That is, particles accelerated by the impeller collide with and rub against other particles and liners in the crushing chamber[24]. Therefore, the essence of stone powder wall shaping process is multiple times of crushing of particles and aggregates with small energy. This paper will focus on the stone powder wall formation, its shaping mechanism, the establishment and validation of the theoretical contact model, and based on all these, discusses the design of the stone powder wall segments.

2. Stone Powder Wall Formation Mechanism

When the crusher is working, the stones enter the machine through the feeder and pass through the divider to the rotor chamber and contact with the high-speed rotating impeller. After being accelerated by the centrifugal force in the impeller, they are ejected from the three flow channels evenly distributed around the rotor chamber at high speed. Part of them are piled up in the crushing chamber by the action of the particle liner to form a physical solid wall. And the particles flying out from the impeller flow passage collide with them again, achieving the shaping effect, hence the stone powder wall (or stone wall) mechanism.

Figure 3 shows two types of contact for sand shaping: rock-metal contact and rock-rock contact. The gray particles and red particles are the first and second group of particles ejected from the rotor, respectively. The gray particles hit the sensor counter inside the impact plate and collide with the red particles again after rebounding from the contact point of the impact plate. The collision of the particles with the impact plate and the chamber forms a rock-metal contact, while the collision between the particles forms a rock-rock contact. A study showed that rock-rock contact is the main way of particle formation[25].

According to Emerson et al[25]. (Figure 4), the inter-particle collision energy is higher in the rock-rock contact mode than in the rock-metal contact mode. In addition, the collision between rock bed and sand not only crushes the particles, but also protects the interior of the crushing chamber from wear and tear, thus extending the service life of the machine.

As shown in Figure 5, most current crushers are equipped with a smooth impact plate with pits on the surface after a period of collision with the particles. These pits deposit rock powders, forming a “rock bed”. As shown in Figure 6(a), the rock particles collide with the impact plate and the rock powder accumulates in the pits, forming a rock bed that not only crushes the particles, but also protects the interior of the crushing chamber from wear and tear. As shown in Figure 6(b), the pits deposit rock particles, and although it accelerates the wear of the impact plate, the rock layer will protect it from abrasion, thus still extending the service life of the machine.

3. Stone Powder Wall Shaping Mechanism

As the impact velocity of particles keeps increasing, the energy gained by particles on the cyclotron accelerator is gradually elevated. The kinetic energy is determined by K_e = 0.5 mv², where m is the mass of particles and v is the velocity of particles.

Residence Time: the length of time that a particle exists in the simulation. “Residence Time = Simulation Time - Particle Creation Time”.

Normal force: it is generated when two objects are in direct contact. It is always perpendicular to the object that exerts the force. The force depends on the contact model, which is usually a spring force with a damping component.

3.1 Analysis and comparison of the data of “rock-rock” and “rock-metal” models

As shown in Figure 7(a), the particles accelerate through the cyclotron accelerator and impact on the flat steel plate, and are ejected from the impact plate to the crushing chamber. With the increase of working time, a layer of stone powders will form on the surface of the flat steel plate, as shown in Figure 7(b).

When the simulation is done on a flat impact plate, as shown in Figure 8, the residence time of particle-stone wall is larger than that of particle-metal in the same group, thus the stone powder wall has a damping effect on the particles. Both have decreasing residence time as the impact velocity of particles increases. However, the range of variation of particle-metal residence time is small, proving that the overall velocity change has a less effect on particle-metal residence time than that of the particle-stone wall. By comparing the simulation data, it is found that under the impact load, the energy of “rock-metal” model is 3 to 8 times larger than that of “rock-rock” model. The stone powder wall stores and decomposes impact energy, having an obvious elastic damping effect.

Figure 9 shows the particle energy when simulated on a flat impact plate, which further verified a significant elastic damping effect of the stone powder wall.

As shown in Figure 10(a), the particles accelerate through the cyclotron accelerator and impact on the arc steel plate, and are ejected from the impact plate once or twice to the crushing chamber. With the increase of working time, a layer of stone powders will form on the surface of the arc steel plate, as shown in Figure 10(b).

As shown in Figure 11, in the simulation on the arc impact plate, the particle-steel plate residence time is smaller than the particle-stone wall residence time at a low to medium speed of 30-50 m/s; between 50-70 m/s, the particle-steel plate residence time gradually becomes larger than the particle-stone wall residence time. However, the particle-stone wall residence time decreases gradually with the increase of impact velocity, and the variation ranges of particle-stone wall residence time and particle-steel plate residence time are both small, proving that the arc has a positive effect on prolonging the particle residence time.

