# Torque Analysis of a Gyratory Crusher with the Discrete Element Method

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## Abstract

**:**

## 1. Introduction

## 2. Gyratory Crusher

## 3. DEM Model

## 4. Simulation

#### 4.1. Geometry

#### 4.2. Material Parameters

#### 4.3. Simulation Conditions

## 5. Results

#### 5.1. Model Validation

#### 5.2. Base Case

#### 5.3. Effect of Open Side Setting

#### 5.4. Effect of Eccentric Speed

#### 5.5. Effect of Non-Uniform Feeding

## 6. Conclusions

- A Metso 60-110 gyratory crusher has been modeled by using the Discrete Element Method with the software Rocky DEM. The diverse comparisons studied clarified that the developed model correctly predicts the performance of the gyratory crusher. The validation was performed in terms of throughput, product size, and crushing power. Nevertheless, there is a lack of availability of complete data sets of crushers due to the challenges related to instrumenting and the very high cost; thus, it is still necessary to validate with data of different crushers in several operational conditions [20].
- It was discussed that the crushing power obtained from torque produced by radial nodal forces is different from the power calculated with all the forces. This difference is because the work of the transverse forces is considered, overestimating the power by approximately 20 percent. This difference is due to the type of movement of the gyratory crusher; thus, using (4) in order to evaluate crushing power in cone crushers is also recommended.
- The proposed change of variable is a suitable tool for analyzing the behavior of loading forces distribution on the mantle. This change of coordinates permits studying the crushing behavior under different operating conditions.
- Regarding the comparison between Bond’s model and the DEM model, both can accurately predict the crushing power. The model of Bond, which is widely used in mining, is more conservative when calculating power under different operating conditions. As it is only an equation, it is convenient for preliminary calculations. For design, optimization, and power analysis in gyratory crushers, it is recommendable to utilize a DEM model, which allows simulations with a high level of detail and under different operating conditions and diverse configurations.
- Under the ideal simulated conditions, the head spin is less than 10% of the rotational speed of the eccentric. As is an ideal case, the presented values of head spin are the lower limit of the head spin and are only produced by the forces of the particles and do not represent a problem or failure in the machine.
- It is recommendable to operate the gyratory crusher with the full crushing chamber, since the following is the case:
- Larger particles are crushing at a higher vertical position that produces lower torque;
- The power and torque have less temporal variation in comparison to the non-uniform case where the crushing power changes from 0 to 2000 kW; avoiding these cycles of load and unload will reduce the fatigue in the machine elements of the gyratory crusher;
- If the crusher is non-uniformly fed, its efficiency can be considerably reduced. The throughput decrease 34% while the total energy consumption increases 20% regarding the ideal case.

- The definition of the ratio of the minimum simulated size will help to configure DEM simulation of crushers. This relationship is focused on the product size, which is essential in crushers.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Symbol | Definition |

