Study on the Variation Characteristics of the Average Velocity of Special-Shaped Flake Particle Systems Moving in Elliptical Drums

Abstract

Researching the moving speed of particles in a drum is helpful to the optimal design of the rotating device. In this paper, the average mixing velocity of flake particles with different shapes in an elliptical drum is studied. In detail, the mixing systems of rectangular, triangular, and circular flake particles and circular particles are studied; the rollers with different eccentricities are studied; the velocity and periodic variation characteristics of the particle system at 15 rpm and 45 rpm are compared and analyzed; the variation curves of particle average velocity, velocity peak fitting curve (PFC), velocity trough fitting curve (TFC), and velocity fitting under various working conditions are drawn. The results show that the average velocity fluctuation range of the particle system increases gradually with the increase of the eccentricity of the rotating device and the volume of flake particles; the fluctuation of particle average velocity fitting curve and PFC curve in the elliptical drum has a certain cycle, while the fluctuation of the above two curves in the circular rotary device has almost no cycle.

1. Introduction

Rotary kilns are widely used in metallurgy, pharmacy, food, and other production fields [1,2]. For example, in the process of developing and utilizing oil shale, it is very important to study the mixing and movement characteristics of irregular oil shale particles in the rotary device for their subsequent production and utilization [3]. The movement of particles will significantly affect the quality and heat transfer of products [4]. Wang et al. [5] used the discrete element method to numerically study the radial mixing and heat transfer of spherical granular materials with different properties in the drum.

Zhang et al. [6] studied the movement of particles in the quadrilateral drum. In the flighted rotating drum (FRD) system, the average speed of particle movement increases with the increase of drum speed and the decrease of filling rate. Li et al. [7] predicted the flow behavior of binary particles in horizontal Rotating Circular Drum (RCD) and Rotating Elliptical Drum (RED). Lo et al. [8] studied the variation law of velocity and kinetic energy of particles with a 50% filling rate in smooth wall drums and proved that the energy distribution of the particle system has a certain correlation with flow density. Han et al. [9] studied the velocity distribution and movement of particles and found that there was a positive correlation between the maximum kinetic energy of particles in different areas of the surface and the duration of the avalanche. At the same time, the change of particle velocity will also significantly affect its motion form, mixing degree, and heat transfer process [10,11,12]. In addition, studying the variation law of particle velocity in the drum through technical means such as particle tracking [13,14,15] can also provide theoretical support for mixing prediction [16] and drum mixing device design [17,18].

Changing the physical shape and inclination of the drum often affects the movement of particles. Li et al. [19,20] simulated different flow patterns and particle movement directions in circular and elliptical drums, observed the mixing and segregation phenomena of binary particle systems at different speeds, and studied the influencing factors of the periodic change of particle velocity. Zhang et al. [21] studied the effects of adding baffles and moving baffles on the periodic properties of particle motion. Widhate et al. [22] studied the velocity distribution of particles in the drum with an inclination angle of 10° and found that the average velocity of particles in the drum with inclined rotation increases with the increase in filling rate. The installation of board reading or other additional structures in the rotary device will also have a great impact on the movement of particles [23,24,25].

In actual production, many particles have irregular physical shapes. Li et al. [26] have made an in-depth study of the wave crest velocity of the motion of special-shaped particles and found that the flow of rice particles often has multiple velocity peaks. Lin et al. [27] used the method of spatial filtering velocimetry to measure the particle flow velocity of irregular particles on the surface in rolling mode. In the particle velocity profile, there are two velocity peaks near 0.6 times the drum diameter from the drum center. Wang et al. [28] studied the influence of variables such as particle shape and rotating speed on the mixing degree, and found that when the particle system is close to the centrifugal state, the smallest particle (1.0~1.67 mm) has the maximum radial velocity.

