Research on Coarse-Grained Discrete Element Model and Optimization for Fine Particles

Abstract

Optimization is important for the performance improvement of mechanical equipment. To advance this approach, a coarse-grained model for the discrete element method (DEM) is proposed with consideration of mechanical structure. This study identified a coarse-grained model that can be used in particle simulation, and designed a mixing equipment model, which was further optimized through combination with the coarse-grained model. The optimization and characteristics of a stirred mill were investigated. The novelty of this study is that the coarse-grained model was used for equipment optimization. Different results were obtained for different model structures. Concentration is related to the model. The average collision energy was obtained from media-to-wall or particle-to-wall collisions. The largest number of collisions that cause different string performance in different models was obtained. The optimized model had the largest average collision energy. The characteristics of different models combined with the coarse-grained model were determined, and useful results regarding the collision energy were obtained for future performance considerations. In summary, a suitable model was established and combined with an appropriate coarse-grained model to achieve performance improvement.

1. Introduction

With the development of science and technology, computer simulation is widely used in many industries owing to its high efficiency [1,2,3,4,5]. Over the past decades, structural optimization has become an important tool in the design process. The structural optimization of equipment during the mixing process is very important [6,7,8]. Nowadays, the discrete element method (DEM) is receiving attention from many researchers and engineers as a simulation method for the forecast of particle behavior during powder processing.

Simulation is an essential tool for process analysis, particularly when experimental data are difficult to obtain. However, to handle the enormous number of particles in engineering-scale problems, it is necessary to reduce the level of detail and accuracy. Within the DEM framework, particle-to-particle and particle-to-wall contacts are important for the prediction of particle behavior. In the DEM, the motion of a single particle is simulated based on Newton’s equations of motion. The DEM simulations track the evolution of particle assemblies by calculating the coordinates, velocities, forces, and torques of individual particles in the system [9,10,11,12,13,14,15]. In recent years, DEM simulation methods have become a powerful tool, and the information obtained by DEM simulations can provide insight into dynamic behavior. Cleary et al. [16] investigated the comminution mechanisms, particle shape evolution, and collision energy partitioning in tumbling mills. Deng et al. [17] worked on the DEM-based analysis of mixing and collision dynamics in the adhesive mixing process. Cisternas et al. [18] investigated trends in the modeling, design, and optimization of multiphase systems in mineral processing. Previous DEM simulations of the mixing process have focused on the mixing mechanisms and mixing equipment [19,20,21].

Coarse-grained modeling originated from the work of Michael Levitt and Ariel Warshel in the 1970s [22,23,24]. Currently, coarse-grained models are often used as components of multiscale modeling protocols in combination with reconstruction tools [25]. Queteschiner et al. [26] used a multi-level coarse-grained model with the DEM. The coarse granulation system can be considered as part of the exact scaling system. The granularity and domain in the original and exact scaling models differ only in terms of the constant scale factor h. Coarse-graining techniques are an approach for solving macro-scale problems. When coarse grain is used in a DEM, the number of particles in the model reduces the number of particles in the system; thus, the computational time and efficiency are improved [27,28,29].

Many studies have reported nanoparticle emissions caused by coatings [30], paints [31], and tiles [32]. Moreover, cases of nanoparticle exposure at coating workplaces have been reported in the field of occupational hygiene [33]. Such exposures can lead to water pollution. Additionally, attention should be given to the field of real-life particle emission [34,35,36,37,38,39], and release scenarios should be considered within the framework of risk analysis. Scenario analysis is a well-established method for developing strategic plans that are more flexible or robust to a range of plausible future states. For damage and failure in the form of crack propagation, local damage has significant influence on the overall structural strength and durability, and the discovery and synthesis of new materials often leads to technological breakthroughs [40,41,42].

This study mainly focused on the standard use of equipment. If DEM particle simulations involve large sizes, the advantages of the coarse-grained model can be better exploited and the efficiency of mixing can be improved. The contact forces acting on the coarse particles are estimated under the assumption that the kinetic energy of the coarse particles is the same as that of the original particles. Therefore, a suitable and reliable coarse-grained system can effectively represent the original system. This study identified a coarse-grained model that can be used to simulate particles, and designed a model of the mixing equipment, which was optimized and combined with the coarse-grained model. The media-to-wall or particle-to-wall collisions and the average collision energy were obtained, and the characteristics of different models combined with the coarse-grained model were determined. Thus, a suitable model was established and an appropriate coarse-grained model was identified for performance improvement.

