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Numerical simulation of tetrahedral particle mixing and motion in rotating drums
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Numerical simulation of tetrahedral particle mixing and motion in rotating drums Nan Gui a , Xingtuan Yang a , Jiyuan Tu a,b , Shengyao Jiang a,∗ , Zhen Zhang a a Institute of Nuclear and New Energy Technology, Collaborative Innovation Center of Advanced Nuclear Energy Technology, Key Laboratory of Advanced Reactor Engineering and Safety of Ministry of Education, Tsinghua University, Beijing 100084, China b School of Engineering, RMIT University, Melbourne, VIC 3083, Australia
a r t i c l e
i n f o
Article history: Received 11 April 2017 Received in revised form 10 July 2017 Accepted 15 August 2017 Available online xxx Keywords: Tetrahedron Non-spherical particle Mixing Drum Flow regime Discrete element method
a b s t r a c t A regular tetrahedron is the simplest three-dimensional structure and has the largest non-sphericity. Mixing of tetrahedral particles in a thin drum mixer was studied by the soft-sphere-imbedded pseudohard particle model and compared with that of spherical particles. The two particle types were simulated with different rotation speeds and drum filling levels. The Lacey mixing index and Shannon information entropy were used to explore the effects of sphericity on the mixing and motion of particles. Moreover, the probability density functions and mean values and variances of motion velocities, including translational and rotational, were computed to quantify the differences between the motion features of tetrahedra and spheres. We found that the flow regime depended on the particle shape in addition to the rotation speed and filling level of the drum. The mixing of tetrahedral particles was better than that of spherical particles in the rolling and cascading regimes at a high filling level, whereas it may be poorer when the filling level was low. The Shannon information entropy is better than the Lacey mixing index to evaluate mixing because it can reflect the real change of flow regime from the cataracting to the centrifugal regime, whereas the mixing index cannot. © 2017 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.
Introduction Particulate materials and their assemblies are far from well understood because they have rather complicated properties and flow behavior. Size and shape are the two most commonly used parameters to describe particles (Taylor, 2002). The scales of particles in different industries may range from nanometer to large gravels (Zhang, Lu, Wang, & Li, 2008). Meanwhile, shape also influences the microscopic behavior and macroscopic properties of granular media, e.g., polarizability and viscosity (Bullard & Garboczi, 2013). However, vast amounts of data are usually needed to characterize particle shape. Various shape descriptors have been used in the modeling of particle flows (Abou-Chakra, Baxter, & Tüzün, 2004). Because of their easy and accurate description, modeling, and measurement, particles with regular shapes, e.g., circle/sphere, square/cube, and triangle/tetrahedron, are of fundamental impor-
∗ Corresponding author. E-mail addresses: [email protected], [email protected] (N. Gui), [email protected] (S. Jiang).
tance in research related to drag (Hölzer & Sommerfeld, 2009; Loth, 2008; Wachs, 2009; Yow, Pitt, & Salman, 2005), mixing (Liu et al., 2017; Nafsun et al., 2017), and packing (Boon, Houlsby, & Utili, 2013; Wachs, Girolami, Vinay, & Ferrer, 2012). In particular, shape must be considered in particle packing, because the packing of non-spherical particles is considerably different from that of spherical particles (Li, Zhao, Lu, & Xie, 2010). The tetrahedron is a fundamental shape with many important features. For example, for common objects, the sphericity is the largest for a sphere (1) and the lowest for a regular tetrahedron (0.671) (Li, Li, Zhao, Lu, & Meng, 2012). In other words, a regular tetrahedron can be regarded as the most non-spherical shape, which means it has the lowest flowability. Modeling a polyhedron using the discrete element method (DEM) is much more complex than the modeling of many other non-spherical shapes like spherocylinders (Meng & Li, 2017) and ellipsoids (Zhao, Zhou, Liu, & Lai, 2015). They also mentioned that the study of tetrahedral materials is important in geotechnical, mining, and transportation engineering. DEM-based methods are the most attractive techniques for modeling non-spherical particle shapes (Lu, Third, & Muller, 2015; Wachs et al., 2012). Wachs et al. (2012) used a variant of DEM named Grains3d to cope with convex non-spherical particles by
https://doi.org/10.1016/j.partic.2017.08.004 1674-2001/© 2017 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.
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Nomenclature D f (·) fi,ms g G Itet kc kr ms Mtet MI ME nA , nB ni,ms Np Ns N P ri,ms Ri,ms s, s0 , sr t ti,ms Ti,ms up
vi Vj,tet Vp ␦xi,ms ı i,ms T j,tet c r ω
Drum diameter Function Contact force vector of member sphere i Gravity acceleration Gravity force of tetrahedron Moment of inertia of tetrahedron Stiffness factor for particle collision Stiffness factor of the restoring force in the relaxation process Mass of member sphere Mass of tetrahedron Mixing index Shannon information entropy Concentrations of two predefined types of particles Unit vector in the normal direction in collision for member sphere i Number of tetrahedrons Number of member spheres in a tetrahedron Number of data points for Overall proportion of particle Position vector of member sphere i with respect to tetrahedron centroid Restoring force of member sphere i pointing to the equilibrium position Standard deviations at unmixed state and randomly mixed state Time Unit vector in the tangential direction in collision for member sphere i Torque of the tetrahedral particle caused by the member sphere i Magnitude of particle velocity components Velocity vector of member sphere i Velocity vector of tetrahedron j Particle velocity vector Displacement vector of member sphere i in collision Displacement vector of member sphere i in the relaxation process Interval Rotational velocity vector of tetrahedron Damping coefficient in collision Damping coefficient in the relaxation process Variance Rotating velocity of drum Coefficient of friction
Subscripts c Collision or contact Indices i, j ms Member sphere p Particle r Restoring Tetrahedron tet x, y Directions of coordinates
using the Gilbert–Johnson–Keerthi algorithm to compute the distance between convex bodies. They demonstrated the capability of their method to model regular shapes including a sphere, cylinder, cube, and tetrahedron. However, they did not compare tetrahedra and spheres to a sufficient extent to provide details of the effects of particle shape on mixing in a drum. Recently, out group proposed the soft-sphere-imbedded pseudo-hard particle model (SIPHPM),
Fig. 1. Sketch of a surface collision between member spheres of two tetrahedral particles.
which has been validated for modeling of non-spherical shapes (Gui, Yang, Jiang, & Tu, 2016). In this work, we use the SIPHPM method to simulate the motion and mixing of tetrahedral particles in a drum mixer. The main purpose of this work is to compare the mixing and motion of regular tetrahedra with those of spheres to determine the effects of shape on particle mixing and motion in drums.
Methodology SIPHPM The SIPHPM (Gui et al., 2016) was used here to simulate the collision between tetrahedral particles. The particles were all uniform regular tetrahedra. Each was composed of 28 member spheres enclosing six surfaces (Fig. 1). Each tetrahedron was assumed to be composed of a pseudo-rigid material that was always undeformable. The collision between tetrahedra was solved through the collision between the member spheres and surfaces enclosed by the member spheres. In addition, the member spheres could deviate slightly from their equilibrium positions. A restoring force was always generated once a deviation occurred. The equilibrium positions of member spheres were on the vertexes and sides of each tetrahedron and determined by the position and orientation of the tetrahedron. The governing equations of the tetrahedra and member spheres were coupled and solved as follows: For the member spheres:
⎧ f i,ms = kc ıxi,ms − c ıx˙ i,ms , ⎪ ⎪ ⎪ ⎪ iff ,t > f i,ms , ni,ms , thenf i,ms , t i,ms = f i,ms , ni,ms ⎪ ⎪ ⎨ i,ms i,ms T i,ms = r i,ms × f i,ms ,
⎪ ⎪ ⎪ R i,ms = kr ı i,ms − r ı˙ i,ms , ⎪ ⎪ ⎪ ⎩
(1)
v˙ i = f i,ms + R i,ms /ms − g,
where ␦xi,ms , ri,ms , and ı i,ms are the deformation displacement, position vector (relative to the center of the tetrahedron), and deviation displacement from the equilibrium position of the ith member sphere, respectively; fi,ms , Ri,ms , and Ti,ms are the damping elastic collision force, restoring force, and torque of the ith member sphere, respectively; n and t are the normal and tangential directions at the collision point, respectively; g is the gravity acceleration; kc , kr , c , and r are the stiffness factors (kx ) and damping coefficients (x ) of collision (denoted by subscript ‘c’) and restoration (denoted by subscript ‘r’), respectively; is the friction coefficient for member spheres; vi and ms are the velocity and mass of the member spheres, respectively; and ‘·’ is the inner operator.