In order to study the motion law and velocity distribution of special-shaped particles in the drum, this paper compares the changes in particle velocity of rectangular, triangular, and circular flakes in circular drums and elliptical drums with different drum speeds and eccentricities. In this paper, the velocity changes of three different shapes of particles in an elliptical cylinder with different eccentricities at 15 rpm and 45 rpm were studied and compared with the circular cylinder. Section 2 shows the establishment of the DEM model for device and irregular particles, as well as the average velocity analysis method of particles. Section 3 mainly presents the average velocity change of special-shaped particles in rollers with different eccentricities. Section 4 summarizes the findings, which provide new ideas for the mixing and movement of particles with different shapes.

2. Numerical Methods

2.1. Establishment of DEM Model for Device and Irregular Particles

In this paper, the discrete element method is used for simulation [29], and the Hertz–Mindlin model is selected as the contact mechanics model [30]. The drum is made of steel, the density is 7800 kg·m⁻³, the elastic modulus is 1.82 × 10⁹ N·m⁻², the shear modulus is 7.5 × 10¹⁰ Pa, and the Poisson’s ratio is 0.3. The structure of the elliptical drum is shown in Figure 1, and the major and minor axis are shown in

Table 1. Structural parameters of rotating drums.

Value	Eccentricity
	e0 = 0	e1 = 0.45	e2 = 0.60	e3 = 0.75
major axis a (mm)	84	94	94	94
minor axis b (mm)	84	76	75.2	60.87

, the parameters of particles are shown in

Table 2. Physical and mechanical properties of particles.

Parameters Value	Parameters Value
Particles’ Poisson ratio (−)	0.25
Particles’ shear modulus (${\mathrm{P}}_{\mathrm{a}}$)	1 × 108
Particle density (kg·m−3)	1800
Drum—Particles’ restitution coefficient	0.1
Particles—Particles’ restitution coefficient	0.5
Drum—Particles’ static friction coefficient	0.9
Particles—Particles’ static friction coefficient	0.1
Rolling friction coefficient	0.05

[21].

In this study, particles are abstractly simplified into three different shapes of flake particles, namely isosceles right triangle flake (TP), rectangular flake (RP), and circular flake (CP). The triangular bevel, rectangular diagonal, and circular diameter are taken as the characteristic dimensions of flake particles. Three flake particle models are spliced with spherical particles with a diameter of 2 mm, and the particles are scaled quantitatively in the simulation software to achieve the target feature size of 3 mm. Although the particle shape is different, its equivalent diameter is the same. The particle model is shown in Figure 2.

In this section, two speeds of 15 rpm and 45 rpm are set to study the motion mixing characteristics of flake particle systems with different shapes in step and rolling modes. In the actual production process, the material filling rate commonly used in the rotary device is 1/3~1/2 [30], so the filling rate of the particle system is set as 1/3, and the volume ratio of 1 mm particles to 3 mm particles is 1:1; see

Table 3. Experimental conditions, data acquisition, and fitting.

Particle Filling Method (Quantity/pcs)	Drum Speed (rpm)	Data Acquisition	Data Fitting
1 mm (6050) + RP (948)	15, 45	Average velocity of particles	Peak fitting cure (PFC)
1 mm (6050) + TP (1090)			Trough fitting cure (TFC)
1 mm (6050) + CP (554)			Average velocity fitting cure

for the experimental conditions and data acquisition.

2.2. Calculation Method of Average Velocity of Special-Shaped Particle System

In order to better explore the motion characteristics of overall special-shaped particles in the rotating device, this section explores the average velocity fluctuation characteristics of overall triangular, rectangular, and circular flake particles.

The average speed v can be expressed as Equation (1):

$$v=\frac{1}{n}{\displaystyle \sum }_1^n{u}_{\mathrm{i}}$$

(1)

The particles will show different motion characteristics from the beginning to the end in the process of moving the rotating device. In the beginning, the average velocity fluctuation frequency and fluctuation range of the particles are not stable. After reaching the stable stage, the velocity fluctuation frequency and fluctuation range gradually stabilize. Therefore, the whole movement process of particles is divided into two stages, namely, the inception phase and stabilizing phase. Because in the inception phase the average velocity fluctuation range of flake particles obviously decreases; in the stabilizing stage, the average velocity of the flake particle system fluctuates within a certain range. The subsection analysis of average velocity is helpful to study the motion characteristics and the motion cycle of the average velocity of the whole flake particle system.

