*3.1. The Yield of Each Section of VFFS*

Figure 8a shows the distribution of product yields between undersized and oversized products with different surface energy levels, where the distribution is related to the surface energy between particles. With the increase in the surface energy level between particles, the yield of the undersized product first increased and then decreased, and at the same time, the yield of oversized products decreased and then increased. This means that for each surface energy level, the sum of the undersized and oversized product is 100%. In the case of the surface energy level of 0–8 J/m<sup>2</sup> , the particle movement speed dominates the movement and separation behavior of the particles. With the increase in the surface energy level within the range of 0–8 J/m<sup>2</sup> , the particle movement speed decreases, increasing the contact time between the particle and the screen surface. Therefore, the amount of material passing through the screen increases. When the cohesive force continues to increase to a certain level, the particles agglomerate together, and the impact of particle agglomeration is greater than the particle movement speed. As more fine particles agglomerate together, their size increases to greater than the aperture size, and the material screening percentages decreases. The stronger the surface energy between the particles, the longer the distance the agglomerated particles need to deagglomerate. More fine particles finally enter the oversized products, so the yield of undersized products drops again. When the surface energy level is 36 J/m<sup>2</sup> , around 70% of the material enters the oversized product.

*3.1. The Yield of Each Section of VFFS* 

oversized product.

**Figure 8.** (**a**) Distribution of the yields between the undersized and oversized product with different surface energy levels; (**b**) The undersized product yields of each section with different surface energy levels. **Figure 8.** (**a**) Distribution of the yields between the undersized and oversized product with different surface energy levels; (**b**) The undersized product yields of each section with different surface energy levels.

The yield of each section of the undersized product is further analyzed, and the results are shown in Figure 8b. It can be seen that with the increase of the surface energy level, the yield of the undersized product in Section 1 shows a trend of increasing and then decreasing. When the surface energy is 8 J/m2, the yield in Section 1 is the largest, which is 17.91%. When the surface energy of particles continues to increase, the yield of Section 1 will gradually decrease. For the yield of Section 2, with the increase of the surface energy, it also shows the law of first increasing and then decreasing. However, the maximum yield appears at the surface energy of 24 J/m2. With the increase of the surface energy, the yield in Section 3 first decreases and then increases to a maximum value when the surface energy is 32 J/m2 and then decreases again. The yield in Section 4 first decreases and then increases. For the surface energy levels of 0, 4, 8, 12 and 16 J/m2, the undersized product yield of each section gradually decreases along the direction of the material flow, and Section 1 accounts for the largest proportion. When the surface energy is 20 J/m2, the yield in Section 2 is greater than the yield of Section 1. When the surface energy level continues to increase to 32 J/m2, the yield of Section 3 is greater than the products of Sections 1 and 2. In Section 1, less than 4% of the particles pass through the screen. As the The yield of each section of the undersized product is further analyzed, and the results are shown in Figure 8b. It can be seen that with the increase of the surface energy level, the yield of the undersized product in Section 1 shows a trend of increasing and then decreasing. When the surface energy is 8 J/m<sup>2</sup> , the yield in Section 1 is the largest, which is 17.91%. When the surface energy of particles continues to increase, the yield of Section 1 will gradually decrease. For the yield of Section 2, with the increase of the surface energy, it also shows the law of first increasing and then decreasing. However, the maximum yield appears at the surface energy of 24 J/m<sup>2</sup> . With the increase of the surface energy, the yield in Section 3 first decreases and then increases to a maximum value when the surface energy is 32 J/m<sup>2</sup> and then decreases again. The yield in Section 4 first decreases and then increases. For the surface energy levels of 0, 4, 8, 12 and 16 J/m<sup>2</sup> , the undersized product yield of each section gradually decreases along the direction of the material flow, and Section 1 accounts for the largest proportion. When the surface energy is 20 J/m<sup>2</sup> , the yield in Section 2 is greater than the yield of Section 1. When the surface energy level continues to increase to 32 J/m<sup>2</sup> , the yield of Section 3 is greater than the products of Sections 1 and 2. In Section 1, less than 4% of the particles pass through the screen. As the surface energy increases, the section with the maximum yield moves toward the discharging end. That is, the higher the adhesion, the longer the distance required for the depolymerization of the agglomerated particles.

