# Stirring Performance Analysis Based on the Influence of Mechanics and Stirred Mill Environment

Authors: Xin Jiang
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The stirring performance analysis has great research significance. In this research, we investigate mechanics press and performance in relative frequencies and energies due to interaction forces between particles. The collisions related to media and particle were obtained. Density influence of the model was obtained. The concentration has a relationship with the air and fluid. We investigate air condition at the center of the stirring shaft concentration. In the center of the stirring shaft, air condition velocity vector is relatively smaller than fluid condition. At the collisions between media or particle and the wall, average collision energy in density of air condition was the largest in particle and wall, and average collision energy in different densities was obtained. Air condition has the largest total media and largest total particle average collision energy. We obtained a reasonable stirring model, a suitable stirring density, and working environment for better stirring performance. This research has great significance for the design of mixing products and the research of working environment.


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## Body

## 1. Introduction

With the progress and development of the modern industry, the performance of industrial equipment and products also has higher requirements [1–5]. Stirring performance analysis is very meaningful [6–9]. The forces between particles can also affect stirring properties. The different mixing environments and conditions in which the mixing particles are present also have a significant impact on the mixing results [10–12]. Particles are important in a variety of industrial applications, including engineering and energy processes. This powerful numerical simulation has received a lot of attention in the last few decades as computers have become very popular [13–15]. It can play an important role in analysis and in the optimal design of processes.The dynamical behavior continues to receive great attentions because of its relevance to a wide range of applications in industries [16–18]. It is of great significance to study the mechanical and pressure interactions between particles. When the particles are being mixed, two types of particle-particle interactions are important, where particles generate energy when they collide. In practice, when particles interact, they deform and generate internal stresses. Most of the deformation is elastic, but part of it is due to plastic or viscous effects. These interactions are described by the field of contact mechanics. Stirring is a complex process, but the complexity at the bulk level is the collected outcome of the interactions among particles, materials, and devices [19–21]. Although this information is difficult to obtain from experiments based on the discrete element method, it is easy to determine the energy distribution based on well-established contact mechanics. Mishra and Rajamani [22] were the first to use DEM to model the electrical movement in a tumbling mill. Since then, DEM have been used to model a wide range of grinding equipment, including roller mills, planetary mills, and stirred mills. Different forms of energy such as collision and dissipated energy, which can be obtained from the DEM, are used in the literature to characterize the damaged particles, and the frequency of particle collision can well reflect the effect of particle collision [23–26]. For example, Datta and Rajamani [27] compared the dissipated energy of the DEM simulation with the particles. The limitations of the study should also be considered in the future, such as the damage and failure in the form of crack propagation, and the damage has significant influence on the structural strength and durability. The discovery and synthesis of new materials often lead to technological breakthroughs [28–31]. The DEM method can be used to simulate a wide variety of granular flow and mechanics situations. This method allows a more detailed study of the dynamics of powder flows. DEM research has significantly improved our knowledge on particle collision.In this research, we investigate mechanics press and performance in relative frequencies and energies due to interaction forces between particles.The collisions related to media and particle were obtained. Density influence of the model was obtained. We investigated air condition of the stirring shaft concentration. This research makes contribution to knowledge in the associated field, and we combined different environments and structures for analysis. We obtained a reasonable stirring model, a suitable stirring density, and working environment for better stirring performance.


