centrifuge-design

name	centrifuge-design
description	Centrifuge design — G-factor, rotor stress (hoop/radial), containment, critical speed, drive system, industrial decanter centrifuge, laboratory ultracentrifuge, separation theory (Sigma factor), ASME Code.
metadata	{"priority":7,"promptSignals":{"phrases":["centrifuge design","centrifuge rotor","G force centrifuge","centrifugal separation","decanter centrifuge","rotor stress","centrifuge critical speed"],"minScore":3}}

Centrifuge Design — Complete Skill

Centrifugal Force and G-Factor

Centrifugal acceleration: a_c = ω² × r = (2πN/60)² × r [m/s²]

G-factor (ratio to gravity): G = a_c / g = (2πN/60)² × r / 9.81 = 1118 × (N/1000)² × r [dimensionless; N in rpm; r in m]

Practical G ranges: Sedimentation (blood): 1,000–3,000 G Dewatering sludge: 500–2000 G Pharmaceutical: 15,000–60,000 G Ultracentrifuge: 100,000–700,000 G

Separation Theory (Sigma Factor)

Settling velocity for spherical particle (Stokes' law): v_s = d_p² (ρ_p - ρ_f) × G / (18 μ) [m/s]

Sigma factor (equivalent area of separation): Σ = V × ω² r_mean / g × C_geometry [m²]

where V = liquid volume in bowl; C_geometry ≈ 0.5 for disk stack

Design equation: Q / Σ = 2 × v_terminal (gravity settling comparison) Q_max = 2 × v_terminal × Σ → capacity scales with both Σ and v_terminal

Disk stack centrifuge: Σ = (2π N²ω² / 3g) × (r₃³ - r₂³) × cot(θ) × n_discs n_discs = number of discs; θ = half-cone angle (30–40°); r₂, r₃ = inner/outer disc radius

Rotor Stress Analysis

Solid Disk Rotor

Radial stress: σ_r(r) = (3+ν)/8 × ρ_rotor × ω² × (R² - r²) [Pa]

Hoop stress: σ_θ(r) = ρ_rotor × ω² × [(3+ν)/8 × R² - (1+3ν)/8 × r²]

Maximum stresses occur at center (r = 0): σ_r,max = σ_θ,max = (3+ν)/8 × ρ_rotor ω² R²

For steel (ν = 0.3): σ_max = 0.413 × ρ × ω² R²

Thin Ring/Hoop Stress

For thin ring (rotor bowl wall): σ_θ = ρ_rotor × ω² × r² [equivalent to hoop pressure p = ρ × ω² × r × t]

Rotating Disk with Central Hole

At hole (r = a): σ_θ,max = ρ_rotor × ω² × [(3+ν)/4 × R² + (1-ν)/4 × a²]

Von Mises criterion: σ_VM = √(σ_r² + σ_θ² - σ_r × σ_θ) ≤ S_y / SF [SF = 1.5–2.0 for centrifuges]

Material Selection for Rotors

Titanium Ti-6Al-4V: best specific strength (S_y/ρ) for ultracentrifuges; lower G_max than aluminum at same ρ Aluminum 7075-T6: high-speed laboratory rotors; excellent S_y/ρ; lower density Carbon fiber (CFRP): gas centrifuges (uranium enrichment); highest specific strength; anisotropic Stainless 316L: process centrifuges; corrosion resistance; heavier → lower speed

Peripheral velocity limit (relates to specific strength): V_max = √(S_y / ρ) [m/s; self-limited by stress at rim] Ti-6Al-4V: V_max ≈ 470 m/s; Al 7075: V_max ≈ 460 m/s; SS 316: V_max ≈ 290 m/s

Critical Speed

Flexural critical speed: N_c = (30/π) × √(k_bearing / m_rotor) [rpm; simplified for rigid mounting]

More accurately: from Rayleigh beam theory N_c1 = (60/2π) × √(EI / (ρ_rotor L⁴)) × π²/L² [rpm; pinned-pinned]

Stiffness-critical vs. inertia-critical: For long rotors: flexural mode limits; for short: shear mode Design rule: operate at N_op < 0.8 N_c1 OR N_op > 1.2 N_c1 (avoid critical region) Subcritical design: below first critical; most laboratory centrifuges Supercritical design: operate above one or more criticals; must pass through quickly with damping

Containment Design

Burst containment (ANSI/ASME): Fragment energy = ½ I_rotor ω_burst² [J] Containment must absorb fragment energy elastically/plastically

Design approach: Containment wall absorbs KE by plastic deformation + cracking Typical: 5–10 mm steel for small lab centrifuges; 25–50 mm for industrial

Over-speed protection: Electronic governor; mechanical over-speed trip Over-speed test: 110% of max rated speed (empty rotor) without structural failure

Decanter (Horizontal Bowl) Centrifuge

Function: continuous separation of slurries into liquid + concentrated solids cake Components: outer bowl (rotating), inner conveyor (rotating slightly faster/slower than bowl)

Differential speed: Δω = ω_bowl - ω_scroll [conveyor turns slightly slower to transport solids toward discharge] Typical Δω = 0.5–10 rpm; controlled by back-drive gear

Torque on scroll: T_scroll = A_cake × τ_cake × r_mean [from cake shear stress] Power = T_scroll × Δω

Capacity: Q_liquid ≤ η_vol × A_dam × √(2 × G × h_dam) [overflow dam controls pool depth] Q_solids depends on cake conveyance speed and Δω

Bowl geometry: L/D ratio: 2–4 (longer → more separation time; better clarity) Cone angle at solids discharge: 8–25° (lower → better dewatering; higher → better conveyance) G: 1000–3500G typical

ASME Code for Centrifuges

ASME Section VIII (pressure vessels): applicable when bowl > 15 psig internal pressure ANSI/ASME B40.100: centrifuge testing requirements Safety factor on rotor burst: typically 4× operating stress (bowl); 2× (rotor disk) Label: rated G, max speed, rotor type, serial number

Ultracentrifuge (Laboratory)

Preparative: separation/pelleting; 10,000–100,000 G; swinging-bucket or fixed-angle Analytical: measure sedimentation coefficient s₂₀,w; Svedberg equation s₂₀,w = sedimentation coefficient at 20°C in water; units: Svedberg (S) = 10⁻¹³ s

Sedimentation coefficient: s = v_s / (ω² r) = m_particle (1 - v̄ρ_medium) / f [s; f = frictional coefficient]

Diffusion coefficient: D = k_B T / f (Einstein-Stokes)

Molecular weight from s and D (Svedberg equation): M = s R T / (D (1 - v̄ρ))

Output

Provide: G-factor [dimensionless], rotor speed N [rpm], peripheral velocity V_tip [m/s], maximum rotor stress σ_VM [MPa] vs. S_y [MPa] with SF, critical speed N_c1 [rpm] and operating regime (sub/super-critical), Sigma factor Σ [m²], maximum throughput Q [m³/hr], material selection (with density and S_y), containment energy requirement [kJ], applicable standard (ASME VIII, ANSI B40.100), and separation efficiency for specified particle size [μm].