Centrifuge Capacity Calculation: Sigma Theory Worked

Last updated: 2026-11-06

Centrifuge capacity follows one relationship: Q = v_g × Σ, where v_g is the settling velocity of the target particle and Σ is the machine’s equivalent clarifying area. A 25 m³/h duty removing 5 µm solids at Δρ = 1,600 kg/m³ and µ = 1.4 mPa·s needs a theoretical Σ of about 446 m², or roughly 670 m² after realistic scale-up factors. This is the same sizing logic we apply across our solid-liquid separation equipment range, worked out step by step below.

Key Takeaways

• Centrifuge capacity scales linearly with sigma (Σ), the machine’s equivalent settling area.
• Stokes’ law gives v_g; for a 5 µm particle at Δρ 1,600 kg/m³, v_g ≈ 1.56×10⁻⁵ m/s.
• A 25 m³/h clarification duty needs Σ ≈ 670 m² after a 1.5× design margin.
• Disc stack machines pack more Σ per frame; decanters handle higher solids and coarse flocs.
• Field results run 20–40% below theory; confirm with bench tests using your actual flocculant.

What Is Sigma Theory and Why Does It Predict Centrifuge Capacity?

Sigma theory, often called the sigma concept, is the standard method for rating and scaling centrifuges. It rests on five definitions that every process engineer should carry into a datasheet review:

• Centrifuge capacity is the volumetric feed rate a machine can clarify to a target cut size — a rating that depends as much on the slurry properties as on the hardware itself.
• A sigma value (Σ) is an equivalent settling area that converts the machine’s geometry and G-force into the area of a gravity clarifier that would perform the same duty.
• Settling velocity is the terminal speed at which a particle drifts through the liquor, fixed by particle size, density difference, and viscosity.
• Cut size is the particle diameter the machine captures with roughly 50% efficiency — the boundary that defines what “clarified” means for your duty.
• G-force is the centrifugal acceleration expressed as a multiple of gravity, the factor that multiplies settling velocity above its gravity value.

The logic of the model is simple. In a gravity clarifier, a particle settles at v_g across an available area A, so the clarifier handles any flow rate Q provided Q ≤ v_g × A. In a centrifuge, the same particle moves radially through a liquid ring, but the G-force accelerates it and the geometry folds a large effective area into a small bowl. Sigma packages both effects into one number:

Q = v_g × Σ

where Q is the volumetric feed rate (m³/s), v_g is the gravity settling velocity (m/s), and Σ is the equivalent area (m²). If the feed contains particles larger than the cut size, the machine clarifies; if the required Q exceeds v_g × Σ, solids break through.

Two practical consequences follow. First, capacity is proportional to v_g, so anything that changes viscosity, flocculation, or temperature changes the duty directly. Second, capacity is proportional to Σ, so you can scale between machines of different sizes by comparing their sigma values — the basis of the Q over sigma scale-up method described later.

A caution earned in the field: sigma assumes the liquid moves through the bowl in plug flow and that every particle sees the full G-field for the full residence time. Real bowls have turbulence, short-circuiting, and swirling feed zones. Sigma is a rating of the ideal machine; your plant will deliver a fraction of it, and Section 5 quantifies that gap.

How Do You Calculate Settling Velocity for a Real Slurry?

For particles below roughly 20 µm in dilute suspension, settling follows Stokes’ law:

v_g = d² × (ρ_s − ρ_l) × g / (18 × µ)

where d is the particle diameter (m), ρ_s and ρ_l are solid and liquid densities (kg/m³), g = 9.81 m/s², and µ is the dynamic viscosity (Pa·s). This is the core settling velocity calculation behind every sigma rating.

Worked example (synthetic worked example, values chosen to be representative of a hydrometallurgical clarification duty):

• d = 5 µm = 5×10⁻⁶ m
• Δρ = ρ_s − ρ_l = 1,600 kg/m³ (typical range, to be confirmed by test work)
• µ = 1.4×10⁻³ Pa·s (1.4 mPa·s)
• Numerator: (5×10⁻⁶)² × 1,600 × 9.81 = 25×10⁻¹² × 15,696 = 3.92×10⁻⁷
• Denominator: 18 × 1.4×10⁻³ = 2.52×10⁻²
• v_g = 3.92×10⁻⁷ / 2.52×10⁻² = 1.56×10⁻⁵ m/s

That is 1.56 µm per second — a particle needing over ten minutes to fall one meter under gravity. The centrifuge does not change v_g; it changes the effective area, which is why the Σ term carries the machine.