Please cite this article in press as: Gui, N., et al. Numerical simulation of tetrahedral particle mixing and motion in rotating drums. Particuology (2017), https://doi.org/10.1016/j.partic.2017.08.004
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For the tetrahedra:
⎧ Ns ⎪ ⎪ ⎪ f i,ms − G, ⎨ Mtet V˙ j,tet = ⎪ ⎪ ˙ T i,ms , ⎪ ⎩ I tet · j,tet = i
3
Bed radius R × depth W (mm) Bed rotating speed, ω ( rad/s)
(2)
i
˙ where Vj,tet , j,tet , Mtet , and Itet are the translational velocity, rotational velocity, mass, and moment of inertia of the jth tetrahedral particle, respectively; G is the gravity force of the tetrahedron; and Ns = 28 is the number of member spheres each tetrahedron contains. After Eqs. (1) and (2) were solved, the collision and motion of both member spheres and tetrahedra could be obtained simultaneously. The tetrahedron-to-wall and tetrahedron vertexto-tetrahedron surface collisions were also incorporated by solving the member sphere-to-wall and member sphere-to-surface collisions. For example, a member sphere-to-surface collision is sketched in Fig. 1, where the member sphere of one tetrahedron contacts the surface of another tetrahedron at point ‘C’. This collision was determined by solving the collisional forces (in Eq. (1)) in the normal (‘n’) and tangential (‘t’) directions. In addition, the tetrahedron-to-wall collision was solved by the member sphereto-wall collision in a similar manner. The main features of the current SIPHPM method that differ from those of the sphere-assembly method (Yan, Yu, & McDowell, 2009) are the following. (1) The member spheres are located discretely on the sides of the non-spherical shape and the surfaces enclosed by the member spheres are also primary elements to be considered in solving the collision of tetrahedra. Therefore, the whole outer surface of the non-spherical shape does not need to be covered by member spheres, which lowers the required number of member spheres considerably. However, the member sphere-to-surface collision must be considered. (2) The overlap between the member spheres is avoided to build in the construction of tetrahedron. A relaxation force is superposed on each member sphere to force it to oscillate slightly about the equilibrium position. Gui et al. (2016) have already validated the SIPHPM method. Simulation configurations and conditions In this work, a drum of diameter D = 800 mm was filled with regular tetrahedra with a side a of 16 mm. The initial filling state (Fig. 2(a)) was formed through a free falling process of all the particles from a certain height above the bottom of the drum. After that, the drums were rotated at the same acceleration (2 rad/s2 ) from the same initial state to different target speeds of ω = (0.25–5) rad/s (Table 1). Two categories (Category A with Np = 1449 and Category B with Np = 924) with different filling levels of tetrahedral particles were studied. The drum was rotated at ω = 0.25, 0.5, 1, 2, and 3 rad/s for Category A and ω = 0.5, 1, 2, 3, 4, and 5 rad/s for Category B. To compare the behavior of tetrahedra with that of spheres, rotating drums filled with the same number of spheres of the same mass and equivalent diameter as those of the tetrahedra were also simulated. The parameters used are listed in Table 1. Results and discussion Mixing of tetrahedra Fig. 2(a) shows the initial static packing state of the tetrahedra, which was the starting point of rotation. The particles were stratified, forming layers. The mixing states of Np = 1449 tetrahedra in the drums for Category A after about 10 s of rotation at different
Initial rotation acceleration, ˇ (rad/s2 ) Side of tetrahedron, a (mm) Number of tetrahedrons, Np
Number of sphere in a tetrahedron Total number of spheres, Ns
Sphere diameter in referential cases, de (mm) Tetrahedron density, t (kg/m3 ) Stiffness factor, kc , kr (N/m) Poisson ratio, Restitution coefficient, e Friction coefficient, Time step, ıt (s) Number of simulation step, Nt
800 × 32 0.25, 0.5, 1, 2, 3, 4, 5 2 16 1449 for Category A, 924 for Category B 28 25,872 for Category A, 40,572 for Category B 13.764 2000 4 × 103 , 1 × 105 0.3 0.95 0.3 1 × 10−6 1 × 107
angular speeds are shown in Fig. 2(b)–(f). Fig. 2(g) and (h) displays the mixing states of spheres at ω = 2 and 3 rad/s, respectively. At ω = 0.25 rad/s (Fig. 2(b)), the particles were almost unmixed between the stratified layers. For ω = (0.5–2) rad/s (Fig. 2(c)–(e)), the tetrahedra were well mixed in the rolling and cascading flow states. At ω = 3 rad/s (Fig. 2(f)), a tumbling/centrifugal flow state of the tetrahedra was established. The flow states of the spheres (Fig. 2(g) and (h)) were the same as those of the tetrahedra. In a similar manner, the mixing of Np = 924 tetrahedra for Category B are shown in Fig. 3(a)–(f) for ω = (0.5–5) rad/s. The flow regimes for Category B at ω = 3 and 4 rad/s are still rolling or cascading, whereas they are centrifugal regime for Category A in Fig. 2(f). Thus, the flow regime of tetrahedra depends on the filling level. A lower filling level of tetrahedra seems to cause the tetrahedra to go into the centrifugal regime later than that at a higher filling level. A similar trend was also observed for spheres; e.g., see the centrifugal regime in Fig. 2(h) at ω = 3 rad/s for Category A, and the cataracting regime in Fig. 3(g) at ω = 3 rad/s for Category B. The filling level may affect the flow regime of particles through the following two mechanisms. (1) A high filling level may increase the frequency of contacts between the particles and rotating walls. (2) A high filling level may increase the space occupied by the particles and decrease the free void space of the drum above the filling level. Thus, the particles in a rotating drum at a high filling level may be more prone to display centrifugal behavior than that containing particles at a lower filling level. Compared with the cascading flow regime of tetrahedra in Fig. 3(d) at ω = 3 rad/s, the spheres are upcast in Fig. 3(g) at ω = 3 rad/s. Moreover, comparing Fig. 3(e) with (h) (both at ω = 4 rad/s), the flow regime of tetrahedra is cascading whereas it is centrifugal for spheres. Thus, the flow regime is also affected by the particle shape. Because spheres have the best flowability whereas tetrahedra have the worst, it is harder for tetrahedra than spheres to enter the centrifugal regime. In other words, particles with lower sphericity may keep to steady regimes more easily than those with higher sphericity, which may have implications to maintain the desired working regime of cascading.
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Fig. 2. (a) Initial packing state at t = 0 s, and the mixing states of tetrahedra at t = 10 s in Category A at ω of (b) 0.25, (c) 0.5, (d) 1, (e) 2, and (f) 3 rad/s. The inset in (a) is a schematic model of a tetrahedron. Mixing states of spheres at t = 10 s are shown in (g) and (h) at ω of 2, and 3 rad/s, respectively, for comparison. The particle indices are shown by different colors.
