Crushers and grinders: jaw, gyratory, roll, ball mill

Why it matters

Choosing the wrong size-reduction machine wastes power, wears out liners and produces the wrong product size. Plant engineers must know which machine suits which feed size, hardness and target product, and must be able to set the operating speed of a mill and the gap of a roll crusher from first principles.

Key ideas

Forces used. Size-reduction machines break solids by compression, impact, attrition (rubbing) and cutting. Coarse crushers mainly use compression; ball mills use impact and attrition; cutters handle tough fibrous materials.

Classification by duty.

• Coarse (primary) crushers — jaw and gyratory crushers. Feed up to about 1–1.5 m lumps; product roughly 50–250 mm.
• Intermediate (secondary) crushers — cone crushers, roll crushers, hammer mills. Product down to a few millimetres.
• Fine grinders — ball, rod and tube mills, attrition mills; product below about 1 mm down to tens of microns.
• Ultrafine grinders — fluid-energy (jet) mills, agitated media mills; product of a few microns.

Jaw crusher (Blake type). Two jaws form a V-shaped chamber: one fixed, one swinging, pivoted at the top so that the greatest movement is at the narrow discharge end. Lumps are nipped and crushed by compression as they work down; the product size is set by the discharge gap ("set"). Simple and robust, it is intermittent in action — it crushes only on the closing stroke.

Gyratory crusher. A conical head gyrates eccentrically inside a fixed conical bowl, so crushing happens continuously at some point around the circumference. For the same feed opening it has a much larger capacity than a jaw crusher and is preferred for very high tonnages of hard ore; jaw crushers are preferred at smaller capacities and for easier maintenance. Both are primary crushers.

Roll crusher. Two rolls rotate towards each other and nip lumps between them. Whether a lump is drawn in depends on the angle of nip 2α, the angle between the two tangents at the points of contact. A lump is gripped only if tan α ≤ μ (friction coefficient between rock and roll); for smooth steel rolls the maximum nip angle is typically about 32°. This fixes the largest particle that can be fed for a given roll diameter and gap. Smooth rolls give a fairly uniform intermediate product with few fines; toothed rolls handle softer, larger feed.

Ball mill. A horizontal rotating cylinder partly filled (roughly 30–50% by volume) with steel or ceramic balls. At low speed the balls cascade (slide and roll — attrition). At moderate speed they are lifted and fall (cataracting — impact). Above the critical speed the balls are pinned to the shell by centrifugal force and grinding stops. At critical speed, for a ball at the top of its path, gravity just equals the centripetal force needed: m·g = m·ω²·(R − r). Mills are operated at about 65–80% of critical speed. Larger balls are used for coarse feed, smaller balls for fine grinding. Ball mills can grind wet or dry, in open circuit or closed circuit with a classifier returning oversize.

Open and closed circuit. In closed-circuit grinding a classifier or screen returns oversize to the mill, so fines leave as soon as they form; this reduces over-grinding and energy use.

Formulas

n_c = (1/2π)·√(g / (R − r)) (rev/s); in rpm n_c = (60/2π)·√(g/(R − r)) ≈ 42.3 / √(D − d)

• n_c: critical speed; g = 9.81 m/s²; R, D: mill radius and diameter (m); r, d: ball radius and diameter (m). If balls are small, n_c ≈ 42.3/√D rpm.

cos α = (R_r + d_g) / (R_r + r_p)

• α: half the angle of nip; R_r: roll radius (m); d_g: half the gap between rolls (m); r_p: radius of the largest particle that can be nipped (m). Valid for smooth rolls with tan α ≤ μ.

v = π·D_r·N / 60

• v: roll peripheral speed (m/s); D_r: roll diameter (m); N: speed (rpm).

ṁ_max = ρ_b·(2d_g)·W·v

• Theoretical roll-crusher capacity (kg/s) if a continuous ribbon of bulk density ρ_b (kg/m³), thickness equal to the gap and width W (m) passes at roll speed; actual capacity is only a fraction of this (take the factor from your data book).

Worked examples

Example 1 (standard): ball mill speed. A ball mill of diameter 1.8 m uses 80 mm balls. Find the critical speed and a suitable operating speed at 75% of critical.

1. R − r = 0.90 − 0.04 = 0.86 m.
2. n_c = (60/2π)·√(9.81/0.86) = 9.549 × 3.377 = 32.3 rpm.
3. Check: 42.3/√(1.8 − 0.08) = 42.3/1.311 = 32.3 rpm.
4. Operating speed = 0.75 × 32.3 = 24.2 rpm.

Example 2 (GATE level): largest feed to a roll crusher. Smooth rolls 600 mm in diameter are set with a gap of 12 mm. The angle of nip is 32°. Find the largest feed particle that can be nipped, and the theoretical capacity if the rolls are 0.4 m wide, run at 100 rpm and the bulk density is 1600 kg/m³.

1. α = 32°/2 = 16°; R_r = 0.30 m; d_g = 0.006 m.
2. R_r + r_p = (R_r + d_g)/cos α = 0.306/0.9613 = 0.3183 m.
3. r_p = 0.3183 − 0.30 = 0.0183 m, so the largest feed diameter = 36.7 mm.
4. v = π × 0.6 × 100/60 = 3.14 m/s.
5. ṁ_max = 1600 × 0.012 × 0.4 × 3.14 = 24.1 kg/s ≈ 86.9 t/h (theoretical; actual is a fraction of this). The reduction ratio is about 36.7/12 ≈ 3, typical of a single pair of smooth rolls.

Common mistakes

• Running a ball mill "as fast as possible". Above critical speed the charge centrifuges and grinding stops.
• Forgetting the ball size in R − r for large balls in small mills.
• Using the full nip angle 2α in cos α instead of half of it.
• Using the roll diameter where the radius belongs in the nip relation.
• Calling roll crushers fine grinders. They are intermediate crushers; fine powders come from ball, rod or jet mills.
• Quoting the theoretical ribbon capacity of rolls as the real capacity.

For GATE CH

Expect numericals on ball-mill critical speed (often with "operating at x% of critical") and the angle of nip or maximum feed size of a roll crusher. Conceptual questions ask which machine suits primary, intermediate or fine reduction, cascading versus cataracting, and why closed-circuit grinding saves energy. Learn the critical-speed derivation, since variants (different units, ball size included) are common.

Quick check

1. What is the critical speed of a 2 m ball mill with small balls?
2. What happens to grinding above the critical speed?
3. Which crusher crushes continuously around its circumference?
4. In cos α = (R + d)/(R + r), what is d? Answers: 1. 42.3/√2 ≈ 29.9 rpm. 2. The balls centrifuge against the shell and grinding stops. 3. The gyratory crusher. 4. Half the gap between the rolls.