Critical speed of a ball mill

Derive equation of critical speed of a ball mill 

The critical speed of a ball mill is the speed at which the centrifugal force exerted by the rotating mill shell on the balls is equal to the gravitational force acting on the balls. At this speed, the balls are held against the mill shell due to centrifugal force and do not fall, which results in no grinding action.

Mathematically, the critical speed is expressed as:

$$N_c = \frac{1}{2\pi} \sqrt{\frac{g}{R - r}}$$

Where:

• $N_c$ = critical speed (in revolutions per second or minute),
• $g$ = gravitational acceleration,
• $R$ = internal radius of the mill,
• $r$ = radius of the grinding ball.

In practical terms, the critical speed is often described in terms of the percentage of this theoretical speed (e.g., 75–90% of the critical speed) to achieve optimal grinding efficiency.

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Critical speed is derived based on the balance of forces acting on a ball in a rotating mill.

The critical speed ($N_c$) in revolutions per minute (RPM) is given by:

$$N_c = \frac{1}{2\pi} \sqrt{\frac{g}{R - r}}$$

Here:

• $g$ = gravitational acceleration,
• $R$ = radius of the mill,
• $r$ = radius of the ball.

Derivation:

1. Force balance: At critical speed, the centrifugal force equals the gravitational force acting on the ball.

   $$m r \omega^2 = m g$$

   where:

   • $m$ = mass of the ball,
   • $\omega$ = angular velocity,
   • $r$ = radius of the circular path.

2. Relating angular velocity to rotational speed:

   $$\omega = 2\pi N$$

   Substituting, the equation becomes:

   $$m r (2\pi N)^2 = m g$$

3. Solving for $N$: Rearranging and simplifying:

   $$N_c = \frac{1}{2\pi} \sqrt{\frac{g}{R - r}}$$

This derivation considers the motion of a single ball at the top of its trajectory, where the centrifugal force exactly counteracts gravitational pull.

The detailed derivation using this framework is discussed in the section on comminution and mill dynamics in R.P. King's book (Chapter 5, Sections 5.7 and 5.15). Pages 160–161 and 202–205 in Modeling and Simulation of Mineral Processing Systems.