2.3. Comparison and Verification of Working Conditions

In order to verify the correctness of the numerical model selected in the simulation, the particle distribution and mixing of the binary particle system in an elliptic drum with an eccentricity of 0.45 and a circular drum with a diameter of 84 mm were compared and analyzed under the filling rate of 1/3 and rotating speed of 15 rpm. It can be observed from the comparison diagram of the circular drum in Figure 3 that with the rotation of the drum, the particle system appears to have obvious zoning. The 3 mm particles with larger particle sizes are distributed in the outer layer of the particle system and mixed with small particles. This phenomenon is in good agreement with the research results of Wang et al. [27]. The 1 mm particles with smaller particle sizes are concentrated in the interior of the particle system, and there is no distribution of large particles in the center.

2.4. Lacey Mixing Index Validation

In order to quantitatively analyze the mixing of the particle system in the drum under the two modes of experiment and simulation, through the meshing, point selection, and corresponding calculation of the experimental image, the particle distribution and mixing in the elliptical drum with a filling rate of 1/3, a rotating speed of 15 rpm, a centrifugal rate of 0.45, and a diameter of 84 mm are compared and analyzed, and the mixing degree of particle mixing in the elliptical drum under the above working conditions is obtained, as shown in Figure 4. The results show that the distribution of the Lacey mixing index between the experimental results and the simulation results is consistent.

The particle system in the drum divides the mixing process into three stages according to the change of mixing degree, namely I: rapid mixing stage; II: slow mixing stage; III: fluctuating mixing stage, as shown in Figure 4. The mixing mechanism is different in different stages.

Through the above comparison conditions, it can be seen that the numerical simulation and experimental results show good consistency from the perspective of particle distribution and the opposite side of mixing evaluation. Therefore, the numerical model used in this paper is more reasonable.

3. Results and Discussions

3.1. Influence of Rollers with Different Eccentricities on Average Speed

During the movement of particles in the rotating device, the particles in the active layer roll down to the bottom of the stratosphere with the rotation of the rotating device under the action of gravity, and the average velocity of particles will appear at velocity peaks and troughs in each cycle.

Take a rotary device with a speed of 15 rpm as an example. At this speed, the rotation cycle of the rotary device is 4 s. As can be seen from Figure 5a–c, under the same particle shape, with the increase of eccentricity, the fluctuation range of the overall average particle velocity increases, the average velocity cycle is almost the same, and the frequency of reaching the peak and trough is almost the same. The peak velocity increases with the increase of eccentricity, and the trough velocity decreases with the increase of eccentricity. The variation trend of the average velocity peak of particles with the same shape under different eccentricities is almost the same. The peak velocity of the average velocity first reaches the maximum velocity, then decreases gradually, and finally stabilizes gradually, and the average velocity fluctuation is smaller and smaller.

To explore the speed fluctuation range and fluctuation characteristics of special-shaped particles at different eccentricities, the different eccentricities of rectangular flake particles at the speed of 15 rpm are taken as an example. The analysis of Figure 5 shows that the average speed and periodic change of particles at 15 rpm are shown in

Table 4. The average particle velocity and periodic variation (RP) at 15 rpm.

	Eccentricity	Peak (PFC)/Trough (TFC)	Inception Phase		Stabilizing Phase
			0 s	8 s	8~20 s	Cycle Length (T/s)
Average velocity of particles (v/m·s−1)	e0	Peak	0.127	0.092	0.086~0.091	1.95
		Trough	0.043	0.074	0.072~0.080
	e1	Peak	0.144	0.094	0.097~0.104	2.19
		Trough	0.043	0.071	0.072~0.075
	e2	Peak	0.140	0.097	0.101~0.106	1.97
		Trough	0.042	0.075	0.072~0.074
	e3	Peak	0.174	0.108	0.115~0.121	1.93
		Trough	0.035	0.065	0.064~0.066