2. Numerical Modeling

2.1. Simulation Method

The DEM is one of the most reliable and popular computer methods for simulating particle behavior through model analysis and calculation [43,44,45,46]. The DEM simulates the mechanical response of systems using discrete elements. Specifically, this method calculates the forces between hypothetical or actual discrete components to determine the motion of the discrete components through dynamic simulation. Thus, useful simulation experiments can be conducted. To calculate the model of contact forces, the force $\overrightarrow{F}$ can be divided into the contact force and fluid force, as follows:

$$\overrightarrow{F}={\overrightarrow{f}}_C+{\overrightarrow{f}}_D$$

(1)

The contact force ${\overrightarrow{f}}_C$ is further divided into the normal force ${\overrightarrow{f}}_{Cn}$ and tangential force ${\overrightarrow{f}}_{Ct}$. The model of the forces proposed by Cundall and Strack is expressed as follows:

$${\overrightarrow{f}}_{Cn}=-k{\overrightarrow{d}}_n-\eta {\overrightarrow{v}}_n$$

(2)

$${\overrightarrow{v}}_n=({\overrightarrow{v}}_r\cdot \overrightarrow{n})n$$

(3)

$${\overrightarrow{f}}_{Ct}=-k{\overrightarrow{d}}_t-\eta {\overrightarrow{v}}_t$$

(4)

$${\overrightarrow{v}}_t={\overrightarrow{v}}_r-{\overrightarrow{v}}_n$$

(5)

where ${\overrightarrow{d}}_n$ and ${\overrightarrow{d}}_t$ are the particle displacements in the normal and tangential direction, respectively; ${\overrightarrow{v}}_r$ is the relative velocity; k is the stiffness of the spring; and $\eta$ is the coefficient of viscous dissipation [47]. The contact force acting between particles or between a particle and the wall is modeled using the Voigt model, as shown in Figure 1.

The force calculated based on linear assumptions is applied when the particles overlap, as follows:

$$\left|f_{Ct}\right|>{\mu }_f\left|f_{Cn}\right|$$

(6)

where ${\mu }_f$ is the friction coefficient. Then, sliding is considered to occur, and the tangential force is expressed as follows:

$$f_{Ct}=-{\mu }_f\left|f_{Cn}\right|\overrightarrow{t}$$

(7)

where $\overrightarrow{t}$ is a unit vector. In dense phase flows, a particle typically touches several other particles at any time. The mass velocity is calculated first; then, the final velocity of the other mass required to conserve momentum is calculated to obtain the kinetic energy that is either gained or lost to make such a collision possible. The collision energy E is defined as the kinetic energy at the collision, and the collision energy is calculated as follows:

$$E=\frac{1}{2}\frac{m_1{m}_2}{m_1+m_2}{v}^2$$

(8)

where $m_1$ and $m_2$ refer to the mass of two colliding objects, and $v$ is the relative speed between the two objects. Coarse-grained simulations are very useful in design and analysis. Moreover, much effort has been put into modeling the force fields, and the force matching method can be used to derive them.

Coarse-grained models aim at simulating the behavior of complex systems using their coarse-grained representation. Figure 2 shows the coarse-grained model. In a large-scale DEM simulation, large particles cannot be used because fine and coarse particles have differences between them, such as in terms of drag and cohesive forces. Therefore, the coarse-grained model, which was developed to simulate non-cohesive particles in large-scale powder systems, only considers the contact, drag, and gravitational forces. This study identified a coarse-grained model and designed a model that can simulate particles. The simulation model was optimized and combined with the coarse-grained model.

2.2. Simulation Conditions and Model

This study conducted numerical simulation and considered the effect of the involved parameters, which are listed in

Table 1. Simulation parameters.

Number of Calculations [–]	500,000
Time increment [s]	1.0 × 10−5
Media diameter [m]	0.002
Media density [kg/m3]	7770
Particle density [kg/m3]	2260
Coefficient of restitution [–]	0.36
Coefficient of friction [–]	0.52
Rotaion speed [rpm]	300

.

