# Rod Mill Product Control and Its Relation to Energy Consumption: A Case Study

^{1}

^{2}

^{*}

## Abstract

**:**

_{300}control parameter, for example), which is a strategic objective of this study. In addition, by controlling other process parameters, such as slurry density or lifter geometry, energy consumption and its subsequent saving and pollution can be controlled, depending on process plant requirements.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Materials and Particle Size Distribution Determination

_{10}, D

_{50}, and D

_{80}, representing the sizes that are under 10%, 50%, and 80%, respectively, of the particle population (Table 1).

_{10}, D

_{50}, and D

_{80}were obtained by linear regression at the target percentage limits, using Excel solver calculation. The F

_{300}is the simple sum of all differential masses below 300 μm mesh size. The same applies to parameter C

_{1050}, but using the population of particles above 1050 μm mesh size (Figure 1).

#### 2.2. Experimental Design

#### 2.3. Mathematical Modeling for Optimization

_{1}is the breakage rate per minute. The parameters α, k, n

_{1}, n

_{2}correspond to the breakage distribution functions, and they are related to material properties such as mineralogy, texture, or in some cases, fracture mechanisms [38,39]. ω

_{1}, β

_{1}, ω

_{2}, and β

_{2}are process model parameters.

^{®}script (R2017a). After that, the model was run to obtain simulations of the different experiments, varying the studied parameters for optimization. They were compared with the experimental values to evaluate the model robustness, by means of RMSE (root-mean-square error) [42].

## 3. Results

#### 3.1. Particle Size Distribution and Energy Consumption Determination

_{300}percentage, reaching undesirable values above 30% in all cases, being critical when the Vc is above 70%, with about 45% of the particle population below 300 μm arising. On the other hand, when the energy consumption is lower, the C

_{1050}parameters become poor, resulting in an insufficient amount of ore being liberated, which is detrimental in terms of material recirculated to the mill (Figure 7A).

_{300}parameter, but the C

_{1050}is still high (Figure 7B). All these observations agree with [45], which presents an energy model based on the grinding media charge and rotational speed of a mill, also affecting mill power draft [1].

_{300}and C

_{1050}are striking. The objectives are to reduce both values, which is not achieved simultaneously when this variation is applied: The F

_{300}reaches the 20% target, but the C

_{1050}is 15% above the expected value, while in all the other cases, the opposite is the case.

_{1050}parameter, reaching about 1%, at the cost of excess grinding, demonstrated by lower D

_{50}and F

_{300}values.

#### 3.2. Numerical Simulation

_{1}normally ranges between 0 and 1.5, but it must necessarily be below the value n

_{2}, which is achieved in this case, reaching average values of 1 and 1.5, respectively. The resulting specific kinetic values vary depending on milling dynamics, and are presented in Figure 8F. The constants ω

_{1}, β

_{1}, ω

_{2}, and β

_{2}are useful for calculating the probability that the particles are under the milling mechanical action, or to be classified by the drag effect of water.

_{1}represents the dynamics under a specific condition. It can be seen how it is inversely proportional to the slurry solid/liquid ratio (Figure 8F), as it affects the simulation of product particles, shown by the displacement of the product curves in Figure 8B. The breakage kinetics were clearly affected by slurry density [17].

_{300}overgrinding and less than 2% in the parameters of the coarsest generation C

_{1050}, and the differences in the D

_{50}values reach 26.6 μm RMSE (Table 7).

## 4. Discussion

_{300}parameter (Figure 9B), from 47% to 20% (Table 5), but with a relatively low 10% energy reduction ratio (Figure 9A). After that, at the same level, feed flow, media charge, and critical rotation speed have an important impact on this issue, with reduction in F

_{300}parameters, from 35% to 45%.

_{300}, the parameter related to finer particle generation. What is also noteworthy is the D

_{50}value (with 20% media charge) being close to 650 μm, far exceeding the 600 μm limit that is considered a well-controlled milling product for this particular material. The fact that in this experiment, the C

_{1050}value was above the target of this study, recommended to be less than 5%, with recirculation to the mill at around 20% of the mass, is not considered a negative option, as recirculation is a common action in ore processing. Considering that the ground material cannot be reprocessed, while coarser particles and overgrinding can be controlled in subsequent recirculation, this case does not represent a crucial problem in this process control study, as the main objective, which is to reduce energy consumption costs, is achieved.

_{50}value, and 20% at F

_{300}, the only failure being at 24% in the C

_{1050}parameter. The energy consumption result is not the lowest in this study.

