Experimental and Modeling Study of the Regular Polygon Angle-spiral Liner in Ball Mills

Abstract

Load behavior is one of the most critical factors affecting mills’ energy consumption and grinding efficiency, and is greatly affected by the liner profiles. Generally, as liner profiles vary, the ball mill performances are extremely different. In order to study the performance of the ball mill with regular polygon angle-spiral liners(RPASLs), experimental and numerical studies on three types of RPASLs, including regular quadrilateral, pentagonal and hexagonal, are carried out. For the fine product of desired size, two critical parameters are analyzed: the energy input to the mill per unit mass of the fine product, $E^{*}$, and the rate of production of the fine product, $F^{*}$. Results show that the optimal structure of RPASLs is Quadrilateral ASL with an assembled angle of 50${^{\circ }}$. Under this condition, the specific energy consumption $E^{*}$ has the minimum value of 303 J per fine product and the production rate $F^{*}$ has the maximum value of 0.323. The production rate $F^{*}$ in the experimental result is consistent with the specific collision energy intensity to total collision energy intensity ratio $E_\mathrm{s}/E_\mathrm{t}$ in the simulation. The relations between the production rate $F^{*}$ and the specific energy consumption $E^{*}$ with collision energy intensity $E_\mathrm{s}$ and $E_\mathrm{t }$ are obtained. The simulation result reveals the essential reason for the experimental phenomenon and correlates the mill performance parameter to the collision energy between balls, which could guide the practical application for Quadrilateral ASL.

1 1 Introduction

Ball mills are widely used in mineral processing and chemical industries for particle size reduction. About one third of total electricity energy consumed by them was used to process powder, and almost half of this energy was wasted [1, 2]. Their high energy consumption and low milling efficiency are still big challenges [3,4,5]. The efficiency of the milling process is highly dependent on load behavior that could lead to different levels of collision energy and to particles being broken into different small pieces in the cylinder [6, 7]. Since the load behavior is greatly affected by the liner profiles [8, 9], interest is growing in optimizing liner design to enhance the load motion and collision behavior in mills. A large amount of work was carried out to study the effect of different liner configurations on load behavior and power consumption combined with mill speed, ball load, particle load, ball diameter, mill diameter and so on. For example, CLEARY [10] studied the effect of the mill speed, lifter shape and lifter pattern on the load behavior and power consumption in a ball mill. He concluded that the steeper lifters lead to more energy consumption at low speed. HLUNGWANI, et al [11], used a simulation model to study the mill power draw and load behavior with different lifter profile in a 2D mill. They found that the energy consumption with the trapezoidal lifters was much more than the square lifters. CLEARY and OWEN [12] analyzed the life cycle performance of two liner designs in a HICOM 110 mill. They pointed out that small geometry differences in new liners could produce large variations in the charge behavior and profiles life cycle behavior. POWELL, et al [9] discussed the influence of changes in the shape of liner lifters on grinding efficiency. They presented a simplified breakage rate model to correlate liner profile to mill performance in an 8 m diameter ball mill. REZAEIZADEH, et al [13], conducted an experimental study to analyze the influence of charge filling, mill speed, number of lifters and geometry of lifter in a laboratory mill. They found that increasing the mill speed, number of lifters, and height of lifter could improve the impact value and impact frequency between balls. However, those mentioned liners kept the load motion and collision behavior constant during the grinding process, which enhanced the collision behavior slightly. While the load behavior in the regular polygon angle-spiral liner(RPASL) changes constantly owing to its unique structure and assembled angle(the angle of each liner sequentially shifted with respect to each other). The release point and impact area are changing all the time in the grinding process to mix the feed and balls effectively [14]. Although theoretical research on the RPASLs with different assembled angles has been done in our previous study [14], the application of RPASL still relies on experience in practice. In order to make it evidence based, experimental and numerical study on the ball mill performance with different types of RPASLs were conducted and discussed with the aid of discrete element method(DEM) in this article.