Figure 8a shows the distribution of product yields between undersized and oversized products with different surface energy levels, where the distribution is related to the surface energy between particles. With the increase in the surface energy level between particles, the yield of the undersized product first increased and then decreased, and at the same time, the yield of oversized products decreased and then increased. This means that for each surface energy level, the sum of the undersized and oversized product is 100%. In the case of the surface energy level of 0–8 J/m2, the particle movement speed dominates the movement and separation behavior of the particles. With the increase in the surface energy level within the range of 0–8 J/m2, the particle movement speed decreases, increasing the contact time between the particle and the screen surface. Therefore, the amount of material passing through the screen increases. When the cohesive force continues to increase to a certain level, the particles agglomerate together, and the impact of particle agglomeration is greater than the particle movement speed. As more fine particles agglomerate together, their size increases to greater than the aperture size, and the material screening percentages decreases. The stronger the surface energy between the particles, the longer the distance the agglomerated particles need to deagglomerate. More fine particles finally enter the oversized products, so the yield of undersized products drops again. When the surface energy level is 36 J/m2, around 70% of the material enters the

#### *3.2. The Yield Accounted for Size Fraction in Different Sections*

Figure 9 shows the yield accounted for different size fractions in different sections, during the screening process of the VFFS. For 8 mm particle screening and different surface energy levels, the 10 mm particles are the oversized product and are all concentrated in Section 5. When the surface energy is 0, 4, and 8 J/m<sup>2</sup> , the 4 and 5 mm particles are mainly concentrated in Section 1, accounting for about 50% of this size fraction. With the increase of particle size, the yield accounts for this size fraction in Section 1 gradually decreases. For the 8 mm particles, the yields of this size fraction are 16.59%, 18.41%, and 19.80%, respectively. In addition, for the case of these surface energy levels, as the surface energy between particles increases, the yield of each size fraction also increases. This is because the increase in the level of adhesion reduces the speed of particle movement. The contact time between the particles and the screen surface is increased, increasing the yield of the particles of each size fraction. For the cases of the surface energy of 12, 16, and 20 J/m<sup>2</sup> , the yield of each size fraction in Section 1 gradually decreased, and the yield of 4 and 5 mm particles decreased to 51.93%, 44.50%, 36.63% and 47.98%, 38.87%, 35.40%, respectively. For the case of the surface energy levels of 24, 28, and 32 J/m<sup>2</sup> , in the product of Section 1, the yield accounted for the size fraction of each sized particle further decreases. Meanwhile, it is worth noting that the yield of 4 mm particles in Section 1 is slightly smaller than that of

*Minerals* **2021**, *11*, x FOR PEER REVIEW 10 of 15

*3.2. The Yield Accounted for Size Fraction in Different Sections* 

merization of the agglomerated particles.

5 mm particles, which is due to the surface energy of particles having a greater influence on fine ones. Meanwhile, it is worth noting that the yield of 4 mm particles in Section 1 is slightly smaller than that of 5 mm particles, which is due to the surface energy of particles having a greater influence on fine ones.

surface energy increases, the section with the maximum yield moves toward the discharging end. That is, the higher the adhesion, the longer the distance required for the depoly-

