Particle Mixing Study in Rotating Wavy Wall Tumblers by Discrete

Article pubs.acs.org/IECR

Particle Mixing Study in Rotating Wavy Wall Tumblers by Discrete Element Method Simulation Nan Gui,*,†,§,∥,⊥ Jianren Fan,‡ Jinsen Gao,† and Xingtuan Yang∥,⊥ †

State Key Laboratory of Heavy Oil Processing and §College of Mechanical and Transportation Engineering, China University of Petroleum, Beijing, 102249, People’s Republic of China ‡ State Key Laboratory of Clean Energy Utilization, Zhejiang University, Hangzhou, 310027, People’s Republic of China ∥ Institute of Nuclear and New Energy Technology, Tsinghua University, Beijing, 100084, People’s Republic of China ⊥ Key Laboratory of Advanced Reactor Engineering and Safety, Ministry of Education, Beijing, 100084, People’s Republic of China ABSTRACT: The mixing of granular particles in rotating wavy wall tumblers is simulated by the discrete element method. The rotating wavy wall is modeled by sinusoidal traveling waves on a fixed circle. Six wavenumbers (kλ = 0, 3, 5, 7, 9, 11) under three rotating velocities (ωd = 0.1, 0.5, and 1.0 π/s) are simulated for detailed analysis. The mixing characteristics are explored by using the difference of dimensionless particle concentrations and the fractal dimension of internal mixing interface, as well as mechanical energies, etc. It is found that the wavy wall tumbler performs more effectively for particle mixing enhancement than the circular tumbler, especially under low rotating velocities. The underlying mechanisms are explained; i.e., the rotating wavy wall can produce a periodic oscillating motion upon the conventional rotating motion of the tumbler. The oscillating motion is especially useful for enhancing mixing under low rotating velocities, although it seems to be overly effective under high rotating velocities since it makes the transition to the centrifugal regime earlier. The results improve the knowledge of particle mixing characteristics in wavy tumblers and are very useful for designing effective mixers.

1. INTRODUCTION For the rich scientific essences and wide engineering applications, many scientists and engineers have been interested in the particle mixing and segregation phenomena in various types of mixers for several decades. Great improvements on the fundamentals and underlying mechanisms of the mixing and segregation of granular particles have been achieved.1−7 Among the many important issues in particle mixing, particular attention has been paid to the effects of the geometric configuration of the mixer on particle mixing dynamics. For example, Vargas et al.8 explored the phenomena of suppression of segregation caused by time modulation of selective baffle placements in rotating drums. The influences of axial and peripheral placements of baffles were compared, and the former was found to reduce segregation drastically. Hu et al.9 performed an experimental study of granular flow in a modified rotating cone, focusing on the effects of geometric parameters and operating conditions. They found the rotating vessel with a stirrer is helpful to enhance the granular fluidity and mixing rate. Sunkara et al.10 developed a mathematical model to study the influence of flight design on the particle distribution in a flighted rotating drum. The particle falling time is a function of the curtain height and can be estimated by geometric analysis. Lekhal et al.11 measured the instantaneous, the averaged, and the fluctuating velocity fields of the dry and wet particles on exposed surfaces in a pitched blade agitated mixer. They found that the flow dynamics change from a regime dominated by individual particle motion to a regime controlled by the small clump motion when moisture is added. Moreover, Juarez et al.12 studied the cutting and shuffling approaches for predicting the mixing degree of particles in three-dimensional granular rotating tumblers. The geometric © 2014 American Chemical Society

aspects of the rotation angles of two axes and the angle between the axes lead to good mixing after a few iterations. Alizadeh et al.13 carried out an experimental study of operating conditions and geometric configurations of tumbling blenders. The segregation was found to be far less important in the tetrapodal blender than in the conventional V-blender. Dury et al.14 studied the boundary effects on the angle of repose in rotating cylinders. It was found that the characteristic range of influence of the wall does not depend on either density or gravity. Besides, many studies on particle mixing analysis are based on geometric methods. For instance, McCarthy et al.15 described the particle mixing dynamics through a geometric wedge−wedge mapping approach to capture the large scale aspects of mixing. The geometric method provides potential benefits for the rational design and optimization of granular mixing devices. Cisar et al.16 studied the radial segregation of distinct particles in various quasi-two-dimensional regular polygonal tumblers. The Poincaré plots of the velocity field were used to model the flow, and the Kolmogorov−Arnol’d− Moser region was found to attract smaller particles. Moreover, Cisar et al.17 used the cellular automata and the region-mapping methods to study the geometric effect of the tumbler on particle mixing. They found the mixing is fastest in a triangular tumbler among triangular, square, hexagonal, and pentagonal tumblers. Similarly, Figueroa et al.18 used the Poincaré sections constructed from the passive scalars advected by velocity fields Received: Revised: Accepted: Published: 13087