DEM, a numerical method, is widely used to simulate the motion and interaction of individual particle in dynamic environment [15]. It allows us to gain a good understanding of the internal dynamics, such as the motion behavior of balls and their collision energy, and to develop liner retrofits with grinding operation [16,17,18,19,20]. DJORDJEVIC, et al [21], used a 3D model to simulate the effect of liner shape and mill speed on the distribution of the impact energy in a 5m diameter autogenous(AG) mill. They demonstrated the interaction of lifter design, mill speed and mill filling through DEM modeling. SEYFI, et al [22] and ROSENKRANZ, et al [23] carried out the experimental and DEM modeling investigation of the balls motion to obtain the collision energy between balls. In order to observe the load behavior and obtain the collision energy in the milling process, the numerical model is ste up in this paper.

The structure of this paper is organized as follows. Section 2 introduces the experimental study. Experimental tests in a pilot scale ball mill at a constant operating condition were carried out in section 2.1 and the results of the performance parameters of the ball mill were discussed in section 2.2. Section 3 presents the numerical study to correlate the mill performance parameters to the collision energy. The numerical model was built in section 3.1 and the simulation results were presented in section 3.2. An optimal profile and configuration of RPASL with proper assembled angle for the fine product of desired size was summarized in section 4.

Fig. 1

Pilot scale ball mill

Table 1 Parameters and characteristics of the ball mill

2 2 Experimental Study

2.1 2.1 Experiment

A pilot scale ball mill was designed as shown in Fig. 1. Its geometry information is listed in Table 1. We carried out a detailed study on the effect of liner profile and assembled angle on the grinding kinetics by conducting experiments with three types of RPASLs (Quadrilateral ASL, Pentagonal ASL, and Hexagonal ASL) and three assembled angles(10${^{\circ }}$, 30${^{\circ }}$, and 50${^{\circ }})$. The geometric structures of RPASLs are displayed in Fig. 2. The volume of Triangle ASL is too small to make the utilization of the energy input to the mill much lower [14], So triangle ASL was not considered in this work. Fig. 3 shows QASL with an assembled angle of 10${^{\circ }}$. The geometrical parameters of RPASL marked in Fig. 2 are listed in Table 2: n is the number of sides of RPASLs, $\sigma$ is the liner thickness at the thinnest point, $h_{n }(n=$ 4, 5, 6) is the liner thickness at the thickness point, $R_{n }(n=$ 4, 5, 6) is the radius of the inscribed circle, and $\varPhi$ is the inner diameter of the shell.

Silica sand was used as the feed material in the experimental tests. Its original particle size distribution is shown in Fig. 4. The horizontal axis is “Particle size /mm”. The vertical axis is the “Cumulative mass fraction passing /%”, which is the sum of the mass percent corresponding to different particle sizes. The vertical axis and horizontal axis are plotted on logarithmic scales. The nominal diameters $D_{10}$, $D_{50}$ and $D_{90}$ of the starting material, corresponding to 10, 50 and 90% cumulative mass fraction of passing material, are 2.5 mm, 6 mm, and 10 mm, respectively, as shown in Fig. 4.

Fig. 2

Geometric structures of RPASLs

Fig. 3

QASL with an assembled angle of 10${^{\circ }}$

Table 2 Geometrical parameters of the liners

Fig. 4

Particle size distribution of feed material

Grinding tests were carried out at a ball load $\lambda _{m}$ of 40%, corresponding to 40% filling of mill volume by static ball charge [24], and a low voidage corresponding to an interstitial filling of void spaces of the ball bed of $\mu$=0.38 in the ball bed. The voidage, $\mu$, is defined as follows:

$$\begin{aligned} \mu =(V_{\hbox {m}} -NV_{\hbox {sm}} )/V_{\hbox {m}} , \end{aligned}$$

(1)

where N is the number of media, $V_\mathrm{m}$ is total volume of media including voidage, and $V_\mathrm{sm}$ is the volume of a single medium. In practice the voidage is usually determined by the media’s packing forms. We suppose all the balls in this cylinder have two packing forms, cube( $\mu _{1}$ $=$0.477) and tetrahedron( $\mu _{2}$ $=$0.283), taking the average ($\mu =({\mu }_{1}+{\mu }_{2})$/2 $=$0.38) as the voidage in the ball bed [25].