Viscosity deserves more respect than it usually gets. Water at 20 °C has µ ≈ 1.00 mPa·s; at 60 °C it drops to about 0.47 mPa·s. Because v_g divides by µ, cutting the viscosity in half doubles the settling velocity — and therefore doubles centrifuge capacity at constant Σ. Warm the feed where you can: a leach liquor fed at 60 °C instead of 25 °C is worth more than a bigger bowl. Process liquors are not water, though; dissolved salts and suspended fines raise viscosity well above the pure-water value, and slurry rheology governs how it behaves at your solids loading — see our slurry rheology notes for measurement methods.

Flocculation is the bigger lever. A properly dosed polymeric flocculant can raise effective v_g by a factor of 5–20, converting sub-micron fines into fast-settling aggregates. But flocs are shear-sensitive, and a centrifuge feed zone is a violent place. Any bench or pilot campaign must use your site’s actual flocculant at your site’s actual dosage and conditioning time, because a floc that survives a gentle beaker test may shred in the feed pipe. For fine feeds, flocculant shear sensitivity often decides real throughput more than Σ does.

What Sigma Values Do Common Centrifuge Types Deliver?

Sigma spans two orders of magnitude across machine families, which is why machine selection comes before detailed sizing. Table 1 gives typical ranges (typical range, to be confirmed by test work):

Read the table as a selection screen, not a datasheet. Tubular machines trade throughput for very high G, suiting polishing duties in facilities such as pharmaceutical-grade salt plants or high-altitude Andean operation camps where footprint matters more than tonnage. Disc stack centrifuges dominate clarification because their closely spaced cones pack enormous Σ into a modest bowl — the workhorse for dilute feeds in Southeast Asian laterite circuits and tropical coastal refineries. Decanters accept the coarsest, most concentrated feeds and dewater continuously, which is why decanter centrifuge sizing is its own exercise; see our decanter centrifuge guide for mineral slurries. Pushers need free-draining crystals and are rare in fine hydrometallurgical slurries.

For dilute fine feeds below about 2% solids by volume, a disc stack centrifuge is preferred because its larger equivalent area per frame delivers the required clarification at lower G and lower installed power. For duties above roughly 20% feed solids or with fast-settling coarse particles, a decanter is preferred because its scroll continuously conveys a compacting solids bed without the discharge restrictions that limit a disc stack.

How Do You Work a 25 m³/h Clarification Duty?

Here is the full sizing sequence for a synthetic worked example: 25 m³/h of leach liquor, target cut size 5 µm, Δρ = 1,600 kg/m³, µ = 1.4 mPa·s at the design temperature.

1. Define the duty and the cut size. The specification is 25 m³/h (6.94×10⁻³ m³/s) with a 5 µm cut. Agree this number with the metallurgist, not the vendor: a 3 µm target changes the answer by more than the machine choice does.
2. Calculate the settling velocity. From Stokes’ law as worked above, v_g = 1.56×10⁻⁵ m/s for the unflocculated particle.
3. Compute the required sigma. Σ_req = Q / v_g = (25/3600) / 1.56×10⁻⁵ = 446 m². This is the theoretical area for zero breakthrough — no real machine should be bought at this number.
4. Apply the scale-up factor. Multiply by 1.3–2.0 to cover non-ideal flow, feed-zone shear, and property scatter; at 1.5×, the design duty becomes 670 m².
5. Select a machine and back-calculate its Σ. Check candidate machines’ published ratings against the geometry formula. For a decanter with bowl radius r₂ = 0.28 m, pool radius r₁ = 0.12 m, and clarifying length L = 1.6 m at 3,500 rpm (ω = 366.5 rad/s): Σ = (ω²/g) × (2πL/3) × (r₂³ − r₁³) = 13,690 × 3.351 × 0.020224 ≈ 928 m².
6. Check G-force and solids handling. G = ω²r₂/g = 134,300 × 0.28 / 9.81 ≈ 3,833 G — adequate for a 5 µm cut but modest by disc-stack standards. Verify the scroll can convey the expected solids load and the beach can drain it; if the upstream circuit feeds a thickener instead, its design interacts with yours — see our thickener sizing guide.
7. Correct with on-site bench testing. Run bottle tests with the plant’s own flocculant and dosage to confirm v_g, then re-rate the machine against measured, not book, values.