## 5. Conclusions

- -
- It was shown that by varying the percentage of critical mill speed or grinding media charge, the impact on reducing energy consumption becomes noticeable.
- -
- This work will take on more significance when it is necessary to reach mineral liberation while preventing overgrinding phenomena. In this sense, varying the solid/water ratio may well control excessive fine particle generation. Without being the best action to reduce energy consumption, some improvement in this respect can be observed.
- -
- It was also demonstrated how it is possible to control the particle size of the product by varying the feed rate, and lifter and rod geometry, while keeping energy utilization constant. By applying one or a combination of these criteria, it is possible to achieve both objectives, to control grinding without excessive energy consumption.
- -
- In the overall process, considering comminution as the most expensive module, an improvement in energy consumption efficiency could lead to subsequent savings for companies. A 20% reduction in media charge could lead to a 12% reduction in energy bill, which would be a remarkable milestone. Modeling and prediction approaches can also be used to manage production and energy issues, as simulation could express the product size, considering the parameters that affect the milling process.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Laboratory tumbling rod mill design. (

**A**) Profile view, (

**B**) 35 mm lifter height configuration, (

**C**) 23 mm lifter height configuration.

**Figure 3.**Scheme of methods used to evaluate main operating conditions that govern rod milling with potash ore.

**Figure 4.**Scheme of the population balance model approach, with a perfect mixed mill combined with the piston flow process.

**Figure 6.**Results of the particle size distribution feed and product of all tests, divided by category, where (

**A**) represents V, changing the critical rotation speed, (

**B**) represents M, where the media charge was changed, (

**C**) represents the curves of S, varying the solid/liquid ratio, (

**D**) represents changes in the solid feed flow, (

**E**) represents two tests where the rod diameters were changed, and (

**F**) represents the lifter heights that were varied.

**Figure 7.**Experimental results, where energy consumption and D

_{50}parameter are on the left axis, while F

_{300}and C

_{1050}parameters are on the right axis, when (

**A**) critical rotational speed is varied, (

**B**) media charge is varied, (

**C**) solid/water ratio is varied, (

**D**) feed flow rate is varied, (

**E**) rod size is varied, and (

**F**) lifter geometry is varied.

**Figure 8.**Partial results of the numerical simulation compared with experimental particle size distribution, where (

**A**) experimental values of the tests varying the grinding media charge, (

**B**) experimental values of the tests varying the slurry solid/liquid ratio, (

**C**) simulated values of the tests varying the grinding media charge, (

**D**) simulated values of the tests varying the slurry solid/liquid ratio, (

**E**) simulated and experimental values of the tests when the solid flow is varied, and (

**F**) behavior of the kinetic function S.

**Figure 9.**(

**A**) Evaluation of the energy consumption reduction ratio when an operating condition is varied, (

**B**) evaluation of the reduction ratio of the parameter F

_{300}when an operating condition is varied, (

**C**) best energy performance of each item studied, compared with the control parameters using rod milling. The selected experiments are Vc (50%), media charge (20%), S/L (60%), feed flow rate (3000 g/min), rod size (30 mm), and lifter height (35 mm).

Parameter | Meaning |
---|---|

D_{10} | 10% of the particles are under this size |

D_{50} | 50% of the particles are under this size |

D_{80} | 80% of the particles are under this size |

F_{300} | Percentage of particles less than 300 μm |

C_{1050} | Percentage of particles larger than 1050 μm |

Feature | Value | Units |
---|---|---|

Internal diameter | 388 | mm |

Internal length | 506 | mm |

Rod media charge | 40, 33, 27, and 20 | % |

Rod diameter | 30 and 40 | mm |

Critical speed (Vc) | 71.2 | rpm |

Engine power | 4 | kW |

**Table 3.**Experimental design of the rod mill experiments for modeling purposes. Vc is the critical rotational speed of the mill, the solid feed is 3 kg/min, rod diameter 30 mm, and lifter height 23 mm.

Tests | Vc | Media Charge | S/L |
---|---|---|---|

(%) | (%) | (%) | |

A | 50 | 40 | 50–55–60–65 |

B | 60 | 40 | 50–55–60–65 |

C | 70 | 40 | 50–55–60–65 |

D | 80 | 40 | 35–55–60–65 |

**Table 4.**Experimental design of the rod mill experiments for product control study. Vc is the critical rotational speed of the mill. The tests are grouped into six categories, where V changes Vc, M changes media charge percentage, S changes solid/liquid ratio, F changes solid feed flow, R changes rod diameter, and L changes lifter height.

Test | Vc | Media Charge | S/L | Solid Feed Flow | Rod Dimensions | Lifter Dimensions |
---|---|---|---|---|---|---|

(%) | (%) | (%) | (g/min) | (mm) | (mm) | |

V1 | 50 | 40 | 60 | 3000 | 30 | 23 |

V2 | 60 | |||||

V3 | 70 | |||||

V4 | 80 | |||||

M1 | 60 | 40 | 60 | 3000 | 30 | 23 |

M2 | 33 | |||||

M3 | 27 | |||||

M4 | 20 | |||||

S1 | 60 | 40 | 30 | 3000 | 30 | 23 |

S2 | 55 | |||||

S3 | 60 | |||||

S4 | 70 | |||||

F1 | 60 | 40 | 60 | 1500 | 30 | 23 |

F2 | 3000 | |||||

R | 60 | 40 | 60 | 3000 | 40 | 23 |

L | 60 | 40 | 60 | 3000 | 40 | 35 |

Evaluated Parameter | Energy Consumption (kWh/t) | D_{50} (μm) | F_{300} (%) | C_{1050} (%) | |
---|---|---|---|---|---|