Particle load ratio U refers to the fraction of interstitial volume of static ball charge occupied by particles. It was found by DENIZ, et al [26] that U=0.6 could provide a better grinding effect. The operational speed is 80% of the cylinder critical speed [27],$N_\mathrm{c}$, which refers to the rotational speed when balls in the cylinder have centrifugal motion. $N_\mathrm{c}$ can be calculated by [28]

$$\begin{aligned} N_{\hbox {c}} =\frac{42.3}{\sqrt{\varPhi -d}}, \end{aligned}$$

(2)

where $\varPhi$ is the inner diameter of the shell, d is the diameter of the media. The values of these parameters and the detail experimental conditions are listed in Table 1.

Production rate of fine product of the desired size, $F^{*}$, whose equivalent diameter was 0.075 mm was used as one of the grinding performance parameters by HAO [29] and BALLANTYNE, et al [30]. Here, the fine product of 0.075 mm was collected in the grinding process. The mill was stopped after grinding 20, 40, 60, 80, 100 and 120 min, and then small samples were prepared. The total mass of the feed material silica sand was about 29 kg, and the small sample taken out each time was 100 g, these samples were not put back into the mill cylinder.

2.2 2.2 Experimental Results

Particle size distributions at different grinding time for the three types of RPASLs are reported in Fig. 5. Both the horizontal and vertical axes are plotted on logarithmic scales to make the figures readable. Figs. 5(a)–(c) show the particle size distributions in QASL with different assembled angles, PASL’s and HASL’s are reported in Figs. 5(d)–(f) and Figs. 5(g)–(i), respectively. In QASL, the distribution of the curves in Fig. 5(a) is similar to the one in Fig. 5(b) or Fig. 5(c), and similar phenomena can also been observed in PASL and HASL. While the distributions of the curves are obviously different in different types although they have the same assembled angle, such as the distribution of the curves in Figs. 5(a)(d)(g), (b)(e)(h) or (c)(f)(i). Thus, the assembled angle does not influence the breakage of coarse particles as much as the liner type.

As stated above, the distribution of the curves in the same type is similar to each other, while the variation of the mass fraction of the fine product is significantly presented by the starting point of each curve. In QASL(Figs. 5(a)–(c)), the starting point of the 120 min curve in Fig. 5(a) exhibits a maximum value of the mass fraction of the fine product, in PASL(Figs. 5(d)–(f)), the maximum value appears in Fig. 5(e) and in HASL(Figs. 5(g)–(i)), the maximum one is in Fig. 5(h).

In addition, the coarse particles become fine gradually as grinding time increases, the curves almost overlap except the starting two points when the time is longer than 60 min, indicating that 60 min may be an effective grinding time for a product size larger than 0.1 mm. During the effective grinding time, the accumulation of the product increases significantly, while longer than it nearly no changes in the accumulation occur, but more energy is consumed by the mill, thus the effective grinding time is necessary to reduce the ball mill’s energy consumption. As can be seen in Fig. 5, the effective grinding time differs significantly for different product sizes in the milling process. For the fine product smaller than 0.1 mm, the time is longer than 120 min.

