Centrifuge (3D): rotor and disc-stack separator

What this is

A centrifuge replaces the pull of gravity by a much larger centripetal acceleration so that particles which would take hours to settle separate in minutes. This page covers two machines: a laboratory swinging-bucket rotor (batch, tube-by-tube) and a continuous disc-stack separator (a steady feed of liquid through a spinning bowl). It is a physics lesson on sedimentation scaling; it is not a guide to operating real equipment, which has strict speed limits and safety interlocks.

Relative centrifugal force

RCF = omega^2 r / g with omega = 2 pi rpm / 60. For r in cm this is the handbook formula RCF = 1.118e-5 x r x rpm^2. Force grows with the square of the speed and linearly with the radius, so the same rpm gives very different forces in different rotors; protocols should be written in x g, not rpm.

Stokes sedimentation

A small sphere in a viscous liquid reaches a terminal speed where drag equals weight minus buoyancy: v = d^2 (rho_p - rho_f) a / (18 eta), valid while the particle Reynolds number stays below about 1. In a centrifuge a = omega^2 r, so dr/dt = s omega^2 r with the sedimentation coefficient s = d^2 (rho_p - rho_f)/(18 eta) (measured in Svedberg units, 1 S = 1e-13 s). The speed grows with radius, so the exact crossing time of a tube is t = ln(r2/r1) / (s omega^2), not simply length over speed. A particle lighter than the liquid (cream in milk) moves the other way and floats inward.

Sigma: the equivalent settling area

The Sigma factor is the area of a gravity settling tank that would clear the same flow: for a tube Sigma = V omega^2 / (g ln(r2/r1)); for a disc stack Sigma = 2 pi n omega^2 (ro^3 - ri^3) cot(theta) / (3 g) with n discs of half-angle theta. A particle with gravity settling speed vg is fully caught when vg Sigma >= Q, giving the full-capture size d100 = sqrt(18 eta Q / (drho g Sigma)); the 50 % cut size is d100 / sqrt(2). The model integrates the grade-efficiency curve over a log-normal size distribution to get the solids captured.

Density gradient (isopycnic) banding

With a liquid whose density rises along the tube, a particle moves until the liquid density equals its own, then stops. The banding position is therefore set by density, not by size, which is how species with similar sizes are separated. Settling is integrated numerically, and the band radius is checked against the analytic position where rho(r) equals the particle density.

Honest simplifications

Laminar, spherical, non-interacting particles in a Newtonian liquid; no hindered settling, wall effects, tube-sector (Coriolis) convection, or pellet compaction; acceleration and braking times are ignored. The disc-stack grade curve is the ideal laminar one. Real separators lose efficiency to turbulence and flow maldistribution. Rotor stress uses a thin-ring estimate (hoop stress = density x tip speed squared) for a titanium rotor with a safety factor of 2.

Safety

A rotor that exceeds its rated speed can burst with huge energy. Real centrifuges are balanced, interlocked, armoured and run only within the rotor's certified limits; samples may be hazardous. The page flags overspeed so you can see the margin, nothing more.

How this differs from the exploded-assembly lab centrifuge

The other lab-centrifuge pages on this site focus on a particular instrument and pelleting time. This pair adds the continuous disc-stack separator with Sigma and cut size, the Stokes-in-a-centrifuge derivation, and the isopycnic mode, all in one place.