BPM | Bonded particle model |

DEM | Discrete element method |

NUF | Non-uniform filling |

PBM | Population balance model |

PRM | Particle replacement method |

PSD | Particle size distribution |

ROM | Run-of-mine |

Variables | |

$\alpha $ | constant parameter |

$\beta $ | constant parameter |

F | force (N) |

r | position |

v | velocity |

ê | unit vector |

$\gamma $ | tilt angle ($\xb0$) |

Ê | Specific energy (kWh/t) |

$\lambda $ | stabilization parameter |

$\mu $ | friction coefficient |

$\omega $ | angular speed (rad/s) |

$\varphi $ | diameter (m) |

$\psi $ | angle (rad) |

$\sigma $${}^{2}$ | variance |

$\theta $ | angular position (rad) |

$\tilde{A}$ | constant parameter |

$\tilde{\gamma}$ | amage accumulation coefficient |

$\tilde{e}$ | restitution coefficient |

$\tilde{F}$ | percentile of the feed size distribution (mm) |

$\tilde{P}$ | percentile of the product size distribution (mm) |

$\epsilon $ | angle (rad) |

$\phi $ | constant parameter |

A | geometric point |

b’ | constant parameter |

css | closed side setting (mm) |

D | damage |

E | energy (J) |

e | eccentricity (mm) |

H | height (m) |

K | stiffness (N/m) |

k | constant parameter or ratio |

M | mass (kg) |

O | geometric point |

O’ | geometric point |

oss | open side setting (mm) |

P | power (kW) |

Q | geometric point |

r | radius or radial coordinate (m) |

s | overlap (m) |

T | torque (Nm) |

t | time (s) |

W | work (W) |

Subindex | |

$\eta $ | normal |

max | maximum |

min | minimum |

$\tau $ | tangential |

$\epsilon $ | transverse |

c | concave |

d | kinetic |

hs | head spin |

i | index |

l | loading |

lo | lower |

m | machine |

ms | minimum simulated size |

mt | mantle |

p | particle |

r | radial |

ref | reference |

s | static |

sim | simulated |

u | unloading |

up | upper |

w | wall |

x | related to x-axis |

Y | vertical |

z | related to z-axis |

50 | 50th percentile |

80 | 80th percentile |

## Appendix A. DEM

#### Appendix A.1. Contact Model

#### Appendix A.2. Breakage Model

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**Figure 1.**Metso 60-110 gyratory crusher. Concave cut in half, mantle, main shaft, and spider are presented. Some geometrical parameters are presented, such as the height, ${H}_{mt}$; upper diameter, ${\varphi}_{up,mt}$; lower diameter ${\varphi}_{lo,mt}$ of the mantle ($mt$) and the upper diameter ${\varphi}_{up,c}$; and lower diameter ${\varphi}_{lo,c}$ of the concave (c). A geometric scale is drawn at the right bottom.

**Figure 2.**Geometric parameters of the gyratory crusher, force decomposition, and moving frame of reference. Tilt angle, $\gamma $; pivot point, Q; eccentricity at the base of the mantle, ${e}_{0}$; open side setting, $oss$; closed side setting, $css$; axial axis of the main shaft, ${y}_{m}$; the fixed $XYZ$ and moving frame of reference $xyz$ are presented for the simulation time t such that $\theta =\omega t$. A force ${\mathit{F}}_{i}$ is applied in point A, showing their radial ${\mathit{F}}_{r,i}$; transverse ${\mathit{F}}_{\epsilon ,i}$; and vertical ${\mathit{F}}_{Y,i}$ components. The position of A is represented by ${\mathit{r}}_{i}$, $\epsilon $, and $\psi $, and the radial direction is represented by ${\widehat{e}}_{r}$. Both points O and ${O}^{\prime}$ belong to the horizontal plane.

**Figure 3.**(

**a**) Mixer with a mobile rectangular plate rotating around the vertical axis. The force ${\mathit{F}}_{\mathbf{i}}$ is applied at the rectangular plate. (

**b**) Cross-section of the mantle with a vector force, ${\mathit{F}}_{i}$, applied at point A. The node i and all the cross-section has velocity ${\mathit{v}}_{i}$. (

**c**) Rectangular decomposition of the force, ${\mathit{F}}_{i}$, in radial and transverse direction. (

**d**) Rectangular decomposition of the force, ${\mathit{F}}_{i}$, at the x-axis and z-axis.

**Figure 5.**Schematic diagram of the primary crusher used in DEM simulation and the complete setup, showing the truck hopper, crusher feeder hopper, and two rectangular inlets.

**Figure 6.**Non-uniform filling simulation setup. Large rocks are inserted covering a feed inlet, and one side of the chamber is fed.

**Figure 7.**Snapshot of the base simulation at $t=9.86$ s showing a frontal section of the crusher. The particles are colored by their particle size, and the concave was cut to observe the particles. The position of the closed side setting, $css$, is also presented.

**Figure 8.**DEM simulated feed and product particle size distributions. The feed is common for all the simulations, and the product is for the base case.

**Figure 9.**Nodal forces at simulation time $t=9.86$ s. The length and the color of the vector represent the magnitude of the force vectors. The position of the closed side setting, $css$, is also presented.

**Figure 10.**Polar distributions of the base case at simulation time $t=9.86$ s of the base case: (

**a**) radial forces and (

**b**) torque of radial forces. $\epsilon $ is the transverse direction, r is the radial direction, and ${y}_{m}$ is the vertical direction. Each surface plot represents a top view of the mantle.

**Figure 11.**Results of simulations changing the open side setting: (

**a**) mass flow rate and specific energy, (

**b**) power comparison with Bond, (

**c**) head spin, and (

**d**) 80th percentile of the product size distribution.

**Figure 12.**Results of simulations changing the eccentric angular speed: (

**a**) mass flow rate and specific energy, (

**b**) power comparison with Bond, (

**c**) head spin, and (

**d**) 80th percentile of the product size distribution.

**Figure 13.**Snapshot of the non-uniform feeding simulation at $t=12.2$ s showing a frontal section of the crusher. The particles are colored by their particle size, and the concave was cut in order to observe the particles. The position of the closed side setting, $css$, is also presented: (

**a**) frontal view and (

**b**) top view.

**Figure 14.**Eightieth percentile of the product size distribution vs. the vertical position of the particles in the crushing chamber.

