New Approach to Effective Dry Grinding of Materials by Controlling Grinding Media Actions

Abstract

The grinding process plays a crucial role in many technological operations. However, the complexity of increasing product fineness and energy efficiency in particle size reduction poses a problem in grinding processes. This study proposes a new approach for increasing grinding efficiency under dry grinding conditions in mills with grinding media. The approach involves a complex impact on the particle, in which it is subjected to two- and one-sided actions by the grinding media in the horizontal and vertical directions, respectively. The efficiency of the approach was tested by mathematical modeling and experimentation. The difference between the theoretical and experimental results was less than 11%, indicating the reliability of the proposed model. The results indicate that the proposed approach enhances the grinding efficiency by nearly fourfold and can be applied in industrial sectors that require high product fineness.

1. Introduction

The grinding process is an essential step in different technological operations [1,2,3]. For example, the processes are used to increase the mineral extraction degree, obtain new building mixture properties, and accelerate chemical reactions in the mining and processing, construction, and chemical industries, etc. [4].

Among the problems present in grinding processes, the restricted possibilities of providing the required product fineness value and the high energy costs in mills can be highlighted. The reasons for these problems relate to the design features of mills and the lower grinding efficiency at finer particle sizes, caused by the increased contact area and reduced probability of effective impacts.

At present, mills using grinding media in their operation procedure are commonly used in grinding processes (for example, ball, stirred, and vibration mills, etc.). Steel balls, ceramic balls, glass beads, and natural stones are typically used as the grinding media. The main function of the grinding media is transferring destruction energy from the working body of the mill to the ground particles [5].

The main factor affecting the grinding process efficiency in the considered mill type is the collision energy of the grinding media [6,7,8]. The grinding media collision energy is proportional to the destruction energy, which is transferred from the grinding media to the ground particles during the action of the working body.

The literature analysis showed that several approaches for increasing the collision energy of grinding media have been developed to provide high product fineness and energy efficiency in the grinding process.

One approach relates to increased grinding media velocities. The collision energy of grinding media is proportional to their kinetic energy, mainly depending on the grinding media velocity. The grinding media velocity increases with increased movement velocity of the working mill body. For example, increased collision energy of grinding media is achieved with increased rotational speeds of the cylindrical chamber in ball mills [9,10,11], oscillation amplitudes of the chamber in vibratory mills [12], and rotational speeds of the stirrer in stirred mills [13,14]. However, there are some problems related to implementing the approach in the specified mill types. For example, in ball mills, increased chamber rotational speeds increased the grinding media concentrations on the chamber wall due to excess centrifugal force over gravitational force [15]. In vibratory mills, increased collision energy from increased amplitudes and frequencies of grinding chamber oscillation is restricted due to mill design features [16]. In stirred mills, increased collision energy from increased stirrer rotational speeds is problematic, as it also decreases the grinding media concentration in the high-kinetic-energy zone (near stirrers) [17,18].

The next approach concerns adjusting the geometrical parameters of the mill. The geometrical characteristics for ball, vibratory, and stirred mills include chamber size, grinding media filling ratio, stirrer shape, grinding media size, and grinding media shape. Numerous studies are dedicated to investigating the effect of the specified geometric parameters on the grinding process efficiency. The investigation results showed that an increased chamber diameter decreases energy consumption in stirred mills [19], which can be explained by the increased gap distance between the chamber wall and the stirrer edge. As the grinding media filling ratio increases, the collision energy of the grinding media increases. However, an increased grinding media filling ratio resulted in decreased free space for grinding media acceleration, decreasing the collision energy of the grinding media [20,21]. Several research works have focused on the effect of stirrer design on grinding efficiency. The investigation results demonstrated that the disk design of stirrers decreases energy consumption. However, the use of wing and cross types of stirrers characterized by increased energy consumption has resulted in increased collision energy [19,22]. Regarding grinding media size, the investigation results showed that the large grinding media size increases energy consumption [23,24]. According to the research results, the optimal grinding media–ground particle size ratio is 20:1 [25]. In addition, spherical grinding media demonstrated higher grinding performance than non-spherical ones [26,27].

Generalizing the two approaches, we noted a problematic trend: increased grinding process efficiency increases the grinding media velocity and geometrical parameter values (especially the grinding media filling ratio).