April 4, 2014 July 29, 2014 July 29, 2014 August 6, 2014 dx.doi.org/10.1021/ie501409f | Ind. Eng. Chem. Res. 2014, 53, 13087−13097

Industrial & Engineering Chemistry Research

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2.2. Wavy Boundary Configurations. As aforementioned, rotating tumblers with different shapes and imbedded blades have been designed and used to enhance mixing or suppress segregation of particles in mixers. In this study, a wavy boundary configuration is designed, which has a sinusoidal wave type of configuration and is expressed in the polar coordinates as

in circular, square, and elliptical mixers to determine the conduction and convection, etc. With regard to other motions apart from rotation, Chen and Wei19 carried out systematic measurements of the strength of particle transport in tubes under various excitation conditions. The capillarity-like phenomenon is observed in granular material under the vertical vibration motion. Wightman et al.20 studied the effects of rotational and rocking motions of the mixer on particle mixing. By comparison with purely rotational motion, the rocking motion is found to enhance mixing drastically. In addition, granular convection, which can drive size segregation, was studied by Liffman et al.21 under horizontal shaking motions. A new granular convection mechanism caused by the horizontal vibration was shown, which is different from the convection mechanism driven by vertical vibration. In addition, compared to the effect of interstitial gas flow in the rotating tumbler, particle mixing dynamics as well as heat transfer is always of primary importance in many practical engineering applications,22 such as the vacuum contact drying process.23,24 Thus, motivated by the importance of the particle mixing enhancement in many industrial applications and the necessity of investigations of tumbler geometry or geometryrelated effects, this study is carried out to study the particle mixing characteristics in a rotating drum with various wavy geometries. It aims to provide some useful indications on the complicated effects of wavy geometry of the tumbler boundary on mixing enhancement and to explore the mixing mechanisms in wavy tumblers.

ρ(t , θ ) = R 0 + A sin(kλθ + ωat )

where ρ and θ are the polar radius and angle, respectively. R0 is the radius of gyration, i.e., the radius of the base circle. A, kλ, and ωa are the amplitude, wavenumber of the wavy boundary, and angular frequency of the sinusoidal wave, respectively. The wavenumber kλ is defined as the number of full waves within a unit radian in the circumferential direction. Hence it is dimensionless (rad−1), which is different from conventional dimensional definitions (m−1). The parameter group (A, kλ, ωa) determines both the geometric configuration of the wavy wall and the angular rotating velocity (ωd = ωa/kλ) of the tumbler. In this study, the amplitude A is fixed, and kλ = 0, 3, 5, 7, 9, and 11 are used, where kλ = 0 denotes the configuration of circular tumbler. At the beginning, the wavy drum is prefilled with spherical particles settled at the bottom. The tumbler is considered as a rigid, motion-controlled large particle with a finite radius, a sufficiently large mass, and frictional wavy walls. When the tumbler rotates, it is accelerated at a constant acceleration of ω̇ = π/s2 until it gets the target speed of rotation. Three target rotating speeds are simulated, i.e., the very slow speed (ωd = 0.1 π/s), the intermediate normal speed (ωd = 0.5 π/s), and the relatively rapid speed (ωd = 1.0 π/s). Thus, it only costs no more than 1 s for the tumbler to be accelerated from static to the largest target speed. Compared to the total simulation time of 150 s, the influence of the acceleration process can be neglected. The time step is dt = 5 × 10−5 s, which is fine enough to maintain the numerically stable condition of simulation. In addition, the parameters are listed in Table 1.