The same impact velocity is applied to the particles so that the initial energy of the particles is the same, and as shown in Figure 12, the maximum value of energy carried by the particles increases with the increase of velocity. The energy of particle- stone wall model decreases more slowly and the frequency of the particle energy exchange is much higher than that of the particle-steel plate model at the same velocity.

3.2 Particle contact model in shaping process

Figure 13 shows the elastic-plastic contact model formed by the stone particles when they act on the stone wall.

As shown in Figure 14, r₁ and r₂ are the diameters of particle 1 and particle 2, respectively, and B is the width of the contact surface, the expression of which is defined as:

The non-dimensional shape parameter ${\displaystyle \sigma }$can be determined by comparing the data results of the internal friction angle parameter obtained from numerical simulations and the particle shape test with the target test results. In Figure 14, a normal basic unit is fixed on the contact surface, which consists of a spring and a damper connected in parallel and is then connected in series with the separator. The stiffness of the spring is kn (N/m²), reflecting the elasticity of particle contact point and the damping coefficient ɳ_n of the damping generator. To achieve energy dissipation, a separator is used to simulate the absence of tension (N/m) when the cement fails. Similarly, a tangential basic unit is defined, which consists of a spring and a damper connected in parallel and is connected in series with the slider. The difference between the tangential basic unit and the normal basic unit is that the spring and damper in the former are on the tangent and the slider replaces the separator in the normal unit. The slider is used to simulate plastic sliding at the contact point under a certain shear strength, which is determined by the normal pressure at a specified point and the Mohr - Coulomb criterion. The result shows that the normal basic unit has a cementing effect at the contact point, which resists inter-particle torsion, while the tangential basic unit does not have this effect.

Assume that the thickness of the contact area is constant and normal and tangential basic units are continuously distributed on the contact surface. The overall normal stiffness K_n and tangential stiffness K_s (N/m) of particles, the damping coefficient µ_n, and the stiffness of normal basic elements k_n and the stiffness of tangential basic elements k_s satisfy the following requirements:

If a finite relative rotation angle θ_r is specified, denote c₁ c or c₂ c by B. The cementing between the particles will produce a force couple that will be transferred from particle 1 to particle 2.

Derive the relationship between the force couple M and θ_r with small rotation angles.

The first step is to consider the absence of sliding, which corresponds to the linear elastic phase. As shown in Figure 15(a), with a constant normal force F_n, particle 1 is rotated counterclockwise by a small angle θ_r relative to particle 2. The concentration of the normal contact force p is linearly distributed along the contact area, as shown in Figure 15(b), and the load intensity at the left p^l and the load intensity at the right p^r are expressed as

${\displaystyle {\overline {p}}}$ is the average intensity of normal forces, defined as

The distribution of the normal force on the contact surface causes the bending moment M:

K_m is the rolling stiffness, in N·m.

K_m depends on δ, r and K_n. If K_m is a constant, then M increases linearly with it.

The second step is to consider the partial sliding, which corresponds to the plastic phase. The plastic mechanism is defined by the separation of the separator. According to Figure 16(a), when particle 1 is rotated counterclockwise relative to particle 2 with a constant normal force F_n, its basic contact unit at the right edge and the separator fail and crack. With the increase of θ, the damage of normal elements and the cracking of the separator continue to move inwards from the edge, so that p at the corresponding point is 0 and is redistributed linearly, as shown in Figure 16(b). Let l be the contact width and P_min (P_max) indicate the minimum (maximum) normal contact force located at the rightmost (leftmost) end of the contact area (the width is l).

The bending moment M and the normal force F_n are:

When l = B, the normal contact point is not damaged and the separator does not separate, i.e., there is no plasticity, and the critical values of the relative rotation angle θ_r and bending moment M are θ⁰ _r and M₀, respectively:

Put the equation into the following equation, the rolling bending moment with ${\textstyle l\leq B}$ can be obtained:

Once the above process is completed, the linear elastic and plastic phases represented by the equations (5) and (11) are combined to establish the rolling contact model. As shown in Figure 16(c), theoretical and numerical analysis for plastic process is more difficult. Two types of simple models for stone powder wall contact are proposed.

