Introduction

The cement manufacturing process consists, basically, of the calcination and in the fusion of a material constituted by, approximately, 94% (w/w) of limestone, 4% (w/w) of clays and 2% (w/w) of iron and aluminum oxides in a rotating oven operating at temperatures of about 1450 °C. In this oven, after the raw materials are subjected to the processes of grinding and dosage, a series of reactions happen, in which the material is sintered and partially cast, originating clinker (Rocha et al. 2011; Andrade et al. 2002). In order to obtain the cement, after the addition of additives, this clinker must be cooled and ground (Andrade et al. 2002; Varela; Vieira 2005).

Grinding is a unit operation with high energetic consumption, consuming around 70% of the electrical energy of a conventional cement production plant (Harder 2003). From the amount of energy consumed in this process, approximately 40% is consumed in the clinker grinding (Jankovic et al. 2004). Despite its energetic effectiveness is very low, ball mills are traditionally the most used devices to the grinding of this material (Camalan and Hoşten 2015; Sridhar, Sankar and Prassad 2016; Mishra et al. 2015).

This device consists of a cylindrical drum that horizontally rotates around its own axis. Inside it, grinding media, which are bigger, heavier, and harder than the ore, are placed. Furthermore, the internal surface of the drum can be equipped with lifters to raise the grinding media to higher heights. The aim of the lifters is to promote the breakage of the mineral by impact of the collisions with these bodies (Rezaeizadeh et al. 2010).

The rotating drums may present different flow regimes (sliding, surging, slumping, rolling, cascading, cataracting and centrifuging), which depend on the rotation speed, degree of filling of the drum, physical properties of the material and geometry of the cylindrical part (Mellmann 2001). Since the diameter of the mineral material must be reduced by the grinding process in the ball mills, it is necessary to operate the drum in the cascading regime, which favors the breakage by abrasion, or cataracting, which favors the breakage by impact. These two regimes have in common the elevation of the material load at higher heights, which provides more energy to the system (Metzger and Glasser 2013).

Cataracting regime holds a very important characteristic: the position of the shoulder and toe points. The shoulder point is the highest angular position that the grinding media can reach before its breakage. The toe point is the lowest angular position where the inside materials contact with the surface of the mill (Powell 2004).

It is of utmost importance that the grinding process may be performed in ideal conditions to provide good effectiveness. However, the experimental analysis of this operation in real scale is costly and time-consuming. Thus, computational modeling comes as an alternative to the determination of ideal conditions, in order to reduce this cost (Lee et al. 2019). The numerical simulations, allied to experimental data, that validate models, can provide satisfactory results for various processes. Some models of ball mills rely on information of collision energy of simulations that do not include mineral particles, only grinding media (Datta and Rajamani 2002; King and Bourgeois 1993; Tavares and Carvalho 2009). In these cases, it must be studied how the collision energy is divided between the particles and the influence of this energy in the product of the process.

To study the movement of grinding media inside the mills, single-phase simulations are used, applying Lagrangean approach, which considers the solid phase as discrete particles (Cleary et al. 2003). A lagrangean method that is widely used for the simulation of granular materials is the Discrete Element Method (DEM). DEM was developed by Cundall and Strack (1979) and uses Newton’s second law to estimate particle motion. A force displacement law for the calculation of contacts between particles and between particle and wall is used to update the unbalanced contact forces (Brandao et al. 2020). From the knowledge of microscopical mechanics properties of the particles and the behavior of interaction between them, DEM is able to model the mechanical, physical and macroscopic behavior of bulk solids (Ramos et al. 2011).

Most of the papers found in literature concerning grinding use DEM (Weerasekara et al. 2013; Cleary and Owen 2019; Rodriguez et al. 2018). Some authors use the energy and impact forces obtained through DEM to the prediction of breakage rates of the particles in grinding studies (Tuzcu and Rajamani 2011; Carvalho and Tavares 2014; Radziszewski and Allen 2014). Recently, Cleary and Owen (2019) explored the spectrum of collisional energy in ball mills. The authors observed that the operational conditions of the mill significantly changed the energy distribution over the load in the equipment. Silva et al. (2019) quantified the number and the intensity of the collisions in a dryer with inert bed, to different speeds of bed rotation. The authors related these variables with the breakage of microalgae adhered to the surface of the inert. It was then observed that the conditions that led to higher number of collisions between the inert particles provided an increase of the performance of the process.