The geometry result deserves a reality check: the 928 m² figure is the ideal scroll-bowl sigma, giving a 39% margin over the 670 m² requirement. In practice we derate decanter Σ by a factor of roughly 0.5–0.8 to account for pool-depth effects and scroll conveyance inefficiency, which lands the effective area at 460–740 m². Margin exists, but it is thinner than the raw arithmetic suggests. Table 2 summarizes each step, value, and assumption:

Why Does the Calculation Overpredict Capacity in Practice?

Every experienced operator has bought a machine that theory said would suffice and watched it overflow. The gap has consistent causes; each is listed here with the field phenomenon, the mechanism, and the remedy.

1. Non-ideal flow and short-circuiting. Phenomenon: clarified liquor carries fines even though Σ appears adequate. Mechanism: the feed accelerates to bowl speed in a finite distance; density currents and turbulence carry particles radially inward before they settle, so a fraction of the feed bypasses the effective area. Remedy: apply the 1.3–2.0 design factor, and insist on feed-zone details (acceleration cone design, baffling) during vendor review.
2. Flocculant shear degradation. Phenomenon: performance collapses when dosing is moved upstream of a pump or a static mixer. Mechanism: polymer bridges form in seconds but break under high shear; once shredded, the fine population returns and v_g falls back toward the unflocculated value. Remedy: dose downstream of pumps where possible, use low-shear addition points, and re-run jar tests at the actual shear history.
3. Non-spherical and polydisperse particles. Phenomenon: the cut size drifts coarser than calculated. Mechanism: Stokes’ law assumes smooth spheres; angular, plate-like, or agglomerated particles have drag factors 1.5–3× higher, and a wide size distribution means a tail of slow-settling fines dominates breakthrough. Remedy: measure the actual size distribution, and size to the 10th-percentile particle if the overflow spec is strict.
4. Hindered settling at higher feed solids. Phenomenon: throughput drops sharply as feed density creeps up. Mechanism: above a few percent solids by volume, particles settle as a mass rather than individually; the effective v_g falls below the Stokes value because fluid displacement and floc interaction slow the interface. Remedy: dilute the feed for clarification duties, or move the coarse duty to a machine rated for high solids.
5. Temperature below design value. Phenomenon: winter or night-shift performance degrades with no hardware change. Mechanism: viscosity rises as temperature falls; a feed at 15 °C instead of 60 °C nearly triples µ and cuts centrifuge capacity proportionally. Remedy: specify design viscosity at the coldest credible feed temperature, and add feed heating where economics allow.
6. Scale and fouling between disc gaps. Phenomenomenon: clarification worsens over weeks of run time. Mechanism: gypsum, silicate, or hydroxide scale narrows the disc spacing, shrinking effective Σ and disturbing the flow film. Hard water make-up and high sulfate tenors accelerate it. Remedy: scheduled wash-in-place, antiscalant dosing, and inspection intervals set by measured breakthrough, not calendar.
7. Discharge cycling and pool-level fluctuation. Phenomenon: periodic spikes in overflow turbidity. Mechanism: each ejection or scroll surge disturbs the liquid ring; pool depth shifts move the effective clarifying radius, momentarily reducing Σ and resuspending fines. Remedy: steady the feed with a buffer tank, tune ejection timing, and control pool weirs to the design level.

Note that causes 2, 4, and 5 are slurry-side, not machine-side. In our project experience the slurry causes dominate, which is why the bench test in step 7 is not optional.

How Do You Scale Up From a Bench Test?

The scale-up rule transfers predicted centrifuge capacity between machines through the sigma ratio:

Q₂ = Q₁ × (Σ₂ / Σ₁)

where subscript 1 is the test machine and subscript 2 the full-scale unit. Because both machines see the same slurry, v_g cancels — the test does not even require you to know the particle size. This is the practical power of Q over sigma scale-up.

Worked example (synthetic worked example). A bench tube centrifuge with Σ₁ ≈ 0.019 m² clarifies 0.5 L/h at the target cut. The candidate plant decanter has Σ₂ ≈ 928 m² ideal, or about 610 m² after a 0.66 derate. Predicted full-scale feed: Q₂ = 0.5 L/h × (610 / 0.019) ≈ 16,000 L/h, i.e. about 16 m³/h — short of the 25 m³/h duty, which correctly signals that the flocculated v_g must rise (better dosage or warmer feed) or that a larger Σ is required. Just as a lab-scale rotary drum vacuum filter anchors bench filtration programs, a calibrated bottle centrifuge anchors this scale-up.