Consequently, another approach proposed in [28] aims to improve the grinding efficiency by organizing impacts from different directions, which enhances particle destruction due to the irregular shape of the particles.

The authors of [28] proposed increasing product fineness as a result of particle size reduction by grinding media action in radial and axial directions. This organization of grinding media movement leads to the Rebinder effect, i.e., a decrease in particle strength due to the periodic compression and tensile stress of the particles [28]. While the grinding media movement in the radial direction is provided by the ellipsoidal rotor, the grinding media movement in the axial direction is achieved via chamber vertical vibration. As shown in Figure 1, the particle is stressed between two grinding balls (grinding media). It should be clarified that while the impact velocity of grinding ball “1” is greater than zero (due to the ellipsoidal rotor) before collision with the particle, the impact velocity of grinding ball “2” is zero in the radial direction (Figure 1a,b). Analogically, in the axial direction, while the impact velocity of grinding ball “3” is greater than zero (due to chamber vertical vibration) before collision with the particle, the impact velocity of grinding ball “4” is zero (Figure 1c,d). Thus, the particle experiences a one-sided impact from the grinding balls.

Another approach has been proposed to increase general collision energy [29]. The idea of the approach is to facilitate the grinding process through the counter-collision of the grinding media (Figure 2).

This is provided by the design of mills with V-shaped chambers in which the movement of the grinding media is provided due to the inclination of the chamber sections. In this case, the one-sided impact presented in [28] changes due to the double-energy impact, increasing the general collision energy.

A new approach for particle size reduction with grinding media is proposed to increase the grinding process efficiency (Figure 3). The approach essentially comprises the joint application of the last two approaches, i.e., organizing the grinding media collision in the radial and axial directions, with counter-collision of the grinding media in the radial direction (providing two-sided impact on the particles).

A new mill design is developed for the proposed approach (Figure 4).

The mill design includes a chamber filled with grinding media (steel balls), a vibrating table (with drive), pistons, and eccentrics with electric motors. The material is loaded into the chamber through an inlet. As a result of eccentrics, the reciprocating movement of the pistons is observed. Therefore, the counter-collision of the grinding media and double-energy impact on the particles are provided. As a result of the vertical effect of the vibrating table, the vibrational movement of the grinding media is conducted in the vertical direction.

The main advancement of the present work compared to [28] is the development of a method that enables two-sided impact on a particle in the horizontal direction while maintaining perpendicular excitation, thereby increasing particle destruction energy and improving product fineness and energy efficiency.

In comparison with the reviewed literature, the main contribution of this work is the development of an approach that enhances grinding process efficiency by increasing the collision energy of particle destruction by grinding media.

This study aimed to theoretically and experimentally investigate the effectiveness of the proposed approach for the dry grinding process and comprised the following steps:

(1)

Conducting a mathematical model for the description of the grinding process and predict the efficiency of the proposed approach;

(2)

Developing the design of the experimental apparatus reproducing the proposed approach;

(3)

Conducting experimental investigations of the proposed approach.

2. Materials and Methods

2.1. Theoretical Description of the Grinding Process

Product fineness calculation is carried out using our own mathematical model presented in [30].

According to the mathematical model, three stages of particle interaction with the grinding media can be identified: (1) the pre-contact stage, characterized by the relative motion and approach of the grinding media toward the particle; (2) the moment of initial contact, involving the onset of mechanical interaction; and (3) the penetration stage, where the grinding balls partially intrude into the particle, leading to stress concentration and subsequent structural changes.

In the first stage, the grinding media movement in the horizontal direction is the result of the reciprocating motion of the pistons (Figure 5a).