Type 1: Elastic-plastic model

Type 2: Elastic-brittle model

When δ= 0, the contact model is a simplified model in the classical particle discrete element method. The dashed lines in Figure 16(c) indicate type 1 and type 2, respectively, where the former is more like the rolling contact model and represents the elastic-plastic action that approximates the Mohr - Coulomb criterion. Type 2 is used to simulate particle crushing, in which particle crushing is achieved by the disappearance of rolling caused by the sudden disappearance of contact width.

Based on the stone powder wall shaping contact model, the overall shear contact model is made to cover the continuously distributed tangential basic units and control the relevant tangential motion on the shear surface; the overall normal contact model is obtained from the continuously distributed normal basic units, which determines the normal motion of the contact surface on the contact surface.

The mechanical contact model of stone powder wall shaping is now derived, but there is a certain correlation between stone powder wall shaping and the particles’ angle of incidence, so based on the above conclusion, an incidence angle factor I is added to improve the contact model of stone powder wall shaping.

3.3 The verification of the contact models

By simulating the incidence angles at different positions of the arc plate (Figure 17), particles can be ejected at a certain position for several times, and hence meeting the demand of the shaping mechanism.

The impact simulation tests were conducted on the arc impact plate at velocities of 30, 50, and 70 m/s, respectively, as shown in Figure 18(a). At the low impact velocity of 30 m/s, the residence time of particles of 20 mm-diameter was the longest, and showed an increasing to decreasing trend. The residence time of particles of 20 mm-diameter in the particle-steel group was the smallest at the impact velocity of 50 m/s, as shown in Figure 18(b). At the impact velocity of 70 m/s, the overall residence time did not fluctuate much, and the residence time of the steel plate group was larger than that of the stone wall group, as shown in Figure 18(c).

The impact simulation tests were conducted on the flat impact plate at velocities of 30, 50, and 70 m/s, respectively, as shown in Figure 19. The analysis showed that the residence time of the steel plate group was less than that of the stone wall group. As shown in Figure 19(a), at low impact velocities of 30 and 50 m/s, the difference of residence time was not great, and the residence time stayed stable as the particle size increased. As shown in Figure 19(b and c), the difference between the residence time of the particles of the steel plate group and the stone wall group is larger at the medium and high impact velocities of 50 and 70 m/s.

Particle shaping is achieved by multiple times of crushing between particles and shaping plates with small energy, as shown in Figure 20. The arc impact plate can effectively increase the time and chance of collision. The mechanical contact model of stone powder wall shaping is now derived, but there is a certain correlation between stone powder wall shaping and the particles’ angle of incidence, so based on the above conclusion, an incidence angle factor I is added to improve the contact model of stone powder wall shaping. Under the impact velocity of 30 m/s, the contact model of particles is mainly type 1, that is, the elastic-plastic model, and the incidence angle factor I ranges reasonably from 0.9 to 1.1. Under the impact velocity of 50-70 m/s, which is conventional, the contact model of particles is mainly type 2, that is, the elastic-brittle model, and the incidence angle factor I ranges from 1.0 to 1.2.

According to Figure 21 shows the effect of average shear force and average extrusion force on the crushing ratio. When only a single force of the two are at its maximum or minimum, the crushing ratio of rock particles will be greatly reduced, while when the two forces act simultaneously and are balanced, the crushing effect is the best.

Based on the cross-sectional scanning electron microscope (SEM) images of the crushed particles (Figure 22), and by comparison, the crushing mechanisms of three typical rocks were analyzed.

According to the comparison of sand aggregates before and after crushing, shown in Figure 22, and the SEM image in Figure 23, it is found that in the process of crushing, when the acceleration exerted by the main impeller on the rock particles causes the impact stress on the particles more than what they can withstand, the particles are prone to crushing. The crushed samples are mostly limestone, and the crushed surface gets thicker from the outer edge, which is generally of arc shape, to the center. The crushed surface is not neat but presents a step-like shape. With the same external factors and the same initial state, larger particles need greater crushing energy, and the crushing effect is better. With the same mass, the number of elongated shapes crushed out of larger particles is also larger.

As shown in Figure 24, the fracture surface has the same fracture along the crystal surface direction, while the overall damage is not very significant, so it can be judged that the particle shaping is mainly achieved by “small energy” of shear force. Based on the above analysis, the essence of the stone powder wall shaping mechanism is “multiple times of small energy crushing”. The contact models involve:

Type 1: Elastic-plastic model

Type 2: Elastic-brittle model

The established mathematical models can reasonably reflect the change of rolling bending moment of sand aggregate under different contact models. The SEM image are in agreement with the theory to a certain extent, indicating that the mathematical models are reasonable.