Critical rotational speed [%] | 50 | 8.0 | 395 | 31 | 9 |

60 | 10.7 | 360 | 35 | 5 | |

70 | 11.7 | 325 | 46 | 5 | |

80 | 13.3 | 320 | 47 | 5 | |

Media charge [%] | 20 | 7.8 | 645 | 24 | 21 |

27 | 8.7 | 500 | 31 | 13 | |

33 | 9.3 | 443 | 31 | 11 | |

40 | 10.6 | 377 | 40 | 6 | |

Solid/water rate S/L [%] | 30 | 9.0 | 315 | 47 | 5 |

55 | 9.4 | 350 | 40 | 5 | |

60 | 8.1 | 368 | 38 | 6 | |

70 | 9.3 | 643 | 20 | 24 | |

Feed flow [g/min] | 1500 | 9.1 | 205 | 66 | 1 |

3000 | 8.8 | 368 | 38 | 4 | |

Rod size [mm] | 30 | 8.1 | 368 | 33 | 6 |

40 | 9.8 | 390 | 32 | 4 | |

Lifter size [mm] | 23 | 9.8 | 390 | 32 | 4 |

35 | 9.1 | 405 | 31 | 5 |

**Table 6.**Parameters of the mathematical approach to predict particle size distribution. All parameters are dimensionless.

Parameter | Value |
---|---|

k | 0.500 |

n_{1} | 1.022 |

n_{2} | 1.500 |

S_{1} | - |

α | 3.204 |

ω_{1} | 0.005 |

β_{1} | 2.970 |

ω_{2} | 0.005 |

β_{2} | 4.372 |

**Table 7.**Comparison between experimental key values and simulated parameters using particle size distribution prediction model.

Experimental | Simulated | |||||
---|---|---|---|---|---|---|

Parameter | F_{300} (%) | C_{1050} (%) | D_{50} (μm) | F_{300} (%) | C_{1050} (%) | D_{50} (μm) |

Critical rotational speed | 31 | 9 | 395 | 34 | 8.5 | 410 |

35 | 5 | 360 | 39 | 5.0 | 365 | |

46 | 5 | 325 | 47 | 4.0 | 320 | |

47 | 5 | 320 | 46 | 4.0 | 325 | |

Media charge [%] | 40 | 6 | 377 | 41 | 4.0 | 340 |

30 | 11 | 443 | 31 | 9.0 | 420 | |

31 | 13 | 500 | 30 | 9.0 | 425 | |

24 | 21 | 645 | 21 | 17.0 | 580 | |

Solid/water Rate S/L [%] | 20 | 24 | 643 | 20 | 21.0 | 640 |

38 | 6 | 368 | 41 | 5.0 | 360 | |

40 | 5 | 350 | 49 | 3.0 | 340 | |

47 | 5 | 315 | 48 | 3.0 | 316 | |

Feed flow [g/min] | 66 | 1 | 205 | 72 | 1.0 | 210 |

38 | 4 | 368 | 40 | 4.5 | 355 | |

Rod size [mm] | 32 | 4 | 390 | 37 | 6.0 | 380 |

33 | 6 | 368 | 40 | 4.5 | 360 | |

Lifter size [mm] | 32 | 4 | 390 | 39 | 4.5 | 380 |

31 | 5 | 405 | 35 | 5.5 | 400 | |

RMSE | 4.1 | 1.9 | 26.6 |

**Table 8.**Energy reduction ratio when the media charge is reduced and the starting point is 40% media charge.

% Media Charge | Energy (kWh/t) | Energy Red. (%) |
---|---|---|

40 | 10.6 | 0 |

33 | 9.3 | 12.1 |

27 | 8.7 | 18.1 |

20 | 7.8 | 26.8 |

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**MDPI and ACS Style**

Anticoi, H.; Guasch, E.; Pérez-Álvarez, R.; de Luis-Ruiz, J.M.; Oliva, J.; Hoffman Sampaio, C.
Rod Mill Product Control and Its Relation to Energy Consumption: A Case Study. *Minerals* **2022**, *12*, 183.
https://doi.org/10.3390/min12020183

**AMA Style**

Anticoi H, Guasch E, Pérez-Álvarez R, de Luis-Ruiz JM, Oliva J, Hoffman Sampaio C.
Rod Mill Product Control and Its Relation to Energy Consumption: A Case Study. *Minerals*. 2022; 12(2):183.
https://doi.org/10.3390/min12020183

**Chicago/Turabian Style**

Anticoi, Hernan, Eduard Guasch, Rubén Pérez-Álvarez, Julio Manuel de Luis-Ruiz, Josep Oliva, and Carlos Hoffman Sampaio.
2022. "Rod Mill Product Control and Its Relation to Energy Consumption: A Case Study" *Minerals* 12, no. 2: 183.
https://doi.org/10.3390/min12020183