Mixing index and entropy analyses Category A: Np = 1449 The Lacey mixing index (Lacey, 1954) was used here to quantify the mixing levels in the drums under different conditions. This index is expressed as: MI (t) =
s02
− s2 (t)
s02 − sr2
number in a cell. In a general mixing process, MI (t) progresses from 0 to 1, where a higher MI implies a better mixing state. The Shannon information entropy (Shannon, 1948) was also used to confirm the results. The Shannon information entropy is defined based on the local concentration of particles:
ME = ,
(3)
where s(t) is the standard deviation of particle concentration, i.e., the square root of the mean squares of the deviations of particle concentration at t; s2 (t) is the variance; s02 is the variance corresponding to the completely unmixed state; sr2 corresponds to the random mixture; and s02 = P(1 − P) and sr2 = P(1−P) , where P is the n overall proportion of one type of particles and n is the particle mean
[−nA (r)log2 (nA (r)) − nB (r)log2 (nB (r))] dr,
(4)
r
where nA (r) and nB (r) are the concentrations of the two predefined types of mixing particles at r, and r is a position vector. In a system composed of two types of particles, the concentration at r can be computed as nA (r) = NA (r)/(NA (r) + NB (r)). When nA is 0 or 1, nA (r)log2 (nA (r)) = 0. Otherwise, nA (r)log2 (nA (r)) = / 0. By integrating over the whole area of a drum, ME can be used to quantify the overall degree of particle mixing in the drum.
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Fig. 3. Mixing states of tetrahedra at t = 10 s in Category B at ω of (a) 0.5, (b) 1, (c) 2, (d) 3, (e) 4, and (f) 5 rad/s; mixing states of spheres at ω of (g) 3 and (h) 4 rad/s.
Using the same data, including the same initial unmixed states, to determine both the mixing index and information entropy may give insights into the difference between them, which could aid evaluation of the real particle mixing level. Therefore, Fig. 4(a)–(d) compare the Lacey mixing index and Shannon information entropy with respect to time and number of revolutions for Category A with Np = 1449 tetrahedra. The following is seen. (1) MI (t) of tetrahedra and spheres are almost the same at ω = 0.25 rad/s (Fig. 4(a) and (c)) because the particles are almost unmixed (see Fig. 2(b)). According to ME (t) (Fig. 4(b) and (d)), the mixing of tetrahedra is slightly greater than that of spheres. Because MI (t) and ME (t) were computed from the same data, the difference between Fig. 4(a) and (b) (or Fig. 4(c) and (d)) for ω = 0.25 rad/s may indicate the difference between the mixing index and entropy for mixing evaluation. MI (t) considers the
fluctuation of particle concentration and indicates the overall uniformity of particle dispersion. ME (t) is based on the formation of local mixing structure. Thus, when MI (t) is the same for tetrahedra and spheres, ME (t) is not necessarily the same. This means that, under the same level of overall dispersion, the local mixing structure may still differ depending on particle shape. (2) For 0.5 ≤ ω ≤ 2 rad/s, MI (t) for tetrahedra with respect to both time (Fig. 4(a) and (b)) and the number of revolutions (Fig. 4(c) and (d)) are greater than those for MI (t). The same behavior was determined for ME (t). Thus, both mixing index and entropy analyses indicate the same conclusion that tetrahedra mix better than spheres in the rolling and cascading regimes. This would be because the interactions between tetrahedra are more complicated than those between spheres. (3) For ω = 3 rad/s, the inverse becomes true for MI (t) (Fig. 4(a) and (c)). The tetrahedra have lower MI (t) than the spheres. This
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Fig. 4. Variations of (a, c) Lacey mixing indices and (b, d) Shannon information entropies for Category A with (a, b) time and (c, d) number of revolutions at different angular speeds. Lines show the data for spheres and scatter plots depict the data for tetrahedra.
means that in the centrifugal regime, the spheres are more uniformly distributed near the wall of the rotating drum than the tetrahedra. However, the mixing entropy in Fig. 4(b) and (d) indicates that the local mixing of tetrahedral particles is still better than that of the spheres when ω = 3 rad/s.
Category B: Np = 924 The mixing index and entropy for Category B are shown in Fig. 5(a) and (b) with regard to time and Fig. 5(c) and (d) in terms of revolutions, respectively. The following is found:
(1) At ω = 0.5 rad/s, MI (t) values of tetrahedra are lower than those of spheres (Fig. 5(a) and (c)), whereas their ME (t) are similar (Fig. 5(b) and (d)). This means that the overall dispersion uniformity of tetrahedra is lower than that of spheres, even though the formation of local mixing structures is equivalent. (2) At ω = 1 rad/s, the MI (t) values of tetrahedra cross over those of spheres. A similar trend was also observed for ME (t), for which the cross point was earlier than that of MI (t). This indicates that the spheres are mixed faster than the tetrahedra, while the ultimate level of mixing of spheres is lower than that of the tetrahedra. This situation is reasonable because the spheres are more flowable than the tetrahedra. In other words, the tetrahedral shape may need to reach a threshold value to start to flow, just like the non-Newtonian fluids. Thus, spheres are easier to mix than tetrahedra. However, as indicated above for Category A, the tetrahedra in the rolling regime are mixed better than the spheres. (3) Based on MI (t) in Fig. 5(a) and (c), the mixing levels of spheres are higher than those of tetrahedra at ω = (2–5) rad/s. This
means that from the cascading to the centrifugal regime, the mixing of tetrahedra is always poorer than that of spheres. (4) However, ME (t) values of spheres in Fig. 5(b) and (d) at ω = 4 and 5 rad/s decrease after they reach the highest points because of the change of flow regime. Note that spheres at ω = 4 rad/s have already entered the centrifugal regime (Fig. 3(h)). Thus, as indicated by ME (t), the mixing level may decrease in the centrifugal regime. In this regard, the mixing entropy ME (t) is better than the mixing index MI (t) to evaluate mixing because the latter cannot reflect the change of mixing regime (see Fig. 5(a) and (c)). Based on the good reflection of flow regime evolution by ME (t), we believe that ME (t) shows the real mixing state. (5) Referring to the curves in Fig. 4 illustrating the most useful flow regimes from rolling to cataracting, the mixing of non-spherical particles is better than that of spherical particles at high filling levels. In contrast, for the same flow regimes, the mixing of nonspherical particles is poorer than that of spherical particles at low filling levels. These results correlate the influences of filling level and particle shape as well as rotation speed with mixing level.
Probability density function of velocity In this section, we calculated probability density functions (PDFs) as follows to explore the characteristics of velocities (up,x , up,y , ωp,z ): f (, t)T =
ıN (T ) N T
|
T ∈ t− 1 T,t+ 1 T 2 2
,
(5)
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Fig. 5. Variations of (a, c) Lacey mixing indices and (b, d) Shannon information entropies for Category B with (a, b) time and (c, d) number of revolutions at different angular speeds. Lines show the data for spheres and scatters plot the data for tetrahedra.