## 2. Numerical Modeling

### 2.1. Simulation Method

In numerical modeling, we consider the force and discrete element theory [32–35]. Modeling of contact forces can be calculated. The force F⟶ can be divided into the contact force and fluid force as follows:(1)F⟶=f⟶C+f⟶D.The contact forcef⟶C is further divided into the normal force f⟶Cn and tangential force f⟶Ct. Cundall and Strack model for the forces is as follows::(2)f⟶Cn=−kd⟶n−ηv⟶n,v⟶n=v⟶r⋅n⟶n,f⟶Ct=−kd⟶t−ηv⟶t,v⟶t=v⟶r−v⟶n.In the equation,d⟶n and d⟶t are the particle displacements in the normal and tangential directions, respectively, v⟶r is the relative velocity, k is the stiffness of the spring, and η is the coefficient of viscous dissipation. The contact forces acting on the particles or between the particles and the wall are modelled using the Voigt model. Figure 1 shows Voigt model.Figure 1
Voigt model in DEM simulation.In the simulations, the collision energy of the medium is calculated to assess the energy provided by the particles to the sample. Various models are proposed to calculate the collision energy. The collision energyE is defined as the kinetic energy at the time of collision, and the collision energy is calculated as follows: (3)E=12m1m2m1+m2v2.In the equation,m1 and m2 refer to the mass of two colliding objects and v is the relative speed between two objects. Some collision-energy distributions were calculated for the particle-particle and particle-wall interactions.We can see the interactions between particles in Figure2. In most applications, two types of particle-particle interactions are important, namely, collisions and persistent interactions when particles are stacked. In practice, when particles interact, they deform and generate internal stresses.Figure 2
Interactions between particles.The coupling of discrete element methods with computational fluid dynamics is a widely used method for modeling particle-fluid interactions. Figure3 shows fluid and particle coupling. The collisions are very fast in fluid flow. Therefore, the effect of the collision is important. These can be characterized by normal and tangential coefficients of restitution. The detailed deformation of the particles is also important.Figure 3
Fluid and particle coupling.


### 2.2. Simulation Condition and Model

We carry out relevant evaluations through numerical simulation. Table1 lists the parameters used in the simulations.Table 1
Simulation parameters.
Number of calculations (−)500000Time increment (s)1.0 × 10−5Media diameter (m)0.002Media density (kg/m3)7770Particle density (kg/m3)2260Coefficient of restitution (−)0.36Coefficient of friction (−)0.52Rotation speed (rpm)300We consider the numerical simulation in the vacuum conditions. Table2 lists the parameters used in the vacuum conditions.Table 2
Vacuum conditions.
VacuumValueDensity (kg/m3)1.225 × 10−3Viscosity (Pa·s)1.79 × 10−5We carry out relevant numerical simulation in the air conditions. Table3 lists the parameters used in the air conditions.Table 3
Air conditions.
AirValueDensity (kg/m3)1.225Viscosity (Pa·s)1.79 × 10−5In model design, different models are subjected to different forces and other factors during the mixing process, and these can affect the performance of different model results. In this study, we consider the nature of the model, the transverse, and other cross-sections. We optimize the models and analyse the characteristics of the different model and compare some physical quantities of models. Figure4 shows the models.Figure 4
Simulation model.


## 2.1. Simulation Method

In numerical modeling, we consider the force and discrete element theory [32–35]. Modeling of contact forces can be calculated. The force F⟶ can be divided into the contact force and fluid force as follows:(1)F⟶=f⟶C+f⟶D.The contact forcef⟶C is further divided into the normal force f⟶Cn and tangential force f⟶Ct. Cundall and Strack model for the forces is as follows::(2)f⟶Cn=−kd⟶n−ηv⟶n,v⟶n=v⟶r⋅n⟶n,f⟶Ct=−kd⟶t−ηv⟶t,v⟶t=v⟶r−v⟶n.In the equation,d⟶n and d⟶t are the particle displacements in the normal and tangential directions, respectively, v⟶r is the relative velocity, k is the stiffness of the spring, and η is the coefficient of viscous dissipation. The contact forces acting on the particles or between the particles and the wall are modelled using the Voigt model. Figure 1 shows Voigt model.Figure 1
Voigt model in DEM simulation.In the simulations, the collision energy of the medium is calculated to assess the energy provided by the particles to the sample. Various models are proposed to calculate the collision energy. The collision energyE is defined as the kinetic energy at the time of collision, and the collision energy is calculated as follows: (3)E=12m1m2m1+m2v2.In the equation,m1 and m2 refer to the mass of two colliding objects and v is the relative speed between two objects. Some collision-energy distributions were calculated for the particle-particle and particle-wall interactions.We can see the interactions between particles in Figure2. In most applications, two types of particle-particle interactions are important, namely, collisions and persistent interactions when particles are stacked. In practice, when particles interact, they deform and generate internal stresses.Figure 2
Interactions between particles.The coupling of discrete element methods with computational fluid dynamics is a widely used method for modeling particle-fluid interactions. Figure3 shows fluid and particle coupling. The collisions are very fast in fluid flow. Therefore, the effect of the collision is important. These can be characterized by normal and tangential coefficients of restitution. The detailed deformation of the particles is also important.Figure 3
Fluid and particle coupling.


