At present, there have been many studies on the theoretical suitable rotation rate of ball mills, but basically they are derived from the energy or impact power of the steel ball when it lands.Impact crushing and grinding crushing are the main crushing forms of materials. Adopting different crushing forms has a great impact on the grinding efficiency. When determining the optimal speed rate, the crushing form of the material should be considered comprehensively.

A. Collision energy and grinding efficiency of ball mill

1. The detachment angle of any layer of steel ball

Assuming that the steel ball moves in layers, regardless of the relative sliding between the steel ball and the liner and the steel ball and the steel ball, the steel ball in motion is divided into dozens of layers. The level of any layer is the distance from the center of the steel ball to the center of the ball mill.

2. Collision speed

When the steel ball lands and collides with the material, the relative tangential velocity between the steel ball and the material is the collision speed.

3. Collision energy per unit time

Assume that the steel ball has a single radius and mass.Since there is no sliding of the steel balls in each layer, and the movement of the steel balls is continuous, the collision energy between the steel balls and the material includes radial collision energy and tangential collision energy.When the ball mill is working, the total radial collision energy of the steel ball and the material per unit time, and the total tangential collision energy of the steel ball and the material per unit time, so the total collision energy of the steel ball and the material per unit time is the ratio of the tangential collision energy per unit time to the total collision energy is called the tangential collision energy ratio.

4. Grinding efficiency

In the work of a ball mill, the radial collision energy plays an impact crushing role on the material, and the tangential collision energy plays a grinding and grinding role on the material.

B. Analysis of various collision energies per unit time of ball mill

Taking the number of layers of steel balls as 11 layers, the filling rate of the ball mill was calculated in detail, and the filling rate corresponding to different speed rates was obtained.

In the range of the rotational speed rate of 0.6~0.9, the inner diameter collision energy, tangential collision energy and total collision energy per unit time all increase and then decrease with the increase of the rotational speed rate; and the tangential collision energy ratio decreases with the increase of the rotational speed rate.

When the rotational speed rate is 0.79, the total collision energy per unit time is large, which is 1.09% and 10.92% higher than the corresponding rotational speed rates of the outer steel ball and the polycondensated steel ball when the throwing height is large, respectively.

When the rotational speed rate is 0.84, the radial collision energy per unit time is large. At the same time, the tangential collision energy ratio is as high as 17.53%. From this, it can be seen that there is still some energy to grind and break the material, and its effect cannot be ignored.

When the rotational speed rate is 0.62, the tangential collision energy per unit time is large. At this time, the tangential collision energy ratio is 62.89%, and there is still a large part of the energy to impact and break the material.

When the rotational speed rate is 0.68, the tangential collision energy ratio exceeds 50%, and most of the collision energy is used for grinding and crushing the material.

C. Excellent speed rate of ball mill

When the impact grinding effect of the ball mill is mainly used, the grinding capacity of the tangential collision energy on the material is much smaller than that of the same radial kinetic energy grinding capacity, but the tangential collision energy ratio at this time is relatively large, so the grinding effect of the tangential collision energy cannot be ignored, but the situation is the opposite when the grinding and grinding effect is the main one.In other words, the same tangential collision energy and radial collision energy have different grinding effects when the material is broken in different forms, so it is not comprehensive enough to optimize the speed rate based on the high value of the collision energy per unit time.

Taking different crushing forms, the optimal speed rate of ball mills varies greatly.As the effect of impact crushing increases, the optimal speed rate of the ball mill increases, and the value of the grinding efficiency factor decreases first and then increases.

The material is mainly broken by impact, and the rotational speed rate corresponding to the grinding efficiency factor is between 0.83 and 0.84, which is the optimal rotational speed rate when the impact is the main force.The material is mainly ground and broken, and the speed rate corresponding to the grinding efficiency factor has a large range of changes.

D. Draw conclusions

Taking a specific ball mill as an example, through the analysis of tangential collision energy, radial collision energy, total collision energy and grinding efficiency per unit time, the specific crushing form of the material is considered comprehensively, and the following conclusions are drawn.

1. When the rotational speed rate is 0.79, the total collision energy per unit time between the steel ball and the material is large; when it is 0.84, the radial collision energy per unit time is large; when it is 0.62, the tangential collision energy per unit time is large.Within the range of commonly used rotational speed rates, the tangential collision energy ratio decreases with the increase of the rotational speed rate.

2. When impact crushing is the main focus, the tangential collision energy ratio per unit time is still large, and the effect of tangential collision energy on the grinding and crushing of the material cannot be ignored; when grinding and crushing are the main focus, the impact crushing effect of the material cannot be ignored.

3. When impact crushing is the main form of crushing, the optimal speed rate of the ball mill is between 0.83 and 0.84; when grinding and crushing is the main form of crushing, the speed rate corresponding to the large grinding efficiency factor has a large range of changes.