where N is the total number of data points of , and ıN is number of data points located within time interval
the t − 12 T, t + 12 T . Here, f(up,x ) (translational velocities up,x and up,y , and rotational velocity ωp,z ) and T = 1 s for Category A and 0.1 s for Category B, respectively. Fig. 6 shows f(ωp,z ), f(up,x ), and f(up,y ) with T = 1 s for Category A at t = 1 and 10 s. At t = 1 s, f(ωp,z ) (or f(up,y )) of tetrahedra and spheres do not have evident differences (Fig. 6(a) and (b)), except for f(up,x ) maxima of tetrahedra and spheres. This confirms that the flowability of tetrahedra is lower than that of spheres, because f(up,x ) of a tetrahedron always has its maximum at up,x = 0, whereas f(up,x ) of a sphere has its maximum at about up,x ≈ −0.15 m/s (Fig. 6(a)). Moreover, f(ωp,z ) values of tetrahedra and spheres are quite different (Fig. 6c). The spheres have larger f(ωp,z ) at larger |ωp,z |, whereas the tetrahedra always have lower f(ωp,z ) at non-zero |ωp,z |. This means that the rotational motions of tetrahedra and spheres during the early stage of mixing are quite different (Fig. 6(f)). In other words, it confirms the previous assumptions that tetrahedra are the most non-spherical particles and thus have the lowest flowability. This is also true at t = 10 s. However, at t = 10 s, f(up,x ) and f(up,y ) or the translational motions of tetrahedra are quite different from those of spheres at almost all rotation speeds. At low rotation speeds (ω ≤ 1 rad/s), the peak values of f(up,x ) and f(up,y ) of spheres are larger and more symmetrical than those of f(up,x ) and f(up,y ) of tetrahedra. In addition, Fig. 7 also shows the case where the time interval T was shortened from 1 to 0.1 s to determine the PDF characteristics of instantaneous velocities. Similar results for f(up,x ), f(up,y ), and f(ωp,z ) were also obtained for Category B. Regarding translational motions, the velocity PDFs of tetrahedra are more widely distributed than those of spheres and have larger skewness, which
means the velocities of tetrahedra are more asymmetrically distributed throughout the bed than spheres. Regarding rotational velocity, the PDFs of tetrahedra are narrower than those of spheres. In other words, most tetrahedra have smaller rotational velocities than those of spheres. In addition, at ω = 5 rad/s, f(ωp,z ) of tetrahedra and spheres both become narrower than those at lower ω, which is caused by the centrifugal regime with near-zero rotational rotation of both tetrahedral and spherical particles. Mean and variance of velocity With the aid of PDFs, more characteristics of particle velocity were quantified. The mean and variance of velocity were directly calculated using the following equations:
+∞
=
f (, t)(t)dt,
(6)
−∞
+∞
() =
2
f (, t)((t) − ) dt,
(7)
−∞ 1/2
where is |V p |(|V p | = (v2p,x + v2p,y ) ) or ωp,z . Taking Category B as an example, Fig. 8(a) and (b) shows |Vp | and ωp,z , respectively. The following is found. (1) At low rotation speeds (ω = 0.5, 1 rad/s), |Vp | of tetrahedra is equivalent or greater than that of spheres. (2) In contrast, at higher rotation speeds (ω ≥ 3 rad/s), |Vp | of tetrahedra is lower than that of spheres. (3) Additionally, ωp,z of tetrahedra is always lower than that of spheres. This is because of the following. (i) The rolling of spherical particles near the wall, which is caused by the friction from the drum, can evidently increase the particle rotational velocity. Thus, the spherical particles, which have greater flowability,
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Fig. 6. Probability density functions of translational velocities (a, d) f(up,x ) and (b, e) f(up,y ), and rotational velocities (c, f) f(ωp,z ) for Category A at (a–c) t = 1 s and (d–f) t = 10 s. Here, T = 1 s.
always have larger rotational velocity than the tetrahedral particles. (ii) In contrast, the greater rolling of particles near the wall may sometimes attenuate the efficiency of kinetic energy transfer from the rotating wall to the spherical particle assembly. In other words, the tetrahedral particle assembly would have larger internal interactions than those of the spherical particle assembly. Thus, when the drum rotates, the friction from the wall of the drum would be transferred more efficiently throughout the tetrahedral particle assembly than throughout the spherical particle assembly, especially at low rotation speeds when there is sufficient time for energy transfer through particle–particle interactions. As a result, the mean translational velocity of tetrahedral particles is greater than that of the spherical particles at low rotation speeds. Finally, Fig. 9 shows the corresponding variances of Vp and ωp,z . The trends in Fig. 9 are similar to those observed in Fig. 8 for the
mean values of Vp and ωp,z . The mean values and variances of Vp of tetrahedra are greater than those of spheres at lower rotation speeds (ω < 2 rad/s), whereas the opposite is the true at higher rotation speeds (ω ≥ 3 rad/s). The variance of ωp,z of tetrahedra is always lower than that of spheres. The reasons for this are similar to those explained above. In addition, when the flow enters the centrifugal regime, the mean rotational velocity and variances of translational and rotational velocity may decrease because of the uniform low rotational velocity for all particles in the centrifugal regime. Brief summary We studied the mixing and motion behavior of regular tetrahedra—the most non-spherical shape—in rotating drums
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Fig. 7. Probability density functions of translational velocities (a, d) f(up,x ) and (b, e) f(up,y ), and rotational velocities (c, f) f(ωp,z ) for Category B at (a, b, c) t = 2.5 s and (d, e, f) t = 7.5 s. Here T = 0.1 s.
through comparison with the behavior of spherical particles using SIPHPM simulations. The main findings were as follows. 1. The flow regime depended on both the shape and filling level of particles. A lower filling level always made the particle flow enter the centrifugal regime later, as did the degree of non-sphericity. Because of its lower flowability, less spherical particles are more favorable than spherical ones to keep the flow regime steady. Thus, the tetrahedral shape exists in a rolling flow regime, whereas the spherical shape enters the centrifugal regime at ω = 4 rad/s at a lower filling level (Np = 924). 2. MI and ME evaluate the mixing state from different viewpoints, that is, the overall uniformity of dispersion and the levels and accumulation of local mixing, respectively. Importantly, the influence of particle shape on mixing was related to the fill-
ing level. In particular, at a high filling level and in the common rolling and cascading regimes, both MI and ME indicated that the tetrahedra mixed better than spheres because of the more complicated interaction between the non-spherical particles than between the spheres. In contrast, at a lower filling level, the mixing of tetrahedra may be poorer than that of spheres in the rolling and cascading regimes. 3. In the centrifugal regime, MI and ME showed different trends for the influence of particle shape on mixing. We believe ME is better than MI to evaluate mixing because of the successful indication of flow regime transition by ME . For example, ME indicated that the influence of particle shape on mixing at different filling levels in the centrifugal regime was the opposite to that in the rolling and cascading regimes.
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Fig. 8. Variations of mean values of (a) particle translational velocity |Vp | and (b) particle rotational velocity ωp,z with time for Category B.
Fig. 9. Variations of variances of (a) particle translational velocity (Vp ) and (b) particle rotational velocity (ωp,z ) with time for Category B.
4. With regard to the mixing efficiency in the rolling and cascading regimes, at low filling levels and low drum rotation speeds, the spheres were mixed faster than the tetrahedra; this is because spheres are more likely to roll and flow than tetrahedra, especially at low rotation speeds. However, the ultimate levels of mixing of spheres were lower than those of tetrahedra. 5. The PDFs of translational velocities of tetrahedra had wider spans, lower peaks, and more asymmetrical distributions than those of spheres. The mean and variance values of translational velocities of spheres were greater than those of tetrahedra at high rotation speeds, whereas they may be equivalent or lower than those of tetrahedra at low rotation speeds. The PDF of rotational velocity ωp,z consolidated the conclusion that the nonspherical tetrahedral particles were harder to rotate than the spherical particles. This is because they had narrower PDF distributions, smaller PDF values at larger ωp,z , and lower values of mean and variance of rotational velocity ωp,z than those of the spherical particles. Additionally, the mean ωp,z and variances of translational velocities may decrease in the centrifugal regime for the tetrahedral particles.
Acknowledgments The authors are grateful for the support of this research by the National Natural Science Foundation of China (Grant Nos. 51406100 & 51576211), the Science Fund for Creative Research Groups of the National Natural Science Foundation of China (Grant No. 51321002), the National High Technology Research and Develop-
ment Program of China (863) (2014AA052701), and the Foundation for the Author of National Excellent Doctoral Dissertation of P.R. China (FANEDD, Grant No. 201438).
