ORIGINAL ARTICLE

Mitigation of Dynamic Stresses of a Ball Mill Using Rubber Coatings

Dr. Arturo Ocampo-Ramirez^{I  *} 

Dra. Dayvis Fernández-Valdés^II 

MSc. María Victoria Gómez-Águila^III 

Dra. Minelkis Machado-Molina^IV 

Ing. María del Carmen Sigler-Muñoz^IV 

Dr. Ernesto Ramos-Carbajal^V 

^IUnidad de Simulación e Ingeniería Mecánica Estructural-GrupoSSC, Ciudad de México, México.

^IISEPI-ESIME Instituto Politécnico Nacional. Ciudad de México, México.

^IIIUniversidad Autónoma Chapingo, Departamento de Ingeniería Mecánica Agrícola, Texcoco, México.

^IVUniversidad Agraria de La Habana, Facultad de Ciencias Técnicas, San José de las Lajas, Mayabeque, Cuba.

^VEscuela de Estudios Agropecuarios Mezcalapa, Universidad Autónoma de Chiapas (UNACH), México.

ABSTRACT

Ball mills are widely used in mining, agriculture and cement industries. There is little information on the design of such equipment from the structural point of view. Some of the main causes of failure are fractures of their walls, due to the impact of steel balls on their internal surfaces. Normally, they are attenuated through the use of wet milling, which acts as a shock absorber and a casing that protects the internal surfaces of the mill. However, it should be noted that this casing is not used in small mills. To avoid failure, the thickness of the mill is over-designed, increasing its cost. The design of small mills improves when considering the direct impact of the steel balls on the inner wall. This leads to the most critical condition of operation. In this work, the resulting stress field was evaluated following a couple of approaches: (I) the dynamic coefficient of impact loads was evaluated with the working energy principle and (II) a numerical analysis was performed with the Finite Elements Method. The operating parameters were calculated with the BM-Crush Program. The results showed that the cyclic stresses were close to the elastic limit. It was proposed to implement a rubber coating on the internal walls of the mill and the stress field was reduced 8.3 times, for a rubber thickness of 3 mm. In this way, a potential fatigue failure could be reduced.

Keywords: Internal Wall; Load Impact; Direct Impact; Finite Elements

INTRODUCTION

Designing a machine capable of performing its task efficiently, reducing the use of energy, sizing it appropriately and considerably reducing the dynamic stress resulting from the impact to avoid fatigue failures, represents an important challenge (^{Ocampo et al., 2015}). The structural integrity of a mill can be rapidly degraded due to the action of constant impact on the mill walls (^{Osorio et al., 2014}; ^{Solórzano et al., 2014}).

Due to the little information in the literature on the designing of ball mills, from the structural point of view, the study of the dynamic phenomena is of great importance and it has not been reported for them. This allows designing equipment beyond static conditions (^{Ocampo et al., 2016}), optimizing manufacture and operating costs and, therefore, extending the useful life of each of their mechanical components.

The impacts on a structure generate dynamic stresses that can be of high intensity and short duration. These can be defined as events in which a rapid transfer of energy takes place. The impact between two bodies begins when the relative distance between them approaches zero and an impact force opposes the penetration into a body (^{Wang & Mason, 1992}; ^{Gilardi & Sharf, 2002}). Ball mills transform large particles into smaller particles through the action of impact, which leads to the weakening of the internal walls and failure due to fracture in many cases.

Transient stress analysis programs are based on the Finite Element Method and they are time dependent. They solve the problems in an explicit process. Besides, during the period in which the impact takes place, contact between surfaces, large deformation and multiple nonlinearities occur. Therefore, the total solution is calculated through small intervals of time. If large time steps are used, the calculation will be expedited, but the end result will be less accurate. However, if small steps are used, the computation time will be increased, but the end results will be very accurate.

Based on the abovementioned, the objective of this work was to study the effect of dynamic stress on the internal walls of the ball mill under critical impact conditions and propose a rubber coating to mitigate stress allowing a longer life of the equipment.