. The eccentricity has little effect on the period of the average speed of overall special-shaped particles, and its fluctuation range is no more than 8.96% of the periodic mean value. However, with the increase of the eccentricity of the rotating device, the average velocity fluctuation range of the overall special-shaped particles increases. Taking the fluctuation range of the wave crest fitting curve in the stabilizing stage of e₀ and e₃ as an example, the average value of the wave crest under the e₃ working condition is 2.28 times that of e₀. This is because with the increase of eccentricity, the length difference between the major and minor axes of the elliptical rotary device increases. Compared with the circular drum (diameter = 84 mm), the particles at both ends of the major axis of the elliptical drum (a = 94 mm) have a higher linear velocity at the same angular velocity. At the same time, the trajectory of particles becomes longer, resulting in drastic changes in the speed of particles. Affected by the rotation cycle of the drum, this change is most obvious at 1/4 T and 3/4 T (peak and trough).

When the rotating speed of the rotary device increases to 45 rpm, the overall fluctuation trend of the average speed of special-shaped particles is almost consistent with that of 15 rpm; see Figure 6 for details. The fluctuation range of the average speed increases with the increase of eccentricity, but the distribution of peak speed and trough speed is obviously different from that of 15 rpm. Next, the velocity distribution characteristics of rectangular flake particles under different eccentricities at 45 rpm will be analyzed as an example.

As can be seen from Figure 6a, in the rotating device with an eccentricity of e₀, the average velocity peak of rectangular flake particles gradually decreases from the highest value of 0.299 m·s⁻¹, and the average velocity trough gradually increases from the lowest value of 0.125 m·s⁻¹. Finally, the fluctuation range of average velocity is stable at 0.203 m·s⁻¹~0.219 m·s⁻¹. When the eccentricity of the rotating device increases, the peak velocity of the particles also increases, but it can be seen from the figure that the peak velocity of the particles is stable at 0.235 m·s⁻¹~0.245 m·s⁻¹ in the final stationary stage regardless of the eccentricity. The trough velocity of particles decreases with the increase of eccentricity in the stable stage.

When the eccentricity is e₁, the trough velocity of rectangular flake particles is stable at 0.181 m·s⁻¹~0.189 m·s⁻¹; when the eccentricity is e₂, the trough velocity of rectangular flake particles is stable at 0.178 m·s⁻¹~0.181 m·s⁻¹; when the eccentricity is e₃, the trough velocity of rectangular flake particles is stable at 0.157 m·s⁻¹~0.161 m·s⁻¹. Because the length of the major axis of the rotating device with different eccentricities is the same, and only the length of the minor axis is different, the maximum linear velocity of the particles moving in a circular motion with the rotating device is the same. Finally, it shows that the peak velocity of the particles in the elliptical rotating device is stable in a certain range. The higher the eccentricity is, the shorter the minor axis of the elliptical rotary device, as a result, the smaller the minimum linear velocity of the particles is, the smaller the trough velocity.

From the above analysis, it can also be seen that the rotating speed has a certain influence on the velocity distribution of particles in rotating devices with different eccentricities. Under the two rotating speed conditions, the particle velocity distribution range increases with the increase of the eccentricity of the rotating device, but there are obvious differences in the distribution of peak velocity, that is, when the rotating speed of the rotating device is 15 rpm, the particle velocity peak increases with the increase of eccentricity, while when the rotating speed of the rotating device is 45 rpm, the particle velocity peak fluctuation range continues to be stable in a certain range. This is because with the increase of rotating speed, the influence of the gravity factor decreases gradually, and the trend of the circular motion of particles with the rotation of the rotating device increases gradually. At this time, the length of the major axis and minor axis of the rotating device has a greater impact on the average speed of particles.

By fitting the average velocity peak of particles under various working conditions in Figure 6, the average velocity peak fitting curve (PFC) of particles under different working conditions at 45 rpm is obtained, as shown in Figure 7. The effects of different roundnesses of particles on the fluctuation period and amplitude (A) of PFC are analyzed.