2. NUMERICAL METHODS 2.1. Discrete Element Method. In this study, particle− particle collision is simulated by the discrete element method (DEM), which was proposed by Cundall and Strack25 and is still widely applied in various areas on granular materials. The soft-sphere approach of DEM deals with interparticle collision by three basic mechanisms, i.e., elastic collision, viscous damping, and friction, which are formulated as follows:

Table 1. Simulation Parameters

fc, ij = −ks δ x ij − γd ẋ ij if |fc, ij·t| > μf |fc, ij·n| ,

then |fc, ij·t| = μf |fc, ij·n|

radius of gyration, R0 (m) amplitude of wavy tumbler, A (m) number of particles, (Na, Nb) particle diameter, dp (mm) particle density, ρp (kg/m3) restitution coefficient, e friction coefficient, μf stiffness factor, k (N/m) wavenumbers on drum boundary, kλ rotating velocity, ωd (π/s) simulation time step, Δt (s) total simulated time, Ts (s)

(1)

where fc and δx are the contact force and interparticle displacement, respectively. ks, γd, and μf are the coefficients of elastic stiffness, damping, and friction, respectively. In x ̇ and elsewhere, the symbol “·” represents the time derivative operator. t and n mean the tangential and normal directions between a pair of colliding particles, respectively. The translational and rotational motions of each particle are traced by Newton’s law of motion, and are expressed respectively as follows: Fi mp, i

(2)

ω̇p, i = Tp, i /Ip, i

(3)

ẍ i =

(4)

1.44 0.06 (10 115, 10 189) 12 7800 0.95 0.3 10 000 0, 3, 5, 7, 9, 11 0.1, 0.5, 1.0 5.0 × 10−5 150

3. SIMULATION RESULTS AND DISCUSSION 3.1. Mixing Visualizations. 3.1.1. For Different Wavenumbers. To observe the mixing process, the particles are predivided into two equivalent parts, called part α and part β, respectively. In this way, particle mixing can be characterized by the evolution of the mixing interface between part α and part β. The interface can be quantified by the difference of the concentrations of particles between part α and part β, i.e.

where F = ∑fc + g is the total force experienced by one particle. mp, Ip, ωp, and Tp are the particle mass, moment of inertia, angular rotating velocity, and torque caused by contact, respectively. g is the gravity acceleration. This model has already been well validated and used for particle mixing studies in tumblers and fluidized beds earlier.26−28 13088

dx.doi.org/10.1021/ie501409f | Ind. Eng. Chem. Res. 2014, 53, 13087−13097


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Figure 1. Snapshots of particle mixing state Mαβ for kλ = (a) 0, (b) 3, (c) 5, (d) 7, (e) 9, and (f) 11, respectively, at t = 25 s and ωd = 0.1 π/s. The rotating direction is anticlockwise.

⎧ cα − β(x) ⎪ when cα(x) > 0 or cβ(x) > 0 ⎪ Mαβ (cα , cβ) = ⎨ cα + β(x) ⎪ ⎪0 when cα(x) = 0 and cβ(x) = 0 ⎩

According to the categorization of flow regimes in the circular rotating tumbler, three basic types and seven subtypes of flow regimes can be identified when the rotating velocity increases from very low to very large values.29 At present, it is found that the flow regime in the circular drum with kλ = 0 is slipping, which is of no use for practical industrial application since the particles are stratified into several stable and almost purely separated layers. It cannot work effectively for mixing. Only a fairly small fraction of particles immediately near the wall can be driven by the tumbler wall from the left bottom side to the right top side within a thin near wall boundary layer. However, in the wavy tumbler, the mixing states are totally different from that in the circular tumbler. For kλ = 3 with the longest wavelength (λw = 2π/kλ = 2.09), the mixing interface of particles is going to be continuously folded and jagged, forming hierarchical stripes and complicated geometric structures. In this way, the stratification structures of the interface are developing, as well as the degree of mixing. Furthermore, for kλ = 5, 7, 9, and 11, mixing is still enhanced by increasing wavenumbers. The structure of the mixing interface becomes more and more complicated, which illustrates the basic evolution characteristics of particle mixing.