4. Experiment verification

4.1 lntroduction of experimental prototype

Figure 25 Schematic diagram of test prototype

4.2Analysis of experimental process and results

The prototype is tested on the self-designed impact-crushing simulation test machine. Since area I is a stone -walled structure in the optimization model, since the angle control is not yet designed in this paper, 3D printing technology is used to print the shape in a manner similar to the strength of the rock wall. Figure 27 below shows the shape of the printed cavity, and the strength of the stone wall is achieved by adjusting the fill rate of the 3D components.

Figure 4.3 3D printed stone wall shape

The surface is then coated with a layer of stone beating raw materials to perform simulation of the surface contact parameters, which can be seen in Figure 28 (limestone, pebble and granite) from the left to the right.

To simulate the different contact properties of different materials, we coated the 3D printed specimen with a certain particle thickness of stone wall material in order to realize the actual contact parameters of the particles. Since the influence of stone wall material thickness on the test cannot be ascertained, to test the influence of stone wall thickness, we designed an orthogonal test set. Select the appropriate orthogonal array based on the number of factors and levels. In most cases the number of factors less than or equal to the number of columns of the orthogonal array, and the number of factor levels should be consistent with the number of levels in the orthogonal table. Assuming the above conditions are satisfied, the orthogonal array with smaller specifications should be chosen. We adopt the three-factor, three- level test for this mud wall thickness test. Due to its three-tiered nature, L₉(3⁴) is adopted. Table 1 shows the factor-level table, and Table 2 shows the trained test table and the final data.

The results of the Analysis of Variance Table 3 show that the impact angle and impact velocity have a large influence on the particle break up ratio, but the thickness of the mud wall of the test target has little impact.

For the purposes of this paper, in order to adjust the use of the test prototype, firstly, to change the shape of the new counter plate, and repeating the test of each contraption plate four times, and calculating the materials broken respectively. Insert this optimized component into a proof machine, the relative position is illustrated in Figure 31, The left side is the optimized cavity shape, the left side is the sand making machine, and the particle impact velocity is adjusted by controlling the rotational speed of the rotor. Figure 32 shows a photograph of the stone wall after impact crushing. The particles can be seen to impact in to the I region of the stone wall, and the stone wall in I area is broken, which is consistent with the definition of the crushing function in I area in the test simulation. A large amount of energy is absorbed into the stone wall and the particles are broken. The material then ejected upwards to Zone II, and the particles from the stone wall on the surface were disrupted by friction, and eventually flew along Zone III of the rock wall with only minor

Figure 31 Installation of test prototype

As can be seen in Table 4, only the advantages and disadvantages of the four shock plate shapes are compared by different methods, focusing on how they differ from the original shock plate shapes. Thus, depending on the grinding ratio, the grinding performance of the three impact plates can be compared.

By means of experimental verification, the structural parameters that can make the best crushing performance of the new counter plate are obtained: when the angle of the I area is 35 degrees, area angle II is 100 degrees, and area angle III is 10 degrees, the shape of the shaped particles is relatively optimum, and the value of the performance index can be improved by about 8%-10% compared to the counter plate commonly used in the industry. These results provide strong support for the machine-made sand shaping mechanism.

5. Conclusions

The following conclusions were obtained:

(1) The numerical analysis of “rock-rock” and “rock-metal” models verifies that the energy of “rock-metal” model under impact is 3 - 8 times greater than that of “rock-rock” model. The stone powder wall plays a damping role in the impact process, which effectively reduces the impact energy.

(2) The residence time of particle-stone wall model is about 0.002s longer than that of particle-metal model at low to medium speed of 30-50 m/s on the arc impact plate, and about 0.01s longer than that of particle-metal model at the speed of 50-70 m/s. The comparison between flat and arc impact plates shows that the arc impact plate has a positive effect on particle residence.

(3) Mathematical models of stone powder wall particles shaping mechanism were established, and the incidence angle factor is considered to analyze particles’ tangential and normal motion. At low impact velocity, it is mainly the elastic-plastic model, while at high impact velocity, it is mainly the elastic-brittle model. It is concluded that the mechanism of stone powder wall shaping is multiple times of low-energy shear crushing in nature.

Data Availability

The data will be provided upon request, and all of them can be used without any conflict of interest.

Acknowledgements

This project is funded by Fujian University of Technology and is a scientific and technological project of Fujian University of Technology. Project number is (GY-Z220201). And the Youth Project of Fujian Natural Science Foundation, with the grant number of 2020J05180.

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