Sage Journals: Discover world-class research

Abstract

Based on analyzing the problem of the screen with circular, linear, and elliptical trajectory, the spatial Lissajous trajectory has been put forward. The article analyzes the dynamic theory of the spatial Lissajous trajectory screen, and the virtual simulation with discrete-element method is applied to study the solid migration rules. A discrete-element model is developed based on the JKR contact model which takes into considerations viscous forces of contacting particles. Two important factors, migration velocity and filter ratio, are selected to describe the working performance of the vibrating screen. The research results show that the spatial Lissajous vibrating screen has approximate solid migration velocity like the linear vibrating screen, but the filter rate of spatial Lissajous trajectory is the highest. Therefore, the spatial Lissajous trajectory vibrating screen can reduce the phenomenon of screening blockage, and the advantage will be more evident in the case of high drilling fluid viscosity, high mesh number, and serious screen blockage.

Keywords

Vibrating screen spatial Lissajous trajectory solid particle migration

Introduction

The vibrating screen is a key equipment for separating granular materials based on size and is often used extensively in mining, building, agriculture, metallurgy, drilling, and other industries.¹ The vibrating trajectory is the most important factor of a vibrating screen. So researchers never stop to explore the appropriate trajectories of the screen. As we know, the common vibrating trajectory is circular, linear, and elliptical as shown in Figure 1(a)–(c). According to the trajectory of movement, screens can be divided into three types: linear vibrating screen, circular vibrating screen, and elliptical vibrating screen.^2,3 These three types of trajectory generate vibration in the longitudinal plane.^4,5 So granular materials are only stressed in the x and y axis direction by the sieving force, but in the z axis direction without force. The difficult sieving and anomalistic critical particles will easily produce the blocking and influence the screening efficiency and processing performance.

Figure 1.

Four types of trajectory of vibrating screen: (a) circular trajectory, (b) linear trajectory, (c) elliptical trajectory, and (d) special Lissajous trajectory.

The discrete-element method (DEM), which was developed by Cundall and Strack⁶ in the 1970s for modeling the behavior of assemblies of disks and spheres, can monitor the movement and interaction of all particles in the granular system.⁷ DEM has been widely used in the study of segregation of granular medium system fields, such as mining, construction, pharmaceuticals, food production, and so on.^8–12 In addition, the successful applications of DEM not only have given an insight to study the granular systems but also have got positive results to promote the development of vibrating screens.^13,14 The reliability of numerical simulation using DEM has been proved by plenty of simulations of industrial particle flows.^9,12,15–17 Simulations and experiments of the linear vibrating screen are relatively mature.^18–20 The characteristics of linear screen based on a two-dimensional (2D) and three-dimensional (3D) model were studied, and the effects of the screen size, frequency, and amplitude on the screening efficiency were analyzed and verified by experiments.^21–24 Most of these studies concentrate on finding the relationship between the microscopic information and macroscopic process using DEM.^25–27

Aiming to develop a new type of screen with better screening efficiency and processing performance, the spatial Lissajous trajectory has been put forward as Figure 1(d). In this study, the article will analyze the dynamic theory of the spatial Lissajous trajectory screen, and the virtual simulation with DEM will be applied to study the solid migration rules.

Dynamic theory of the spatial Lissajous trajectory vibrating screen

The structure of screen box, exciting motor, and spring are simplified, and the corresponding coordinate system shown in Figure 2 is established. Based on the way of parametric design, the 3D model of the spatial Lissajous trajectory vibrating screen is established, which is shown in Figure 3.

Figure 2.

Simplified figure of the spatial Lissajous vibrating screen.

Figure 3.

The 3D model of the spatial Lissajous trajectory vibrating screen.

Neglecting the damping force and supposing that there is no phase difference between the two linear vibration motors, the differential equation of motion of the sieve box along the three directions of x, y, and z is as follows

{\begin{matrix} Mx ″ + 4 k_{x} x = m_{c} r_{c} ω_{c}^{2} \sin ω_{c} t \\ My ″ + 4 k_{y} y = 2 m_{l} r_{l} ω_{l}^{2} \cos ω_{l} t \\ Mz ″ + 4 k_{z} z = m_{c} r_{c} ω_{c}^{2} \cos ω_{c} t \end{matrix}

(1)

where $M$ is the calculated mass of the screen box (including the mass of the slurry), $m_{c}$ is the mass of the eccentric block of the circular vibration exciter, $m_{l}$ is the mass of the eccentric block of the rectilinear vibration exciter, $k_{x}, k_{y}, k_{z}$ are the stiffness of single damping spring in each direction, $r_{c}$ is the eccentricity of the circular vibration exciter, $r_{l}$ is the eccentricity of the rectilinear vibration exciter, $ω_{c}$ is the angular velocity of the circular vibration exciter, and $ω_{l}$ is the angular velocity of the rectilinear vibration exciter.