(5)

where c(x) is the local number concentration of particles at position x, and cα±β(x) = cα(x) ± cβ(x). It is noticed that it is within −1 ≤ Mαβ ≤ 1, where Mαβ = ±1 denotes the purely separated state and Mαβ = 0 denotes the fully mixed state. For example, Figure 1 shows the mixing states evaluated by Mαβ at t = 25 s for tumblers with wavenumbers kλ = (a) 0, (b) 3, (c) 5, (d) 7, (e) 9, and (f) 11, respectively, under ωd = 0.1 π/ s. It is noticed that the operating conditions, e.g., rotation velocity, evolution time, amplitude of wavy wall, particle size, and radius of gyration, are the same except for the wavenumber. Thus, Figure 1 illustrates the effect of the wavenumber of the tumbler on the mixing characteristics of particles. After comparison, it is clearly observed that mixing is enhanced from kλ = 0 (circular tumbler) to kλ = 11 (highly wavy tumbler). 13089



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Figure 2. Snapshots of particle mixing state Mαβ for kλ = (a) 0, (b) 3, (c) 5, and (d) 7, respectively, at t = 150 s and ωd = 0.1 π/s.

Moreover, Figure 2 shows the results of Mαβ at t = 150 s for kλ = (a) 0, (b) 3, (c) 5, and (d) 7, respectively. Under a long time of duration (with more than seven revolutions), it is confirmed that the mixing state for kλ = 0 remains almost not developed. The particles in part α and part β stay at clearly separated states. For kλ = 3, the mixing is developing faster than that for kλ = 0, although fairly slowly. However, for more large wavenumbers, the mixing states are nearly fully developed, such as those illustrated in parts c and d of Figure 2 for kλ = 5 and 7, respectively. In Figure 2d, the regions of Mαβ = ±1 almost totally disappear, which means the purely separated states almost do not exist. The results remain the same for even larger wavenumbers. Thus, the wavy tumblers are no longer useless for particle mixing at very low rotating speeds as is the circular tumbler under the same low speed of rotation. In other words, the wavy configuration extends the useful range of flow regimes for particle mixing, especially under fairly low speeds of rotation. 3.1.2. For Different Rotating Speeds. On the other hand, for example, Figure 3 shows the comparison of the typical results of particle mixing states under different rotating velocities at t = 15 s with kλ = 3. As shown in Figure 3, the mixing is enhanced from ωd = 0.1 π/s (Figure 3a) to ωd = 1.0 π/s (Figure 3c) through ωd = 0.5 π/s (Figure 3b). As the rotating velocity increases, the mixing interface becomes more and more complicated and with more and more fine structures. At the low rotation speed, ωd = 0.1 π/s (Figure 3a), the mixing interface is clear, and does not have many fine striations. At the intermediate speed, ωd = 0.5 π/s (Figure 3b), the mixing is greatly enhanced. The mixing interface is no longer a smooth curve. More complicated and finer structures begin to occur. Moreover, at the high rotation speed, ωd = 1.0 π/s (Figure 3c), the mixing state is nearly fully developed. A cloud-like fully mixed area is formed, and the purely separated regions (Mαβ = ±1) are reduced significantly.

Figure 3. Snapshots of particle mixing state Mαβ at t = 15 s and kλ = 3 for ωd = (a) 0.1, (b) 0.5, and (c) 1.0 π/s, respectively.

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Thus, in conclusion, either increasing the wavenumber or increasing the rotation velocity can result in mixing enhancement. However, the details on mixing enhancement are still not clear, and the relative efficiency of mixing enhancement either by increasing wavenumber or by increasing rotation velocity needs to be explored. 3.2. Fractal Mixing Interface. 3.2.1. Under Low Rotation Speed (ωd = 0.1 π/s). By visualizing the mixing states in Figures 1−3, the mixing interface seems to be fractal. Based on the earlier indications of fractal characteristics,27,30 the length of the mixing interface L(ε) is proportional to ε−Sl:

L(ε) ∝ C iε−Sl

(6)