MATERIALS AND METHODS

The impact can be divided in two phases: the compression and the expansion. The first one begins when the relative distance between the bodies is equal to zero and the relative velocity between them is greater. This phase ends when the relative velocity between the bodies is zero, leading to the expansion phase, which ends when the impact force is zero. The resultant force of the impact occurs over the contact surfaces of the bodies, resulting in a certain distribution of stresses, which depend on the geometry of the contact surfaces according to Hertz Theory (^Hertz,1882, ^{Goldsmith,1960} and ^{Gilardi and Sharf, 2002}).

Velocity is the simplest parameter to classify the different types of impacts.

1. Subsonic (v<50 m/s). Elastic and plastic deformations take place.

2. Transonic (50 m/s < v < 500 m/s) There is generalized plastic deformation.

3. Supersonic (500 m/s < v < 2000 m/s) There is viscous resistance of the material.

4. Hipersonic (2000 m/s < v) The material is considered as a hydrodynamic fluid.

In the case of this paper, a subsonic problem was considered, following the approach of ^{Meoa et al. (2003)}; ^{Kubair (2004)} and ^{Flores et al. (2014)}. The analytic solution was based on the principle of work-energy (Figure 1).

In this paper, a dynamic structural analysis was performed (Table 1), in which the impacts were considered to occur in the interior of a ball mill. In addition, a coating on the interior walls of the mill was considered, allowing a significant reduction of the impact stresses and enabling a longer life of the equipment.

TABLE 1 Dynamic analysis with the Finite Element Method 

Solutions	Explicit
Loading Conditions	Dynamic
Geometrical and Mechanical Characteristics	Nonlinear (material, geometry, time and combination)
Process time	Forces applied for short periods of time
Time and computational resources	Relatively long Relatively expensive
Resolution	Multiple iterations Total time divided Δt very small process

Method of Bond

This method was used to determine the mill size in the BM-Crush Program. The mill diameter was obtained with Equation 1. The length depends on the ratio (Length/Diameter, L/D) that was supposed from the beginning of the calculations. K _B is a constant which is equal to 4.456x10^-5 for dry milling and Φ _c is the critical velocity.

$D={\left[\frac{{0.94*P}_a}{K_B{\left(J\right)}^{0.461}{\left({\varphi }_c\right)}^{1.505}\left(\frac{L}{D}\right)}\right]}^{\frac{1}{3.5}}$ (1)

The mechanical power P _a was obtained with Equation 2, given by Bond, where; (d ₈₀ and D ₈₀ ) are the particle size at the beginning and at the end of the mill process, respectively. They are expressed in meters. C is the capacity of the mill in tons/hour (^{Duda, 2003}).

$P_a=10w_i\frac{1}{0.907}\left[\frac{1}{d_{80}}-\frac{1}{D_{80}}\right]*C$ (2)

The Bond index (w _iD ) must be adjusted by means of (w _i ) for other operating conditions using Equation 3, where K _j depends on diverse factors that can be obtained from tables and other numerical procedures (^{Duda, 2003}; ^{Wills, 2006}; ^{Neikov et al., 2009}).

$wi=KjwiD$ (3)

w _iD depends on the diameter of the ball mill and it is corrected with equation 4.

$W_{iD}=\left\{\genfrac{}{}{0pt}{}{{\left[\frac{2.44}{D}\right]}^{0.2}D\le 3.81m}{0.914D>3.81m}\right.$ (4)

Finally, the maximum size of the balls was determined by the following expression developed by Bond (Equation 5) (^{Wills, 2006}).

$d_{B1}=25.4\left[{\left(\frac{D_{80}}{cbm}\right)}^{1/2}{\left(\frac{{\rho }_s{w}_i}{100{\phi }_c{\left(3.281D\right)}^2}\right)}^{1/3}\right]\left[mm\right]$ (5)

where:

ρ s =	density of the material to be milled in gr/cm^3
cbm=	constant ball mill = 350

The initial operating conditions (Table 2) for sizing mill (length, diameter, velocity, mechanical power, size, distribution of steel balls and dry milling) were obtained using the flowchart that shown in the Figures 2 and 3.