According to the solution method of ternary quadratic differential equation, the steady state solution is as follows

{\begin{matrix} x = A_{x} \sin (ω_{c} t) \\ y = A_{y} \cos (ω_{l} t) \\ z = A_{z} \cos (ω_{c} t) \end{matrix}

(2)

$A_{x}, A_{y}, and A_{z}$ represent the amplitude in each direction, and the concrete expressions are

{\begin{matrix} A_{x} = \frac{m_{c} r_{c} ω_{c}^{2}}{4 k_{x} - M ω_{c}^{2}} \\ A_{y} = \frac{2 m_{l} r_{l} ω_{l}^{2}}{4 k_{y} - M ω_{l}^{2}} \\ A_{z} = \frac{m_{c} r_{c} ω_{c}^{2}}{4 k_{z} - M ω_{c}^{2}} \end{matrix}

(3)

DEM simulation

Model of vibrating screen

The simple model of the vibrating screen is established in the simulation, and the value of mesh aperture is 0.83 mm × 0.83 mm, which is shown in Figure 4.

Figure 4.

DEM model of the vibrating screen.

Simulation parameters

In this work, the simulation is achieved based on the JKR contact model. The main parameters are shown in Table 1, which come from the experimental data and other research results.

Table 1.

Simulation parameters.

Parameter name		Value
Vibration parameters	Screen obliquity (°)	0
	Stimulation direction angle (°)	45
	Amplitude (mm)	Circle trajectory: 0.61			Line trajectory: 2.57
	Frequency (Hz)	Circle motor: 12.5			Line motor: 25
Material character	Poisson ratio	Particles: 0.2			Screen: 0.3
	Shear modulus (Pa)	Particles: 5 × 10⁷			Screen: 7 × 10¹⁰
	density (kg/m³)	Particles: 2600			Screen: 7800
Collision character	Restitution coefficient	Particle-particle: 0.03			Particle-screen: 0.4
	Static friction coefficient	Particle-particle: 0.03			Particle-screen: 0.5
	Rolling friction coefficient	Particle-particle: 0.01			Particle-screen: 0.002
Particle distribution	Particle diameter (mm)	0.3	0.5	0.7	0.8	1.8
Particle distribution	Particle numbers	3000	6000	8000	4000	5000

Results and comparison

Migration velocity

Figure 5 is the particles migration on the spatial Lissajous trajectory vibrating screen by throwing motion, and it can be seen that particles can migrate forward, which shows that the principle of spatial Lissajous trajectory vibrating screen is correct.

Figure 5.

The particles migration on the spatial Lissajous trajectory vibrating screen.

Particle migration velocity is an important index of the performance of vibrating screen. Because of the randomness of particle-to-particle collision, the migration velocity of single particle is irregular. So the average migration velocity of particle group is used as the index.

In order to compare the migration velocity of linear, elliptical, and spatial Lissajous trajectory vibrating screen, the three migration velocity curves are obtained, which are shown in Figure 6. The fluctuant lines are the dynamic curves, conveying velocity of particle group, while the smooth lines are the curves disposed.

Figure 6.

The particles migration velocity with three different vibrating trajectories.

It can be seen from Figure 6 that the migration velocity of solid particles in the elliptical trajectory is the lowest compared with that in the linear and spatial Lissajous trajectories. The migration velocity difference of the linear and spatial Lissajous trajectories in migration direction is very small. Relatively speaking, the solid particle migration velocity of spatial Lissajous trajectory is easier to reach a stable state.

Filter ratio

In the process of screening, not all of the particles can pass through the screen. Only partly small particles can be transported out from the right side outlet by the particle-to-particle and particle-to-screen interactions.

When there is a viscous force on the surface of the particles, the particles will form clusters between them, as shown in Figure 7. Small particles will adhere to large particles. Under the influence of the viscous force, medium-sized particles are easy to penetrate through sieve holes. While smaller particles are easier to form clusters, which will hinder their passing through the screen holes.

Figure 7.

The particles viscous situation.

In this work, the research on filter ratio is mainly aimed at those particles whose diameter is smaller than the mesh of the screen. The filter ratio is defined as $η_{i} = n_{di} / N_{di}$ , where $n_{di}$ is the number of particles (diameter $d_{i}$ ) that pass through the screen. $N_{di}$ is the total number of particles (diameter $d_{i}$ ) in the simulation. Table 2 is the statistic of particles filter ratio with different vibrating trajectory.

Table 2.

Four kinds of particles’ filter ratios with different vibrating trajectories.

Particlesdiameter (mm)	0.3	0.5	0.7	0.8
Lissajousfilter ratio (%)	35.3	59.27	41.6	20.15
Linear filterratio (%)	32.2	57.17	41.36	18.7
Ellipticalfilter ratio (%)	30.1	53.42	35.13	16.2

From Table 2, it can be seen that the filter rate of spatial Lissajous trajectory is the highest, while that of elliptical trajectory is the lowest. Because of the increase of a circumferential exciting force, there exists a circumferential acceleration vector when particles contact the screen surface, which makes it easier for particles to penetrate through the screen hole and reduces the phenomenon of screening blockage.

Conclusion

A new-type spatial Lissajous trajectory is put forward, and the 3D model of the spatial Lissajous trajectory vibrating screen is established.