Figure 9 shows the yield accounted for different size fractions in different sections, during the screening process of the VFFS. For 8 mm particle screening and different surface energy levels, the 10 mm particles are the oversized product and are all concentrated in Section V. When the surface energy is 0, 4, and 8 J/m2, the 4 and 5 mm particles are mainly concentrated in Section Ⅰ, accounting for about 50% of this size fraction. With the increase of particle size, the yield accounts for this size fraction in Section 1 gradually decreases. For the 8 mm particles, the yields of this size fraction are 16.59%, 18.41%, and 19.80%, respectively. In addition, for the case of these surface energy levels, as the surface energy between particles increases, the yield of each size fraction also increases. This is because the increase in the level of adhesion reduces the speed of particle movement. The contact time between the particles and the screen surface is increased, increasing the yield of the particles of each size fraction. For the cases of the surface energy of 12, 16, and 20 J/m2, the yield of each size fraction in Section 1 gradually decreased, and the yield of 4 and 5 mm particles decreased to 51.93%, 44.50%, 36.63% and 47.98%, 38.87%, 35.40%, respectively. For the case of the surface energy levels of 24, 28, and 32 J/m2, in the product of Section 1, the yield accounted for the size fraction of each sized particle further decreases.

**Figure 9.** Comparison of the particle size distributions captured into Sections 1 (**a**), 2 (**b**), 3 (**c**), 4 (**d**), and 5 (**e**) with different surface energy levels. **Figure 9.** Comparison of the particle size distributions captured into Sections 1 (**a**), 2 (**b**), 3 (**c**), 4 (**d**), and 5 (**e**) with different surface energy levels.

The yield of each sized particle in Section 2 was observed. For the case of 0, 4, and 8 J/m2 surface energy, the yield of the undersized product in Section 2 was significantly lower than Section 1. Within the range of 0–8 J/m2, as the surface energy increased, the yield of fine particles increased, and this phenomenon was more obvious when the adhesion level was higher. When the surface energy was 24 J/m2, the yields of each size fraction in Section 2 begin to exceed those in Section 1. The yields of large-sized materials in Sections 3 and 4 are generally higher than in small-sized materials. As the surface energy increases, more fine particles are deagglomerated under the movement of the sieve mat in Sections 3 and 4, and the yield of fine particles in the product begins to exceed that of The yield of each sized particle in Section 2 was observed. For the case of 0, 4, and 8 J/m<sup>2</sup> surface energy, the yield of the undersized product in Section 2 was significantly lower than Section 1. Within the range of 0–8 J/m<sup>2</sup> , as the surface energy increased, the yield of fine particles increased, and this phenomenon was more obvious when the adhesion level was higher. When the surface energy was 24 J/m<sup>2</sup> , the yields of each size fraction in Section 2 begin to exceed those in Section 1. The yields of large-sized materials in Sections 3 and 4 are generally higher than in small-sized materials. As the surface energy increases, more fine particles are deagglomerated under the movement of the sieve mat in Sections 3 and 4, and the yield of fine particles in the product begins to exceed that of

coarse particles, becoming the dominant product in Sections 3 and 4. For particles with a

supplement, is required to complete the depolymerization of agglomerated particles. The higher the adhesive force level, the closer the maximum yield section in the undersized product is to the discharging end. Moreover, the smaller the particle size, the more obvi-

The screening percentages of various size fractions in different Sections of VFFS with different surface energy levels are shown in Figure 10. In the products of Section 1, for the particles at the surface energy of 24, 28, 32 J/m2, and 36 J/m2, it can be seen that the screening percentages of various size fractions are significantly lower than that of other surface energy levels. Taking 4 mm particles as an example, the screening percentages of 4 mm particles are 21.93%, 15.41%, 9.00%, and 7.10%, respectively. The main effect of Section 1 is to promote the depolymerization of agglomerated particles. The longer the transporting distance of agglomerated particles, the better the depolymerization effect. It can be observed that for 4 mm particles under the case of the surface energy level of 20 J/m2, the screening percentage in Section 1 is 36.63%, in Section 2 increases to 54.89%, and the screening percentages in Sections 3 and 4 are 52.45% and 50.74%, respectively. When the cohesive force level continues to increase to 28 J/m2, the screening percentage in Section 1 is 15.41%, in Section 2 it is 38.08%, and increases to 48.23% and 48.89% in Sections 3 and 4, respectively. When the surface energy is 36 J/m2, the screening percentage in Section 1 is only 7.10%, and further increases to 18.19%, 32.24%, and 41.55%, respectively. Compared with the coarse particles, the surface energy level has a more significant effect on fine particles. After the depolymerization of fine particles, the screening percentage of fine parti-

*3.3. The Screening Percentage of Different Size Fractions of Different Sections* 

ous this phenomenon.

cles will be significantly improved.

coarse particles, becoming the dominant product in Sections 3 and 4. For particles with a higher level of adhesion, a longer movement distance, that is, a higher external energy supplement, is required to complete the depolymerization of agglomerated particles. The higher the adhesive force level, the closer the maximum yield section in the undersized product is to the discharging end. Moreover, the smaller the particle size, the more obvious this phenomenon.