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Numerical simulation of tetrahedral particle mixing and motion in rotating drums Nan Gui a , Xingtuan Yang a , Jiyuan Tu a,b , Shengyao Jiang a,∗ , Zhen Zhang a a Institute of Nuclear and New Energy Technology, Collaborative Innovation Center of Advanced Nuclear Energy Technology, Key Laboratory of Advanced Reactor Engineering and Safety of Ministry of Education, Tsinghua University, Beijing 100084, China b School of Engineering, RMIT University, Melbourne, VIC 3083, Australia
a r t i c l e
i n f o
Article history: Received 11 April 2017 Received in revised form 10 July 2017 Accepted 15 August 2017 Available online xxx Keywords: Tetrahedron Non-spherical particle Mixing Drum Flow regime Discrete element method
a b s t r a c t A regular tetrahedron is the simplest three-dimensional structure and has the largest non-sphericity. Mixing of tetrahedral particles in a thin drum mixer was studied by the soft-sphere-imbedded pseudohard particle model and compared with that of spherical particles. The two particle types were simulated with different rotation speeds and drum filling levels. The Lacey mixing index and Shannon information entropy were used to explore the effects of sphericity on the mixing and motion of particles. Moreover, the probability density functions and mean values and variances of motion velocities, including translational and rotational, were computed to quantify the differences between the motion features of tetrahedra and spheres. We found that the flow regime depended on the particle shape in addition to the rotation speed and filling level of the drum. The mixing of tetrahedral particles was better than that of spherical particles in the rolling and cascading regimes at a high filling level, whereas it may be poorer when the filling level was low. The Shannon information entropy is better than the Lacey mixing index to evaluate mixing because it can reflect the real change of flow regime from the cataracting to the centrifugal regime, whereas the mixing index cannot. © 2017 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.
Introduction Particulate materials and their assemblies are far from well understood because they have rather complicated properties and flow behavior. Size and shape are the two most commonly used parameters to describe particles (Taylor, 2002). The scales of particles in different industries may range from nanometer to large gravels (Zhang, Lu, Wang, & Li, 2008). Meanwhile, shape also influences the microscopic behavior and macroscopic properties of granular media, e.g., polarizability and viscosity (Bullard & Garboczi, 2013). However, vast amounts of data are usually needed to characterize particle shape. Various shape descriptors have been used in the modeling of particle flows (Abou-Chakra, Baxter, & Tüzün, 2004). Because of their easy and accurate description, modeling, and measurement, particles with regular shapes, e.g., circle/sphere, square/cube, and triangle/tetrahedron, are of fundamental impor-
∗ Corresponding author. E-mail addresses: [email protected], [email protected] (N. Gui), [email protected] (S. Jiang).
tance in research related to drag (Hölzer & Sommerfeld, 2009; Loth, 2008; Wachs, 2009; Yow, Pitt, & Salman, 2005), mixing (Liu et al., 2017; Nafsun et al., 2017), and packing (Boon, Houlsby, & Utili, 2013; Wachs, Girolami, Vinay, & Ferrer, 2012). In particular, shape must be considered in particle packing, because the packing of non-spherical particles is considerably different from that of spherical particles (Li, Zhao, Lu, & Xie, 2010). The tetrahedron is a fundamental shape with many important features. For example, for common objects, the sphericity is the largest for a sphere (1) and the lowest for a regular tetrahedron (0.671) (Li, Li, Zhao, Lu, & Meng, 2012). In other words, a regular tetrahedron can be regarded as the most non-spherical shape, which means it has the lowest flowability. Modeling a polyhedron using the discrete element method (DEM) is much more complex than the modeling of many other non-spherical shapes like spherocylinders (Meng & Li, 2017) and ellipsoids (Zhao, Zhou, Liu, & Lai, 2015). They also mentioned that the study of tetrahedral materials is important in geotechnical, mining, and transportation engineering. DEM-based methods are the most attractive techniques for modeling non-spherical particle shapes (Lu, Third, & Muller, 2015; Wachs et al., 2012). Wachs et al. (2012) used a variant of DEM named Grains3d to cope with convex non-spherical particles by
https://doi.org/10.1016/j.partic.2017.08.004 1674-2001/© 2017 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.
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Nomenclature D f (·) fi,ms g G Itet kc kr ms Mtet MI ME nA , nB ni,ms Np Ns N P ri,ms Ri,ms s, s0 , sr t ti,ms Ti,ms up
vi Vj,tet Vp ␦xi,ms ı i,ms T j,tet c r ω
Drum diameter Function Contact force vector of member sphere i Gravity acceleration Gravity force of tetrahedron Moment of inertia of tetrahedron Stiffness factor for particle collision Stiffness factor of the restoring force in the relaxation process Mass of member sphere Mass of tetrahedron Mixing index Shannon information entropy Concentrations of two predefined types of particles Unit vector in the normal direction in collision for member sphere i Number of tetrahedrons Number of member spheres in a tetrahedron Number of data points for Overall proportion of particle Position vector of member sphere i with respect to tetrahedron centroid Restoring force of member sphere i pointing to the equilibrium position Standard deviations at unmixed state and randomly mixed state Time Unit vector in the tangential direction in collision for member sphere i Torque of the tetrahedral particle caused by the member sphere i Magnitude of particle velocity components Velocity vector of member sphere i Velocity vector of tetrahedron j Particle velocity vector Displacement vector of member sphere i in collision Displacement vector of member sphere i in the relaxation process Interval Rotational velocity vector of tetrahedron Damping coefficient in collision Damping coefficient in the relaxation process Variance Rotating velocity of drum Coefficient of friction
Subscripts c Collision or contact Indices i, j ms Member sphere p Particle r Restoring Tetrahedron tet x, y Directions of coordinates
using the Gilbert–Johnson–Keerthi algorithm to compute the distance between convex bodies. They demonstrated the capability of their method to model regular shapes including a sphere, cylinder, cube, and tetrahedron. However, they did not compare tetrahedra and spheres to a sufficient extent to provide details of the effects of particle shape on mixing in a drum. Recently, out group proposed the soft-sphere-imbedded pseudo-hard particle model (SIPHPM),
Fig. 1. Sketch of a surface collision between member spheres of two tetrahedral particles.
which has been validated for modeling of non-spherical shapes (Gui, Yang, Jiang, & Tu, 2016). In this work, we use the SIPHPM method to simulate the motion and mixing of tetrahedral particles in a drum mixer. The main purpose of this work is to compare the mixing and motion of regular tetrahedra with those of spheres to determine the effects of shape on particle mixing and motion in drums.
Methodology SIPHPM The SIPHPM (Gui et al., 2016) was used here to simulate the collision between tetrahedral particles. The particles were all uniform regular tetrahedra. Each was composed of 28 member spheres enclosing six surfaces (Fig. 1). Each tetrahedron was assumed to be composed of a pseudo-rigid material that was always undeformable. The collision between tetrahedra was solved through the collision between the member spheres and surfaces enclosed by the member spheres. In addition, the member spheres could deviate slightly from their equilibrium positions. A restoring force was always generated once a deviation occurred. The equilibrium positions of member spheres were on the vertexes and sides of each tetrahedron and determined by the position and orientation of the tetrahedron. The governing equations of the tetrahedra and member spheres were coupled and solved as follows: For the member spheres:
⎧ f i,ms = kc ıxi,ms − c ıx˙ i,ms , ⎪ ⎪ ⎪ ⎪ iff ,t > f i,ms , ni,ms , thenf i,ms , t i,ms = f i,ms , ni,ms ⎪ ⎪ ⎨ i,ms i,ms T i,ms = r i,ms × f i,ms ,
⎪ ⎪ ⎪ R i,ms = kr ı i,ms − r ı˙ i,ms , ⎪ ⎪ ⎪ ⎩
(1)
v˙ i = f i,ms + R i,ms /ms − g,
where ␦xi,ms , ri,ms , and ı i,ms are the deformation displacement, position vector (relative to the center of the tetrahedron), and deviation displacement from the equilibrium position of the ith member sphere, respectively; fi,ms , Ri,ms , and Ti,ms are the damping elastic collision force, restoring force, and torque of the ith member sphere, respectively; n and t are the normal and tangential directions at the collision point, respectively; g is the gravity acceleration; kc , kr , c , and r are the stiffness factors (kx ) and damping coefficients (x ) of collision (denoted by subscript ‘c’) and restoration (denoted by subscript ‘r’), respectively; is the friction coefficient for member spheres; vi and ms are the velocity and mass of the member spheres, respectively; and ‘·’ is the inner operator.