DEM simulation results show that the solid particles can migrate forward, which shows that the principle of spatial Lissajous trajectory vibrating screen is correct. And the migration velocity approaches a stable value gradually.

The spatial Lissajous vibrating screen has approximate solid migration velocity like the linear vibrating screen, but it can reduce the phenomenon of screening blockage. Therefore, the advantages of spatial Lissajous trajectory vibrating screen will be more evident in the case of high drilling fluid viscosity, high mesh number and serious screen blockage.

Footnotes

Handling Editor: James Baldwin

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors are grateful for the financial support from the Hubei Provincial Department of Education in China (grant no. D20171303).

ORCID iD

Zhipeng Lyu

References

Jiang

Zhao

Duan

et al. Kinematics of variable-amplitude screen and analysis of particle behavior during the process of coal screening. Powder Technol 2016; 306: 88–95.

Yin

Zhang

Han

. Simulation of particle flow on an elliptical vibrating screen using the discrete element method. Powder Technol 2016; 332: 432–454.

Liu

Wang

Zhao

et al. DEM simulation of particle flow on a single deck banana screen. Int J Min Sci Technol 2013; 23: 273–277.

Kichkar

. Analysis of an assigned oscillatory trajectory of a vibratory drilling screen. Chem Pet Eng 2010; 46: 69–71.

Kichkar

. Positioning of shale-shaker drive with an assigned vibration trajectory. Chem Pet Eng 2010; 46: 3–4.

Cundall

Strack

ODL

. Discrete numerical model for granular assemblies. Int J Rock Mech Min 1979; 29: 47–65.

Lim

EWC

. Density segregation of dry and wet granular mixtures in vibrated beds. Adv Powder Technol 2016; 27: 2478–2488.

Dong

Esfandiary

. Discrete particle simulation of particle flow and separation on a vibrating screen: effect of aperture shape. Powder Technol 2016; 314: 195–222.

Cleary

. DEM prediction of industrial and geophysical particle flows. Particuology 2010; 8: 106–118.

10.

Cleary

. DEM simulation of industrial particle flows: case studies of dragline excavators, mixing in tumblers and centrifugal mills. Powder Technol 2000; 109: 83–104.

11.

Zhu

Zhou

Yang

et al. Discrete particle simulation of particulate systems: a review of major applications and findings. Chem Eng Sci 2008; 63: 5728–5770.

12.

Combarros

Feise

Zetzener

et al. Segregation of particulate solids: experiments and DEM simulations. Particuology 2014; 12: 25–32.

13.

Tong

Zhou

et al. Modeling and parameter of optimization for the design of vibrating screens. Miner Eng 2015; 83: 149–155.

14.

Chen

Tong

. Modeling screening efficiency with vibrational parameters based on DEM 3D simulation. Min Sci Technol 2010; 20: 615–620.

15.

Delaney

Cleary

Hilden

et al. Testing the validity of the spherical DEM model in simulating real granular screening processes. Chem Eng Sci 2012; 63: 215–226.

16.

Elakamp

Kruggel-Emden

. Review and benchmarking of process models for batch screening based on discrete element simulations. Adv Powder Technol 2015; 26: 679–697.

17.

Dong

. Numerical simulation of the particle flow and sieving behavior on sieve bend/low head screen combination. Miner Eng 2015; 31: 2–9.

18.

Wang

Tong

. Screening efficiency and screen length of a linear vibrating screen using DEM 3D simulation. Min Sci Technol 2011; 21: 451–455.

19.

Katarzyna

Piotr

Remigiusz

. Verification of the mathematical model of the screen blocking process. Powder Technol 2014; 256: 506–511.

20.

Zhao

Liu

Yan

. A virtual experiment showing single particle motion on a linearly vibrating screen-deck. Min Sci Technol 2010; 20: 276–280.

21.

Jiao

Zhao

. Screen simulation using a particle discrete element method. J China Univ Min Technol 2007; 36: 232–236.

22.

Xiao

Tong

. Particle stratification and penetration of a linear vibrating screen by the discrete element method. Int J Min Sci Technol 2012; 22: 357–362.

23.

Webb

Pandiella

et al. Discrete particle motion on sieves—a numerical study using the DEM simulation. Powder Technol 2003; 133: 190–202.

24.

Zhao

Zhang

Jiao

et al. Simulation of discrete element of particles motion on the vibration plane. Int J Min Sci Technol 2006; 35: 586–590.

25.

Harald

Frederik

. Modeling of screening processes with the discrete element method involving non-spherical particles. Chem Eng Sci 2014; 37: 847–856.

26.

Zhao

Liu

et al. Simulation of the screening process on a circularly vibrating screen using 3D-DEM. Min Sci Technol 2010; 21: 677–680.

27.

Xiao

Tong

. Characteristics and efficiency of a new vibrating screen with a swing trace. Particuology 2013; 11: 601–606.

Research on spatial Lissajous trajectory vibrating screen

Abstract

Keywords

Introduction

Dynamic theory of the spatial Lissajous trajectory vibrating screen

DEM simulation

Model of vibrating screen

Simulation parameters

Results and comparison

Migration velocity

Filter ratio

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References