In order to carry out the DEM modeling, it is necessary to determine the input parameters such as the coefficient of restitution, the coefficient of static friction and the coefficient of rolling friction. According to Horabik and Molenda (2016), the methods for determining the parameters are usually for an isolated particle and the set of particles in the bed present a different behavior. Although calibration could lead to parameters values that are not the real ones, the method is widely used as a methodology, whose measurement of coefficients is performed indirectly and represent the real behavior of the system in various conditions (Silvério et al. 2014; Cunha et al. 2016). In rotating drums operating at cascading and cataracting regimes, the calibration can be done adjusting the parameters aiming to find the shoulder and foot points closest to the experimental ones.

The objective of this paper was to evaluate the final granulometric distribution of the clinker in a ball mill operating in different rotation speeds, varying the filling degrees of grinding media and clinker. Then, through DEM simulations, it was analyzed the dynamic of grinding media and the collision forces engaged between them and between the grinding media and the wall. These results were related to the final granulometry of the product obtained in the experimental work.

Materials and methods

Experimental device

The mill consisted in a rotating cylinder with diameter of 24.5 cm and internal length of 47 cm, and four lifters with 3.8 cm width and inclined 45° related to the cylindric wall (Fig. 1). The grinding media used were steel spheres of 35 mm diameter, with density and bed voidage in values of 7890 kg/m3 and 0.368. The clinker used in the grinding had density and bed voidage corresponding to 3039 kg/m3 and 0.557.

Fig. 1
figure 1

Experimental module used in the tests and interior of the mill

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To analyze the influence of the proposed variables, two 3k factorial designs were used. In both designs, drum rotation speeds of 15, 31 and 47 rpm were evaluated, in grinding times of 5, 10 and 15 min. Nevertheless, in the first design, clinker filling degree was varied in 2.5, 5 and 7.5% of the drum volume and filling degree of grinding balls was kept to 4% of the drum capacity. In the second design, clinker filling degree was fixed in 5% of the drum volume and the filling degree of grinding media varied in 2, 4 and 6% of the drum volume.

Numerical study

The simulations were performed using DEM, which tracks the individual course of all particles and considers the interactions that occur between them as much as between them and the rigid walls of the equipment. It is worth to mention that in this paper only simulations with grinding media were performed, without clinker.

Although in some conditions the clincker load is higher than the grinding media load. The density and diameters of the grinding media are much greater than the ores ones. Consequently, the energy of collisions between the grinding media and between grinding media and wall are much bigger than between them and the clinker, and will not affect the grinding media trajectory.

All the simulations were performed using the open software Liggghts®.

Applying Newton’s second law, the movement of each individual i particle may be described as shown in Eqs. (1 , 2), respectively.

$${m_i}\frac{{d{u_i}}}{{dt}}=\sum\limits_{{j=1}}^{{{N_c}}} {\left( {{F_{n,ij}}+{F_{t,ij}}} \right)+{m_i}g}$$
(1)
$${I_i}\frac{{d{w_i}}}{{dt}}=\sum\limits_{{j=1}}^{{{N_c}}} {\left( {{R_i}+{F_{t,ij}}} \right)+{T_{ij}}}$$
(2)

Where ui and wi are, respectively the rotational and translational motion of the particle i; mi e Ii are the mass and the moment of inertia of the particle; Fn,ij, Ft,ij e Tij represent the normal motions, tangencies and the torque between each pair of i-j particles respectively; Nc is the number of particles in contact with i particle; g é the gravity acceleration and Ri is the bolt of i particle.

To the calculation of the normal and tangential motions used in this paper, it was used the Herz-Mindlin non-linear contact model. The main constitutive equations of the model are presented in Table 1, where ess is the coefficient of restitution, Y is Young modulus, G is the shear modulus, v is the Poisson ratio.