Considering that the source of the velocity of ${\upsilon }_x$ is unbalanced vibrator action and, consequently, the pistons perform a vibratory movement, then, the grinding media movement in the horizontal direction can be described by the following oscillation equation:

$$x=A_x{\omega }_x\cos {\omega }_x{t}_x,$$

(1)

where $A_x$ is the horizontal amplitude of oscillations (mm), ${\omega }_x$ is the horizontal angular frequency (rad/s), and $t_x$ (s) is the oscillation time in the horizontal direction. The grinding media velocity before collision with a particle in the horizontal direction can be determined as the first derivative of Equation (1) as follows:

$${\upsilon }_x={\displaystyle \frac{\partial x}{\partial t}}=A_x{\omega }_x,$$

(2)

Considering the grinding media movement in the vertical direction is caused by the vibrating table, the grinding media motion can be described by the following oscillation equation:

$$y=A_y{\omega }_y\cos {\omega }_y{t}_y,$$

(3)

The grinding media velocity can be determined as the first derivative of Equation (3) as follows:

$${\upsilon }_y={\displaystyle \frac{\partial y}{\partial t}}=A_y{\omega }_y,$$

(4)

where $A_y$ is the vertical amplitude of oscillations (mm), ${\omega }_y$ is the vertical angular frequency (rad/s), and $t_y$ (s) is the time of oscillation in the vertical direction.

Considering the grinding process is realized in two directions (horizontal and vertical) by actions of horizontal ${\upsilon }_x$ and vertical ${\upsilon }_y$ velocities, particle size reduction is modeled to occur via the resultant velocity ${\upsilon }_{xy}$.

Because the velocities of ${\upsilon }_x$ and ${\upsilon }_y$ are conducted in mutually perpendicular directions, the velocities are presented as a resultant velocity ${\upsilon }_{xy}$ in further calculations as follows:

$${\upsilon }_{xy}=\sqrt{{\upsilon }_x^2+{\upsilon }_y^2}=\sqrt{A_x^2{\omega }_x^2+A_y^2{\omega }_y^2},$$

(5)

In the second stage, grinding media–particle contact occurs (Figure 5b). When the grinding media and particle make contact, the potential grinding media energy is equal to zero $E_p=0$, and the kinetic energy is maximal $E_k=\max$.

When the grinding media and particle make contact, the grinding media velocity is determined using the theorem on the equality of the momentum of interacting bodies before and after collision:

$$m_E{\upsilon }_{xy}=\left(m_E+m_{GM}\right)\cdot {\upsilon }_C,$$

(6)

where $m_E$ is the external mass acting on the grinding media, $m_{GM}$ is the grinding media mass, and ${\upsilon }_C$ is the grinding media velocity after collision with a particle.

The following can be derived from Equation (6):

$${\upsilon }_C={\displaystyle \frac{m_E}{m_E+m_{GM}}}\cdot {\upsilon }_{xy}={\displaystyle \frac{\frac{F_{xy}}{g}\cdot {\upsilon }_{xy}}{\frac{F_{xy}}{G}+\frac{G}{g}}}={\displaystyle \frac{{\upsilon }_{xy}}{1+\frac{4\pi \cdot \gamma \cdot d_{GM}^3}{24F_{xy}}}},$$

(7)

where $\gamma$ is the grinding media density, $d_{GM}$ is the grinding media diameter, $F_{xy}$ is the resultant comminution force, $G$ is the grinding media weight, and g is the acceleration of gravity.

In the third stage, the penetration of the grinding media into the particle and, consequently, particle deformation occur (Figure 5c). In this stage, the conversion of kinetic energy of grinding media movement to potential energy of the particle destruction is conducted. As particles experience dual-energy impact due to counter-grinding media collision, the particle receives dual kinetic energy as follows:

$$E_K+E_p=2E_K.$$

(8)

Accordingly, dual potential energy is received as follows:

$$E_p+E_p=2E_p.$$

(9)

According to the law of energy conservation, kinetic and potential energies can be equated as follows:

$$2E_K={\displaystyle \frac{4F_{xy}}{g}}\cdot \left({\displaystyle \frac{{\upsilon }_{xy}^2}{1+\frac{\pi \cdot \rho \cdot d_{GM}^3}{6F_{xy}}}}\right)=2E_p={\displaystyle \frac{8Y}{3\left(1-{\mu }^2\right)}}\cdot {\left({\displaystyle \frac{d_{p(i)}\cdot d_{GM}}{2\cdot \left(d_{p(i)}+d_{GM}\right)}}\right)}^{0.5}\cdot {\displaystyle \frac{2}{5}}\cdot u_{\max }^{2.5},$$

(10)

where $\mu$ is Poisson’s ratio, $Y$ is the elastic modulus, $\rho$ is the grinding media density, $d_{p(i)}$ is the initial particle size, and $d_{GM}$ is the grinding media diameter.

