*Article* **A Numerical Study of Separation Performance of Vibrating Flip-Flow Screens for Cohesive Particles**

**Chi Yu, Runhui Geng and Xinwen Wang \***

School of Chemical and Environmental Engineering, China University of Mining and Technology (Beijing), Beijing 100083, China; bqt1800301011@student.cumtb.edu.cn (C.Y.); sqt1900301005@student.cumtb.edu.cn (R.G.) **\*** Correspondence: xinwen.w@cumtb.edu.cn

**Abstract:** Vibrating flip-flow screens (VFFS) are widely used to separate high-viscosity and fine materials. The most remarkable characteristic is that the vibration intensity of the screen frame is only 2–3 g (g represents the gravitational acceleration), while the vibration intensity of the screen surface can reach 30–50 g. This effectively solves the problem of the blocking screen aperture in the screening process of moist particles. In this paper, the approximate state of motion of the sieve mat is realized by setting the discrete rigid motion at multiple points on the elastic sieve mat of the VFFS. The effects of surface energy levels between particles separated via screening performance were compared and analyzed. The results show that the flow characteristics of particles have a great influence on the separation performance. For 8 mm particle screening, the particle's velocity dominates its movement and screening behavior in the range of 0–8 J/m<sup>2</sup> surface energy. In the feeding end region (Sections 1 and 2), with the increase in the surface energy, the particle's velocity decreases, and the contact time between the particles and the screen surface increases, and so the passage increases. When the surface energy level continues to increase, the particles agglomerate together due to the effect of the cohesive force, and the effect of the particle's agglomeration is greater than the particle velocity. Due to the agglomeration of particles, the difficulty of particles passing through the screen increases, and the yields of various size fractions in the feeding end decrease to some extent. In the transporting process, the agglomerated particles need to travel a certain distance before depolymerization, and the stronger the adhesive force between particles, the larger the depolymerization distance. Therefore, for the case of higher surface energy, the screening percentage near the discharging end (Sections 3 and 4) is greater. The above research is helpful to better understand and optimize the screening process of VFFS.

**Keywords:** vibrating flip-flow screen; DEM; wet stick material; JKR model; separation performance

## **1. Introduction**

Flip-flow screening technology is a new concept of screening technology that has been widely used and promoted in recent years. The VFFS has a wide range of applications in many fields, such as the fine coal screening process, cyclic screening of ore grinding products by high-pressure roller mill, and resource utilization of building solid waste [1,2]. Compared to traditional vibrating screens, such as linear vibrating screens and circular vibrating screens, the VFFS has the following advantage: small vibration intensity of main screen frame (2–3 g), therefore the dynamic load on the foundation is small; high vibration intensity of the sieve mat (up to 30–50 g). Furthermore, the VFFS is extremely friendly to the screening of viscous and wet fine-grained material, and it is not easy to block apertures on the screen surface while ensuring high screening efficiency and processing capacity. Due to the existence of water content between viscous and wet particles, there is a liquid bridge force between particles; particles will gather into clusters when the cohesion between particles is strong enough. When using traditional vibrating screens to process the wet and fine particles, the vibration intensity is not enough to make the agglomerated

**Citation:** Yu, C.; Geng, R.; Wang, X. A Numerical Study of Separation Performance of Vibrating Flip-Flow Screens for Cohesive Particles. *Minerals* **2021**, *11*, 631. https:// doi.org/10.3390/min11060631

Academic Editors: Daniel Saramak, Marek Pawełczyk and Tomasz Niedoba

Received: 14 May 2021 Accepted: 8 June 2021 Published: 14 June 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

particles depolymerized, and the screens are extremely prone to blockage, adhesion, and compaction, which deteriorates the screening process [3]. The vibration frequency of VFFS is generally lower than a traditional screen, but through the large deformation of the elastic sieve mat, the peak acceleration is easy to produce. The vibration response of the sieve mat agitates the particle bed to deagglomerate the agglomerated particles. This drives the fine particles to flow down the bed and then pass through the screen to become the undersized product. The elastic sieve mat agitates the bed to depolymerize the agglomerated particles.

