Centrifugation and centrifugal separators

Why it matters

Particles of a few microns, or liquids whose densities differ only slightly, settle far too slowly under gravity for any practical tank. Spinning the suspension multiplies the driving acceleration by thousands, so centrifuges clarify fermentation broths, separate cream from milk, dewater crystals and sludges, and break oil–water emulsions in a fraction of the volume a gravity settler would need.

Key ideas

Centrifugal acceleration. A body rotating at angular velocity ω at radius r experiences an acceleration a_c = ω²·r directed outward (in the rotating frame). The ratio a_c/g is the relative centrifugal force (RCF) or "g-number"; industrial centrifuges run from a few hundred to over 10 000 g. ω in rad/s = 2π·N/60 with N in rpm.

Centrifugal settling. Replace g by ω²r in the terminal-velocity force balance. In the Stokes region the radial velocity is u = ω²·r·D_p²(ρ_p − ρ)/(18μ), so it increases with radius and the particle accelerates outward as it moves. Check Re_p as in gravity settling. If ρ_p < ρ (oil droplets, cream), the particle moves inward.

Time to cross the bowl. Integrating dr/dt = ω²r·D²Δρ/(18μ) from r₁ (liquid surface) to r₂ (bowl wall) gives t = 18μ·ln(r₂/r₁)/(ω²·D²·Δρ). A particle is removed if this is shorter than its residence time V/q, where V is the liquid volume held in the bowl and q the throughput.

Sigma (Σ) concept. Writing the throughput as q = 2·u_g·Σ, where u_g is the particle's gravitational settling velocity, isolates a single number Σ (m²) that depends only on the machine geometry and speed — the area of a gravity settler with the same performance. Σ lets you compare and scale centrifuges of different types handling the same feed. (Definitions of Σ differ slightly between texts according to which particle — completely removed or 50% removed — is chosen; use the one in your textbook consistently.)

Liquid–liquid separation. In a two-liquid bowl, the heavy liquid forms the outer layer and the light liquid the inner layer. The interface position is set by the radii of the two overflow dams, through a hydrostatic balance in the centrifugal field; the radius of the "neutral zone" r_i follows from ρ_A(r_i² − r_A²) = ρ_B(r_i² − r_B²). Moving the dams moves the interface, which is how a cream separator is adjusted.

Equipment.

• Tubular bowl centrifuge: long narrow bowl, very high speed (often 15 000–50 000 rpm in small units), for clarifying liquids or separating two liquids with small solids content; solids collect on the wall and are removed batchwise.
• Disc-stack centrifuge: a stack of conical discs splits the flow into thin layers so particles travel only a short distance; very large Σ. Used for milk, edible oils, yeast.
• Decanter (scroll) centrifuge: horizontal solid bowl with an inner screw conveyor turning slightly slower than the bowl, which continuously scrolls settled solids up a conical beach. Handles high-solids slurries such as sludges.
• Centrifugal filters (basket, pusher): perforated bowl with a filter medium; the centrifugal head drives filtrate through the cake. Used to dewater crystals.

Formulas

ω = 2π·N/60, a_c = ω²·r, RCF = ω²·r/g F_c = m·ω²·r Stokes, centrifugal: u = ω²·r·D_p²·(ρ_p − ρ) / (18·μ) Time from r₁ to r₂: t = 18·μ·ln(r₂/r₁) / (ω²·D_p²·(ρ_p − ρ)) Tubular bowl, complete removal: q = V·ω²·D_p²·(ρ_p − ρ) / (18·μ·ln(r₂/r₁)), with V = π·b·(r₂² − r₁²) Sigma form: q = 2·u_g·Σ, u_g = g·D_p²·(ρ_p − ρ)/(18μ), Σ = ω²·V / (2·g·ln(r₂/r₁)) on the same basis Neutral zone: r_i² = (ρ_A·r_A² − ρ_B·r_B²) / (ρ_A − ρ_B)

• N: speed (rpm); ω: angular velocity (rad/s); r: radius (m); m: mass (kg); u: radial settling velocity (m/s); D_p: particle diameter (m); μ: liquid viscosity (Pa·s); b: bowl length (m); r₁, r₂: radius of liquid surface and bowl wall (m); q: throughput (m³/s); Σ: sigma value (m²); ρ_A, ρ_B: heavy and light liquid densities; r_A, r_B: radii of heavy- and light-liquid overflow dams (m).

Worked examples

Example 1 (standard): g-number and settling velocity. A centrifuge runs at 10 000 rpm with a bowl radius of 0.1 m. Particles of D_p = 5 µm, density 1500 kg/m³, are in water (1000 kg/m³, μ = 10⁻³ Pa·s). Compare settling at the wall with gravity settling.

1. ω = 2π × 10 000/60 = 1047.2 rad/s.
2. RCF = ω²r/g = 1047.2² × 0.1/9.81 = 11 180 g.
3. Gravity: u_g = 9.81 × (5 × 10⁻⁶)² × 500/(18 × 10⁻³) = 6.81 × 10⁻⁶ m/s.
4. Centrifugal at r = 0.1 m: u = 11 180 × 6.81 × 10⁻⁶ = 0.0762 m/s.
5. Check Re_p = 1000 × 0.0762 × 5 × 10⁻⁶/10⁻³ = 0.38 < 1, so Stokes holds.

Example 2 (GATE level): tubular bowl capacity. A tubular bowl (b = 0.2 m, r₂ = 25 mm, liquid surface r₁ = 10 mm) runs at 15 000 rpm. Find the throughput at which 2 µm particles with Δρ = 200 kg/m³ are completely removed from a liquid of viscosity 1.2 × 10⁻³ Pa·s.

1. ω = 2π × 15 000/60 = 1570.8 rad/s; ω² = 2.467 × 10⁶ s⁻².
2. V = π × 0.2 × (0.025² − 0.010²) = 3.30 × 10⁻⁴ m³.
3. Time to cross: t = 18 × 1.2 × 10⁻³ × ln 2.5/(2.467 × 10⁶ × (2 × 10⁻⁶)² × 200) = 0.01979/1.974 × 10⁻³ = 10.0 s.
4. q = V/t = 3.30 × 10⁻⁴/10.0 = 3.29 × 10⁻⁵ m³/s (about 118 L/h).
5. Σ = ω²V/(2g·ln 2.5) = 2.467 × 10⁶ × 3.30 × 10⁻⁴/(2 × 9.81 × 0.916) = 45.3 m², i.e. this small bowl matches a 45 m² gravity settler.

Common mistakes

• Using rpm directly for ω; convert to rad/s.
• Keeping g in the settling formula for a centrifuge; replace it by ω²r.
• Using a single radius when the particle travels from r₁ to r₂; integrate to get the ln(r₂/r₁) term.
• Forgetting that light droplets (ρ_p < ρ) move inward.
• Comparing Σ values computed on different bases (complete removal versus cut size).

For GATE CH

Expect: RCF or centrifugal force from speed and radius; settling velocity in a centrifugal field; time for a particle to reach the wall or maximum throughput of a tubular bowl; interface (neutral zone) radius in a liquid–liquid separator; scale-up using Σ. Practise the ln(r₂/r₁) derivation and the neutral-zone balance.

Quick check

1. What is the RCF at r = 0.2 m and 5000 rpm?
2. How does centrifugal settling velocity vary with radius in the Stokes region?
3. What does the Σ value of a centrifuge represent?
4. Which centrifuge continuously discharges high-solids sludge? Answers: 1. About 5590 g. 2. It is proportional to r. 3. The area of an equivalent gravity settler. 4. The decanter (scroll) centrifuge.