Equation (10) is followed by the formula for determining the maximal displacement $u_{\max }$, which is expressed as follows:

$$u_{\max }={\left({\displaystyle \frac{15}{4}}\cdot \left({\displaystyle \frac{1-{\mu }^2}{Y}}\right)\cdot {\displaystyle \frac{F_{xy}}{g\left(1+\frac{\pi \cdot \rho \cdot d_{GM}^3}{6F_{xy}}\right)}}\right)}^{-0.4}\cdot {\left({\displaystyle \frac{d_{p(i)}\cdot d_{GM}}{2\cdot \left(d_{p(i)}+d_{GM}\right)}}\right)}^{0.2}\cdot {\upsilon }_{xy}^{0.8}.$$

(11)

The formula for calculating product fineness can be derived from the grinding media velocity equation, leading to the beginning of particle destruction, as demonstrated below:

$${\upsilon }_{\min }=2.0858\cdot d_{GM}\cdot \sqrt{{\displaystyle \frac{\left(d_{GM}+d_{p(i)}\right)\cdot \rho \cdot \left(1-2\mu \right)}{d_{p(i)}\cdot \sigma }}}.$$

(12)

Based on Equation (12), product fineness can be calculated as

$$d_{p{(f)}_1}={\displaystyle \frac{d_{GM}\cdot \rho \cdot \left(1-2\mu \right)}{{\left(\frac{{\upsilon }_x}{6.84\cdot {10}^6}\right)}^2\cdot \sigma -\rho \cdot \left(1-2\mu \right)}}.$$

(13)

The product fineness resulting from the grinding process in Figure 3 should be compared with the product fineness obtained using the approach depicted in Figure 1. The formula for determining product fineness based on the approach shown in Figure 1 was derived in [28] as follows:

$$d_{p{(f)}_2}={\displaystyle \frac{d_{GM}\cdot \rho \cdot \left(1-2\mu \right)}{{\left(\frac{{\upsilon }_x}{13.31\cdot {10}^6}\right)}^2\cdot \sigma -\rho \cdot \left(1-2\mu \right)}}.$$

(14)

Subsequently, the derived formulas are validated by substituting specific mill parameters, the material being processed, and the grinding media. The calculation procedure is conducted corresponding to the two following cases:

• Case 1: The grinding process is conducted as a result of two- and one-sided grinding media actions in the horizontal and vertical directions, respectively (using Equation (13)).
• Case 2: The grinding process is conducted as a result of one- and two-sided grinding media actions in the horizontal and vertical directions (using Equation (14)).

The mill, material, and grinding media parameter values used in the calculation are presented in

Table 1. The input data for the calculation.

Parameter	Symbol	Value
Mill parameters
Amplitude in horizontal direction	$A_x$	4 mm
Oscillation frequency in horizontal direction	${\omega }_x$	50 Hz
Amplitude in vertical direction	$A_y$	2 mm
Oscillation frequency in vertical direction	${\omega }_y$	50 Hz
Material parameters (sand)
Poisson’s ratio	$\mu$	0.27
Elastic modulus	$Y$	$36\cdot {10}^9\left(\hspace{0.17em}\mathrm{N}\mathrm{/}{\mathrm{m}}^2\right)$
Initial size (diameter) of the particle	$d_{p(i)}$	100 µm
Ultimate stress	$\sigma$	1.15 MPa
Grinding media parameters
Grinding media density (steel)	$\rho$	$7.7\cdot {10}^{-5}\left(\hspace{0.17em}\mathrm{N}\mathrm{/}\mathrm{m}\hspace{0.17em}{\mathrm{m}}^3\right)$
Diameter of the grinding media	$d_{GM}$	10 mm

.

According to Equations (2) and (4), respectively, since the collision velocity of the grinding media and, hence, the collision energy depend on the amplitude and oscillation frequency in the horizontal and vertical directions, these parameters were chosen as inputs for the calculations.

The calculation results are presented in

Table 2. The calculation results.

