Empirical sizing for ball milling relates the ball diameter (Db, metres) to the feed size (Df, metres) via a grindability constant K (≈ 0.9 to 1.4 for increasing hardness).\nWhich relation is commonly used?

Introduction / Context:
Selecting initial ball size is key for efficient breakage in a ball mill. A practical empirical guideline links ball diameter to the feed top size and grindability, ensuring sufficient impact energy without excessive wear.

Given Data / Assumptions:

• Db = grinding media diameter (m), Df = characteristic feed size (m).
• K is a grindability constant (≈ 0.9–1.4, increasing with hardness).
• We seek a commonly cited proportionality.

Concept / Approach:
Energy transfer in impact breakage scales with ball mass and velocity; larger feed requires larger balls to generate required fracture energy. A frequently used rule is Db proportional to sqrt(Df), which can be written as Db^2 proportional to Df, with K capturing material effects.

Step-by-Step Solution:

Verification / Alternative check:
Industrial practice often begins with a top ball size derived from the square-root relation, refined by trials and Bond Work Index studies.

Why Other Options Are Wrong:

Common Pitfalls:
Using oversized balls that increase wear and reduce fines production; ignoring material grindability (K).

Final Answer:
Db^2 = K * Df