Fig. 5

Particle size distributions for different RPASLs and assembled angles

Fig. 6

Mass fraction of different particles when grinding time longer than 60 min

The variation of the mass fraction of different particle sizes is presented in Figs. 6(a)–(c). Based on the result in Fig. 5, the case that has the maximum value of the fine product in each type is collected in Fig. 6. It can be seen that the mass fraction of the fine product increases significantly with the increasing time. While the fraction of the coarse particle of 0.1 mm remains constant from 80 min to120 min, and its value is still the largest, up to 70%. This suggests that the RPASLs are good at accumulating the particles of about 0.1 mm, and the effective grinding time is within 80 min. As the main purpose of grinding operation is to produce fine particles of desired size, the curve of QASL-50${^{\circ }}$ exhibits the maximum mass fraction of the fine product.

Further, Figs. 7(a)–(c) show the mass fraction of the fine product of 0.075 mm,$\lambda _\mathrm{p}$, in different cases. In Fig. 7(a), the curve of 50${^{\circ }}$ assembled angle shows the maximum value of both mass fraction $\lambda _\mathrm{p}$ and production rate$F^{*}$; In Figs. 7(b)–(c), the curves of 30${^{\circ }}$ assembled angle show the similar trend. So the three curves of QASL-50${^{\circ }}$, PASL-30${^{\circ }}$ and HASL-30${^{\circ }}$ are discussed in Fig. 8. It can be seen that the curve of QASL-50${^{\circ }}$ remains the maximum values of the mass fraction and production rate as well.

Fig. 7

Mass fraction of the fine product of 0.075 mm in different cases

Fig. 8

Comparison of the fine product mass in following three cases: QASL-50${^{\circ }}$, PASL-30${^{\circ }}$ and HASL-30${^{\circ }}$

Fig. 9

Energy consumption in different cases

Besides the rate of production of the desired size$F^{*}$, the energy consumption is also an equally important parameter for characterization of the mill performance [30, 31]. For a batch mill drawing constant power, the specific energy input to the mill, $E^{*}$, is given by [31]

$$\begin{aligned} E^{{*}}=\frac{Pt}{M_{\hbox {p}} }, \end{aligned}$$

(3)

and

$$\begin{aligned} M_{\hbox {p}} =\lambda _{\hbox {p}} \cdot M_{\hbox {s}} , \end{aligned}$$

(4)

where $E^{*}$ is energy input to the mill per unit mass of the fine product, P is the net power drawn by mill, t is the grinding time in one test, $M_\mathrm{p}$ is the mass of the fine product. $M_\mathrm{s}$ is the mass of the feed particles.

Fig. 9 reports the specific energy consumption of the ball mill, including the RPASLs and non-RPASL cases simultaneously. Comparing to the RPASL cases, the energy consumption in the non-RPASL one is the largest. Among the RPASLs cases, the case of QASL-50${^{\circ }}$ has the lowest of about 303 J per fine product. Besides, it should be noted that the assembled angles of RPASL may be compensated by the interior angles of the regular polygon. For example, the interior angle of HASL is 60${^{\circ }}$, the angles of 10${^{\circ }}$ and 50${^{\circ }}$ assembled in the mill will form the same structure. Their energy consumption, therefore, shows the similar value.

3 3 Numerical Study

The simulation models corresponding to the pilot scale mill with the three cases of QASL-50${^{\circ }}$ PASL-30${^{\circ }}$ and HASL-30${^{\circ }}$ were built in SolidWorks, one of them is shown in Fig. 10, then imported into the workspace of EDEM [32]. The grinding balls in the cylinder were generated in the particle factory to observe the ball motion and collision behavior.

Fig. 10

The simulation model of the ball mill

3.1 3.1 EDEM Simulation

The simulation was conducted in the same condition with the experiment, as listed in Table 1. Although EDEM cannot provide direct information about particle breakage, the intensity of breakage can be derived from the collision energy. Four contact models, including ball-ball, ball-particle, ball-liner and particle-particle, are defined in it. The collision energy between particles is so small that its contribution to particle breakage can be ignored [14]. The milling process was simulated without feed material, that is, only ball-ball and ball-liner were considered in the processing. The details of the numerical setting are presented in Table 3.