Table 1 Constructive equations of the Hertz-Mindlin model
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It was used a software of open code Liggghts® in the numerical simulations. In order to guarantee the stability, the time step in the simulation used was 20% of Rayleigh time which consists of a function of the steel shear modulus. In the simulations of this paper 1 time-step corresponded approximately to 0.085 s.

Calibration of parameters

According to Coetzee (2016), one of the forms to determine the parameters used in a DEM simulation is the variation of the parameters until the response of any bulk property acquired through the simulation corresponds to the experimentally measured result. To determine the values of the static friction coefficient, the rolling friction coefficient, and the restitution coefficient, it was used a central composed design with 3 variables and two replicas in the central point, totaling 16 simulations, which levels are presented in Table 2. To the definition of the levels of the experimental design, parameters found in the literature were used to the simulations of steel spheres using DEM (Cleary, Morrison, Morrell 2003; Cleary and Owen 2019; Owen and Cleary 2015).

Table 2 Coded and real values of the parameters used in the simulations for the grinding media
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The calibration simulations were performed based on the analysis of shoulder and toe points of the grinding media. The objective was to find parameters that led to these angles close to the experimental ones.

In all the simulations it was used shear modulus of \(3.57\times {10}^{6}\) Pa, Young modulus \(5\times {10}^{6}\) and Poisson ratio of 0.2 (Brandao et al. 2020).

Results and discussions

With the purpose of evaluating the final granulometric classification of the clinker for the drum operating in different operational conditions, it was determined the Sauter mean diameter of the particles in different grinding times. The initial mean diameter of the particles was 0.87 mm. Table 3 presents the experimental results of the first factorial design, in which the grinding media filling degree was kept constant, in 4% of the drum volume.

Table 3 Experimental results of average diameter of the particles after 5, 10 and 15 min of grinding, in different clinker filling degree (2.5, 5 and 7.5%), and different rotation speeds (15, 31 and 47 rpm) keeping the grinding media filling degree at 4% of the drum volume
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Analyzing Table 3, it is observed that, when clinker filling degree inside the drum is lower, keeping grinding media filling degree constant, lower are Sauter mean diameters of the clinker, for the rotation speeds of 31 and 47 rpm. In the conditions with less clinker related to the quantity of steel spheres, lower is the probability of effective collisions between the clinker and the grinding media. Different behaviors have been observed for 15 rpm. The condition with clinker filling degree of 5% presented Sauter mean diameter lower than the condition with only 2.5% of clinker filling degree. As the rotation speed is the main variable that changes the flow regime of the drum, this difference may be connected to the change of the flow regime that must be investigated by DEM simulations.

Furthermore, when evaluating the effect of the rotation speed, after the 15 min of grinding, it is observed that the increase of the rotation speed from 31 to 47 rpm did not promote an effective improvement in the grinding as the increase from 15 to 31 rpm. The results of the second design were, then, analyzed, where the clinker filling degree is kept constant, 5% of the drum volume, which are presented in Table 4.

Table 4 Experimental results of average particle diameter after 5, 10 and 15 min of grinding, in different grinding media filling degree (2, 4 and 6%) and different rotation speeds (15, 31 and 47 rpm) with a clinker filling degree corresponding to 5% of the drum volume
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Observing Table 4, among all the quantities of grinding media inside the drum, the condition in which the volume of grinding balls corresponds to 4% of the drum volume and rotation speed of 15 rpm presented the highest decrease of the diameter, after 15 min of grinding, if compared to the initial.

For all clinker and grinding media filling degrees, after 10 and 15 min of grinding, the mean diameters are smaller than at 5 min of grinding. In addition, the largest part of the particles are ground in the first 5 minutes. Up to a certain point, grinding shows effectiveness, however, when it reaches a certain value of diameter of particles, this process is constant.