TABLE 2 Operating Conditions of the ball mill 

Required production	50-100	kg/h
Bond index	16-20	kWh/ton
Density of the material to be milled	2500	kg/m^3
Input ratio	99-127	------
Electromechanical efficiency	90-94	%

Impact Forces

The analytical study of the impact forces considered in this paper was based on the principle of work-energy. In Figure 1, the mass m falls from its initial position until the spring is deformed to a certain length δ _m , reaching a momentary stop. Subsequently, the potential energy of spring is released taking as a baseline the lowest position of mass m, the equation of the energy balance is:

$P\left(h+{\delta }_m\right)=\frac{1}{2}{k}_r{\delta }_m^2$ (6)

where, k _r is the spring stiffness, g is equal to 9.81 m/s², P is the weight of the body and h is the height of fall of the body.

$\frac{1}{2}{k}_r{\delta }_m^2-P{\delta }_m-Ph=0$ (7)

Solving this quadratic equation and taking only the positive root:

${\delta }_m=\frac{P}{k_r}+\sqrt{{\left(\frac{P}{k_r}\right)}^2+2h\frac{P}{k_r}}$ (8)

The term P/k _r is the spring deflection under static load which can be replaced by the static deformation δ _st , leaving the following equation.

${\delta }_m={\delta }_{st}\left[1+\sqrt{1+\frac{2h}{{\delta }_{st}}}\right]$ (9)

FIGURE 1 Balance of energy in accordance with the principle of "Work - Energy". 

Finally, the dynamic load coefficient (Equation 10) is obtained by making δ _st = 1. It is valid in the elastic range. In this case, it was necessary to represent the dynamic load coefficient in function of the velocity of the ball (Equation 11). These equations are like those presented by ^{Budynas & Nisbett (2015)}.

${Kd}_i=1+\sqrt{1+\frac{2h}{{\delta }_{st}}}$ (10)

${Kd}_i=1+\sqrt{1+\frac{v_{fi}^2}{{g\delta }_{st}}}$ (11)

The velocity of the impact (V _fi ) is obtained by the following equation:

$V_{fi}^2=V_{ii}^2+2gh$ (12)

with;

$V_{ii}=\sqrt{\frac{sen\left(\phi \right)}{kr}}$ (13)

$sen\left(\phi \right)=k{N}^{2}r$ (14)

$k=\frac{4{\pi }^2}{g{60}^2}$ (15)

where, N ² is 43.89 rpm, $sen\left(\phi \right)$ represents the camber angle and, finally, r is the radius of the mill. In this case, it is 0.297 m.

Characterization of the Models Used in the Analysis

The maximum deflection due to static conditions must be calculated in order to determine the dynamic impact factor. For the determination of the design parameters, the BM-Crush Program was initially developed by Ocampo et al. (²⁰¹⁵, ²⁰¹⁶). It is based on the equations developed by ^{Bond (1960)} and ^{Wang et al. (2014)}. Its purpose is to calculate the dimensions of the mill, the rotational speed, the number and size of balls and their thickness. These parameters were the initial conditions in the numerical analysis using the Finite Elements Method. The flowchart of the code is illustrated in Figure 2. A subroutine that estimates the magnitude of the impact has been added in this paper (Figure 3).

FIGURE 2 Implementation of the BM-Crush Program. 

The Finite Elements Method (FEM) was used to evaluate the dynamic stresses in the internal walls of the ball mill. The initial conditions were carried out through the development of a numerical program implemented in a Matlab code. Table 3 shows the design parameters estimated in accordance with the flowchart of the program BM-Crush (Figure 2), considering the conditions set out in Table 3. Structural Steel was considered in all the analysis. Its Young´s modulus and Poisson were 260 MPa and 0.3, respectively.

The Mooney Rivlin formulation was used. It is based on a linear combination of two constants (C ₁ =1.65 MPa and C ₂ =-0.44 MPa), described as a function of strain energy. These constants were obtained experimentally following the methodology available in the open literature (^{Wood, 1977}; ^{Yu & Zhao, 2008}).

