Ball mill operation:\nRelative to the critical speed, how should the operating speed of a ball mill be chosen for efficient grinding?

Introduction / Context:
Ball mills rely on cascading and cataracting of grinding media to produce impact and attrition on the charge. The critical speed is the rotational speed at which centrifugal force equals gravitational force on a mill charge at the shell, preventing media from falling. Understanding the relation between operating and critical speed is central to mill performance.

Given Data / Assumptions:

• Critical speed N_c is defined for a given mill diameter.
• Grinding effectiveness requires media to fall and slide, not stick to the shell.

Concept / Approach:
Operating a ball mill below the critical speed (commonly 60–80% of N_c) promotes cascading and cataracting motions, enabling effective breakage. At or above N_c, media cling to the shell and cease to fall, drastically reducing grinding. Excessively low speeds can lead to poor lift and low energy input. Therefore, “less than the critical speed” is the correct qualitative choice.

Step-by-Step Solution:

Verification / Alternative check:
Mill manuals and plant practice confirm setpoints below N_c to obtain desired media trajectories and power draw characteristics.

Why Other Options Are Wrong:

• Equal to or greater than N_c: media centrifuge; grinding drops sharply.
• “Much more than” or “slightly more than”: both imply centrifuging.

Common Pitfalls:
Confusing “critical” with “optimal”; optimal is typically a fraction of N_c, not at or above it.

Final Answer:
Less than the critical speed