Standish constructed a single-particle model to investigate particle motion base on the reaction kinetics and probability theory. However, the collision between particles is not considered [4,5]. Soldinger developed a semi-mechanical phenomenological model of a linear vibrating screen, taking into account the stratification and passage [6]. Soldinger further extended the model after considering the material loading effect and the screening efficiency of different size particles [7]. The actual screening process is very complicated, and particle movement is affected by many conditions. At present, the discrete element method (DEM) simulation is an effective method for the simulation of granular systems, which has been used in various industrial processes. Cleary et al. quantitatively investigated the particle flow and screening performance of an industrial double-deck banana screen with different accelerations based on DEM simulation [8,9]. Davoodi et al. reported the effect of the aperture shape and the material on the particle flow and sieving performance [10]. Dong et al. simulated the screening process with the discrete element method and studied the influence of rectangular aperture shapes, with different aspect ratios, on material movement and screening efficiency [11]. Zhao et al. studied the influence of the motion parameters of the linear and circular vibration screens on the screening performance [12]. Wang et al. used the discrete element and the finite element methods to study the influence of vibration parameters on the screening efficiency of the vibrating screen. In addition, the distribution of stress and deformation on the screen surface under different vibration conditions has also been reported [13].

The above studies are mostly focusing on dry particulate systems, which are based on the Hertz–Mindlin model. In the actual screening process, due to the small particle size, large specific surface area, and external moisture, the fine particles easily agglomerate with each other to form large-size particles. The particles agglomerate together and move as a whole, making the screening process difficult. Limtrakul et al. reported that fine particles in a fluidized bed have particle agglomeration and stagnation regions due to high cohesion and confirmed the influence of vibration on improving fluidization through experiments [14]. Yang et al. investigated the influence of surface energy on the transition behavior of Geldart A-type particles from a fixed bed to a bubbling bed through a twodimensional DEM-CFD simulation [15]. Cleary et al. reported the effect of cohesion between particles on particle flow over a double-deck banana screen [16]. At present, there are few numerical simulation studies on the movement and separation of viscous and wet material on VFFS.

In this study, the elastic sieve mat of the VFFS is discretized into multiple units by testing the movement of each unit body. According to the phase relationship of the unit body, it can describe the kinematics of the entire elastic sieve mat. The motion of each point on the sieve mat can be transformed into a function form by the Fourier series, which is used as the basis for setting the motion of the VFFS model. The effects of different adhesion levels on particle flow and screening performance on VFFS were compared and analyzed, which is helpful to better understand and optimize the screening process of the VFFS.

#### **2. Simulation Methods**

#### *2.1. Contact Model of Particles*

Due to clay and water present on the particle surface, there is a cohesive force between particles. The commonly used Hertz–Mindlin contact model struggles to comprehensively analyze the mechanical behavior between wet particles and between particles and the screen surface. The Hertz–Mindlin with JKR contact model, which considers the cohesive

force, can better simulate the behavior of viscous and wet particles. Taking into account the effect of the surface energy (adhesion force) between the particles on the movement and screen penetration, the calculation of the normal elastic contact force is based on the Johnson–Kendall–Roberts theory [17,18]. cohesive force, can better simulate the behavior of viscous and wet particles. Taking into account the effect of the surface energy (adhesion force) between the particles on the movement and screen penetration, the calculation of the normal elastic contact force is based on the Johnson–Kendall–Roberts theory [17,18].

Due to clay and water present on the particle surface, there is a cohesive force between particles. The commonly used Hertz–Mindlin contact model struggles to comprehensively analyze the mechanical behavior between wet particles and between particles and the screen surface. The Hertz–Mindlin with JKR contact model, which considers the

Figure 1 shows the contact process of two cohesive particles. *R*<sup>1</sup> and *R*<sup>2</sup> represent the radius of Particle 1 and 2, respectively (mm). *a* stands for the contact radius between the particles (mm), and *a*<sup>0</sup> is the radius of the contact surface considering the adhesion (mm). *δ<sup>n</sup>* is the amount of normal overlap (mm). Due to the cohesive force on the contact surface, the contact radius of these two particles extends from *a* to *a*0. Figure 1 shows the contact process of two cohesive particles. ଵ and ଶ represent the radius of Particle 1 and 2, respectively (mm). stands for the contact radius between the particles (mm), and is the radius of the contact surface considering the adhesion (mm). is the amount of normal overlap (mm). Due to the cohesive force on the contact surface, the contact radius of these two particles extends from to .