TABLE 3 Characteristics of models utilized in the analysis 

Case number	Case One		Case Two
Model number	Model 1	Model 2	Model 1	Model 2
Impact velocity	3.46 m/s	3.46 m/s	3.46 m/s	3.46 m/s
Radius of the ball	32 mm	(90 balls) 25 mm	25 mm	25 mm
Thickness of the wall	(12.7, 15.9, 19.1) mm	12.7 mm	12.7 mm	12.7 mm
Thickness of the coating				3 mm

Two cases were analyzed: (1). In first instance, two FEM models were developed using one half of the ball mill. In the first one, the impact of a single ball over the inner wall was simulated. Three different wall thicknesses were evaluated. In the second model, the simulation was done when ninety balls impacted the inner wall simultaneously. A single thickness was selected from previous model; (2). In the second case, two additional FEM analyses were developed. One eighth of the complete ball mill model was considered. The first evaluation was done without any coating and, in the second one, the structural integrity of the mill with a rubber coating was evaluated, which contributes greatly to the mitigation of dynamic stress. The thickness of this coating was 3mm.

Once those operation parameters were determined, the three preliminary numerical simulations were performed (model 1 of the first case). The purpose was the evaluation of the dynamic stress field on the mill wall. Three different thicknesses (12.7 mm, 15.9 mm and 19.1 mm.) were evaluated. In accordance with the results, the thickness of 12.7 mm was chosen, since the stress is within the elastic range. More detailed analyses were carried out. The transient stress field was obtained, when ninety balls impact simultaneously on the inner wall. In accordance with the requirements of the milling process and the calculation made with the Bond Method, it was estimated that 900 steel balls were required (^{Ocampo et al., 2015}). Based on experimental and numerical tests (^{Sun et al., 2009}), a critical condition is developed when 90 steel balls fall at the same time. For this reason, this situation was considered in the stress analysis.

The relevant parameters of the finite element evaluations are summarized in Table 4.

TABLE 4 Parameters considered in the numerical analyses 

	Case One		Case Two
Element type	Shell-163	Shell-163	Solid-164	Solid-164
Integration formulation	Belytschko-Tsay	Belytschko-Tsay	3-D	3-D
Mesh type	Lagrangian	Lagrangian	Lagrangian	Lagrangian
Number of elements	80000	57800	32000	64000
Time elapsed (Δt) (seconds)	9e-5	4e-5	1e-4	1e-4
Simulation Time	0.004 sec	0.00176 sec	0.004 sec	0.006 sec

In a second step, the stress field, when a ball impacts the internal wall, was evaluated. Such wall was uncoated. Finally, the performance of a coated internal wall was evaluated. Such coating was made of rubber and its thickness was 3 mm. The general procedure of this methodology is shown in Figure 3. All the stress analyses were done with ANSYS-LS-Dyna Code Ver. 2020. and BM-Crush Program. Figures 4, 5 and 6 show the boundary conditions.

FIGURE 3 Implementation of initial and boundary conditions for numerical simulation. 

In the first evaluation of case one (Figures 4 and 5), the stress field was obtained when a single ball impacted the center of the model. As the geometry of the cylindrical part is symmetric, only one half was modelled with Shell 163 elements, in order to save computational resources. Their edges were fixed.

The initial velocity of the ball was 3.43 m/s. For all cases, the inner radius of the ball mill was of 297 mm and the thickness was varied (12.7 mm, 15.9 mm, and 19.1 mm.) (Figure 4). This result was compared analytically.

FIGURE 4 Initial and boundary conditions of the mill for different thicknesses. 

In the next analysis, the dimension of the dominium of analysis was the same as the one mentioned above. The edges were fixed. The structural integrity was evaluated when the wall of the mill was impacted with 90 steel balls simultaneously (Figure 5). The velocity of the balls was 3.453 m/s.

FIGURE 5 Ninety balls impacted simultaneously the inner wall. 

For both models of case two (Figures 6(a) and (b)), the initial velocity of the ball and the element type selected were 3.46 m/s and Solid 164, respectively. Besides, the symmetry of the geometry studied was taken into account and only one eight was evaluated. In the first evaluation, the metallic wall was only considered as shown in Figure 6a.