## 2.2. Simulation Condition and Model

We carry out relevant evaluations through numerical simulation. Table1 lists the parameters used in the simulations.Table 1
Simulation parameters.
Number of calculations (−)500000Time increment (s)1.0 × 10−5Media diameter (m)0.002Media density (kg/m3)7770Particle density (kg/m3)2260Coefficient of restitution (−)0.36Coefficient of friction (−)0.52Rotation speed (rpm)300We consider the numerical simulation in the vacuum conditions. Table2 lists the parameters used in the vacuum conditions.Table 2
Vacuum conditions.
VacuumValueDensity (kg/m3)1.225 × 10−3Viscosity (Pa·s)1.79 × 10−5We carry out relevant numerical simulation in the air conditions. Table3 lists the parameters used in the air conditions.Table 3
Air conditions.
AirValueDensity (kg/m3)1.225Viscosity (Pa·s)1.79 × 10−5In model design, different models are subjected to different forces and other factors during the mixing process, and these can affect the performance of different model results. In this study, we consider the nature of the model, the transverse, and other cross-sections. We optimize the models and analyse the characteristics of the different model and compare some physical quantities of models. Figure4 shows the models.Figure 4
Simulation model.


## 3. Model Grinding Performance

By design suitable model, we make grinding simulation. Figure5 shows the model grinding with particles, and we can obtain the related particle collision frequency and energy.Figure 5
Model grinding.Through the stress intensity theory, high fluid velocity gradients in tangential direction appear especially in the vicinity of the stirrer. These zones with high velocity gradients are zones of high energy density in which most of the actual comminution process takes place. In these zones, grinding media with different velocities collide with each other and lose part of their kinetic energy. If there are product particles captured between the grinding media, part of the kinetic energy is transferred from the grinding media to the product particles.We can try to get some pressure-related results, like solid pressure. Figure6 shows different solid pressures mainly concentrated near the mixing shaft.Figure 6
Model solid pressure.Different models leave different mixing spaces. Pressure will affect the flow rate of particles. Some zones with high-velocity gradients are zones of high concentration. In these zones, grinding media with different velocities collide with each other. They will produce better stirring characteristics.


## 4. Results and Discussion

### 4.1. Comparison of Collision Energy

We make comparison of collision energy. The rate of the reaction depends on the frequency, and frequency influences the collision energy and stirring characteristics. Figure7 shows the comparison of frequency.Figure 7
Comparison of frequency.In the comparison of frequency, particle-media maximum frequency of air is larger. The density of 1.225 × 10−3 kg/m3 around −6 J curve is smoother than air condition. The frequency of particle-media affects the stirring performance. Particle-total maximum frequency is much larger. The overall trend between the energies is similar, which makes better stirring.