#### *3.3. The Screening Percentage of Different Size Fractions of Different Sections*

The screening percentages of various size fractions in different Sections of VFFS with different surface energy levels are shown in Figure 10. In the products of Section 1, for the particles at the surface energy of 24, 28, 32 J/m<sup>2</sup> , and 36 J/m<sup>2</sup> , it can be seen that the screening percentages of various size fractions are significantly lower than that of other surface energy levels. Taking 4 mm particles as an example, the screening percentages of 4 mm particles are 21.93%, 15.41%, 9.00%, and 7.10%, respectively. The main effect of Section 1 is to promote the depolymerization of agglomerated particles. The longer the transporting distance of agglomerated particles, the better the depolymerization effect. It can be observed that for 4 mm particles under the case of the surface energy level of 20 J/m<sup>2</sup> , the screening percentage in Section 1 is 36.63%, in Section 2 increases to 54.89%, and the screening percentages in Sections 3 and 4 are 52.45% and 50.74%, respectively. When the cohesive force level continues to increase to 28 J/m<sup>2</sup> , the screening percentage in Section 1 is 15.41%, in Section 2 it is 38.08%, and increases to 48.23% and 48.89% in Sections 3 and 4, respectively. When the surface energy is 36 J/m<sup>2</sup> , the screening percentage in Section 1 is only 7.10%, and further increases to 18.19%, 32.24%, and 41.55%, respectively. Compared with the coarse particles, the surface energy level has a more significant effect on fine particles. After the depolymerization of fine particles, the screening percentage of fine particles will be significantly improved. *Minerals* **2021**, *11*, x FOR PEER REVIEW 12 of 15

**Figure 10.** Comparison of the screening percentages of various size fractions in Sections 1 (**a**), 2 (**b**), 3 (**c**), and 4 (**d**) of VFFS with different surface energy levels. ergy levels, the screening efficiency of particles increased with the increase in screening **Figure 10.** Comparison of the screening percentages of various size fractions in Sections 1 (**a**), 2 (**b**), 3 (**c**), and 4 (**d**) of VFFS with different surface energy levels.

Figures 11 and 12 show the screening efficiency and misplaced material of various size fractions in different sections and screen lengths of the VFFS with different surface energy levels. The particles shape in the simulation are spherical, so in the actual simulation process, no coarse particles enter the undersized products. The effective placement efficiency of the coarse particles is 100%, and the misplaced material of coarse particles is 0%. Therefore, the screening efficiency is equal to the effective placement efficiency of fine particles, and the total misplaced material is equal to the misplaced material of fine particles. For 8 mm particle screening, when the surface energy between particles is 0 J/m2, the screening efficiency in Section 1 reaches the maximum of 29.15%, and the total misplaced material is 39.50%. With the flow of material, the screening efficiency in Sections 1–4 decreases gradually. When the surface energy increases to 5 and 8 J/m2, the screening efficiency increases in Sections 1 and 2, and the total misplaced material decreases, which is mainly due to the surface energy between particles reducing the movement speed of particles and increasing the residence time of particles on the screen surface, thus increasing the screening efficiency. After the surface energy of 16 J/m2, the influence of particle agglomeration begins to be greater than particle velocity, and the screening efficiency starts to decrease. The screening efficiency of Section 2 begins to be greater than that of Section 1. In addition, the screening efficiency of Sections 3 and 4 are higher than those of the levels 0, 5, 8, and 12 J/m2. For the case of 24 J/m2, the maximum screening efficiency appears in Section 3, which is 30.54%. When the surface energy increases to 32 J/m2, the screening efficiency of Section 4 is the highest, which is 27.88%. For different surface en-