3.2 3.2 Numerical Results

The grinding balls rolled or slid on the liner surface after they reached the highest points [33], then they began to free fall until they encountered the toe positions [34]. MULENGA and MOYS [35] demonstrated a simplified description of charge shape inside the mill. They proposed some positions to describe ball motion states: the impact point was assumed to be the point with the highest concentration of impact on the liner by cataracting balls, and the shoulder was chosen to be the point where the trajectories of the bulk balls near the liners began to depart from the shell. Figs. 11 (a–c) show the motion behavior of the grinding balls in the mill cylinder. The balls in the elliptic circle are cataracting balls thrown from the shoulder position to the toe position. The most cataracting balls are found in the case of QASL-50${^{\circ }}$, followed by the PASL-30${^{\circ }}$, then the HASL-30${^{\circ }}$. The vertical distance of the positions between the impact point and shoulder are presented in Fig. 11. The result is $\Delta h_{Q} \,>\Delta h_{P} \, >\Delta h_{H}$. $\Delta h_{Q}$ in Fig. 11 (a) is the largest, and thus has the greatest collision energy.

It was stated that the value of collision energy between balls within a specific range (the specific collision energy) could result in a high probability of breakage [36]. For collision energy above this specific range the particle breaks heavily, and for others the particle does not break [37]. This specific collision energy was usually used to evaluate the particle breakage energy [36, 37]. The total collision energy intensity, $E_{t}$, and specific collision energy intensity, $E_\mathrm{s}$, are proposed to quantify the ball motion and collision behavior in the simulation, they are defined as

$$\begin{aligned} E_{\hbox {t}} =\frac{\sum \limits _{j=1}^N {E_j } }{M_{\hbox {b}} }, \end{aligned}$$

(5)

and

$$\begin{aligned} E_{\hbox {s}} =\frac{\sum {\Delta E_i } }{M_{\hbox {b}} } \quad i\leqslant j, \end{aligned}$$

(6)

where $E_\mathrm{t}$ is the sum of total collision energy per unit mass of the balls, $E_\mathrm{s}$ is the sum of the specific collision energy per unit mass of the balls, $M_\mathrm{b}$ is the total mass of the grinding balls, $E_{j}$ is the energy of collision j, N is the total number of the collisions between balls. $\Delta E_{i}$ is the energy of collision i within the specific range. Larger values of $E_\mathrm{s}$ and $E_\mathrm{t}$ are expected to produce higher production of the fine product [38]. The specific range of the collision energy intensity$E_\mathrm{s}$ of the feed material in our work is within 0.01–0.1 J/g [38]. The collision energy intensity in the grinding process is presented in Fig. 12.

Table 3 Parameters setting in numerical model

Fig. 11

Images of the motion behavior of grinding balls in the three types

Fig. 12

Collision energy intensity versus simulation time

Table 4 Parameters of the curves in Fig. 12

The curves in Fig. 12 can be expressed by linear functions of $y={ax}+b$, where y is the collision energy per unit mass of the balls, that is, $E_\mathrm{t}$ or $E_\mathrm{s}$, and x is the simulation time. The values of the coefficients are reported in Table 4, the coefficient a represents the growth rate of the collision energy intensity. It can be seen that the growth rate of $E_\mathrm{t}$ is larger than that of $E_\mathrm{s}$, but they both have the maximum values in the cases of QASL-50${^{\circ }}$.