A basic model A was designed by considering factors related to the mechanical behavior, stirring performance, and coarse-grained model. After optimization, the optimized model B was obtained. Optimization was carried out by analyzing the characteristics of different models. Various physical quantities of the two models were compared. The two models are shown in Figure 3.

3. Comparison of Different Model Conditions

Models with different structure were combined with the coarse-grained model. The model was designed and optimized, and various physical quantities of the basic model A and optimized Model B, such as the media concentration and sample concentration, were compared. Figure 4 shows grinding by different models.

The media concentration and sample concentration in cross-sectional view in the transverse direction were compared. Figure 5 shows the media concentration and Figure 6 shows the sample concentration.

The media concentration is also related to the model. In the surrounding area, Model B has the largest media concentration, as indicated by the black line in the Figure 5. In Model A (5 times) the media concentration was larger in some locations compared with Model A (20 times). Near the central axis of rotation, there was a certain sample concentration difference, as indicated by the black line in the Figure 6, and the sample concentration in Model B was larger.

4. Results and Discussion

4.1. Comparison of Collision Energy

A comparison was made between Model A (0.4 mm, 5 times), Model B (0.4 mm, 5 times), and Model A (0.1 mm, 20 times). Figure 7 shows the collision energy comparison.

Model B (0.4 mm, 5 times) had the highest number of total media (Figure 7a), total particles (Figure 7b), media-to-wall (Figure 7c), particle-to-particle (Figure 7e), and media-to-particle collisions (Figure 7f). Model A (0.4 mm, 5 times) had the highest number of media-to-media (Figure 7d) collisions. The location with the highest number of total particle collisions (Figure 7b) of Model B was much larger compared with Model A. With other conditions being equal, a larger number of collisions is very useful for the improvement of particle collision. Greater energy is helpful to the entire stirring process and contributes to the improvement of stirring performance. The results revealed that model B achieved better performance in terms of the number of collisions. The largest number of collisions may be responsible for rapid grinding in different models, such as Model B.

Model A (20 times) had the highest number of media-to-media collisions (Figure 7d). The location with the highest number of total particle collisions (Figure 7b) of Model B is much larger compared with the conventional model. Optimized Model B (0.4 mm, 5 times) had the highest number of collisions, such as total media, total particle, media-to-wall, particle-to-particle, and media-to-particle collisions. The results revealed that model B (0.4 mm, 5 times) achieved better performance in terms of the number of collisions. The highest number of collisions may be responsible for fine grinding in different models. Model B (0.4 mm, 5 times) has many advantages in terms of the number of collisions, which is important for improving the stirring performance.

4.2. Comparison of Average Collision Energy

Different models had different average collision energy. The average collision energy between media or between a particle and the wall was obtained. Figure 8 shows the comparison of the average collision energy.

For media-to-media or particle-to-wall collisions, the average collision energy in Model B (5 times) was the largest for particle-to-particle, media-to-particle, and particle-to-wall collisions. Some properties are different in different models. The most suitable stirring model and working situation were determined through combined analyses.

5. Conclusions

This study investigated the optimization of equipment using a coarse-grained model. Models with different structure produced different results. Moreover, the concentration is related to the model. In some locations, the concentration in Model A (5 times) was larger compared with Model A (20 times). For optimized Model B, the concentration at the center of the stirring shaft was relatively larger. For Model A, the coarse-grained model of 5 times achieved a better performance compared with the model of 20 times; therefore, the coarse-grained model of 5 times is optimal under this condition.

For the coarse-grained model, different model structures produced different results. Model B (0.4 mm, 5 times) had the highest number of total media, total particle, media-to-wall, particle-to-particle, and media-to-particle collisions. Model A (0.4 mm, 5 times) had the highest number of media-to-media collisions. For Model B, the location with the highest number of total particle collisions is much larger compared with the conventional model. The highest number of collisions may be responsible for rapid grinding. The characteristics of different models combined with the coarse-grained model were determined. Thus, a suitable model and an appropriate coarse-grained model were obtained for performance improvement.