**Figure 15.**Comparison between base and non-uniform feeding simulations: (

**a**) crushing power and (

**b**) cumulative mass of the product.

**Figure 16.**Polar distributions of the non-uniform feeding case at simulation time $t=12.2$ s: (

**a**) radial forces, and (

**b**) torque of radial forces. $\epsilon $ is the transverse direction, r is the radial direction, and ${y}_{m}$ is the vertical direction. Each surface plot represent a top view of the mantle.

Variable | Value |
---|---|

Length of the crusher feeder hopper (m) | 18.0 |

Height of the crusher feeder hopper (m) | 9.0 |

Height of the mantle, ${H}_{mt}$ (m) | 4.0 |

Upper mantle diameter, ${\varphi}_{up,mt}$ (m) | 1.4 |

Lower mantle diameter, ${\varphi}_{lo,mt}$ (m) | 3.3 |

Upper concave diameter, ${\varphi}_{up,c}$ (m) | 4.9 |

Lower concave diameter, ${\varphi}_{lo,c}$ (m) | 3.5 |

Eccentricity at the base of the main shaft, ${e}_{0}$ (mm) | 46.6 |

Density (kg/m${}^{3}$) | 7800 |

Inclination, $\gamma $ ($\xb0$) | 0.35 |

Main shaft’s mass (kg) | 176,760 |

Main shaft’s inertia (kg m${}^{2}$) | 126,323.4 |

**Table 2.**Material parameters of the copper ore [18].

Variable | Value |
---|---|

Inlet mass flow rate (t/h) | 70,320 |

Number of particles in the crushing chamber | 75,000 |

Particle shape | Polyhedral |

Particle size (m) | 1.22, 0.732, 0.5, 0.122, 0.07 |

Cumulative particle size distribution (%) | 100, 86.1, 79.39, 50, 39.42 |

Density (kg/m${}^{3}$) | 2930 |

Restitution coefficient | 0.3 |

Static friction coefficient, ${\mu}_{s,p,p}$; ${\mu}_{s,p,w}$ | 0.8; 0.5 |

Dynamic friction coefficient, ${\mu}_{d,p,p}$; ${\mu}_{k,p,w}$ | 0.8; 0.5 |

**Table 3.**Breakage parameters of the copper ore [18].

Variable | Value |
---|---|

${E}_{\infty}$ (J/kg) | 213.5 |

${d}_{0}$ (mm) | 8.07 |

$\phi $ | 1.22 |

$\sigma $ | 0.799 |

${\alpha}_{1.2}/{\beta}_{1.2}$ | 0.51/11.95 |

${\alpha}_{1.5}/{\beta}_{1.5}$ | 1.07/13.87 |

${\alpha}_{2}/{\beta}_{2}$ | 1.01/8.09 |

${\alpha}_{4}/{\beta}_{4}$ | 1.08/3.03 |

${\alpha}_{25}/{\beta}_{25}$ | 1.01/0.53 |

${\alpha}_{50}/{\beta}_{50}$ | 1.03/0.36 |

${\alpha}_{75}/{\beta}_{75}$ | 1.03/0.30 |

$\tilde{\gamma}$ | 5.0 |

$\tilde{A}$ (%) | 67.7 |

${b}^{\prime}$ | 0.029 |

${d}_{min}$ (mm) | 10 |

**Table 4.**Ratio of the minimum simulated size, ${k}_{ms}$ used in DEM simulations of gyratory and cone crushers.

Reference | $\mathit{css}$ (mm) | ${\mathit{d}}_{min}$ (mm) | ${\mathit{k}}_{\mathit{ms}}$ |
---|---|---|---|