*Minerals* **2021**, *11*, x FOR PEER REVIEW 3 of 15

**2. Simulation Methods**  *2.1. Contact Model of Particles* 

γ, therefore, = 2γ.

**Figure 1.** Deformation surface (rough line) considering cohesive force. **Figure 1.** Deformation surface (rough line) considering cohesive force.

The cohesive force between the wet and viscous particles is set as (J/m2), which can be obtained by Equation (1). The cohesive force between the wet and viscous particles is set as *W*(J/m<sup>2</sup> ), which can be obtained by Equation (1).

$$\mathcal{W} = \gamma\_1 + \gamma\_2 + \gamma\_{12} \tag{1}$$

=γଵ + γଶ + γଵଶ (1) where γଵ is the surface energy of Particle 1 (J/m2); γଶ is the surface energy of Particle 2 (J/m2); γଵଶ stands for the interface energy between Particles 1 and 2 (J/m2). When the material of the particles is the same, the interface energy is 0 J/m2, that is, γଵଶ = 0,γଵ = γଶ = where *γ*<sup>1</sup> is the surface energy of Particle 1 (J/m<sup>2</sup> ); *γ*<sup>2</sup> is the surface energy of Particle 2 (J/m<sup>2</sup> ); *γ*<sup>12</sup> stands for the interface energy between Particles 1 and 2 (J/m<sup>2</sup> ). When the material of the particles is the same, the interface energy is 0 J/m<sup>2</sup> , that is, *γ*<sup>12</sup> = 0, *γ*<sup>1</sup> = *γ*<sup>2</sup> = *γ*, therefore, *W* = 2*γ*.

$$a = \sqrt{\delta\_n R^\*} \tag{2}$$

$$\delta\_n = \frac{a\_0^2}{R^\*} - \sqrt{\frac{4\pi\gamma a\_0}{E^\*}}\tag{3}$$

$$\frac{1}{R^\*} = \frac{1}{R\_1} + \frac{1}{R\_2} \tag{4}$$

$$\frac{1}{E^\*} = \frac{1 - \upsilon\_1^2}{E\_1} + \frac{1 - \upsilon\_2^2}{E\_2} \tag{5}$$
 
$$\text{between wet particles (J/m}^2\text{); } \mathbb{R}^\* \text{ is the equivalent contact}$$

ଵ ଶ Here *γ* is the surface energy between wet particles (J/m<sup>2</sup> ); *R* ∗ is the equivalent contact radius (mm); *E* ∗ is the equivalent elastic modulus (N/m<sup>2</sup> ); *E*1, *E*<sup>2</sup> represent the elastic modulus of Particle 1 and 2, respectively (N/m<sup>2</sup> ); *υ*<sup>1</sup> *υ*<sup>2</sup> are the Poisson's ratio of these two particles, respectively (-).

Then, the normal elastic contact force FJKR(N) between the wet particles can be calculated by Equation (6):

$$\mathcal{F}\_{\text{JKR}} = -2\sqrt{2\pi W E^\* a\_0^3} + \frac{4E^\* a\_0^3}{3R^\*} \tag{6}$$

When the surface energy of the viscous particle is 0 J/m<sup>2</sup> , the model FJKR is simplified to the contact force FHertz.

#### *2.2. The DEM Model Setting of VFFS 2.2. The DEM Model Setting of VFFS*

*Minerals* **2021**, *11*, x FOR PEER REVIEW 4 of 15

two particles, respectively (-).

culated by Equation (6):

to the contact force Fୌୣ୰୲.

