Laboratory Centrifuge

What is modelled

In a centrifuge the rotor replaces gravity with a much larger radial acceleration. A small particle in a liquid reaches a terminal speed where the net outward force balances the viscous drag (Stokes drag, valid at low Reynolds number), and settles outward until it meets the tube bottom and forms a pellet.

Equations

• Relative centrifugal force RCF = ω² r / g = 1.118×10−5 rcm rpm².
• Terminal speed v = 2 rp² (ρp − ρm) ω² r / (9 μ), from (4/3)πrp³ Δρ ω² r = 6πμ rp v.
• Sedimentation coefficient s = v / (ω² r) = 2 rp² Δρ / 9μ (1 Svedberg = 10⁻¹³ s).
• Pelleting time t = ln(rmax / rtop) / (s ω²); k-factor k = 2.53×1011 ln(rmax/rtop) / rpm², t = k / s.

Checks run in the page

• RCF from ω² r / g against the bench formula.
• RK4 integration of dr/dt against the closed-form pelleting time.
• Stokes force balance at the tube bottom, with the particle Reynolds number.
• The k-factor time against the Stokes pelleting time.

Honest simplifications

Single non-interacting spherical particles in a uniform liquid, with no hindered settling, diffusion, wall effects, convection, rotor acceleration or braking. Tubes are shown as a straight radial channel rather than an angled or swing-out holder. The rotor ring stress readout is a thin-ring estimate with a generic aluminium density and is not a rotor rating; never use it to judge real equipment.

• Choose Fast bench run: the RCF rises with the square of speed and the pelleting time falls by almost 9 times.
• Choose Fine silica and see how little a half-micron particle moves at modest speed.
• Choose Viscous medium and watch the pelleting time scale with viscosity.