By comparing and analyzing Figure 7, the influence of particle roundness [29] on PFC is studied. With the increase of particle roundness, the cyclicity and curve fluctuation amplitude of PFC in the elliptical drum increase with the increase of particle roundness. The roundness of rectangular particles is the worst, so the fluctuation of PFC under different eccentricities shown in Figure 7 almost does not show cyclicity. When the roundness of particles gradually increases (RP → TP → CP), the cyclic characteristics of PFC increase, and its fluctuation amplitude increases by about 23.56% on average.

The cyclicity of the average velocity peak fluctuation of TP and CP particles in 4.5~17.8 s under different eccentricities of the elliptical drum is highly consistent, which indirectly proves the scientificity of the above conclusion. As CP particles with the most prominent fluctuation cycle, it is found that the average fluctuation cycle is about 3 s under any eccentricity (2.25 times the rotation period of the drum).

The cyclic variation of PFC is limited to elliptical drums, and the above cyclic variation characteristics have not been found for circular drums.

3.2. Comparison of Average Velocity of Particles with Different Shapes

The average velocities of flake particles with different shapes in the rotating device with the same operating parameters are compared and analyzed. When the rotating speed of the rotary device is 15 rpm and the eccentricity is e₀, it can be clearly observed from Figure 8a that all particle velocity fluctuation ranges gradually decrease with the increase of time, and there is no stable cycle for the velocity fluctuation of the device. The velocity fluctuation range of circular flake particles is the largest, followed by square flake particles, and finally triangular flake particles. This shows that the change of particle shape has an influence on the fluctuation range of its average velocity.

As can be seen from Figure 8c, the particle system speed rapidly rises to 0.04 m/s in the start-up stage from 0 to 0.04 s, and then enters the stage of velocity stability. Under the influence of the internal friction force of the system, the particle system still maintains the shape of 0 s, and the average particle velocity is maintained at about 0.04 m/s. At this stage, the original strong chain breaks, and a new force chain is generated. Then, the particle system was supported by the new strong chain at 0.52 s, and the average particle velocity rapidly increased again. At this time, the acceleration time was about 0.2 s, and the average particle velocity increased from 0.04 m/s to about 0.14 m/s. Subsequently, the originally stable strong chain is rapidly broken by the impact of load, violent movement, and excessive rotation, resulting in a sudden drop in the average speed of the particle system. Such a rapid rise and fall of the average speed is caused by the sudden rotation of the drum in the starting stage. Tang et al. [31] study the effect of drum speed on rotating normal force between particles, the rotating speed shows little influence on the contact force distribution. This conclusion is similar to that obtained in this paper.

By comparing (c) in Figure 5, Figure 6 and Figure 8, with the continuous rotation of the drum, the velocity period and amplitude of the particle system become gradually stable. Taking Figure 8c as an example, in the start-up phase of 0.52–0.72 s, the average velocity change rate is 0.3667, while in the stabilizing phase of 11.67–12.50 s, the average particle velocity change rate is only 0.0362. Influenced by the start-up phase, the average speed change rate at the initial stage is about 10 times (at 15 RPM) or even 20 times (at 45 RPM) of that at the stable stage. By longitudinal comparison of the average speed change rate at the starting stage (Figure 5 and Figure 6), it is not difficult to see that the change in speed has no great influence on the speed change rate at the starting stage. The average speed change rate at 45 rpm is only 0.2278 (0.37 times) larger than that at 15 rpm.

In general, it can be concluded from Figure 5, Figure 6 and Figure 8 that in the starting stage, the difference between the drum shape, speed, and particle shape has little influence on the change rate of the particle average velocity.

When the shape of the rotary device changes from circular to elliptical, that is, the eccentricity of the rotary device increases, and the overall fluctuation range of flake particles with different shapes is significantly higher than that of the circular rotary device. The average velocity fluctuation range of flake particles is basically not affected by the particle shape. Under the same eccentricity, the average velocity of particles with different shapes is stable within a certain range, as shown in

Table 5. The average velocity and the average period of particulate matter under different working conditions at 15 rpm.