### 4.2. Comparison of Collision Energy

We make a comparison of model vacuum and air condition collision energy.Density of 1.225E - 04 had the largest number of frequency between total media (Figure8(a)), particle, and particle (Figure 8(f)). Density of 1.225E - 03 had the largest number of frequency between media and the wall (Figure 8(c)). Density of 1.225E - 02 had the largest number of frequency between total particle (Figure 8(b)), particle, and the wall (Figure 8(d)). Air condition had the largest number of frequency between media and media (Figure 8(e)) and media and particle (Figure 8(g)).Figure 8
Comparison of different condition frequencies. (a) Total media, (b) total particle, (c) media-wall, (d) particle-well, (e) media-media, (f) particle-particle, and (g) media-particle.
(a)(b)(c)(d)(e)(f)(g)Figure 9
Comparison of average collision energy.We make a comparison of average collision energy of different conditions. Figure9 shows the comparison of average collision energy.Table4 shows some detailed comparison of average collision energy.Table 4
Average collision energy of different densities.
1.225E - 021.225E - 031.225E - 04AirAverage collision energy mtotal (J/s)6.515E - 066.535E - 066.411E - 067.844E - 06Average collision energy ptotal (J/s)1.658E - 071.661E - 071.656E - 072.509E - 07At the collisions between media or particle and the wall, average collision energy in density of air condition was the largest in particle and wall, and average collision energy in the density of 1.225E - 03 was the smallest in media and wall. Air condition has the largest total media and largest total particle average collision energy.


## 4.1. Comparison of Collision Energy

We make comparison of collision energy. The rate of the reaction depends on the frequency, and frequency influences the collision energy and stirring characteristics. Figure7 shows the comparison of frequency.Figure 7
Comparison of frequency.In the comparison of frequency, particle-media maximum frequency of air is larger. The density of 1.225 × 10−3 kg/m3 around −6 J curve is smoother than air condition. The frequency of particle-media affects the stirring performance. Particle-total maximum frequency is much larger. The overall trend between the energies is similar, which makes better stirring.


## 4.2. Comparison of Collision Energy

We make a comparison of model vacuum and air condition collision energy.Density of 1.225E - 04 had the largest number of frequency between total media (Figure8(a)), particle, and particle (Figure 8(f)). Density of 1.225E - 03 had the largest number of frequency between media and the wall (Figure 8(c)). Density of 1.225E - 02 had the largest number of frequency between total particle (Figure 8(b)), particle, and the wall (Figure 8(d)). Air condition had the largest number of frequency between media and media (Figure 8(e)) and media and particle (Figure 8(g)).Figure 8
Comparison of different condition frequencies. (a) Total media, (b) total particle, (c) media-wall, (d) particle-well, (e) media-media, (f) particle-particle, and (g) media-particle.
(a)(b)(c)(d)(e)(f)(g)Figure 9
Comparison of average collision energy.We make a comparison of average collision energy of different conditions. Figure9 shows the comparison of average collision energy.Table4 shows some detailed comparison of average collision energy.Table 4
Average collision energy of different densities.
1.225E - 021.225E - 031.225E - 04AirAverage collision energy mtotal (J/s)6.515E - 066.535E - 066.411E - 067.844E - 06Average collision energy ptotal (J/s)1.658E - 071.661E - 071.656E - 072.509E - 07At the collisions between media or particle and the wall, average collision energy in density of air condition was the largest in particle and wall, and average collision energy in the density of 1.225E - 03 was the smallest in media and wall. Air condition has the largest total media and largest total particle average collision energy.


## 5. Conclusions

In this research, a reasonable stirring model was designed. We investigated mechanics press and performance in relative frequencies and energies due to interaction forces between particles. The collisions between media or particle and the wall and average collision energy were obtained. Density influence of the model was obtained.The concentration has a relationship with the air and fluid. About sample, air condition at the center of the stirring shaft concentration is relatively larger. In the center of the stirring shaft, air condition velocity vector is relatively smaller than fluid condition.At the collisions between media or particle and the wall, average collision energy in the density of air condition was the largest in particle and wall, and average collision energy in the density of 1.225E - 03 was the smallest in media and wall. Air condition has the largest total media and largest total particle average collision energy. We obtained a reasonable stirring model, a suitable stirring density, and working environment for better stirring performance. This research has great significance for the design of mixing products and the research of working environment.



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