after three-dimensional vibration, the aspect ratios from left to right are 0.25, 0.5, 2.0 and 3.5, respectively.
Declaration Competing Interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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Please cite this article as: C.X. Li, R.P. Zou, D. Pinson, et al., An experimental study of packing of ellipsoids under vibrations, Powder Technol., https://doi.org/10.1016/j.powtec.2019.10.115