Generally, it is noticed a decrease in Sauter mean diameter throughout the time. This was expected, once the grinding promotes the reduction of the size of the particles. The results proved the need of evaluating the dynamics of the grinding medias, which were then performed through simulations.

Simulations using DEM

The simulation of the breakage of clinker requires knowledge of population balance and kinetics of breakage of particles, running away from the scope of this paper. As shown by Rodriguez et al. (2018), in the simulations with the presence of only grinding media, at first, the grinding media presents the same collision behavior presented on the simulations with ore. Therefore, it was evaluated only the dynamics of the grinding media inside the equipment, excluding clinker.

For calibration of the input variables of the model, 16 simulations were performed and the simulated values of toe and shoulder angles were compared with experimental data. It was found that the simulation with static friction coefficient of 0.7, restitution coefficient of 0.3 and rolling friction coefficient of 0.05 better corresponded to experimental angles. A toe and shoulder angles of 231.5° and 26.5°, respectively, was observed experimentally and through this simulation (Fig. 2). The input parameters of this simulation were used to the later simulations of the present paper.

Fig. 2
figure 2

Comparison between toe and shoulder angles obtained experimentally and through simulation

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It was initially evaluated the dynamics of the grinding media inside the ball mills, analyzing the effect of the rotation speed. From DEM simulation, data about the number and intensity of the collisions between grinding medias were obtained, as well as impact forces of these collisions, as shown in Fig. 3a, b, respectively.

Fig. 3
figure 3

a Number of collisions and b collisions forces between the grinding medias (particle-particle) per time step, for a drum filling degree of 4% operating at rotation speeds of 15, 31 e 47 rpm

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Analyzing Fig. 3, to the rotation speed of 15 rpm, there are more collisions between the grinding medias if compared to the other speeds. This result was not expected when it was compared to the diameter obtained in the end of the grindings (Table 4). Due to the higher diameters of the particles obtained in the rotation speed of 15 rpm, it was expected less collisions between the particles. It was also expected that with higher rotation speed, higher would be the impact forces among the particles. Nevertheless, it was observed that the collision forces were higher to 15 and 47 rpm. Facing this result, it was then analyzed the number and particle-wall collisions, which were presented in Fig. 4.

Fig. 4
figure 4

a Number of collisions and b collisions forces between the grinding media and wall (particles-wall) per time step, for a drum filling degree of 4% operating at rotation speeds of 15, 31 e 47 rpm

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It can be noted in Fig. 4a that the number of particle-wall collisions is higher for the rotation speed of 15 rpm than for 31 and 47 rpm. However, the highest particle-wall collision force is noted for the rotation speed of 47 rpm, as shown in Fig. 4b. To a greater comprehension of these results, it is necessary the understanding of the behaviors of the grinding medias throughout the time in the experiment. Figure 5 presents the grinding media behavior through a drum cycle.

Fig. 5
figure 5

Images obtained from simulations at five intervals during a drum cycle with 4% v/v and speeds equivalent to a 15, b 31 and c 47 rpm represented horizontally

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From Fig. 5a, it is observed that for the drum operating at 15 rpm, the quantity of particles of grinding media released by the lifters is higher than for other rotation speeds. Consequently, for 15 rpm there is a higher number of particle-particle and particle-wall collisions. Although the number of collisions for this rotation speed is higher, the forces among them are lower, justified due to the damping caused by the first contact with the grinding media before the wall, as shown at t4 from Fig. 5a. This does not happen in the rotation speeds of 47 rpm and 31 rpm, since the drop of grinding media occurs directly on the drum. Furthermore, the shoulder points are lower for lower rotation speeds, reducing the path traveled by the grinding media before suffering the impact. As a result, there is lower energy in the impact. These two facts explain the results in Table 4, in which the grinding, for the drum operating with 15 rpm, is less effective than in the other rotation speeds, despite the higher number of particle-particle collisions and the higher forces among them.

Thereafter, the grinding media filling degree was varied and the number and forces of impact were evaluated. Figure 6 presents the number and the forces of particle-particle collisions in each step of simulation time for the drum operating at 15 rpm and grinding media filling degrees of 2, 4 and 6%.