FIGURE 6 Impact of one ball in one eighth of the cylindrical geometry of the ball mill. 

The second case evaluated the mitigation of the transient stresses over the internal wall of the ball mill. For this purpose, a rubber coating was considered. Its thickness was 3 mm and it was considered as a hyper elastic model (Figure 6b). Since the assessed stress is a localized phenomenon, it was considered that the stiffness does not greatly affect the results, when a model of one-half and one-eighth of the ball mill was used.

RESULTS AND DISCUSSION

Case One (Model 1)

The thickness of the inner wall plays an important role. As it was increased, the stresses were reduced. The lowest von Mises equivalent stress (110 MPa) was obtained when a plate of 19.1 mm was used. However, it was decided to use a plate of 12.7 mm, because it makes easier the mill manufacture. In this case, the peak stress (142 MPa) must be mitigated (Figure 7). The stress is considered between the instant impact on the inner wall of the mill and when the relative velocity between the bodies is zero. It is in this time that the maximum stress is obtained.

FIGURE 7 Stresses of von Mises for Case One (model 1). 

The stress field is reduced when wall thickness is increased. In all the analyzed cases, the yield stress (260 MPa) was not exceeded. The variation of the peak stresses with respect to the thickness is illustrated in Figure 8.

FIGURE 8 von Mises stress in dynamic conditions. 

Case One (Model 2)

During the mill process, several balls impact the inner wall simultaneously following a random pattern. This situation is difficult to model. The diameter and weight of each one of the balls were 50 mm and 0.514 kg, respectively. The severity of this loading condition was evaluated without any coating. In this case, the distribution of the stresses shown is divided into small periods of time (Figure 9a-Figure 9f) where the stress is maximum for each impact of 90 steel balls, after the relative velocity between mill wall and balls is zero. It can be seen in Figure 9g-Figure 9l that stresses are spread creating areas of high stress concentrations that are located in the mill border.

FIGURE 9 Propagation of the stress wave on the internal walls of the ball mill. 

The results showed that the peak stresses were around 194 MPa, as shown in Figure 10.

FIGURE 10 Maximum and punctual stress of von Mises. 

The simultaneous impact of the steel balls on the internal wall of the mill, generated significant stress, when the plate thickness was 12.7 mm. The maximum stress was 169 MPa. It resulted from the propagation of the stress waves. Figure 11 shows the stress field around the impact area. However, a transient von Mises peak stresses is also shown. It was 194 MPa and is the 75% of yield stress. It was estimated that the peak stresses take place around 2400 times in one hour.

FIGURE 11 Evolution of von Mises stress in dynamic conditions 

Case Two (Model 1)

The impact of a ball over the internal wall (thickness 12.7mm.) was simulated (Figure 12). Only one eighth of the complete cylindrical wall of the mill was evaluated. It was expected that this reduced model will give simplified data about the impact process. At the same time, computation time is saved. The final evaluation was not affected, when the maximum von Mises stress was obtained in the contact area. In other words, this transient stress field is generated at every impact point. Such field is generated randomly on the inner wall. Therefore, the idea is to evaluate the transitory stress field at every point, which takes place repetitively during the milling process.

FIGURE 12 von Mises stress for an uncoated wall. 

This approach also increased the possibility to generate a finer mesh on point of impact over the inner wall. In this circumstance, the maximum stress of the von Mises was 115 MPa. Differences between the stresses of 142 MPa and 115 MPa are due to the size of ball used (r ₁ = 32 mm) and (r ₂ = 25 mm). The purpose was to observe the stresses distribution, when the size of the balls was changed.

Case Two (Model 2)

In order to minimize the effect of cyclic stresses, it was proposed to cover the inner mill walls with a 3 mm rubber layer. This will absorb and dissipate the impact energy, avoiding the need to increase the thickness of the steel mill. The impact over coated wall with rubber was simulated (Figure 13). This material was considered as a hyper-elastic. Its thickness was 3 mm. The Mooney Rivlin formulation was used. It is based on a linear combination of two constants (C ₁ =1.65 MPa and C ₂ = -0.44 MPa). They are described as a function of strain energy. These constants were obtained experimentally. The maximum von Mises stress was 13.8 MPa.