where Sl is the fractal dimension of the mixing interface and ε is the measuring size. log L(ε) = log Ci − Sl log(ε) and Sl = −[d log L(ε)/(d log ε)], where the Napierian base for the logarithm operation is used. Sl indicates the geometric complexity of the mixing interface, and log Ci indicates the coefficient for measuring the interface length. Sl and Ci can be calculated by the slope and the y-axis intercept of the regression line from the log L(ε)−log(ε) plot, respectively. The fractal dimension of mixing interface can be computed by the box-counting method.27 For example, parts a and b of Figure 4 show the relationships between log L(ε) and log ε corresponding to Figures 1 and 2, respectively, i.e., at t = 25 and 150 s, respectively, under various wavenumbers and fixed low rotating velocity, ωd = 0.1 π/s. In general, a perfectly linear relationship between log L(ε) and log ε is observed. At t = 25 s (Figure 4a), the slope Sl is increased from 0.98 to 1.40 when kλ increases from 0 to 11. In other words, the dimension of the mixing interface is increased as the wavenumber increases. Moreover, the intercepts Ci may increase (at least do not decrease) when kλ increases from 0 to 7. However, from kλ = 7 to 11, both Sl and Ci increase fairly slowly (Ci = 2.20−2.28, and Sl = 1.37−1.40), or even not to be changed clearly. At t = 150 s (Figure 4b), when the wavenumber increases, although the intercept Ci seems to be decreased from about 2.03 to 1.33, the fractal dimension of mixing interface increases from Sl = 1.17 to 1.88. Moreover, similar to Figure 4a, the data for kλ = 5−11 are overlapped with each other. From Figure 4, the following are indicated: (1) The overlapped data and agreed fitting curves imply the common fully developed state of mixing, i.e., the limit of the mixing state. (2) The evolution stage of mixing can be illustrated by the upward movement of the fitting curves obtained by the linear regression. (3) The moving speed depends on the wavenumbers. In other words, using a larger wavenumber makes the mixing state reach the limit state of mixing more rapidly. Thus, it explains the following: (i) It can make the particle mixing be quite near the fully developed state within only 25 s at the low speed of ωd = 0.1 π/s by using 7 wavenumbers (Figure 4a). (ii) However, it needs about 150 s to get the fully developed state at the same rotating speed by using 5 wavenumbers (Figure 4b). (iii) The trends are similar for other wavenumbers. It gets earlier for particles to reach the fully developed mixing state by using a larger wavenumber, or vice versa. Using the circular tumbler seems to be the least effective one for getting the fully developed state. Thus, it confirms that increasing the wavenumber can speed up the evolution process of particle mixing in the tumbler. 3.2.2. Under Intermediate and High Rotation Speeds (ωd = 0.5 and 1.0 π/s). On the other side, taking kλ = 0, 5, and 11

Figure 4. Relationship between log L(ε) and log ε under rotating velocity ωd = 0.1 π/s with different wavenumbers at t = (a) 25 and (b) 150 s, respectively.

for a case study, Figure 5 shows the distributions of particles at the same time of t = 25 s under the intermediate and high rotation speeds, where parts α and β are colored differently. Figure 5a−c is for ωd = 0.5 π/s, and Figure 5d−f is for ωd = 1.0 π/s. For the circular tumbler, ωd = 0.5 and 1.0 π/s are within the range of normal operation conditions for particle mixing. In general, using a larger rotation speed can enhance and speed up particle mixing. It is observed from Figure 5 that it is true for almost all the cases, provided the flow regime remains to be working. Thus, the mixing state for ωd = 1.0 π/s (Figure 5d−f) is better than that for ωd = 0.5 π/s (Figure 5a−c), respectively. For quantitative evaluation, Figure 6 illustrates the fractal dimensions under the intermediate rotation speed (ωd = 0.5 π/ s). In the early stage (Figure 6a for t = 5 s), it is demonstrated that the fractal dimensions Sl under different wavenumbers are quite close to each other. In this case, the close values of fractal dimensions mean that the complexities of mixing interfaces are equivalent to each other, or the performances of mixing by using different wavenumbers are generally equivalent to each other. However, the intercepts are different; i.e., the intercepts Ci for kλ = 0, 3, and 5 are close to each other, and Ci for kλ = 9 and 11 are near each other, with Ci for kλ = 7 at an intermediate value. This indicates that the absolute lengths of the mixing 13091



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Figure 5. Distribution of particles with kλ = (a, d) 0, (b, e) 5, and (c, f) 11 for ωd = (a−c) 0.5 and (d−f) 1.0 π/s, respectively, at time t = 25 s.

