A decanter centrifuge does the work of a settling pond the size of several tennis courts, in a bowl you can stand next to. The number that captures that — and that lets you compare two machines or scale from a pilot to the field — is the sigma factor. It is the most useful sizing concept in centrifugation, and almost nobody on a rig has heard of it.
Sigma is an equivalent settling area
Sigma (Σ), derived by Ambler in 1952, is the calculated area of a plain gravity settling tank that would achieve the same separation as the centrifuge in a unit gravitational field. It rolls the bowl's speed, radius and length into a single number with units of area — typically hundreds or thousands of square metres for a field machine. It lets you describe a centrifuge's clarifying power independently of the fluid, and it is the basis for every credible scale-up from lab to plant.
The scale-up law: keep Q over sigma constant
The practical power of sigma is one relationship: for the same separation (the same cut point), the ratio of feed rate to sigma stays constant. Double the sigma and you can double the feed and hold the same cut. This is how you scale a result from a pilot unit to a full-size field decanter without guesswork, and how you compare two machines fairly — not on RPM or bowl diameter alone, but on Q/Σ.
Because Q/Σ is set by the Stokes settling velocity of the particle you want to cut, the same law ties sigma back to the physics: to move the cut finer, you lower Q/Σ — either slow the feed or use a machine with more sigma. Sigma theory assumes idealised settling and ignores turbulence, cake build-up and non-Newtonian effects, so it is a sizing and comparison tool, not a precise predictor — but as a scale-up backbone it is unmatched.
The formula, and the drilling caveat
For a tubular bowl the sigma factor has a clean closed form built from angular velocity, bowl length and radius. A decanter's geometry is more complex, so it uses a geometry-specific Ambler sigma rather than the tubular form — but the tubular expression is the canonical way to see what drives sigma: it grows with the square of speed and with bowl length and the square of radius. That is the same reason a bigger, faster bowl clarifies finer: more sigma, lower Q/Σ for a given feed.
The equation
Sigma factor for a tubular bowl (canonical form, Ambler):
Σ = π · ω² · b · r² ÷ (2g)
ω = angular velocity (rad/s); b = bowl length (m); r = bowl radius (m); g = 9.81 m/s². Scale-up for equal cut point:
Q1 ÷ Σ1 = Q2 ÷ Σ2
Decanters use a geometry-specific Ambler Σ; the Q/Σ scale-up law holds for geometrically similar machines.
Compare and size centrifuges on Q/Σ, not RPM. Two machines at the same speed can have very different sigma, and therefore very different feed capacity at the same cut. Treat sigma as a scale-up and comparison backbone — then confirm on your own fluid, because it idealises the settling.
Common questions
What is the sigma factor of a centrifuge?
It is the equivalent area of a gravity settling tank that would achieve the same separation as the centrifuge. Derived by Ambler, it captures bowl speed, radius and length in one number (units of area) and is the standard basis for scaling up centrifuges.
How do I scale up a centrifuge with sigma?
Keep the ratio of feed rate to sigma constant for the same cut point: Q1/Σ1 = Q2/Σ2. A machine with twice the sigma can take twice the feed at the same cut. Use it for geometrically similar machines and confirm on your own fluid.
Is sigma exact for a decanter?
No. Sigma theory idealises settling and ignores turbulence, cake build-up and non-Newtonian behaviour, and decanters use a geometry-specific sigma rather than the tubular form. It is an excellent sizing and comparison tool, not a precise performance predictor.