1

∗ <sup>=</sup> 1−ଵ

Fୖ = −2ට2π∗

ଵ

ଶ

Here γ is the surface energy between wet particles (J/m2); <sup>∗</sup> is the equivalent contact radius (mm); ∗ is the equivalent elastic modulus (N/m2); ଵ, ଶ represent the elastic modulus of Particle 1 and 2, respectively (N/m2); ଵ ଶ are the Poisson's ratio of these

Then, the normal elastic contact force Fୖ(N) between the wet particles can be cal-

When the surface energy of the viscous particle is 0 J/m2, the model Fୖ is simplified

+

1−ଶ ଶ

ଶ

<sup>ଷ</sup> +

4∗ ଷ

3∗ (6)

(5)

The structures of the VFFS and elastic sieve mat are presented in Figure 2. Different from the traditional vibrating screens, the VFFS consists of two vibrating frames, including the main screen frame and the floating screen frame. The beams of the two frames are arranged in a staggered layout. When the exciter mounted on the main screen frame is operated, both the screen frames move relative to each other through the effect of rubber shear springs. The elastic sieve mats are periodically stretched and slackened to generate peak acceleration, typically 30–50 times gravity. The structures of the VFFS and elastic sieve mat are presented in Figure 2. Different from the traditional vibrating screens, the VFFS consists of two vibrating frames, including the main screen frame and the floating screen frame. The beams of the two frames are arranged in a staggered layout. When the exciter mounted on the main screen frame is operated, both the screen frames move relative to each other through the effect of rubber shear springs. The elastic sieve mats are periodically stretched and slackened to generate peak acceleration, typically 30–50 times gravity.

**Figure 2.** Structures of the VFFS and elastic sieve mat. **Figure 2.** Structures of the VFFS and elastic sieve mat.

For traditional screening equipment such as circular vibrating screens and linear vibrating screens, the vibration parameters of the screen surface are consistent with the vibration response of the screen frame. Therefore, it is relatively easy to set the model of the traditional vibrating screen in the discrete element simulation. Many scholars have already done many in-depth studies in these fields [19–21]. For the VFFS, the vibration response of each position on the elastic sieve mat is different. The accelerometer is used to test the amplitude response at different positions on the elastic sieve mat. Figure 3 shows the measuring displacement of the midpoint of the sieve mat. For traditional screening equipment such as circular vibrating screens and linear vibrating screens, the vibration parameters of the screen surface are consistent with the vibration response of the screen frame. Therefore, it is relatively easy to set the model of the traditional vibrating screen in the discrete element simulation. Many scholars have already done many in-depth studies in these fields [19–21]. For the VFFS, the vibration response of each position on the elastic sieve mat is different. The accelerometer is used to test the amplitude response at different positions on the elastic sieve mat. Figure 3 shows the measuring displacement of the midpoint of the sieve mat. *Minerals* **2021**, *11*, x FOR PEER REVIEW 5 of 15

**Figure 3.** Fourier series analysis of amplitude at the midpoint of the sieve mat. **Figure 3.** Fourier series analysis of amplitude at the midpoint of the sieve mat.

The displacement signal of the measuring point on the screen surface is not a regular simple harmonic function but periodic. Fortunately, any periodic function of time can be represented by the Fourier series as an infinite sum of sine and cosine terms [22]. Its Fourier series representation is given by Equation (7). The displacement signal of the measuring point on the screen surface is not a regular simple harmonic function but periodic. Fortunately, any periodic function of time can be represented by the Fourier series as an infinite sum of sine and cosine terms [22]. Its Fourier series representation is given by Equation (7).

$$\mathbf{u}(t) = \mathbf{a} + \sum\_{n=1}^{\infty} \left[ a\_n \cos(n\omega t) + b\_n \sin(n\omega t) \right] \tag{7}$$

න ()

න () ()

න () ()

where = 2π/τ is called the fundamental frequency (rad/s) and , ଵ, ଶ,…, ଵ, ଶ,… are constant coefficients (-). The M (intercepted order of Fourier series) has a direct impact on the accuracy of the calculation results. The larger the M, the closer the analysis result is to the accurate value [23], but it will also affect the solution efficiency. Then, we take the amplitude of the midpoint as an example for the Fourier analysis. Within a motion cycle, the peak value of the amplitude is 30.27 mm. In contrast to the signals analyzed by the Fourier series with the measured values, the results are shown in Figure 3. The mean square error (MSE) of a period and the relative error (RE) of maximum amplitude in the time domain are used to evaluate the change between the measured amplitude and the

ஶ

ୀଵ

 = 2

ఛ

ఛ

<sup>=</sup> <sup>2</sup>

<sup>=</sup> <sup>2</sup>

Fourier series analysis result. The results are shown in Table 1.

