Critical speed of ball mill

First, the critical speed, the transfer rate

As mentioned above, when the mill is lifted to the point A with the steel ball at the linear speed, since the normal component N and the centrifugal force C of the weight G of the steel ball are equal, the steel ball is parabolic. If the speed of the mill increases, the point at which the steel ball begins to fall is also increased. When the speed of the mill is increased to a certain value Ï… _C , the centrifugal force is greater than the weight of the steel ball, the steel ball rises to the apex Z of the mill and no longer falls, and centrifugal operation occurs. It can be seen that the critical condition of the centrifugal operation is

Figure 1 Stress condition of steel ball during centrifugal operation

C≥G

Let m be the mass of the ball, g be the acceleration of gravity, n is the number of revolutions per minute of the mill, R is the distance from the center of the ball to the center of the mill, and a is the clip of the OA and the vertical axis when the ball is out of the circular path. angle. When the linear speed of the mill is Ï…, when the steel ball rises to point A,

Because G=mg, substitute into the above formula, get

because , substituted into the above formula, get

Take g = 9.81 m / s ² , then ,then

The unit of R is meters.

This is the most basic formula for studying steel ball movement, and it will be used frequently in the future.

When the rotation speed is υ _c and the corresponding number of revolutions per minute is n _C , the steel ball rises to the apex Z and no longer falls, and centrifugalization occurs. At this time, C=G, a=0°, cosa=1, thus

Here, D = 2R, the units are all meters. For the outermost layer of the lining, since the diameter of the ball is much smaller than the inner diameter of the ball mill , R can be regarded as the inner radius of the mill, and D is its inner diameter.

It can be seen from equation (3) that the critical number of revolutions required to centrifuge the steel ball is determined by the distance from the center of the ball to the center of the sail. The outermost ball is furthest from the center of the mill, making it the least number of revolutions required for centrifugation; the innermost ball is closest to the center of the mill, making it the most revolutions required for centrifugation. If the radius of the mill is calculated by the formula (3) as the rotation speed of the mill, although the outermost sphere has been centrifuged, the other spheres can still be thrown, or the ore can be ground. Only when the number of revolutions is much higher than that obtained by the outermost ball according to formula (3), all the spheres are centrifuged, and the useful work of grinding the ore is equal to zero. However, it is desirable that all of the loaded steel balls can drop the ground ore, and if a part is centrifuged, the useful work is reduced. Therefore, the result of calculating the inner radius of the mill by the formula (3) indicates that the limit of the mill speed is not to be made when the outermost sphere is not centrifuged, and it is not necessary to calculate the mill for centrifuging the other layers. Count it. As can be seen from the mountains, the critical number of revolutions of the mill is the maximum speed (rev/min) at which the outermost sphere will not be centrifuged.

Although formula (3) is derived without considering the ball loading rate and sliding, it is still in line with the actual situation when the non-smooth lining is used and the ball loading rate is 40-50%. Therefore, formula (3) is used in the production to calculate the critical number of revolutions of the mill, and most of the mills do not exceed the speed.

Let n be the actual speed of the mill, and its ratio to n _C is expressed as a percentage, called the rate of rotation (Ñ„), ie

Substituting the formula (2) into the above formula, and getting [next]

That is, the angle a marks the position at which the steel ball has risen when it starts to be thrown, called the escape angle. Equation (5) indicates that the higher the rate of rotation, the smaller the angle of departure and the higher the position at which the steel ball rises. When the disengagement angle is 0°, the rotational speed is 1, that is, the actual rotational speed is equal to the critical rotational speed, and the steel ball reaches the apex of the mill, and centrifugalization begins.

Second, supercritical speed operation

When the critical revolution number formula is derived from the front, the frictional force is equal to the tangential component of the weight of the steel ball, and the steel ball does not slide. This assumption is realistic when using a non-smooth liner and a ball loading rate of 40 to 50%. If a smooth lining with a small coefficient of friction is used and the amount of ball is reduced to reduce the positive pressure, then the friction is small enough to balance the tangential component of the ball load, and the steel ball then slides. In this case, although the rotation speed of the mill is several times higher than the n _C calculated by the formula (3), the steel ball does not centrifugally move due to the severe sliding movement. This is the essence and necessary conditions for the mill's supercritical speed operation. RT Huji's research in the 1960s pointed out that under the right conditions, the rotation speed of the mill exceeds 20 times the value of n _C , the steel ball will not be centrifuged, and there is still grinding effect.

The supercritical speed operation not only theoretically breaks the limitations of formula (5), but also improves the processing capacity of the mill. Production practice states that supercritical speed operation can increase mill productivity if applied properly, although power consumption increases correspondingly, but the specific power consumption (ie, kilowatt-hours per ton) is often reduced, as exemplified in the table below.

The speed of the mill and the amount of ball loaded are two key factors affecting the productivity of the mill. Exceeding the value of n _C can increase the productivity of the mill, but it is required to reduce the amount of ball loading, and the reduction of the ball loading will reduce the productivity. Therefore, it is not possible to adopt a method of greatly increasing the rotational speed and greatly reducing the amount of ball loading, which in turn causes a decrease in productivity, as reflected in Figure 2 below. When the ball loading rate is reduced to less than 25%, even if the rotation rate is increased to 145%, the productivity does not reach the ball loading rate of 35 to 40%, and the rotation rate is 110 to 120%. And the rotation speed is too high, the vibration of the mill is very powerful, and it will also cause danger.

The following table Nanfen beneficiation plant to improve industrial mill speed test results

Mills that operate at supercritical speeds are rare after all, and even if they are, they are mostly small mills. In Finland alone, RT Huji has not only conducted systematic experiments and theoretical discussions, but they also used 2700×3600 months of millimeter ball mill to produce ore self-grinding at supercritical speed.