A small sphere moves through a liquid at v = d² × Δρ × a ÷ (18 × μ), where d is its diameter, Δρ the density difference from the liquid, a the acceleration (gravity or centrifugal) and μ the liquid’s viscosity. The speed rises with the square of the particle size and in direct proportion to acceleration, so halving the particle size cuts the speed to a quarter, while doubling the acceleration only doubles it.
Part of Centrifuge Building
Separation theory is the physics of how fast a particle or droplet moves through a liquid under gravity or centrifugal force, which sets how long a centrifuge must run and how small a particle it can remove.
Four Quantities Control How Fast A Particle Moves
The carrier article lists the controlling factors: the density difference between particle and fluid, the particle size (separation rate proportional to diameter squared), the viscosity of the fluid (lower viscosity gives faster separation) and the g-force applied. Stokes’ law ties them together:
v = d² × (ρ_particle − ρ_fluid) × a ÷ (18 × μ)
It gives the steady speed, outward if the particle is denser than the liquid and inward if it is lighter. Each factor has a plain consequence:
- Size: speed goes as d². A particle half the size takes four times as long.
- Density difference: if it is zero, nothing separates at any speed. This is why dissolving things cannot be spun out and why two oils of similar density are hard to part.
- Viscosity: a thicker liquid slows everything. Warming a liquid often lowers its viscosity, which is why milk is separated warm.
- Acceleration: speed goes in direct proportion to a. At a = 9.81 m/s² it is the gravity settling of a pan of milk; at 5,366 g it is the same law with every speed 5,366 times larger.
Worked Example 1: Fat Globules In Milk
The inputs are standard dairy values from general knowledge, not figures from this wiki: a fat globule of 4 µm (real globules span roughly 1-10 µm), a fat density of 920 kg/m³, skim milk at 1,035 kg/m³ and a viscosity of 1.5 mPa·s at the 35-40°C separation temperature. The globule is lighter than the liquid, so it moves inward.
- d² = (4 × 10⁻⁶)² = 1.6 × 10⁻¹¹ m². Δρ = 1,035 − 920 = 115 kg/m³. 18μ = 0.027 Pa·s.
- Under gravity: v = 1.6 × 10⁻¹¹ × 115 × 9.81 ÷ 0.027 = 6.7 × 10⁻⁷ m/s, or 2.4 mm/h. A layer 5 cm deep needs about 21 hours, in line with the 12-24 hours of a gravity cream pan.
- In the reference bowl at 5,366 g: v = 6.7 × 10⁻⁷ × 5,366 = 3.6 mm/s.
- Check the law applies. Stokes’ law needs slow, smooth flow around the particle, meaning a Reynolds number below about 1. Re = ρ × v × d ÷ μ = 1,035 × 0.0036 × 4 × 10⁻⁶ ÷ 0.0015 = 0.0099, far below 1.
- Halve the globule to 2 µm: d² falls to a quarter, so the speed in the bowl falls to 0.9 mm/s. Small globules are the ones that stay in the skim milk, which is why a separator’s quality is judged by the fat left in the skim.
Worked Example 2: Red Cells In Blood
Inputs from general knowledge: a red cell of about 7 µm, a density of about 1,100 kg/m³, plasma at about 1,025 kg/m³ and a plasma viscosity of about 1.5 mPa·s. The cell is denser, so it moves outward.
- d² = 4.9 × 10⁻¹¹; Δρ = 75; v at 1 g = 4.9 × 10⁻¹¹ × 75 × 9.81 ÷ 0.027 = 1.3 × 10⁻⁶ m/s, or 4.8 mm/h.
- At 1,000 g that is 1.3 mm/s, so a cell crosses a 5 cm tube in 5 ÷ 0.13 = about 38 seconds in this ideal.
The ideal is optimistic. Blood is nearly half cells by volume, so cells hinder one another, and real spins run for minutes. The point is the ratio: 1,000 g turns an overnight settling into tens of seconds to minutes. See Tube Centrifuge.
Where Stokes’ Law Stops Being True
- Crowded suspensions settle slower than the law predicts because particles block one another (hindered settling).
- Droplets that merge (emulsions) get larger as they separate and so speed up; a stable emulsion has droplets that do not.
- Very fine particles (below roughly a micrometer) are pushed about by thermal motion and do not separate usefully at village-scale g-forces.
- Non-spherical or deforming particles move at a different speed from a sphere of the same volume.
None of these cancels the lesson, which is how the four factors scale.
What You Need To Apply It
- An estimate of the smallest particle you must remove, which sets the g-force and the residence time
- The densities and viscosity of the liquid at the working temperature
- The distance a particle must travel, which is where the disc stack helps; see Disc-Stack Design
Common Mistakes
- Doubling the RPM and expecting double the speed. The acceleration goes with RPM squared, so the speed goes four times.
- Ignoring temperature. A cold liquid is more viscous and separates slower.
- Applying the law to dissolved material. Dissolved substances need chemistry (crystallization, precipitation), not a centrifuge; see Recrystallization.
Safety
A faster separation is not worth an overspeed rotor
The temptation is to raise the speed to make a stubborn separation work. Stress in the rotor rises with the square of speed too; check Speed Limits and Material Stress first. Blood and milk samples are infectious: seal the tubes and keep spills away from skin.
What’s Next
- Learn next: Centrifugal Force Calculations — getting the acceleration right
- Combine with: Cream Separation Techniques — the dairy use of the same law