The relation of $E_\mathrm{t}$ and $E_\mathrm{s}$ in the three cases is presented graphically in Fig. 13. The linear function is expressed as $y={cx}+d$, where yis the specific collision energy intensity $E_\mathrm{s}$, and x is the total collision energy intensity$E_\mathrm{t}$. The coefficient c means the specific collision energy intensity $E_\mathrm{s}$ to the total collision energy intensity $E_\mathrm{t}$ ratio, $E_\mathrm{s}/E_\mathrm{t}$. Meanwhile Fig. 14 shows the fitted results for the mass fraction of the fine product in Fig. 8. The regressive function is expressed as $y={ex}+f$, where y is the mass fraction of the fine product, and x is the grinding time. The coefficient e is the rate of production of the fine product $F^{*}$. The fitted lines in Fig. 14 are generally consistent with the curves in Fig. 13, then the coefficients of these functions are compared in Table 5. We can see that the values of $c (E_\mathrm{s}/E_\mathrm{t})$ and $e (F^{*})$ are very close to each other in the same case, and in the case of QASL-50${^{\circ }}$, both of them are the largest. Coincidentally, the mass fraction of the fine product $\lambda _\mathrm{p}$ has the same value with the specific collision energy intensity $E_\mathrm{s}$. Based on these findings, it can be see that the mass fraction $\lambda _\mathrm{p}$ and production rate $F^{*}$ of the fine product are related closely to the values of $E_\mathrm{s}$ and $E_\mathrm{s}/E_\mathrm{t}$. The relational equations can be expressed as

$$\begin{aligned} \lambda _{\hbox {p}} =c_1 \cdot E_{\hbox {s}} , \end{aligned}$$

(7)

and

$$\begin{aligned} F^{{*}}=c_2 \cdot \frac{E_{\hbox {s}} }{E_{\hbox {t}} }. \end{aligned}$$

(8)

Combing Eqs. (3), (4), and (7) we have the specific energy input to the mill, $E^{*}$, as

$$\begin{aligned} E^{{*}}=c_3 \cdot \frac{Pt}{E_{\hbox {s}} \cdot M_{\hbox {s}} }, \end{aligned}$$

(9)

where $c_{1}$, $c_{2 }$and$c_{3 }$ are the correlation coefficients.

Fig. 13

Relations of $E_\mathrm{t}$ and $E_\mathrm{s}$ in the three cases

Fig. 14

Fitted result for the mass fraction of the fine product in Fig. 8

Eqs. (8)–(9) show that the mill performance parameters, the production rate the fine product $F^{*}$and the specific energy consumption $E^{*}$, have a direct correlation with the collision energy intensity between balls, which is possible to infer the characteristic of particle breakage and mill energy consumption during grinding process from the collision energy spectrum in the simulation in the future work.

Table 5 Comparison of the coefficients of the lines in Figs. 13 and 14

4 4 Conclusions

• (1) The effects of the type and assembled angle of RPASLs on the breakage of coarse particles are significantly different. The type of RPASLs exhibits a larger degree of variation of particle size distribution than the assembled angle.

• (2) For the silica sand utilized in the experiment, the effective grinding time is necessary to improve the mill grinding efficiency and it differs significantly for different product sizes. For example, the effective grinding time of the coarse product larger than 0.1 mm should be within 60 min to avoid the waste of ball mill energy; The time of the product of 0.1 mm is within 80 min; The time of the fine product of 0.075 mm may be longer than 120 min.

• (3) The mass fraction of 0.1 mm increases up to the maximum value of about 70% with the help of RPASLs, indicating that the RPASLs are effective to collect the coarse particles of 0.1 mm.

• (4) Both the experimental and numerical results show that the optimal structure of RPASLs is the QASL with an assembled angle of 50${^{\circ }}$, which is exhibited by the minimum mill power draw of 303 J per fine product, the maximum production rate of 0.323, and the maximum values of $E_\mathrm{s}$ and$E_\mathrm{s}$/$E_\mathrm{t}$ in the simulation as well.

• (5) The value of the production rate of the fine product $F^{*}$ in the experimental result is consistent with the value of $E_\mathrm{s}$/$E_\mathrm{t}$ in the simulation and the relational equations of the mill performance parameters, $F^{*}$and $E^{*}$, with the collision energy, $E_\mathrm{s}$ and $E_\mathrm{t}$, from the simulation are formulated in the end.