*Length* 

#### *3.4. The Screening Performance of Various Size Fractions in Different Sections and Screen Length*

Figures 11 and 12 show the screening efficiency and misplaced material of various size fractions in different sections and screen lengths of the VFFS with different surface energy levels. The particles shape in the simulation are spherical, so in the actual simulation process, no coarse particles enter the undersized products. The effective placement efficiency of the coarse particles is 100%, and the misplaced material of coarse particles is 0%. Therefore, the screening efficiency is equal to the effective placement efficiency of fine particles, and the total misplaced material is equal to the misplaced material of fine particles. For 8 mm particle screening, when the surface energy between particles is 0 J/m<sup>2</sup> , the screening efficiency in Section 1 reaches the maximum of 29.15%, and the total misplaced material is 39.50%. With the flow of material, the screening efficiency in Sections 1–4 decreases gradually. When the surface energy increases to 5 and 8 J/m<sup>2</sup> , the screening efficiency increases in Sections 1 and 2, and the total misplaced material decreases, which is mainly due to the surface energy between particles reducing the movement speed of particles and increasing the residence time of particles on the screen surface, thus increasing the screening efficiency. After the surface energy of 16 J/m<sup>2</sup> , the influence of particle agglomeration begins to be greater than particle velocity, and the screening efficiency starts to decrease. The screening efficiency of Section 2 begins to be greater than that of Section 1. In addition, the screening efficiency of Sections 3 and 4 are higher than those of the levels 0, 5, 8, and 12 J/m<sup>2</sup> . For the case of 24 J/m<sup>2</sup> , the maximum screening efficiency appears in Section 3, which is 30.54%. When the surface energy increases to 32 J/m<sup>2</sup> , the screening efficiency of Section 4 is the highest, which is 27.88%. For different surface energy levels, the screening efficiency of particles increased with the increase in screening length. When the surface energy is 8 J/m<sup>2</sup> , the screening efficiency of VFFS is the highest, which is 72.52%, and the total misplaced material is 23.81%. *Minerals* **2021**, *11*, x FOR PEER REVIEW 13 of 15 length. When the surface energy is 8 J/m2, the screening efficiency of VFFS is the highest, which is 72.52%, and the total misplaced material is 23.81%.

**Figure 11.** (**a**) Screening efficiency of different sections of the screen with different surface energy levels, (**b**) screening efficiency of different screen lengths with different surface energy levels. **Figure 11.** (**a**) Screening efficiency of different sections of the screen with different surface energy levels, (**b**) screening efficiency of different screen lengths with different surface energy levels.

**Figure 12.** (**a**) Misplaced material of the different sections of the screen with different surface energy levels, (**b**) misplaced

The following conclusions can be drawn from the above research.

of VFFS is realized by setting the multi-point rigid motion of the sieve mat.

end (Sections 3 and 4), the screening percentages of the material are greater.

(1) Due to the amplitude at each point on the sieve mat changing periodically, the motion can be transformed into a function form by the Fourier series. The DEM simulation

(2) When the surface energy level is in the range of 0 to 8 J/m2, the particle velocity in the feeding end region (Sections 1 and 2) dominates the movement behavior of particles passing through the screen. In Sections 1 and 2, the particle movement speed decreases, which increases the contact time between the particles and the screen surface, increasing the screening percentages. When the level of surface energy continues to increase, more fine particles are agglomerated together, which increases the screening difficulty. The effect of particle agglomeration in the feeding end is greater than its movement speed, and the screening percentages of each particle size in the feeding end have been reduced. Agglomerated particles need a certain transporting distance to deagglomerate. The stronger the surface energy between particles, the greater the distance the particles need to deagglomerate. Therefore, for the case of a higher surface energy level, close to the discharging

material of different screen lengths with different surface energy levels.

**4. Conclusions** 

which is 72.52%, and the total misplaced material is 23.81%.