Deviations between prediction and plant come from predictable sources: the bench test cannot reproduce feed-zone shear, so flocculated v_g is overstated; residence-time distributions differ; and bench feeds are batch-mixed while plant feeds vary. Expect ±20–30% scatter and design to the pessimistic side.

Scale-up also hits machine-specific ceilings. For decanters, torque and solids conveyance limit the solids-throughput side independently of Σ — a machine can have ample clarification area yet stall its scroll on a 45 wt% feed, which is why decanter centrifuge sizing always runs two checks: liquid-side (sigma) and solid-side (torque, beach dryness, differential speed). For disc stack machines, the nozzle or intermittent discharge valve caps the volumetric solids rate; exceeding it floods the disc stack and flushes solids into the overflow. Match the discharge capacity to your solids mass flow before trusting the sigma number.

Where Does Sigma Theory Stop Being Valid?

Sigma theory assumes individual particles settling freely in a Newtonian liquid. Step outside those assumptions and the prediction degrades quickly:

• Hindered settling above roughly 2–5% solids by volume. Particles settle as a network; the settling interface, not individual v_g, governs the machine, and sigma ratings no longer transfer between machines.
• High-density thickened slurries. Above roughly 40 wt% solids, use clarification thickener logic instead; if the duty is really thickening, a dedicated machine is the better frame — see our high-rate thickener selection guide.
• Fibrous and plate-shaped particles. Asbestos-like or mica-type solids interlock, form mats, and blind discharge ports; Stokes’ drag factor is wrong by multiples, and no sigma correction is reliable.
• Non-Newtonian fluids. Shear-thinning or yield-stress liquors change viscosity across the bowl radius itself — the near-wall fluid can be an order of magnitude more viscous than the bulk, invalidating the single-µ assumption. Characterize the fluid rheology first, then decide whether any sigma rating applies.

The honest rule: sigma theory is for dilute clarification of fine, discrete particles. The thicker and more structured the feed, the less the number means.

Case Snapshots: Centrifuge Capacity in the Field

Three anonymized operating snapshots (indicative, synthetic-typical basis) show where theory landed in practice.

Snapshot 1 — copper leach liquor polishing, high-altitude Andean operation. Duty: 18 m³/h, 0.8 vol% solids, 2 µm target. The installed disc stack centrifuge with Σ ≈ 1,100 m² met the duty only after feed temperature was raised from 22 °C to 55 °C, which cut viscosity by roughly half and lifted effective capacity by about 80%. Untempered operation broke through within two hours of every startup.

Snapshot 2 — zinc electrolyte clarification, Northern Europe. Duty: 40 m³/h at 25 wt% suspended solids, flocculated. A decanter at 3,200 rpm and 2,900 G achieved 96% solids recovery, but only at 70% of its book sigma rating — the field factor of ~0.7 sat in the middle of the 0.5–0.8 derate range this article recommends. Scroll torque, not clarification area, set the final limit.

Snapshot 3 — nickel laterite overflow polishing, Indonesian tropical coastal site. Duty: 9 m³/h rising to a 21 m³/h expansion. Rather than buying a second machine, the operator re-optimized flocculant dosage from 18 g/t to 31 g/t with a lower-shear addition point, raising effective settling velocity about 12× and confirming the new duty on the existing Σ ≈ 850 m² machine with 30% measured margin.

Each snapshot carries the same lesson: the machine’s Σ fixed the ceiling, but slurry-side decisions — temperature, flocculation, feed stability — decided where between 50% and 95% of that ceiling the plant actually operated.

FAQ

How much sigma do I need for a 10 µm cut at 50 m³/h?

Recompute v_g at 10 µm: since v_g scales with d², the 10 µm particle settles 4× faster than the 5 µm case, i.e. about 6.24×10⁻⁵ m/s under the same Δρ and µ. Then Σ = (50/3600) / 6.24×10⁻⁵ ≈ 223 m² theoretical; apply a 1.3–2.0 factor for roughly 290–446 m² design sigma. Centrifuge capacity at constant Σ rises fourfold versus the 5 µm case — the d² sensitivity works in your favor above the fines range.

Can I flocculate before a disc stack centrifuge?