FIGURE 13 Maximum stress of von Mises with coated. 

The dynamic behavior of the inner wall was compared with the cases analyzed in case two (Figure 14). In first instance, the direct impact of a ball over the inner wall generated fluctuant stresses. This situation was diminished when the rubber coating was used.

FIGURE 14 Evolution of von Mises stress for steel versus rubber. 

In this case, some of the impact energy was absorbed, and the variation of the stresses was reduced notably. In order to get a better idea, the impact over a plate of rubber was analyzed. The level of stresses was reduced. The von Mises stresses, for the condition in which the walls of the mill are uncoated, were sub-damped with three peaks (115, 45 and 35) MPa (Figure 14). It took place at the boundary of the impact point.

Regarding the inner wall coated with rubber, the maximum von Mises stress was reduced 8 times in comparison with the stresses that took place with uncoated conditions. The peak stress was 5% of the yield strength of the shell. The resultant stress fields were compared in Figure 15.

FIGURE 15 Resultant von Mises stress field. (a). Uncoated inner wall. (b) Inner wall coated with rubber. 

Finally, the numerical solution was compared with the results of a simplified analytical procedure. It is based on the "Work-Energy" principle (Figure 16). The results show that the numerical calculations are 23% higher than the analytical estimates. It has to be kept in mind that the analytical model is based on the deformation. The simplifications imply that greater deformations were estimated at the impact area with the finite element method. Therefore, the stresses will be underestimated with an analytical procedure.

FIGURE 16 Comparison of numerical and analytical von Mises stress. 

CONCLUSIONS

• With the implementation of the BM-Crush Program, the design of a ball mill is more efficient, allowing the evaluation of different operating conditions.

• The thickness of the mill was estimated at 12.7 mm under dynamic impact conditions according to the Work-Energy principle.

• The structural integrity of the ball mill was considerably improved with the application of a rubber coating on its internal walls.

• Rubber significantly absorbs and dissipates impact energy, achieving an 8-fold reduction in von Mises stresses compared to stresses in uncoated conditions and a decreased risk of fatigue failure.

ACKNOWLEDGEMENTS

The authors kindly acknowledge the support given by CONACyT and the Instituto Politécnico Nacional.

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⁸The mention of trademarks of specific equipment, instruments or materials is for identification purposes, there being no promotional commitment in relation to them, neither by the authors nor by the publisher.

Received: December 12, 2019; Accepted: September 25, 2020

^*Author for correspondence: Arturo Ocampo-Ramirez^I, e-mail: [e-mail]

Arturo Ocampo-Ramirez, Dr, Investigador, Unidad de Simulación e Ingeniería Mecánica Estructural-GrupoSSC, Ciudad de México, México, e-mail: [e-mail]

Dayvis Fernández-Valdés, Dra, Investigador, SEPI-ESIME Instituto Politécnico Nacional. Ciudad de México, México, e-mail: [e-mail]

María Victoria Gómez-Águila, MSc, Profesora-Investigador, Universidad Autónoma Chapingo, Departamento de Ingeniería Mecánica Agrícola, Texcoco, México, e-mail: [e-mail]

Minelkis Machado-Molina, Dra, Profesora, Universidad Agraria de La Habana, Facultad de Ciencias Técnicas, San José de las Lajas, Mayabeque, Cuba, e-mail: [e-mail]

María del Carmen Sigler-Muñoz, Ing, Profesora, Universidad Agraria de La Habana, Facultad de Ciencias Técnicas, San José de las Lajas, Mayabeque, Cuba, e-mail: [e-mail]

Ernesto Ramos-Carbajal, Dr, Profesor-Investigador, Escuela de Estudios Agropecuarios Mezcalapa, Universidad Autónoma de Chiapas (UNACH), México, e-mail: [e-mail]

The authors of this work declare no conflict of interests.