Maximum amplitude

**Table 1.** Measured amplitude and analyzed amplitude by Fourier series on the midpoint.

**Intercepted Order M M = 1 M = 2 M = 3 M = 4 M = 5** 

(mm) 23.23 28.10 28.79 29.37 29.88 RE (%) 17.19 2.63 2.0 1.57 1.10 MSE 13.06 0.99 0.59 0.40 0.20

When the M is equal to one, the MSE is 13.06. When the M is equal to five, the MSE reduces to 0.2. Meanwhile, the RE is only 1.1%. Therefore, in this paper, the intercepted order of all amplitudes analyzed by the Fourier series is taken as five. The testing vibration

$$\boldsymbol{a} = \frac{2}{\tau} \int\_0^\tau \mathbf{x}(t) dt$$

$$a\_{\boldsymbol{n}} = \frac{2}{\tau} \int\_0^\tau \mathbf{x}(t) \cos(n\omega t) dt$$

$$b\_{\boldsymbol{n}} = \frac{2}{\tau} \int\_0^\tau \mathbf{x}(t) \sin(n\omega t) dt$$

where *ω* = 2π/*τ* is called the fundamental frequency (rad/s) and *α*, *a*1, *a*2, . . . , *b*1, *b*2, . . . are constant coefficients (-). The M (intercepted order of Fourier series) has a direct impact on the accuracy of the calculation results. The larger the M, the closer the analysis result is to the accurate value [23], but it will also affect the solution efficiency. Then, we take the amplitude of the midpoint as an example for the Fourier analysis. Within a motion cycle, the peak value of the amplitude is 30.27 mm. In contrast to the signals analyzed by the Fourier series with the measured values, the results are shown in Figure 3. The mean square error (MSE) of a period and the relative error (RE) of maximum amplitude in the time domain are used to evaluate the change between the measured amplitude and the Fourier series analysis result. The results are shown in Table 1.

**Table 1.** Measured amplitude and analyzed amplitude by Fourier series on the midpoint.


When the M is equal to one, the MSE is 13.06. When the M is equal to five, the MSE reduces to 0.2. Meanwhile, the RE is only 1.1%. Therefore, in this paper, the intercepted order of all amplitudes analyzed by the Fourier series is taken as five. The testing vibration amplitudes of each point on the elastic sieve mat are shown in Figure 4. It can be seen that the vibration amplitudes on the elastic sieve mat are symmetrically distributed, the midpoint has a large amplitude, and the edge measuring point has a relatively small amplitude. The movement of each point can be transformed into a function by the abovementioned Fourier analysis method. *Minerals* **2021**, *11*, x FOR PEER REVIEW 6 of 15 amplitudes of each point on the elastic sieve mat are shown in Figure 4. It can be seen that the vibration amplitudes on the elastic sieve mat are symmetrically distributed, the midpoint has a large amplitude, and the edge measuring point has a relatively small amplitude. The movement of each point can be transformed into a function by the above-mentioned Fourier analysis method.

**Figure 4.** Vibration amplitude of each measuring point of the elastic sieve mat. **Figure 4.** Vibration amplitude of each measuring point of the elastic sieve mat.

Further, the sieve mat is discretized into multiple units, and the simulation of the approximate continuous flexible motion is realized through the setting of multi-point rigid motion, as shown in Figure 5. Further, the sieve mat is discretized into multiple units, and the simulation of the approximate continuous flexible motion is realized through the setting of multi-point rigid motion, as shown in Figure 5.

Figure 6a shows the DEM modeling schematic of the VFFS system. The VFFS used in the simulation process is specifically composed of eight elastic sieve mats, each with a size of 328 × 650 mm. The screen aperture is 8 × 25 mm, and the inclination angle of the screen is 15°. In the simulation, the undersized product is divided into four parts equally by using 633.6 mm as the length interval unit, namely Sections 1–4. Section 5 is used to collect the oversized product. The feeding system is composed of a silo and a vibrating feeder. The material properties are shown in Figure 6b, and the simulation parameters in the DEM are shown in Table 2 [24,25]. It is worth noting that the impact of particle shape on the

**Figure 5.** DEM model of the elastic sieve mat.

*2.3. Simulation Conditions* 

VFFS parameters

Further, the sieve mat is discretized into multiple units, and the simulation of the approximate continuous flexible motion is realized through the setting of multi-point

**Figure 4.** Vibration amplitude of each measuring point of the elastic sieve mat.

amplitudes of each point on the elastic sieve mat are shown in Figure 4. It can be seen that the vibration amplitudes on the elastic sieve mat are symmetrically distributed, the midpoint has a large amplitude, and the edge measuring point has a relatively small amplitude. The movement of each point can be transformed into a function by the above-men-

tioned Fourier analysis method.

rigid motion, as shown in Figure 5.