interface are larger for kλ = 9 and 11, smaller for kλ = 0, 3, and 5, and intermediate for kλ = 7. A bit later (Figure 6b for t = 25 s), both Sl and Ci for almost all the cases are close to each other, respectively. This means that the mixing performances for different wavenumbers are nearly equivalent to each other under ωd = 0.5 π/s (within the normal operation condition). In other words, it is only a bit more advantageous to use the wavy wall tumbler than to use circular ones under the normal condition of rotation. Only weak improvement of mixing performance or slight speedup of the mixing process can be obtained by using wavy tumblers. Moreover, Figure 7 illustrates the fractal dimension for ωd = 1.0 π/s, an even larger rotating speed. The results seem to be more complex than the former. In the early stage (Figure 7a for t = 5 s), the fractal dimension seems to be generally increased by the increasing wavenumber. However, in the stage a bit later (Figure 7b for t = 25 s), the reverse trends become true; i.e., the fractal dimension seems to be generally decreased by even more increase of the wavenumber. Detailed analysis shows that the reverse variation of the fractal dimension is caused mainly by the large scale structures of particle mixing. As shown in Figure 7b, the lengths of mixing interface for different wavenumbers are close to each other under small scales (e.g., ε ∼ 2d), whereas they are different from each other under large scales (e.g., ε ∼ 8d). This means that the large scale structure of the mixing interface is lengthened by the large wavenumber since

the large scale distribution states of particles are somewhat changed inside the highly wavy tumbler (see Figure 5f). As aforementioned, the large scale distribution states within the highly wavy tumbler at the high rotating speed (Figure 5f for kλ = 11 and ωd = 1.0 π/s) are different from those in the circular tumbler or in the wavy tumbler with low wavenumbers and low speeds. It is necessary to explain the mechanisms for these differences. As sketched in Figure 8a, the near wavy wall region could be characterized by a near wall granular boundary layer (GBL). Based on the wavy wall configuration, the GBL could be divided into a number of element parts. Each element part contains a wavenumber and is composed of two sections, i.e., the positive section and the negative section (Figure 8a). In the positive section (PS), particle motion is driven by the frictional force, the normal force from the wall, and the gravity force. The resultant driven force is in the direction near the direction of rotation. However, in the negative section (NS), the resultant force, if it exists, is in the direction counter to the direction of rotation. Thus, in the direction of rotation, the particles in the GBL could be accelerated in the PS but decelerated in the NS. For demonstration, Figure 8b shows the temporal evolutions of the total driven forces from the wall in the circumferential direction for the particles in the positive (PS) and negative sections (NS) at kλ = 3 and 11. Remember that, in the wavy tumbler, the tumbler motion is controlled by a 13092



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Figure 8. (a) Sketch of driven forces of particles in the near wavy wall region within a wavenumber. (b) Comparison of the total driven forces from the wall in the circumferential direction for the particles in the positive (PS) and negative sections (NS) at kλ = 3 and 11 under ωd = 0.1 π/s.

traveling wave on the base circle, which could produce a periodic variation of the driven force on the particles in the GBL. Thus, the processes of acceleration in the PS and deceleration in the NS will take place by turns, as the driven forces are periodically oscillating. This periodic oscillation is clearly observed in Figure 8b. For kλ = 3, the circumferential component of the total driven force of particles in both the PS and the NS are oscillated in time regularly. The magnitude and frequency of periodic variation of the driven forces could be very high when the rotation velocity or wavy number is very large. For example, Figure 8b shows that the oscillations become irregular and vary highly frequently when kλ = 11. Moreover, the absolute magnitudes of the average driven forces are raised from 77.34 to 261.61 N in the PS, and from 51.59 to 198.48 N in the NS. In other words, the difference of absolute magnitudes of driven forces between the PS and the NS are enlarged when the wavenumber increases (from 25.75 to 63.13 N). Thus, under the highly frequent and periodic oscillation of large driven forces, particles may prefer to or have to stay together rather than be dispersed in the near wall region as in normal circular tumblers. Thus, the particle number concentration is differentiated. In other words, the differentiated distribution of particle is caused by the highly frequently and

largely varied driven force from the wall in highly wavy tumblers with a high speed of rotation. In addition, although ωd = 1.0 π/s is a bit large, the flow regime in the circular tumbler is still under normal condition, i.e., cataracting, not centrifuging. However, the flow regime in the wavy tumbler for kλ = 11 and ωd = 1.0 π/s (Figure 5f) is about to transit from the cataracting to the centrifuging regime. Thus, the flow regime in wavy tumblers may take place earlier than the corresponding counterpart in circular tumblers, which means that the performance efficiency of the wavy tumbler is better than that of the circular tumbler. 3.2.3. Temporal Evolution. It is noticed that the aforementioned analyses are based on several fixed time points. Thus, it is necessary to show the temporal evolution characteristics of fractal dimensions, as they may depict a more complete comparison of the evolution characteristics and performance of mixing dynamics in wavy tumblers. They are shown in parts a, b, and c of Figure 9 for ωd = 0.1, 0.5, and 1.0 π/s, respectively. Under the slow rotation speed (Figure 9a for ωd = 0.1 π/s), wavy tumblers seem to perform much better than circular 13093



