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## 1 Calculation of ball mill capacity

The production capacity of the **ball mill** is determined by the amount of material required to be ground, and it must have a certain margin when designing and selecting. There are many factors affecting the production capacity of the ball mill, in addition to the nature of the material (grain size, hardness, density, temperature and humidity), the degree of grinding (product size), the uniformity of the feeding material, and the portion of loaded, , and the mill structure (the mill barrel length, diameter ratio, the number of bins, the shape of the partition plate and the lining plate).

It is difficult to theoretically determine the productivity of the mill. The grinding mill’s production capacity is generally calculated based on the newly generated powder ore of less than 0.074 mm (-200 mesh).

V — Effective volume of ball mill, m3;

G2 — Material less than 0.074mm in product accounts for the percentage of total material, %;

G1 — Material less than 0.074mm in ore feeding accounts for 0.074mm in the percentage of the total material, %;

q’m — Unit productivity calculated according to the new generation grade (0.074mm), t/(m3.h).

The values of q’m are determined by experiments or are calibrated in production with similar ore physical properties and the same equipment and working conditions. When there is no test data and production calibration value, it can be calculated by formula (1-3).

Di1- Standard mill diameter, m;

K’4 — feed size and product size coefficient of mill.

G3 G4 — The production capacity of existing or experimental mills with newly designed and parameters (feed size or product size calculated according to the new generation 0.074mm level) is shown in Table 1-6.

The values of G1 and G2 above should be calculated according to actual data. If there is no actual data, they can be selected according to tables 1-7 and 1-8.

Table 1-4 Ore Grinding Difficulty Coefficient (K’1)

Ore hardness | Ore Grinding Difficulty Coefficient (K’1) | Ore hardness | Ore Grinding Difficulty Coefficient (K’1) | ||

Protodyakonov coefficient | Hardness level | Protodyakonov coefficient | Hardness level | ||

<2 | very soft | 1.4-2.0 | 8-10 | hard | 0.75-0.85 |

2-4 | soft | 1.25-1.5 | >10 | very hard | 0.5-0.7 |

4-8 | medium | 1.0 |

Table 1-5 Mill Correction Coefficient (K’2)

Mill type | Grate ball mill | Overflow ball mill | Rod mill |

K’2 | 1.0 | 0.9 | 0.85 |

Table 1-6 Relative production capacity of feed size and product size (G3 or G4)

Feeding size /mm | Product size / mm | |||||

0.4 | 0.3 | 0.2 | 0.15 | 0.10 | 0.074 | |

Content of -0.074mm (%) | ||||||

40 | 48 | 60 | 72 | 85 | 95 | |

40-0 | 0.77 | 0.81 | 0.83 | 0.81 | 0.80 | 0.78 |

Feeding size /mm | Product size / mm | |||||

0.4 | 0.3 | 0.2 | 0.15 | 0.10 | 0.074 | |

Content of -0.074mm (%) | ||||||

40 | 48 | 60 | 72 | 85 | 95 | |

0.77 | 0.81 | 0.83 | 0.81 | 0.80 | 0.78 | |

20-0 | 0.89 | 0.92 | 0.92 | 0.88 | 0.86 | 0.82 |

10-0 | 1.02 | 1.03 | 1.00 | 0.93 | 0.90 | 0.85 |

5-0 | 1.15 | 1.13 | 1.05 | 0.95 | 0.91 | 0.85 |

3-0 | 1.19 | 1.16 | 1.06 | 0.95 | 0.91 | 0.85 |

Table 1-7 Particle size of crushed products and G1 value of 0.074 mm grade content

Viscosity of crushed ore | 40-0 | 20-0 | 10-0 | 5-0 | 3-0 | |

0.074mm grade content G1 (%) | Refractory ore | 2 | 5 | 8 | 10 | 15 |

Medium Refractory ore | 3 | 6 | 10 | 15 | 23 | |

easy crush ore | 5 | 8 | 15 | 20 | 25 |

## 2. Calculation of Power, Speed and Medium Loading of Ball Mill

### 2.1 Power calculation

G’ — the quantity of loading medium and material, T;

Dm — effective inner diameter of mill barrel, m;

K’5 — grinding medium coefficient, check table 1-9.

Table 1-9 Grinding Medium Coefficient K’ 5

Medium type | Filling rate | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 |

Siliceous stone | 13.3 | 12.25 | 11.0 | 9.5 | 7.8 |

Large steel ball | 11.9 | 11.0 | 9.9 | 8.5 | 7.0 |

small steel ball | 11.5 | 10.6 | 9.5 | 8.2 | 6.8 |

When the filling rate of grinding medium is less than 35% in dry grinding operation, the power can be calculated by formula (1-7).

n —- mill speed, r/min;

G” —- Total grinding medium, T;

η —- Mechanical efficiency, when the center drive, η = 0.92-0.94; when the edge drive, η = 0.86-0.90.