**Figure 5.** DEM model of the elastic sieve mat. **Figure 5.** DEM model of the elastic sieve mat.

#### *2.3. Simulation Conditions 2.3. Simulation Conditions*

Figure 6a shows the DEM modeling schematic of the VFFS system. The VFFS used in the simulation process is specifically composed of eight elastic sieve mats, each with a size of 328 × 650 mm. The screen aperture is 8 × 25 mm, and the inclination angle of the screen is 15°. In the simulation, the undersized product is divided into four parts equally by using 633.6 mm as the length interval unit, namely Sections 1–4. Section 5 is used to collect the oversized product. The feeding system is composed of a silo and a vibrating feeder. The material properties are shown in Figure 6b, and the simulation parameters in the DEM are shown in Table 2 [24,25]. It is worth noting that the impact of particle shape on the Figure 6a shows the DEM modeling schematic of the VFFS system. The VFFS used in the simulation process is specifically composed of eight elastic sieve mats, each with a size of 328 mm × 650 mm. The screen aperture is 8 mm × 25 mm, and the inclination angle of the screen is 15◦ . In the simulation, the undersized product is divided into four parts equally by using 633.6 mm as the length interval unit, namely Sections 1–4. Section 5 is used to collect the oversized product. The feeding system is composed of a silo and a vibrating feeder. The material properties are shown in Figure 6b, and the simulation parameters in the DEM are shown in Table 2 [24,25]. It is worth noting that the impact of particle shape on the screening process is not considered in this study. The particles used in this simulation are all homogeneous spherical particles. *Minerals* **2021**, *11*, x FOR PEER REVIEW 7 of 15 screening process is not considered in this study. The particles used in this simulation are all homogeneous spherical particles.

**Figure 6. (a)** Schematic of the DEM model of the VFFS system; (**b**) material properties of the sample. **Figure 6.** (**a**) Schematic of the DEM model of the VFFS system; (**b**) material properties of the sample.

**Table 2.** Modeling condition in EDEM.

**Material Property Poisson's Ratio (-) Shear Modulus (Pa) Density (kg/m3)**  Particle 0.250 2.200e + 08 2456 Polyrethane 0.499 1.157e + 06 1200 Steel 0.300 7.692e + 10 7850 Collision property Coefficient of restitution Coefficient of static friction Coefficient of rolling friction

**3. Effect of Surface Energy Level on Separation Performance** 

this paper. The calculation formulas are as follows [26,27]:

During the screening process, there are always some fine particles existing in the oversized products and some coarse particles in the undersized products. The screening efficiency and total misplaced material were used to assess the screening performance in

× 100

× 100

(8)

(9)

 − × 

 = + = 100 × ௨ = 100 ×

= + − 100

 <sup>=</sup> × 

<sup>=</sup>

Particle-polyrethane 0.25 0.500 0.01 Particle-steel 0.30 0.154 0.01

Vibration parameter The vibration frequency of 776 r/min, screen inclination of 15° Screen parameters Screen length and width with 2624 and 650 mm, respectively