Eccentricity	Particle Shape	e0	e1	e2	e3
Average velocity (v/m·s−1)	RP	0.080~0.087	0.069~0.106	0.068~0.109	0.063~0.121
	TP	0.080~0.087
	CP	0.078~0.089
Average cycle (T/s)		--	2.05	2.03	2.00

.

The change of eccentricity leads to the continuous change of the average velocity of particles. This phenomenon is due to the change of the major and minor axes in the elliptical rotary device, which leads to the lengthening of the motion trajectory of the particle system. The motion of sheet particles is affected by the motion of the overall particle system, and its velocity also changes continuously. This principle is consistent with the principle described above.

The average velocity fluctuation cycle of particles in the circular drum is extremely unstable. It can be clearly observed that the average velocity cycle of particle system motion decreases gradually after 10~16 s, and this decreasing trend changes nonlinearly with time. After 18~20 s, it is difficult to distinguish a complete cycle. The cycle of this stage is very short and changeable, which does not have regular research significance.

Compared with the circular drum, the fluctuation cycle of the average velocity of particles in the elliptical drum is relatively stable. Considering the variability of the fluctuation period of the average velocity of particles in the circular drum, the average cycle is solved by calculating the mean of the three standard cycles in the stabilizing stage. It is found that with the increase of drum eccentricity, the average cycle of particle average velocity fluctuation gradually decreases in the stabilizing stage, but this phenomenon is not obvious (about 0.0125 times the drum rotating cycle).

When the speed of the rotating device reaches 45 rpm, as shown in Figure 9; when the rotating device is circular, the speed fluctuation range of flake particles is almost the same, that is, the shape of particles has little influence on its speed fluctuation. The closing curves of the average speeds of different particles are almost coincident in both the inception stage and the stabilizing stage. When the shape of the rotating device is elliptic, the peak velocity of the circular flake particles in the stabilizing stage is significantly lower than the average peak velocity of the triangular and rectangular flake particles. The trough velocity is also smaller than the other two particles, and this phenomenon becomes more and more obvious with the increase of eccentricity. For the average velocity of rectangular flake particles and triangular flake particles, whether peak velocity or trough velocity, triangular flake particles are slightly higher than rectangular flake particles.

In the circular drum with eccentricity e₁ and e₂, the PFC has almost no fluctuation after 4 s. In the drum with an eccentricity of e₃, PFC has been in a cycle decreasing fluctuation process since 9.12 s, which is quite different from (a) in Figure 9. The PFC of the former decreases gradually after the change of cycle (fluctuation cycle of PFC), while the PFC of the latter fluctuates little with the change of cycle (fluctuation cycle of particle average velocity change), and the concepts of the two cycles are not the same, therefore, it has no comparative significance.

The fluctuation cycle of the particle average system in the circular drum has strong cyclicity only in 3~4 s of the inception stage, and its cyclicity is weak after 4 s and even the whole stabilizing stage. For the elliptical drum, the cycle of the particle system hardly changes with the change of eccentricity, and its cycle is about 0.65 s.

Although the equivalent diameters of the three flake particles are the same, their shapes are different. The equivalent areas of the three particles are calculated as S_TP = 1.01 × 10⁻⁵ mm², SRP = 1.75 × 10⁻⁵, mm², and S_CP = 2.83 × 10⁻⁵ mm². Because the equivalent area of flake particles is different, the number of collisions between flake particles and 1 mm small particles is different in the process of movement. The larger the equivalent area is, the more the number of collisions between large and small particles will be. Small particles will inhibit the separate movement of flake particles, that is, affect the speed of flake particles and make them smaller.

4. Conclusions

(1) With the increase of the eccentricity of the rotating device and the volume of flake particles, the average velocity fluctuation range of the flake particle system increases gradually;

(2) The fluctuation of particle average velocity fitting curve and PFC curve in an elliptical drum has a certain cyclicity, while the fluctuation of the above two curves in a circular rotary device has almost no cyclicity;

(3) In the elliptical drum, with the increase of particle roundness, the cyclicity and curve fluctuation amplitude of PFC gradually increase. Under different eccentricities, the cyclicity of the average velocity peak fluctuation of TP and CP particles at 4.5~17.8 s is in good agreement.