Roll crushers: give the correct geometric relationship for half the angle of nip α in terms of roll diameter d_r, feed particle diameter d_p, and roll gap d_f.

Introduction / Context:
The angle of nip determines whether a particle can be nipped and drawn into the roll gap. It depends on roll geometry and the relative sizes of the particle and the set (gap). Correct geometry is essential for predicting operating windows and selecting roll diameters.

Given Data / Assumptions:

• d_r: roll diameter; R = d_r/2.
• d_p: feed particle diameter.
• d_f: roll gap (distance between rolls).
• Rigid spheres and smooth rolls assumed for the geometric relation.

Concept / Approach:
From the triangle formed by roll center, particle center at contact, and the line of centers, the cosine of the half angle of nip equals the ratio of the distance from roll center to the gap midline over the distance from roll center to particle center at contact. This yields cos α = (R + d_f/2) / (R + d_p/2).

Step-by-Step Solution:
Set R = d_r/2.Distance to gap midline = R + d_f/2.Distance to particle center at contact = R + d_p/2.Therefore cos α = (R + d_f/2) / (R + d_p/2).

Verification / Alternative check:
For a very small gap (d_f → 0), α decreases; for a very small particle (d_p → 0), α → 0, consistent with geometry.

Why Other Options Are Wrong:
Using sin or tan here is inconsistent with the geometric construction.Expressions without the 1/2 factors do not reduce correctly in limiting cases.

Common Pitfalls:
Confusing the friction criterion tan α ≤ μ (separate condition) with the geometric relationship for α.

Final Answer:
cos α = (d_r/2 + d_f/2) / (d_r/2 + d_p/2)