Material properties The total mass of 5.81 kg


**Table 2.** Modeling condition in EDEM.

## **3. Effect of Surface Energy Level on Separation Performance**

During the screening process, there are always some fine particles existing in the oversized products and some coarse particles in the undersized products. The screening efficiency and total misplaced material were used to assess the screening performance in this paper. The calculation formulas are as follows [26,27]:

$$\begin{array}{l} \eta = E\_c + E\_f - 100\\ E\_c = \frac{\gamma\_o \times O\_c}{F\_c'} \times 100\\ E\_f = \frac{F\_f' - \gamma\_o \times O\_f}{F\_f'} \times 100 \end{array} \tag{8}$$

$$\begin{aligned} M\_{\ o} &= M\_{\text{c}} + M\_{f} \\ M\_{\text{c}} &= 100 \times \gamma\_{\mu} \text{U}\_{\text{c}} \\ M\_{f} &= 100 \times \gamma\_{\text{o}} \text{O}\_{f} \end{aligned} \tag{9}$$

where the *η* is the screening efficiency (%), *E<sup>c</sup>* and *E<sup>f</sup>* stand for the effective placement efficiency of the coarse particles (%) and the effective placement efficiency of fine particles (%), respectively. The *M<sup>o</sup>* is the total misplaced material (%), *M<sup>c</sup>* and *M<sup>f</sup>* are the misplaced material of coarse particles (%) and the misplaced material of fine particles (%), respectively. The *γ<sup>o</sup>* represents the yield of oversized product (%), *γ<sup>u</sup>* is the yield of undersized product (%), *O<sup>f</sup>* is the ratio of fine particles in the oversized product (%), *O<sup>c</sup>* is the ratio of coarse particles in the oversized product (%), *F r c* is the ratio of coarse particles in the feeding (%), and *F r f* is the ratio of fine particles in the feeding (%).

Figure 7 shows the flow characteristics of material on VFFS with three surface energy levels (4, 20, and 36 J/m<sup>2</sup> ). In the case of the surface energy of 4 J/m<sup>2</sup> , when the particles enter the screen, the vibration of the sieve mat quickly enables the material to spread on the screen surface. A larger amount of material pass through the screen in Section 1. As the screening process progresses along the direction of material flow, the amount of penetration in other sections gradually decreases. When the surface energy is 20 J/m<sup>2</sup> , compared to the case of 4 J/m<sup>2</sup> , the yield of material in Section 1 is reduced. This section mainly promotes the depolymerization of agglomerated particles. Meanwhile, the yield of material in Section 2 is increased. When the surface energy is 36 J/m<sup>2</sup> , there is a great cohesion force between the particles, and the agglomerated particles need a longer movement distance to complete the depolymerization process. In Sections 1 and 2, which near the feeding end, the yield of the undersized product is low, and more particles are concentrated in Sections 3 and 4, near the discharging end. To further deepen the understanding of the screening process of VFFS, quantitative analysis was carried out on the products of each section.

the ratio of coarse particles in the oversized product (%),

in the feeding (%), and

section.

where the is the screening efficiency (%), and stand for the effective placement efficiency of the coarse particles (%) and the effective placement efficiency of fine particles (%), respectively. The is the total misplaced material (%), and are the misplaced material of coarse particles (%) and the misplaced material of fine particles (%), respectively. The represents the yield of oversized product (%), ௨ is the yield of undersized product (%), is the ratio of fine particles in the oversized product (%), is

 is the ratio of fine particles in the feeding (%). Figure 7 shows the flow characteristics of material on VFFS with three surface energy levels (4, 20, and 36 J/m2). In the case of the surface energy of 4 J/m2, when the particles enter the screen, the vibration of the sieve mat quickly enables the material to spread on the screen surface. A larger amount of material pass through the screen in Section 1. As the screening process progresses along the direction of material flow, the amount of penetration in other sections gradually decreases. When the surface energy is 20 J/m2, compared to the case of 4 J/m2, the yield of material in Section 1 is reduced. This section mainly promotes the depolymerization of agglomerated particles. Meanwhile, the yield of material in Section 2 is increased. When the surface energy is 36 J/m2, there is a great cohesion force between the particles, and the agglomerated particles need a longer movement distance to complete the depolymerization process. In Sections 1 and 2, which near the feeding end, the yield of the undersized product is low, and more particles are concentrated in Sections 3 and 4, near the discharging end. To further deepen the understanding of the screening process of VFFS, quantitative analysis was carried out on the products of each

is the ratio of coarse particles

**Figure 7.** The flow behavior of material on VFFS with different surface energy levels. (**a**) 4, (**b**) 20, and (**c**) 36 J/m2 **Figure 7.** The flow behavior of material on VFFS with different surface energy levels. (**a**) 4, (**b**) 20, and (**c**) 36 J/m. <sup>2</sup> .