Parameters	Case 1	Case 2
$\mathrm{Collision}\mathrm{velocity},{\upsilon }_{xy}$	${\upsilon }_{x{y}_1}=1404.97$ mm/s	${\upsilon }_{x{y}_2}=1404.97$ mm/s
$\mathrm{Product}\mathrm{fineness},d_{p(f)}$	$d_{p(f)1}=7.3$ µm	$d_{p(f)2}=27.7$ µm

.

Table 2 shows that product fineness in case 1 is nearly four times greater than in case 2. Thus, based on the theoretical investigation results presented in Table 2, the efficiency of the proposed approach for the comminution process is concluded.

2.2. Experimental Investigation of the New Mill Design

2.2.1. Description of Experimental Apparatus

To verify the theoretical investigation results of the proposed approach for the grinding process, an experimental apparatus for experimental research was developed. The experimental apparatus design is presented in Figure 6.

The experimental apparatus includes a fixed frame (1), a chamber (7) filled with grinding media (located inside the chamber), pistons (located inside the chamber), a vibrating table (2), unbalanced vibrators (6), and horizontal (5) and vertical (3) springs, as numbered in Figure 6.

The chamber (7) is a cylindrical container with a diameter of 100 mm and a length of 300 mm. As the grinding media, steel balls with a diameter of 10 mm were selected. The pistons are connected with unbalanced vibrators (6) through a rod. Pistons are returned to their original position with the horizontal springs (5) located between the unbalanced vibrators (6) and pistons. The vibrating table (2) is the platform placed on vertical springs (3). The table is vibrated using an unbalanced rotor driven by the electric motor (4).

As the additional elements of the experimental apparatus, a frequency converter (8) and amplitude meter (9) are included.

2.2.2. Description of Experiment

The experimental procedure is divided into two main parts corresponding to the two cases considered in the theoretical investigations.

The first part is the experimental investigation into the grinding process with two-sided horizontal and one-sided vertical vibrations of the grinding media on the tested material particles. This is achieved by activating both unbalanced vibrators, which generate horizontal vibrations of two pistons. In addition, the electric motor of the vibrating table is activated to vibrate in the vertical direction.

The second part is the experimental research of the grinding process with one-sided horizontal and vertical vibrations of the grinding media on the tested material particles. This is achieved by activating one of the unbalanced vibrators and the electric motor.

Before the start of the first and second parts of the experiment, the chamber is filled with steel balls (grinding media). The grinding media filling ratio is accepted as 80% based on [18]. After that, the chamber is filled with sand as the experimental ground material. Then, the sand was sieved to yield an initial material size of 100 µm [28].

At the beginning of the first part of the experiment, unbalanced vibrators and an electric motor are switched on to start the comminution process. The parameters of the unbalanced vibrators and electric motor have been set analogously to the data used in mathematical modeling (${\omega }_x=50$ Hz, ${\omega }_y=50$ Hz, $A_x\approx 4$ mm, and $A_y\approx 2$ mm).

Product fineness is measured at one-minute intervals. As an instrument for measuring product fineness, a PSH-10 (manufacturer: Labnauchpribor LLC, Moscow, Russia) device was used (Figure 7).

The sand weight was measured using a weigher, and the material layer height was determined using the scale on the flask. Then, using the clock face, the measured parameters were entered. Finally, sand product fineness was displayed on the device screen.

Analogously, the procedures were conducted for the second part of the experiment. The only difference was that in case 2, only one of the unbalanced vibrators was activated to generate a one-sided vibrational impact of the grinding media on the material particles.

3. Results and Discussion

The experimental results compared with the theoretical data are presented in

Table 3. Comparison of theoretical and experimental results.

Case	Description	Experimental Data		Theoretical Data	Difference, %
		Maximal Product Fineness	Maximal Processing Time	Maximal Product Fineness
1	Both of the unbalanced vibrators for the creation of the horizontal vibrational actions are activated; the vibrating table drive is activated	8.1 µm	12	7.3 µm	10.96%
2	One of the unbalanced vibrators for the creation of the horizontal vibrational actions is activated, and the other one is not activated; the vibrating table drive is activated	28.4 µm	17	27.7 µm	2.53%

.

From analyzing the experimental results in Table 3, it can be concluded that the periodic two- and one-sided grinding media collisions in the horizontal and vertical directions (case 1) lead to higher product fineness (8.1 µm) than the grinding process with one-sided grinding media collision in the horizontal and vertical directions (28.4 µm) (Figure 8).