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For the tetrahedra:
⎧ Ns ⎪ ⎪ ⎪ f i,ms − G, ⎨ Mtet V˙ j,tet = ⎪ ⎪ ˙ T i,ms , ⎪ ⎩ I tet · j,tet = i
3
Bed radius R × depth W (mm) Bed rotating speed, ω ( rad/s)
(2)
i
˙ where Vj,tet , j,tet , Mtet , and Itet are the translational velocity, rotational velocity, mass, and moment of inertia of the jth tetrahedral particle, respectively; G is the gravity force of the tetrahedron; and Ns = 28 is the number of member spheres each tetrahedron contains. After Eqs. (1) and (2) were solved, the collision and motion of both member spheres and tetrahedra could be obtained simultaneously. The tetrahedron-to-wall and tetrahedron vertexto-tetrahedron surface collisions were also incorporated by solving the member sphere-to-wall and member sphere-to-surface collisions. For example, a member sphere-to-surface collision is sketched in Fig. 1, where the member sphere of one tetrahedron contacts the surface of another tetrahedron at point ‘C’. This collision was determined by solving the collisional forces (in Eq. (1)) in the normal (‘n’) and tangential (‘t’) directions. In addition, the tetrahedron-to-wall collision was solved by the member sphereto-wall collision in a similar manner. The main features of the current SIPHPM method that differ from those of the sphere-assembly method (Yan, Yu, & McDowell, 2009) are the following. (1) The member spheres are located discretely on the sides of the non-spherical shape and the surfaces enclosed by the member spheres are also primary elements to be considered in solving the collision of tetrahedra. Therefore, the whole outer surface of the non-spherical shape does not need to be covered by member spheres, which lowers the required number of member spheres considerably. However, the member sphere-to-surface collision must be considered. (2) The overlap between the member spheres is avoided to build in the construction of tetrahedron. A relaxation force is superposed on each member sphere to force it to oscillate slightly about the equilibrium position. Gui et al. (2016) have already validated the SIPHPM method. Simulation configurations and conditions In this work, a drum of diameter D = 800 mm was filled with regular tetrahedra with a side a of 16 mm. The initial filling state (Fig. 2(a)) was formed through a free falling process of all the particles from a certain height above the bottom of the drum. After that, the drums were rotated at the same acceleration (2 rad/s2 ) from the same initial state to different target speeds of ω = (0.25–5) rad/s (Table 1). Two categories (Category A with Np = 1449 and Category B with Np = 924) with different filling levels of tetrahedral particles were studied. The drum was rotated at ω = 0.25, 0.5, 1, 2, and 3 rad/s for Category A and ω = 0.5, 1, 2, 3, 4, and 5 rad/s for Category B. To compare the behavior of tetrahedra with that of spheres, rotating drums filled with the same number of spheres of the same mass and equivalent diameter as those of the tetrahedra were also simulated. The parameters used are listed in Table 1. Results and discussion Mixing of tetrahedra Fig. 2(a) shows the initial static packing state of the tetrahedra, which was the starting point of rotation. The particles were stratified, forming layers. The mixing states of Np = 1449 tetrahedra in the drums for Category A after about 10 s of rotation at different
Initial rotation acceleration, ˇ (rad/s2 ) Side of tetrahedron, a (mm) Number of tetrahedrons, Np
Number of sphere in a tetrahedron Total number of spheres, Ns
Sphere diameter in referential cases, de (mm) Tetrahedron density, t (kg/m3 ) Stiffness factor, kc , kr (N/m) Poisson ratio, Restitution coefficient, e Friction coefficient, Time step, ıt (s) Number of simulation step, Nt
800 × 32 0.25, 0.5, 1, 2, 3, 4, 5 2 16 1449 for Category A, 924 for Category B 28 25,872 for Category A, 40,572 for Category B 13.764 2000 4 × 103 , 1 × 105 0.3 0.95 0.3 1 × 10−6 1 × 107
angular speeds are shown in Fig. 2(b)–(f). Fig. 2(g) and (h) displays the mixing states of spheres at ω = 2 and 3 rad/s, respectively. At ω = 0.25 rad/s (Fig. 2(b)), the particles were almost unmixed between the stratified layers. For ω = (0.5–2) rad/s (Fig. 2(c)–(e)), the tetrahedra were well mixed in the rolling and cascading flow states. At ω = 3 rad/s (Fig. 2(f)), a tumbling/centrifugal flow state of the tetrahedra was established. The flow states of the spheres (Fig. 2(g) and (h)) were the same as those of the tetrahedra. In a similar manner, the mixing of Np = 924 tetrahedra for Category B are shown in Fig. 3(a)–(f) for ω = (0.5–5) rad/s. The flow regimes for Category B at ω = 3 and 4 rad/s are still rolling or cascading, whereas they are centrifugal regime for Category A in Fig. 2(f). Thus, the flow regime of tetrahedra depends on the filling level. A lower filling level of tetrahedra seems to cause the tetrahedra to go into the centrifugal regime later than that at a higher filling level. A similar trend was also observed for spheres; e.g., see the centrifugal regime in Fig. 2(h) at ω = 3 rad/s for Category A, and the cataracting regime in Fig. 3(g) at ω = 3 rad/s for Category B. The filling level may affect the flow regime of particles through the following two mechanisms. (1) A high filling level may increase the frequency of contacts between the particles and rotating walls. (2) A high filling level may increase the space occupied by the particles and decrease the free void space of the drum above the filling level. Thus, the particles in a rotating drum at a high filling level may be more prone to display centrifugal behavior than that containing particles at a lower filling level. Compared with the cascading flow regime of tetrahedra in Fig. 3(d) at ω = 3 rad/s, the spheres are upcast in Fig. 3(g) at ω = 3 rad/s. Moreover, comparing Fig. 3(e) with (h) (both at ω = 4 rad/s), the flow regime of tetrahedra is cascading whereas it is centrifugal for spheres. Thus, the flow regime is also affected by the particle shape. Because spheres have the best flowability whereas tetrahedra have the worst, it is harder for tetrahedra than spheres to enter the centrifugal regime. In other words, particles with lower sphericity may keep to steady regimes more easily than those with higher sphericity, which may have implications to maintain the desired working regime of cascading.
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Fig. 2. (a) Initial packing state at t = 0 s, and the mixing states of tetrahedra at t = 10 s in Category A at ω of (b) 0.25, (c) 0.5, (d) 1, (e) 2, and (f) 3 rad/s. The inset in (a) is a schematic model of a tetrahedron. Mixing states of spheres at t = 10 s are shown in (g) and (h) at ω of 2, and 3 rad/s, respectively, for comparison. The particle indices are shown by different colors.