### 2.2 Rotation Speed Calculation of Ball Mill

\

Critical Speed_

When the ball mill cylinder is rotated, there is no relative slip between the grinding medium and the cylinder wall, and it just starts to run in a state of rotation with the cylinder of the mill. This instantaneous speed of the mill is as follows:

N0 —- mill working speed, r/min;

K’b — speed ratio, %.

There are many layers of grinding media in the mill barrel. It is assumed that the media will be concentrated in one layer, called the “polycondensation layer”, so that the grinding media of this layer will be in the maximum drop, i.e. the calculating speed of the mill when the total impact energy is the largest nj.

Therefore, it is theoretically deduced that the reasonable working speed is

The working speeds of various mills are shown in Table 1-10.

Table 1-10 Working speeds of various mills

Mill type | Ball mill | Rod mill | Tube mill |

Working speed n0 | (0.76-0.88)nj | (0.65-0.70)nj | (0.68-0.76)nj |

In production practice, there are many factors affecting the motion state of grinding media. Therefore, the appropriate working speed should be selected according to the actual situation. In determining the actual working speed of the mill, the influences of the mill specifications, production methods, liner forms, grinding media types, filling rate, physical and chemical properties of the ground materials, particle size of the grinding materials and grinding fineness of the products should be taken into account. The actual working speed of the mill should be determined by scientific experiments, which can reflect the influence of these factors more comprehensively.

### 2.3 Quantity of loading medium

Ball loading capacity

The volume of the grinding medium is the percentage of the effective volume of the mill, which is called the filling rate of the grinding medium. The size of filling directly affects the number of shocks, the area of grinding and the load of grinding medium in the grinding process. At the same time, it also affects the height of the grinding medium itself, the impact on the material and the power consumption. A kind of

The ball loading capacity of the mill can be calculated according to the formula (1-14).

Gra — Quantity of Grinding Medium, T.

Rho s — loose density of grinding medium, t/m3. Forged steel balls; P=s=4.5-4.8t/m3 cast steel balls P=4.3-4.6t/m3; rolling steel balls P=6.0-6.8t/m3; steel segments P=4.3-4.6t/m3_-filling ratio of grinding medium, When wet grinding: lattice ball mill pi = 40% – 45%; overflow ball mill phi = 40%; rod mill phi = 35%. Dry grinding: When material is mixed between grinding media, the grinding medium expands, and when dry grinding is adopted, the material fluidity is relatively poor, material flow is hindered by abrasive medium, so filling rate is low, and the filling rate is between 28% and 35%. The pipe mill is 25%-35%. The void fraction of grinding medium_k=0.38-0.42 and the quality of crushed material accounts for about 14% of the quality of grinding medium.

Size and Proportion of Grinding Medium

In the ball mill, the size and proportion of steel balls have a great influence on the productivity and working efficiency of the mill. For coarse and hard materials, larger steel balls should be selected, for fine and brittle materials, with smaller diameter steel balls, the impact times of steel balls in the mill increase with the decrease of ball diameter, and the grinding between balls increases. The clearance is dense with a decrease of spherical diameter. Therefore, it is better to choose the ball with a larger mass and smaller diameter (loose density) as the grinding medium. The size of the ball mainly depends on the particle size of the material to be ground, and the diameter and speed of the mill can be considered appropriately. Formula (1-15) is an empirical formula for spherical diameter and feed size.

dmax — The maximum diameter of steel ball, mm;

amax — the maximum size of feeding granularity, mm.

After calculating the maximum steel ball diameter, the steel ball ratio in the mill can be calculated with reference to Fig. 2-1 (suitable for cement mill, other mills can refer to).

After choosing the maximum diameter and minimum diameter of steel balls according to technological requirements, material properties, mill specifications and various parameters, and then matching grade, using curves, the accumulative percentage of the mass of each corresponding steel balls loaded into the mill can be found, the actual percentage of the mass can be calculated, and the loading quality of steel balls at all levels can be obtained.

According to the production practice of production enterprises, the relationship between ball diameter and material size is shown in Table 1-11. A kind of

Steel balls are gradually worn out in the process of grinding materials. The wear of drop steel ball is related to its impact force. The wear of grinding steel balls is related to the surface area of steel balls. In general, the steel ball in the grinder has both impact and abrasion effects, so the wear is proportional to the n power of the diameter of the steel ball, and the value of n is between 2 and 3.

Table 1-11 The Relation between Steel Ball Diameter and Material Size

Steel ball diameter db/mm | 120 | 100 | 90 | 80 | 70 | 60 | 50 | 40 |

Feeding size /mm | 12-20 | 10-12 | 8-10 | 5-8 | 2.5-6 | 1.2-4 | 0.6-2 | 0.3-1 |

The quality and surface area of forged steel balls of various sizes are shown in Table 1-12. A kind of

Because of the wear of steel balls in the mill production process, in order to keep the mill stable. Steel balls need to be added regularly.

The maximum diameter of additional steel balls is still determined by the method mentioned above. In addition to the addition of additional steel balls, several smaller diameter steel balls should be added according to production experience.

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