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rotating velocity, i.e., within the normal operation condition of mixing. However, for the high rotation speed (Figure 9c for ωd = 1.0 π/s), clear reverse trends become true, i.e., Sl(kλ=0) > Sl(kλ=3) > Sl(kλ=5) > Sl(kλ=7) > Sl(kλ=9) > Sl(kλ=11) at almost any time. This means that the highly wavy tumbler performs worse than the lowly wavy tumbler for particle mixing, and even worse than the circular tumbler. This is mainly caused by the transition of particle distribution from the uniform state to the concentration-differentiated state, as the driven forces of particles are influenced by the augmented and intensified periodic oscillation. In conclusion, it is clearly indicated that the normal operating condition of the rotation speed is shifted from the high speed to low speed ranges by using wavy tumblers. A larger wavenumber may cause a larger degree of shift of operation conditions, or vice versa. 3.3. Energy Analysis. As aforementioned, the main advantage for using the wavy tumbler is caused by shifting the range of normal operation condition of the rotation speed. From an energy-saving point of view, it can reduce the power demands for some specific types of mixing device by working at a fairly low rotating velocity to achieve a sufficient level of mixing. To compare the energy-related characteristics quantitatively, it is necessary to compute the mean kinetic energy and mean gravitational potential energy of particles as follows. (The particle mass is a constant and is omitted here.) EG(t ) =

E K (t ) =

Np

1 Np

∑ gz j

1 Np

∑

(7a)

j=1 Np j=1

1 (vj·vj) 2

(7b)

where Np, g, zj, and vj are the total number of particles, the gravity acceleration, the height, and the velocity vector of the jth particle, respectively. Figure 10 shows the temporal variations of the mean kinetic energy and mean gravitational potential energy under different rotation speeds and wavenumbers. As shown in Figure 10a, under ωd = 0.1 π/s, the mean kinetic energies EK for kλ = 7−11 are about 0.1−0.2 J/kg. However, it is nearly not larger than 0.1 J/kg for kλ = 0−5, which corresponds to the less effective or ineffective performance of mixing. Under ωd = 0.5 π/s, the values of EK are about 0.4 J/kg for kλ = 0, which is more than 2 times the values of EK for kλ = 7−11 under ωd = 0.1 π/s. Based on the aforementioned results and analysis, it is appropriate to consider the circular tumbler as an overall mixing-refined device where the mixing efficiency is manipulated by the rotating speed and the mixing levels are changed totally. In contrast, the wavy tumbler could be considered as a locally refined mixing device where the mixing level is modulated mainly in the GBL region, and it may produce finer mixing enhancement without changing the overall level of the kinetic energy of all particles largely. Thus, it is estimated that using a wavy tumbler under a low rotating speed may reach the equivalent level of mixing as with using a circular tumbler under a high rotating speed. However, in general, using a low speed wavy tumbler could save more energy than using a high speed circular tumbler. Moreover, EK is about 1 J/kg for kλ = 0 at ωd = 1.0 π/s, which is 2 times even larger than those for ωd = 0.5 π/s. In addition, the kinetic energy is increased by increasing

Figure 9. Temporal evolution of fractal dimensions for ωd = (a) 0.1, (b) 0.5, and (c) 1.0 π/s, respectively.

tumblers for enhancing particle mixing and speeding up mixing processes. The larger the wavenumber is, the better the tumbler will perform for enhancing and speeding up particle mixing, i.e., Sl(kλ=0) < Sl(kλ=3) < Sl(kλ=5) < Sl(kλ=7) ≤ Sl(kλ=9,11) at almost any time. This confirms that the wavy tumbler is always much more effective than the circular tumbler for particle mixing enhancement under the low speed of rotation. For the intermediate rotation speed (Figure 9b for ωd = 0.5 π/s), the trends are similar to the former case. However, the degree of performance differences between the circular tumbler and the wavy tumbler, as well as the degree of mixing enhancement and speedup between the wavy tumblers, is reduced, since the performance of the circular and low wavenumber tumblers are improved greatly under the moderate 13094