Fig. 6
figure 6

a Number of collisions and b collisions forces between the grinding media (particle-particle) per time step, for the drum with 2, 4 e 6% by volume of filling degree of grinding media operating with speed of 15 rpm

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Analyzing Fig. 6a, it is observed that the number of collisions between the grinding particles increases as the grinding media filling degree increases. This was an expected result, since higher quantities of grinding spheres inside the drum elevates the probability of them to crash among themselves during the movement. Despite the peaks of collision forces presented in Fig. 6b when the grinding media filling degree is 2%, on the average, the collision forces among the particles are higher for the drum operating with 4% of the volume filled by grinding media. To investigate the reason for this, it was observed the dynamics of the grinding particles in the mill in one cycle of the drum. The graphic representation of the three simulations is shown in Fig. 7.

Fig. 7
figure 7

Images obtained from the simulations at five intervals during a drum cycle with a speed of 15 rpm and a drum filling degree of a 2, b 4 and c 6% v/v

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It can be seen in Fig. 7 that when the drum operates with 2 and 4% of grinding media filling degree, the spheres tend to fall over each other, from a higher point up to a lower one, characterizing the collision forces higher in these cases. For the drum filled with 6% of grinding media filling degree, the spheres presented a different behavior, since there is not a drop, but a rolling among the grinding media when the mill completes its cycle. This explains the lower collision forces in this case, corroborating with the results presented in Table 4 for the rotation speed of 15 rpm.

It was also analyzed the number and intensity of collision forces among the grinding media and the wall, shown in Fig. 8a, b.

Fig. 8 
figure 8

a Number of collisions and b collisions forces between the grinding media and the wall per time step, for the drum with 2, 4 e 6% by volume of filling degree of grinding media operating with rotation speed of 15 rpm

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In Fig. 8a, it can be noted that while the grinding media filling degree increases, the number of particle-wall collisions also increases. As in the case of particle-particle collisions, in this situation it was also expected that the collisions among the grinding media and the drum wall proportionally varied with the quantity of spheres. In Fig. 8b, it is noticed that when the drum operates with 2% of grinding media, some peaks of force of particle-wall collision are observed. This is due to the fact that the grinding media, in lower quantity, tend to drop, mostly, over the drum wall, causing lower particle-particle contact than in the other rotation speeds. Nevertheless, it was observed that the grinding media filling degree of 2% presents the lower efficiencies of grinding. It is plausible to assume that if compared, the particle-wall collisions are less effective related to the particle-particle collisions for the clinker grinding.

Conclusion

From the presented results, it can be concluded that the movement of the particles influenced directly in the granulometry of the final product. In this paper, in which the bed was not completely filled with particles, the increase of rotation speed and the grinding media filling degree did not lead to higher grinding rates. It was possible to observe that, when the drum operated at 15 rpm, lower grinding rates were obtained. An increment of the rotation speed for 31 rpm, increased the grinding rate. Nevertheless, it is worth highlighting that this same increment was not noticed when rotation speed was of 47 rpm. In concern of grinding media filling degree, it was observed better grinding rates when the mill operated at 15 rpm and with a grinding media filling degree of 4%, if compared to loading of 2 and 6%. These results were confronted with the number and the intensity of collisions, analyzed through DEM.

It was observed that the number and the collision forces among the grinding media and between them and drum wall are influenced by the drum rotation speed and grinding media filling degree. The number of collisions, particle-particle as well as particle-wall, is inversely proportional to the rotation speed and directly proportional to the grinding media filling degree. On the whole, analyzing the effect over the diameter of the clinker, it is plausible to attribute that the higher part of the effective collisions to the grinding are from the particle-particle collisions and not from particle-wall collisions.

Comparing the results of the forces and intensity of the collisions through DEM, it was observed that the simulations with higher numbers of collisions did not often lead to the highest grinding rates. Through the analysis of the bed flow regime, it was possible then, to observe the influence of the regime in the final granulometry, as well as to explain the collision forces in each simulated condition, relating with the grinding tests.