Litcher et al. [13] | 5 | 1.5 | 3.33 |

Li et al. [14] | 12 | 4 | 3 |

Delaney et al. [15] | 11 | 8 | 1.375 |

Quist et al. [16] | 34 | 4.8 | 7.08 |

Johansson et al. [3] | 2.2 | 1 | 2.2 |

Chen et al. [17] | 120 | 30 | 4 |

Andre et al. [18] | 4 | 1.9 | 2.11 |

Cleary et al. [20,42] | 11 | 6.7 | 1.64 |

This paper | 111.62 | 10 | 11.16 |

Variable | Value |
---|---|

Open side setting, $oss$ (mm) | 175, 190, 200, 215, 230, 240, 250 |

Closed side setting, $css$ (mm) | 111.62, 126.98, 137.23, 152.51, |

167.72, 177.90, 188.11 | |

Angular speed of the eccentric, $\omega $ (rpm) | 100, 125, 150, 175, 200 |

**Table 6.**Simulation results at different open side settings and $\omega =150$ rpm and crushing power calculated with Bond’s equation. The throughput of the manufacturer (ref.) and the simulated ones with DEM (sim.) are presented. The power calculated by DEM with all the forces (5) and only radial forces (4) calculated by Bond’s equation are tabulated.

$\mathit{oss}$ (mm) | $\dot{\mathit{M}}$ (t/h) | P (kW) | T (kNm) | ${\tilde{\mathit{P}}}_{80}$ (mm) | |||
---|---|---|---|---|---|---|---|

ref. | sim. | DEM Equation (5) | DEM Equation (4) | Bond Equation (1) | |||

175 | 5535 | 4626.5 | 3464.3 | 2910.0 | 1231.3 | 185.3 | 131.7 |

190 | 6945 | 5304.4 | 2378.4 | 2302.7 | 1415.2 | 146.6 | 122.7 |

200 | 7335 | 5910.3 | 2573.2 | 2167.4 | 1411.9 | 138.0 | 159.2 |

215 | 7570 | 6897.5 | 2212.9 | 1863.9 | 1340.5 | 118.7 | 179.0 |

230 | 8280 | 7758.3 | 1619.8 | 1366.0 | 1351.2 | 87.0 | 178.0 |

240 | 8595 | 8473.0 | 1703.3 | 1430.8 | 1329.4 | 91.3 | 197.1 |

250 | 8890 | 9044.9 | 1512.4 | 1277.5 | 1303.8 | 81.3 | 207.9 |

**Table 7.**Simulation results at different eccentric speed and $oss=240$ mm and crushing power calculated with Bond’s equation.

$\mathit{\omega}$ (rpm) | $\dot{\mathit{M}}$ (t/h) | P (kW) | T (kNm) | ${\tilde{\mathit{P}}}_{80}$ (mm) | ||
---|---|---|---|---|---|---|

DEM Equation (5) | DEM Equation (4) | Bond Equation (1) | ||||

100 | 8079.9 | 947.9 | 804.8 | 1104.1 | 76.9 | 210.7 |

125 | 8199.6 | 1316.2 | 1111 | 1183.1 | 84.5 | 202.5 |

150 | 8473.0 | 1703.3 | 1430.8 | 1266.8 | 91.254 | 197.1 |

175 | 8327.3 | 2229.3 | 1883.6 | 1221.6 | 102.76 | 200.0 |

200 | 10,485.0 | 2590.6 | 2187.4 | 1561.5 | 104.3 | 197.7 |

Variable | Base | Non-Uniform Feeding |
---|---|---|

Power draw (kW) | 1979.3 | 1422.1 |

Throughput, $\dot{M}$ (t/h) | 8473.0 | 5556.6 |

Specific energy (kWh/t) | 0.2377 | 0.2559 |

80th percentile of the product size distribution (mm) | 223.52 | 239.27 |

Head spin (rpm) | 2.718 | 6.2621 |

Total crushing time (min) | 4.11 | 6.46 |

Total energy consumption (kWh) | 0.0377 | 0.0455 |

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**MDPI and ACS Style**

Moncada, M.; Toledo, P.; Betancourt, F.; Rodríguez, C.G. Torque Analysis of a Gyratory Crusher with the Discrete Element Method. *Minerals* **2021**, *11*, 878.
https://doi.org/10.3390/min11080878

**AMA Style**

Moncada M, Toledo P, Betancourt F, Rodríguez CG. Torque Analysis of a Gyratory Crusher with the Discrete Element Method. *Minerals*. 2021; 11(8):878.
https://doi.org/10.3390/min11080878

**Chicago/Turabian Style**

Moncada, Manuel, Patricio Toledo, Fernando Betancourt, and Cristian G. Rodríguez. 2021. "Torque Analysis of a Gyratory Crusher with the Discrete Element Method" *Minerals* 11, no. 8: 878.
https://doi.org/10.3390/min11080878