Two decanters can run at the same RPM and give different separation results. Before asking for more speed, check what is entering the machine, how much is being fed, and what is leaving through both outlets. Sigma helps explain the machine's theoretical clarification capacity. It does not replace those checks.
Sigma is an equivalent settling area
Sigma (Σ) is a theoretical equivalent settling area, expressed in square metres. It represents the area of a gravity settling tank that would provide equivalent clarification under the assumptions of the selected centrifuge model.
The calculation brings together rotational speed and the geometry of the settling region. At a fixed geometry, Sigma increases with the square of bowl speed. Two machines running at the same RPM can therefore have different Sigma values.
The geometric Sigma value does not contain the mud properties. The separation result does: particle size, density difference, fluid viscosity, solids concentration and particle behaviour all matter. A Sigma number alone cannot tell you how much drilled solids will be removed or how much barite will be recovered.
The scale-up law: keep Q over sigma constant
Q/Σ is the volumetric feed rate divided by the equivalent settling area. Its units are velocity. In an ideal settling model, it relates the hydraulic loading to the gravity settling velocity of the particles at the model's stated capture criterion.
Conditional scale-up relationship
Q1 / Σ1 = Q2 / Σ2
Use this relationship with comparable feed, a consistent Sigma definition, similar machine geometry and comparable separation efficiency. It is a starting estimate to check against trials, not a guaranteed field capacity.
Within that model, reducing Q/Σ allows particles with a lower settling velocity to be captured. For particles of the same density in the same fluid, that can mean a finer separation. It does not establish the actual cut size in a drilling mud without measurements.
The formula, and the drilling caveat
There is no single Sigma expression that can be applied unchanged to every centrifuge. Check the machine type, the geometry included and the capture criterion before comparing numbers. A value based on complete capture is not automatically interchangeable with one based on a 50% cut.
For a decanter, use a documented calculation appropriate to its settling region and operating pool depth. Bowl diameter, overall length and RPM alone do not fully describe that region. The scroll, feed acceleration, liquid flow and accumulated solids also affect how closely the machine approaches an ideal model.
The equation
Ideal cylindrical settling region: complete-capture basis
Σ = π · L · ω2 · (ro2 − ri2) / [g · ln(ro / ri)]
- L: effective axial settling length (m).
- ro: outer radius of the modelled settling region (m).
- ri: inner liquid-surface radius (m), with 0 < ri < ro.
- ω: angular velocity (rad/s), calculated as RPM × 2π/60.
- g: gravitational acceleration, approximately 9.81 m/s2.
- ln: natural logarithm; use the same length units for both radii.
This expression assumes uniform axial flow, liquid rotating with the bowl, and dilute Stokes settling of spherical particles denser than a Newtonian liquid. It represents a particle travelling from the inner liquid surface to the outer boundary during its residence time. It is an ideal model, not a complete decanter performance calculation.
What “complete capture” means here: the ideal model captures all particles with the specified settling characteristics at or below the limiting flow rate. It does not mean that the centrifuge removes 100% of every solid in a real mud. Finer or slower-settling particles can still leave with the centrate.
Real drilling mud can depart substantially from these assumptions. Non-Newtonian behaviour, hindered settling, flocculation, flow disturbances and solids accumulation can change the result. Do not replace the mud's behaviour with one convenient viscosity number and call the predicted cut size proven.
What to check on the rig
- Define the duty. Are you recovering weighting material, removing drilled solids or dewatering a waste stream? State which outlet is the product and what it must meet.
- Check the actual feed. Record flow rate, solids concentration, particle-size distribution and relevant fluid properties. Include dilution added before the centrifuge in the feed assessment.
- Record the operating condition. Bowl speed, pool setting, differential speed and torque belong alongside the sample results. Respect the manufacturer's operating limits.
- Sample all three streams. Take representative feed, centrate and cake samples during stable operation. Use measured stream rates and solids concentrations to check the solids mass balance; particle-size distributions alone do not establish recovery.
- Check both outlets against the duty. Clearer centrate does not by itself prove better barite recovery, acceptable cake moisture or lower total fluid loss.
Use Sigma to understand and compare theoretical clarification capacity. Use feed, centrate and cake measurements to judge the separation you are actually getting. A steady RPM reading answers neither question on its own.
Common questions
What is the sigma factor of a centrifuge?
Sigma is a theoretical equivalent gravity settling area, expressed in square metres. It combines rotational speed and the geometry of the settling region under a defined model. It does not by itself predict separation performance on a particular fluid.
How do I scale up a centrifuge with sigma?
Use Q1/Σ1 = Q2/Σ2 as an initial estimate for comparable feed, consistent Sigma definitions, similar geometry and comparable separation efficiency. Confirm the result with trials and check solids transport and product quality. Doubling Sigma does not guarantee twice the field throughput.
Is sigma exact for a decanter?
No. Ideal Sigma theory does not fully represent real flow, feed acceleration, solids accumulation, hindered settling or non-Newtonian mud behaviour. Use a documented geometry-specific calculation and verify performance using feed, centrate and cake measurements.
Technical references
- Richard Holdich — Fundamentals of Particle Technology, Chapter 8: Centrifugal separation, section 8.1, equations 8.7–8.10 — ideal geometry, capture criterion and efficiency correction.
- Mathematical model for solids transport power in a decanter centrifuge — cylindrical clarification model and solids transport.
- Scale-Up of Decanter Centrifuges for the Particle Separation and Mechanical Dewatering in the Minerals Processing Industry by Means of a Numerical Process Model (2021).
