Question

In a ball mill, centrifugal force will be exactly balanced by the weight of the ball when mill runs at

• A. Minimum speed
• B. Maximum speed
• C. Critical speed
• D. Optimum speed

Hint:

Understanding the concept of critical speed is crucial for operating ball mills efficiently. Operating too close to or above the critical speed reduces grinding efficiency.

Answer 1

The Correct Option is C

Step 1: Understand the forces acting on a ball in a ball mill.
In a rotating ball mill, the balls are lifted up along the inner wall due to the rotation. Two primary forces act on these balls: Centrifugal Force (\(F_c\)): This force acts radially outward from the center of rotation and is given by \(F_c = m \omega^2 r\), where \(m\) is the mass of the ball, \(\omega\) is the angular velocity of the mill, and \(r\) is the radius of the mill (or the distance of the ball from the center).
Weight of the ball (\(W\)): This force acts vertically downward due to gravity and is given by \(W = mg\), where \(g\) is the acceleration due to gravity.
Step 2: Define the condition for centrifugal force to balance the weight of the ball.
The question states that the centrifugal force is exactly balanced by the weight of the ball. This occurs when the outward centrifugal force component perpendicular to the mill's inner surface at the highest point of the ball's trajectory equals the weight of the ball. However, the most direct balancing occurs conceptually at a specific speed related to the point where the ball would just start to centrifuge or remain stuck to the wall due to centrifugal force overcoming gravity.
The critical speed is defined based on the condition where the centrifugal force on the balls is just sufficient to hold them against the inner wall of the mill at the highest point of their trajectory, preventing them from falling freely.
Step 3: Relate this condition to the critical speed of the ball mill.
The critical speed (\(N_c\)) of a ball mill is the speed at which the centrifugal force on the balls is equal to the gravitational force. At this speed, the balls will start to centrifuge, meaning they will stick to the inner wall of the mill and not fall back down, thus ceasing to perform the grinding action effectively.
The angular velocity at critical speed (\(\omega_c\)) can be found by equating the centrifugal force to the gravitational force (considering the ball at the inner surface of the mill of radius \(R\)): $$m \omega_c^2 R = mg$$$$\omega_c^2 = \frac{g}{R}$$$$\omega_c = \sqrt{\frac{g}{R}}$$
The critical speed in revolutions per minute (RPM) is then: $$N_c = \frac{60 \omega_c}{2 \pi} = \frac{60}{2 \pi} \sqrt{\frac{g}{R}}$$
At this critical speed, the centrifugal force exactly balances the weight of the ball in the sense that it's the threshold where the ball transitions from tumbling/cascading motion (effective grinding) to centrifuging motion (ineffective grinding).
Step 4: Evaluate the given options.
(1) Minimum speed: At minimum speed, the centrifugal force is much less than the weight, and the balls will simply roll or slide at the bottom of the mill, resulting in poor grinding.
(2) Maximum speed: Operating significantly above the critical speed leads to excessive centrifuging, where most balls are stuck to the wall, and grinding is again ineffective.
(3) Critical speed: As explained above, critical speed is the condition where the centrifugal force is just sufficient to balance the effect of gravity on the balls to the point of causing centrifuging.
(4) Optimum speed: The optimum operating speed of a ball mill is typically slightly lower (around 65-80\%) than the critical speed to achieve the best balance of cascading and attrition for effective grinding. At optimum speed, centrifugal force is a significant component but not exactly balanced by weight in the sense of centrifuging.

Therefore, the centrifugal force will be exactly balanced by the weight of the ball in the context of reaching the centrifuging condition at the critical speed.