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Figure 10. Temporal variation of mean kinetic energy EK and mean gravitational potential energy EG of each particle under different rotation speeds and wavenumbers.

wavenumbers from ωd = 0.1 to 1.0 π/s. Thus, increasing either the wavenumber or the rotation velocity can increase the kinetic energy of particles, whereas using the former approach may possibly not need as much external power input as using the latter. On the other hand, Figure 10b shows the variations of potential energies which are similar to those of kinetic energies in Figure 10a. Using kλ = 7−11 under ωd = 0.1 π/s can achieve a similar level of potential energy as using kλ = 0−3 under ωd = 0.5 π/s. Thus, this once again confirms that the particles can obtain as high a level of mechanical energies in highly wavy tumblers under a low rotation velocity as that in circular tumblers under a high rotation velocity. Thus, using a wavy tumbler can save external energy input for operating mixing devices. In addition, under the same rotation velocity, the mechanical energies of particles are larger within a highly wavy tumbler than within a lowly wavy tumbler. Thus, it is

reasonable to enhance mixing by using highly wavy tumblers under the same rotation velocity, provided the flow regimes are maintained.

4. CONCLUSION This study shows the effects of wavenumber and rotating speed of wavy tumblers on the enhancement of mixing characteristics of large particles (dp = 12 mm) in a flatted mixer by DEM simulation and fractal dimension analysis. The main findings can be briefly summarized as follows: 1. The distribution of the particle mixing interface shows the particle mixing can be effectively evolved in wavy tumblers under a low rotation velocity, especially for the conditions when the circular tumbler is of no practical use for mixing. The mixing can be speeded up when the wavenumber is increased. 2. Under a low rotation velocity, using a high wavenumber can not only speed up the mixing process but also enhance the 13095



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Vectors

mixing degree. Under an intermediate rotation velocity, the general trends are similar. However, the degrees of mixing enhancement and process speedup are reduced. However, under a high rotation velocity, the reverse trends become true; i.e., the degree of mixing is no longer enhanced but attenuated. 3. The mechanism of the reverse trend of mixing enhancement by the high wavenumbers is explained by the intensive and periodic oscillation of the driven forces, which leads to the differentiation of particle concentrations into relatively dense and dilute regions. Moreover, using a large wavenumber under a high rotation speed makes the particle flow regime transit from the cataracting to the centrifuging regime earlier than using the circular tumbler. 4. The wavy tumbler extends or shifts the normal operation condition of the circular tumbler toward the low rotation velocity direction. Detailed energy analysis shows that using wavy tumblers may reduce the demand of external power input for operating the tumbler mixer effectively, which is potentially beneficial for industrial applications.

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fc,ij = particle−particle contact force F = particle total force g = gravity acceleration n, t = normal and tangential vectors, respectively Tp = torque v = particle velocity x = particle position vector δxij = particle displacement vector ωp = particle angular velocity Subscripts

AUTHOR INFORMATION

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Corresponding Author

*Fax:+86-010-89739027. E-mail: [email protected]. Notes

REFERENCES

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The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors are grateful for the support of this research by the National Natural Science Foundations of China (Grants 51106180 and 51136006).

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α, β = particle part index a = angular rotation c = contact d = damping G = gravity i, j = indices l = length p = particle s = simulation w = wavelength

NOMENCLATURE

Scalars

A = amplitude of wavy boundary (m) cα, cβ = local number concentration of particles (m−2) Ci = coefficient for measuring the interface length dt = numerical stepping time (s−1) e = restitution coefficient EG = mean gravitational potential energy of particles (J) EK = mean kinetic energy of particles (J) g = gravitational acceleration constant (m·s−2) Ip = moment of inertia of particle (kg·m2) ks = normal stiffness factor (N/m) kλ = wavenumber of wavy boundary L = length of mixing interface (m) mp = particle mass (kg) Mαβ = concentration difference on interface Np = number of particles Rg = radius of gyration (m) Sl = fractal dimension of mixing interface t = time (s) Ts = total simulation time (s) z = particle height (m) γd = damping coefficient ε = measuring size (m) θ = polar angle (rad) λw = wavelength μf = friction coefficient ρ = polar radius (m) ωa = angular frequency of sinusoidal wave (rad·s−1) ωd = rotation speed of tumbler (rad·s−1) 13096



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Particle Mixing Study in Rotating Wavy Wall Tumblers by Discrete

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