Mixing index and entropy analyses Category A: Np = 1449 The Lacey mixing index (Lacey, 1954) was used here to quantify the mixing levels in the drums under different conditions. This index is expressed as: MI (t) =
s02
− s2 (t)
s02 − sr2
number in a cell. In a general mixing process, MI (t) progresses from 0 to 1, where a higher MI implies a better mixing state. The Shannon information entropy (Shannon, 1948) was also used to confirm the results. The Shannon information entropy is defined based on the local concentration of particles:
ME = ,
(3)
where s(t) is the standard deviation of particle concentration, i.e., the square root of the mean squares of the deviations of particle concentration at t; s2 (t) is the variance; s02 is the variance corresponding to the completely unmixed state; sr2 corresponds to the random mixture; and s02 = P(1 − P) and sr2 = P(1−P) , where P is the n overall proportion of one type of particles and n is the particle mean
[−nA (r)log2 (nA (r)) − nB (r)log2 (nB (r))] dr,
(4)
r
where nA (r) and nB (r) are the concentrations of the two predefined types of mixing particles at r, and r is a position vector. In a system composed of two types of particles, the concentration at r can be computed as nA (r) = NA (r)/(NA (r) + NB (r)). When nA is 0 or 1, nA (r)log2 (nA (r)) = 0. Otherwise, nA (r)log2 (nA (r)) = / 0. By integrating over the whole area of a drum, ME can be used to quantify the overall degree of particle mixing in the drum.
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Fig. 3. Mixing states of tetrahedra at t = 10 s in Category B at ω of (a) 0.5, (b) 1, (c) 2, (d) 3, (e) 4, and (f) 5 rad/s; mixing states of spheres at ω of (g) 3 and (h) 4 rad/s.
Using the same data, including the same initial unmixed states, to determine both the mixing index and information entropy may give insights into the difference between them, which could aid evaluation of the real particle mixing level. Therefore, Fig. 4(a)–(d) compare the Lacey mixing index and Shannon information entropy with respect to time and number of revolutions for Category A with Np = 1449 tetrahedra. The following is seen. (1) MI (t) of tetrahedra and spheres are almost the same at ω = 0.25 rad/s (Fig. 4(a) and (c)) because the particles are almost unmixed (see Fig. 2(b)). According to ME (t) (Fig. 4(b) and (d)), the mixing of tetrahedra is slightly greater than that of spheres. Because MI (t) and ME (t) were computed from the same data, the difference between Fig. 4(a) and (b) (or Fig. 4(c) and (d)) for ω = 0.25 rad/s may indicate the difference between the mixing index and entropy for mixing evaluation. MI (t) considers the
fluctuation of particle concentration and indicates the overall uniformity of particle dispersion. ME (t) is based on the formation of local mixing structure. Thus, when MI (t) is the same for tetrahedra and spheres, ME (t) is not necessarily the same. This means that, under the same level of overall dispersion, the local mixing structure may still differ depending on particle shape. (2) For 0.5 ≤ ω ≤ 2 rad/s, MI (t) for tetrahedra with respect to both time (Fig. 4(a) and (b)) and the number of revolutions (Fig. 4(c) and (d)) are greater than those for MI (t). The same behavior was determined for ME (t). Thus, both mixing index and entropy analyses indicate the same conclusion that tetrahedra mix better than spheres in the rolling and cascading regimes. This would be because the interactions between tetrahedra are more complicated than those between spheres. (3) For ω = 3 rad/s, the inverse becomes true for MI (t) (Fig. 4(a) and (c)). The tetrahedra have lower MI (t) than the spheres. This
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Fig. 4. Variations of (a, c) Lacey mixing indices and (b, d) Shannon information entropies for Category A with (a, b) time and (c, d) number of revolutions at different angular speeds. Lines show the data for spheres and scatter plots depict the data for tetrahedra.
means that in the centrifugal regime, the spheres are more uniformly distributed near the wall of the rotating drum than the tetrahedra. However, the mixing entropy in Fig. 4(b) and (d) indicates that the local mixing of tetrahedral particles is still better than that of the spheres when ω = 3 rad/s.
Category B: Np = 924 The mixing index and entropy for Category B are shown in Fig. 5(a) and (b) with regard to time and Fig. 5(c) and (d) in terms of revolutions, respectively. The following is found:
(1) At ω = 0.5 rad/s, MI (t) values of tetrahedra are lower than those of spheres (Fig. 5(a) and (c)), whereas their ME (t) are similar (Fig. 5(b) and (d)). This means that the overall dispersion uniformity of tetrahedra is lower than that of spheres, even though the formation of local mixing structures is equivalent. (2) At ω = 1 rad/s, the MI (t) values of tetrahedra cross over those of spheres. A similar trend was also observed for ME (t), for which the cross point was earlier than that of MI (t). This indicates that the spheres are mixed faster than the tetrahedra, while the ultimate level of mixing of spheres is lower than that of the tetrahedra. This situation is reasonable because the spheres are more flowable than the tetrahedra. In other words, the tetrahedral shape may need to reach a threshold value to start to flow, just like the non-Newtonian fluids. Thus, spheres are easier to mix than tetrahedra. However, as indicated above for Category A, the tetrahedra in the rolling regime are mixed better than the spheres. (3) Based on MI (t) in Fig. 5(a) and (c), the mixing levels of spheres are higher than those of tetrahedra at ω = (2–5) rad/s. This
means that from the cascading to the centrifugal regime, the mixing of tetrahedra is always poorer than that of spheres. (4) However, ME (t) values of spheres in Fig. 5(b) and (d) at ω = 4 and 5 rad/s decrease after they reach the highest points because of the change of flow regime. Note that spheres at ω = 4 rad/s have already entered the centrifugal regime (Fig. 3(h)). Thus, as indicated by ME (t), the mixing level may decrease in the centrifugal regime. In this regard, the mixing entropy ME (t) is better than the mixing index MI (t) to evaluate mixing because the latter cannot reflect the change of mixing regime (see Fig. 5(a) and (c)). Based on the good reflection of flow regime evolution by ME (t), we believe that ME (t) shows the real mixing state. (5) Referring to the curves in Fig. 4 illustrating the most useful flow regimes from rolling to cataracting, the mixing of non-spherical particles is better than that of spherical particles at high filling levels. In contrast, for the same flow regimes, the mixing of nonspherical particles is poorer than that of spherical particles at low filling levels. These results correlate the influences of filling level and particle shape as well as rotation speed with mixing level.
Probability density function of velocity In this section, we calculated probability density functions (PDFs) as follows to explore the characteristics of velocities (up,x , up,y , ωp,z ): f (, t)T =
ıN (T ) N T
|
T ∈ t− 1 T,t+ 1 T 2 2
,
(5)
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Fig. 5. Variations of (a, c) Lacey mixing indices and (b, d) Shannon information entropies for Category B with (a, b) time and (c, d) number of revolutions at different angular speeds. Lines show the data for spheres and scatters plot the data for tetrahedra.