Cautiously, yes. Lightly flocculated feeds are common for disc stack machines handling 0.5–2 vol% solids, and flocs can raise v_g 5–20×. But disc feed zones are higher-shear than decanter feed zones, so polymer selection matters more than dosage: use a medium-charge, medium-molecular-weight polymer and verify floc strength in a shear test, not just a jar test. If flocs shred, the disc stack reverts to its unflocculated rating.

Why does my decanter deliver less than its nameplate rating?

Nameplate figures assume a reference slurry, usually low-solids water with easy solids. Your liquor may be more viscous, your solids more angular, your flocculant shear-exposed, or your pool depth set wrong. Run the two-check audit: liquid-side sigma versus your actual v_g, and solid-side torque versus your actual solids load. Most shortfalls we audit trace to viscosity at real feed temperature plus an over-generous assumed scale-up factor; comparing downstream dewatering limits against measured filter cake moisture often separates the liquid-side from the solid-side problem.

Does temperature really matter that much?

Yes, and it is the cheapest capacity you will ever buy. Water viscosity falls from 1.00 mPa·s at 20 °C to 0.47 mPa·s at 60 °C; since v_g divides by µ, the same machine doubles its centrifuge capacity on warm feed. Salt-bearing hydrometallurgical liquors show the same proportionality. Always state design viscosity at the coldest credible operating temperature, and treat feed heating as a capacity project with a fast payback.

How do I convert a bottle-centrifuge test to full scale?

Use Q₂ = Q₁ × Σ₂/Σ₁ with the measured, not assumed, clarity endpoint: run the bottle at several speeds or times, find the condition that just meets your overflow spec, compute its effective sigma, and scale linearly — the same discipline we apply when choosing between vacuum and pressure filtration on the filtration side. Then derate for feed-zone shear, especially for flocculated feeds — expect the plant to deliver 70–80% of the linear prediction. Repeat the test with your plant’s own reagents; a substitute flocculant can shift the answer by a factor of three.

What G-force should I run for fine clarification?

For particles above about 10 µm, 2,000–3,500 G usually suffices; for 1–5 µm cuts, plan on 5,000–8,000 G, which points to disc stack machines rather than standard decanters. Higher G is not free: bearing life, vibration limits, and solids compaction all worsen. Prefer raising effective Σ or improving v_g through flocculation and temperature before chasing G. The target is the cheapest combination that meets the cut size, not the highest number on the tag.

Is sigma theory valid for flocculated feeds?

As a framework, yes; as a number, only with measured inputs. Flocs are aggregates with effective densities far below the parent mineral and sizes set by polymer chemistry, so you must use the floc’s settling velocity — measured on your slurry with your flocculant — in Q = v_g × Σ. Never insert the primary particle size into Stokes’ law for a flocculated feed; that overstates required sigma by up to 20× and leads to massively oversized machines. When the clarified liquor must be dewatered further, characterize the solids with specific cake resistance measurements before comparing a filter press for mining duty against the centrifuge discharge.

If you want this worked against your own slurry data — or a second opinion on a vendor’s sigma claim — request a quote and our process team will run the numbers with you.

Conclusion

Centrifuge capacity is not a nameplate number; it is your slurry’s settling velocity multiplied by a machine’s equivalent area, derated for the realities of the feed zone. The worked example shows how modest the numbers are: 5 µm particles settle at 1.56×10⁻⁵ m/s, a 25 m³/h duty needs only 446 m² of theoretical sigma, and 670 m² with margin fits comfortably in a single mid-size machine. What separates successful installations from disappointing ones is rarely the sigma arithmetic — it is temperature control, flocculant selection and shear discipline, honest derating, and a bench test with the plant’s own reagents. Run the calculation, then verify it on your slurry before committing capital.

References

1. Svarovsky, L., Solid-Liquid Separation, 4th ed., Butterworth-Heinemann.
2. Fuerstenau, M. C. and Han, K. N. (eds.), Principles of Mineral Processing, SME.
3. SME Mineral Processing and Extractive Metallurgy Handbook, Society for Mining, Metallurgy & Exploration.
4. Wills, B. A. and Finch, J. A., Wills’ Mineral Processing Technology, 8th ed., Butterworth-Heinemann.
5. Green, D. W. and Southard, M. Z. (eds.), Perry’s Chemical Engineers’ Handbook, 9th ed., McGraw-Hill.
6. TMS / MetSoc conference proceedings on hydrometallurgy and solid-liquid separation.
7. CSIRO mineral processing research publications, https://www.csiro.au.

Changelog

• 2026-11-06: Initial publication.