For a rotating trommel, how does the critical speed N vary with drum diameter D? Choose the correct proportionality.

Introduction / Context:
Critical speed is the rotational speed at which material just begins to centrifuge. For rotating cylinders (mills, trommels), it depends on gravity and characteristic radius.

Given Data / Assumptions:

• Trommel approximated as a rotating cylinder.
• Critical condition derived from centripetal balance.

Concept / Approach:
At critical speed, ω^2 R ≈ g. Hence ω ∝ 1/√R and N (rpm) ∝ 1/√D because R ∝ D. Therefore, as diameter increases, critical speed decreases with the square root relationship.

Step-by-Step Solution:
Set ω^2 R = g.Solve ω ∝ 1/√R.Convert to rpm and replace R with D: N ∝ D^-0.5.

Verification / Alternative check:
Mill design rule Nc ≈ 42.3 / √D (D in metres) shows the same dependence.

Why Other Options Are Wrong:
Linear or √D dependence contradicts physics.1/D is too strong; correct dependence is 1/√D.

Common Pitfalls:
Confusing trommel screening speed optimization with true “critical” speed.

Final Answer:
N ∝ D^-0.5