where N is the total number of data points of , and ıN is number of data points located within time interval
the t − 12 T, t + 12 T . Here, f(up,x ) (translational velocities up,x and up,y , and rotational velocity ωp,z ) and T = 1 s for Category A and 0.1 s for Category B, respectively. Fig. 6 shows f(ωp,z ), f(up,x ), and f(up,y ) with T = 1 s for Category A at t = 1 and 10 s. At t = 1 s, f(ωp,z ) (or f(up,y )) of tetrahedra and spheres do not have evident differences (Fig. 6(a) and (b)), except for f(up,x ) maxima of tetrahedra and spheres. This confirms that the flowability of tetrahedra is lower than that of spheres, because f(up,x ) of a tetrahedron always has its maximum at up,x = 0, whereas f(up,x ) of a sphere has its maximum at about up,x ≈ −0.15 m/s (Fig. 6(a)). Moreover, f(ωp,z ) values of tetrahedra and spheres are quite different (Fig. 6c). The spheres have larger f(ωp,z ) at larger |ωp,z |, whereas the tetrahedra always have lower f(ωp,z ) at non-zero |ωp,z |. This means that the rotational motions of tetrahedra and spheres during the early stage of mixing are quite different (Fig. 6(f)). In other words, it confirms the previous assumptions that tetrahedra are the most non-spherical particles and thus have the lowest flowability. This is also true at t = 10 s. However, at t = 10 s, f(up,x ) and f(up,y ) or the translational motions of tetrahedra are quite different from those of spheres at almost all rotation speeds. At low rotation speeds (ω ≤ 1 rad/s), the peak values of f(up,x ) and f(up,y ) of spheres are larger and more symmetrical than those of f(up,x ) and f(up,y ) of tetrahedra. In addition, Fig. 7 also shows the case where the time interval T was shortened from 1 to 0.1 s to determine the PDF characteristics of instantaneous velocities. Similar results for f(up,x ), f(up,y ), and f(ωp,z ) were also obtained for Category B. Regarding translational motions, the velocity PDFs of tetrahedra are more widely distributed than those of spheres and have larger skewness, which
means the velocities of tetrahedra are more asymmetrically distributed throughout the bed than spheres. Regarding rotational velocity, the PDFs of tetrahedra are narrower than those of spheres. In other words, most tetrahedra have smaller rotational velocities than those of spheres. In addition, at ω = 5 rad/s, f(ωp,z ) of tetrahedra and spheres both become narrower than those at lower ω, which is caused by the centrifugal regime with near-zero rotational rotation of both tetrahedral and spherical particles. Mean and variance of velocity With the aid of PDFs, more characteristics of particle velocity were quantified. The mean and variance of velocity were directly calculated using the following equations:
+∞
=
f (, t)(t)dt,
(6)
−∞
+∞
() =
2
f (, t)((t) − ) dt,
(7)
−∞ 1/2
where is |V p |(|V p | = (v2p,x + v2p,y ) ) or ωp,z . Taking Category B as an example, Fig. 8(a) and (b) shows |Vp | and ωp,z , respectively. The following is found. (1) At low rotation speeds (ω = 0.5, 1 rad/s), |Vp | of tetrahedra is equivalent or greater than that of spheres. (2) In contrast, at higher rotation speeds (ω ≥ 3 rad/s), |Vp | of tetrahedra is lower than that of spheres. (3) Additionally, ωp,z of tetrahedra is always lower than that of spheres. This is because of the following. (i) The rolling of spherical particles near the wall, which is caused by the friction from the drum, can evidently increase the particle rotational velocity. Thus, the spherical particles, which have greater flowability,
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Fig. 6. Probability density functions of translational velocities (a, d) f(up,x ) and (b, e) f(up,y ), and rotational velocities (c, f) f(ωp,z ) for Category A at (a–c) t = 1 s and (d–f) t = 10 s. Here, T = 1 s.
always have larger rotational velocity than the tetrahedral particles. (ii) In contrast, the greater rolling of particles near the wall may sometimes attenuate the efficiency of kinetic energy transfer from the rotating wall to the spherical particle assembly. In other words, the tetrahedral particle assembly would have larger internal interactions than those of the spherical particle assembly. Thus, when the drum rotates, the friction from the wall of the drum would be transferred more efficiently throughout the tetrahedral particle assembly than throughout the spherical particle assembly, especially at low rotation speeds when there is sufficient time for energy transfer through particle–particle interactions. As a result, the mean translational velocity of tetrahedral particles is greater than that of the spherical particles at low rotation speeds. Finally, Fig. 9 shows the corresponding variances of Vp and ωp,z . The trends in Fig. 9 are similar to those observed in Fig. 8 for the
mean values of Vp and ωp,z . The mean values and variances of Vp of tetrahedra are greater than those of spheres at lower rotation speeds (ω < 2 rad/s), whereas the opposite is the true at higher rotation speeds (ω ≥ 3 rad/s). The variance of ωp,z of tetrahedra is always lower than that of spheres. The reasons for this are similar to those explained above. In addition, when the flow enters the centrifugal regime, the mean rotational velocity and variances of translational and rotational velocity may decrease because of the uniform low rotational velocity for all particles in the centrifugal regime. Brief summary We studied the mixing and motion behavior of regular tetrahedra—the most non-spherical shape—in rotating drums
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Fig. 7. Probability density functions of translational velocities (a, d) f(up,x ) and (b, e) f(up,y ), and rotational velocities (c, f) f(ωp,z ) for Category B at (a, b, c) t = 2.5 s and (d, e, f) t = 7.5 s. Here T = 0.1 s.
through comparison with the behavior of spherical particles using SIPHPM simulations. The main findings were as follows. 1. The flow regime depended on both the shape and filling level of particles. A lower filling level always made the particle flow enter the centrifugal regime later, as did the degree of non-sphericity. Because of its lower flowability, less spherical particles are more favorable than spherical ones to keep the flow regime steady. Thus, the tetrahedral shape exists in a rolling flow regime, whereas the spherical shape enters the centrifugal regime at ω = 4 rad/s at a lower filling level (Np = 924). 2. MI and ME evaluate the mixing state from different viewpoints, that is, the overall uniformity of dispersion and the levels and accumulation of local mixing, respectively. Importantly, the influence of particle shape on mixing was related to the fill-
ing level. In particular, at a high filling level and in the common rolling and cascading regimes, both MI and ME indicated that the tetrahedra mixed better than spheres because of the more complicated interaction between the non-spherical particles than between the spheres. In contrast, at a lower filling level, the mixing of tetrahedra may be poorer than that of spheres in the rolling and cascading regimes. 3. In the centrifugal regime, MI and ME showed different trends for the influence of particle shape on mixing. We believe ME is better than MI to evaluate mixing because of the successful indication of flow regime transition by ME . For example, ME indicated that the influence of particle shape on mixing at different filling levels in the centrifugal regime was the opposite to that in the rolling and cascading regimes.
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Fig. 8. Variations of mean values of (a) particle translational velocity |Vp | and (b) particle rotational velocity ωp,z with time for Category B.
Fig. 9. Variations of variances of (a) particle translational velocity (Vp ) and (b) particle rotational velocity (ωp,z ) with time for Category B.
4. With regard to the mixing efficiency in the rolling and cascading regimes, at low filling levels and low drum rotation speeds, the spheres were mixed faster than the tetrahedra; this is because spheres are more likely to roll and flow than tetrahedra, especially at low rotation speeds. However, the ultimate levels of mixing of spheres were lower than those of tetrahedra. 5. The PDFs of translational velocities of tetrahedra had wider spans, lower peaks, and more asymmetrical distributions than those of spheres. The mean and variance values of translational velocities of spheres were greater than those of tetrahedra at high rotation speeds, whereas they may be equivalent or lower than those of tetrahedra at low rotation speeds. The PDF of rotational velocity ωp,z consolidated the conclusion that the nonspherical tetrahedral particles were harder to rotate than the spherical particles. This is because they had narrower PDF distributions, smaller PDF values at larger ωp,z , and lower values of mean and variance of rotational velocity ωp,z than those of the spherical particles. Additionally, the mean ωp,z and variances of translational velocities may decrease in the centrifugal regime for the tetrahedral particles.
Acknowledgments The authors are grateful for the support of this research by the National Natural Science Foundation of China (Grant Nos. 51406100 & 51576211), the Science Fund for Creative Research Groups of the National Natural Science Foundation of China (Grant No. 51321002), the National High Technology Research and Develop-
ment Program of China (863) (2014AA052701), and the Foundation for the Author of National Excellent Doctoral Dissertation of P.R. China (FANEDD, Grant No. 201438).
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