**Figure 12.** (**a**) Misplaced material of the different sections of the screen with different surface energy levels, (**b**) misplaced material of different screen lengths with different surface energy levels. **Figure 12.** (**a**) Misplaced material of the different sections of the screen with different surface energy levels, (**b**) misplaced material of different screen lengths with different surface energy levels.

#### **4. Conclusions 4. Conclusions**

efficiency of different screen lengths with different surface energy levels.

The following conclusions can be drawn from the above research. The following conclusions can be drawn from the above research.

(1) Due to the amplitude at each point on the sieve mat changing periodically, the motion can be transformed into a function form by the Fourier series. The DEM simulation of VFFS is realized by setting the multi-point rigid motion of the sieve mat. (1) Due to the amplitude at each point on the sieve mat changing periodically, the motion can be transformed into a function form by the Fourier series. The DEM simulation of VFFS is realized by setting the multi-point rigid motion of the sieve mat.

length. When the surface energy is 8 J/m2, the screening efficiency of VFFS is the highest,

(2) When the surface energy level is in the range of 0 to 8 J/m2, the particle velocity in the feeding end region (Sections 1 and 2) dominates the movement behavior of particles passing through the screen. In Sections 1 and 2, the particle movement speed decreases, which increases the contact time between the particles and the screen surface, increasing the screening percentages. When the level of surface energy continues to increase, more fine particles are agglomerated together, which increases the screening difficulty. The effect of particle agglomeration in the feeding end is greater than its movement speed, and the screening percentages of each particle size in the feeding end have been reduced. Agglomerated particles need a certain transporting distance to deagglomerate. The stronger the surface energy between particles, the greater the distance the particles need to deagglomerate. Therefore, for the case of a higher surface energy level, close to the discharging (2) When the surface energy level is in the range of 0 to 8 J/m<sup>2</sup> , the particle velocity in the feeding end region (Sections 1 and 2) dominates the movement behavior of particles passing through the screen. In Sections 1 and 2, the particle movement speed decreases, which increases the contact time between the particles and the screen surface, increasing the screening percentages. When the level of surface energy continues to increase, more fine particles are agglomerated together, which increases the screening difficulty. The effect of particle agglomeration in the feeding end is greater than its movement speed, and the screening percentages of each particle size in the feeding end have been reduced. Agglomerated particles need a certain transporting distance to deagglomerate. The stronger the surface energy between particles, the greater the distance the particles need to deagglomerate. Therefore, for the case of a higher surface energy level, close to the discharging end (Sections 3 and 4), the screening percentages of the material are greater.

end (Sections 3 and 4), the screening percentages of the material are greater. (3) The screening efficiency increases with the increase in screen length for different surface energy levels. When the surface energy is 8 J/m<sup>2</sup> , the screening performance of VFFS is better, with a screening efficiency of 72.52% and a total misplaced material of 23.81%.

Since the shape of the screen apertures of the elastic sieve mat is a straight slot, in the actual screening process, there is a situation that the strip particles pass through the screen. In future work, the influence of the shape characteristics of the particles on their movement and screening performance should be considered. Furthermore, we still need to carry out some full-scale screening experiments of wet particles based on the experimental VFFS.

**Author Contributions:** Conceptualization, C.Y. and X.W.; methodology, C.Y. and X.W.; software, C.Y.; validation, C.Y., R.G. and X.W.; formal analysis, C.Y.; investigation, C.Y.; resources, X.W.; data curation, C.Y.; writing—original draft preparation, C.Y. and R.G.; writing—review and editing, X.W.; visualization, X.W.; supervision, X.W.; project administration, X.W.; funding acquisition, X.W. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by "the Fundamental Research Funds for the Central Universities" (No. 2020YJSHH17) (No. 2021YJSHH32).

**Data Availability Statement:** All data and models generated or used during the study appear in the submitted article.

**Acknowledgments:** The authors would like to thank the company of TianGong technology for its support in enabling this research.

**Conflicts of Interest:** The authors declare no conflict of interest.