Moreover, comparing the maximum processing time values, it can be concluded that the proposed organization of the grinding process (case 1) increases the grinding process speed by 29.4% compared with case 2 (12 min vs. 17 min, respectively).

The effectiveness of the proposed approach is substantiated by the applied loading scheme, in which material particles are subjected to two- and one-sided impacts in the horizontal and vertical directions. The two-sided impact increases the particle destruction energy in the horizontal direction. As the impacts are applied in mutually perpendicular directions, increased horizontal destruction energy increases the resultant destruction energy acting on the particles.

Comparing the experimental and theoretical results, the experimental product fineness values are notably higher than the theoretical ones. The explanation is that many physical and mechanical processes occurring in a real grinding procedure cannot be considered in mathematical modeling.

In Figure 9, the graphs present the particle size reduction dynamics.

In Figure 9a, the graphical dependence of the product fineness on the processing time corresponding to case 1 is presented. The initial point is 100 µm, corresponding to the beginning of the grinding process. Over time, a gradual decrease in particle size is observed. At a range between 0 and 6 min, the graph shows a rapid decrease in the product fineness value; at a range between 7 and 11 min, particle size reduction speed decreases. At the 12 min mark, a straight line is observed, indicating the cessation of the material particle grinding process. In Figure 9b, a similar trend is observed for the grinding process carried out in case 2. However, a rapid decrease in product fineness is observed between 0 and 7 min. Between 8 and 16 min, the grinding rate slows down, and then, from 17 min, the product fineness remains constant.

Table 3 shows that the difference between the experimental and theoretical results accounted for 10.96% and 2.53% in cases 1 and 2, respectively. This can be explained by characterizing the grinding procedure as a complex process. Consequently, the difference between the theoretical and experimental results can reach up to 15% [28].

Despite the successful validation of the proposed grinding method, several limitations of the experimental setup should be noted:

• The limited power of the drive motor restricted the maximum torque, which prevented the testing of harder or highly compressed materials under realistic industrial conditions.
• The dimensions of the grinding chamber constrained the size of input materials, which may limit the scalability and generalizability of the obtained results.
• The absence of a cooling system during operation may have led to uncontrolled temperature increases, potentially affecting the material properties and grinding efficiency.

These factors should be considered when interpreting the experimental outcomes and will be addressed in future improvements in the setup.

4. Conclusions

This study presents the investigation results of a new approach for the grinding process in mills using grinding media. The idea of the approach is the complex impact of the grinding media on the particle in the horizontal and vertical directions. The organizational peculiarity is that particles experience periodic two- and one-sided collisions of the grinding media in the horizontal and vertical directions, respectively. The approach is implemented by developing the mill design, including a chamber filled with grinding media (steel balls), a vibrating table (with drive), pistons, and unbalanced vibrators with electric motors. While the counter-collision of the grinding media in the horizontal direction is provided by periodic countermovement of the pistons, the one-sided movement in the vertical direction is provided by the vibrating table.

This study theoretically evaluated the effectiveness of the proposed approach using our developed mathematical model. Using the mathematical model, the product fineness value of a tested material (sand) for two cases has been calculated. In case 1, the grinding process is conducted as a result of two- and one-sided grinding media actions in the horizontal and vertical directions; in case 2, the grinding process is performed as a result of one-sided grinding media actions in the horizontal and vertical directions. In cases 1 and 2, the theoretical product fineness values accounted for 7.3 µm and 27.7 µm, respectively.

Experimental studies were conducted to validate the results of theoretical research and showed that the maximum product fineness of the sand particle size reduction accounted for 8.1 µm and 28.4 µm (maximum processing times are 12 and 17 min) for cases 1 and 2, respectively. The differences between the theoretical and experimental studies are 10.96% and 2.53% for the two cases, respectively.

Based on the theoretical and experimental results, it can be concluded that the proposed approach for the grinding process is characterized by high effectiveness. This is proved by the relatively high and low values of the product fineness and processing time when comparing case 1 with case 2.

Thus, the proposed approach for the grinding process can be used in industrial operations that require high product fineness values.