Technical publication · Harvest clarification and scale-up

Disc-Stack Centrifuge Scale-Up Calculation: Sigma Factor, Q/Σ and Throughput

Abstract. This method note scales a qualified disc-stack centrifuge separation using the flow-to-effective-Sigma ratio Q/(cΣ). It calculates matched and selected operating throughput, harvest duration, solids load, usable bowl capacity, discharge interval and total elapsed time. An optional Stokes-equivalent diameter is reported only as a diagnostic of the idealized basis. A synthetic worked example, interactive calculator and explicit shear and validity gates keep a hydraulic scale-up screen separate from clarification performance, cell damage and equipment qualification.

Acatian EngineeringPublished and reviewed 14 September 202629 min technical publication

Qualified qΣ0.375 L/m²·h Operating flow3.60 m³/h Feed time5.00 h Total elapsed5.25 h

Synthetic deterministic screen. Feed properties, clarification endpoint, Sigma convention, correction factors, bowl capacity and discharge downtime are project-specific evidence inputs.

Scope and decision question

What production flow preserves a qualified separation challenge—and can the bowl finish the harvest window?

Question answered

Given a representative reference run that met a declared clarification endpoint, documented Sigma inputs, machine correction factors and a selected operating fraction, what candidate flow preserves Q/(cΣ)? The same basis then converts harvest volume and solids concentration into feed time, bowl loads, discharge interval and total elapsed time.

Why Sigma is useful

Ambler defined Sigma as the equivalent gravity-settling area of a sedimentation centrifuge, derived under explicit settling assumptions [1]. Holding a qualified flow per effective Sigma constant is a transparent first comparison across geometrically different machines.

Why Sigma is not enough

Feed-zone shear can break mammalian cells and shift the particle-size distribution. Hutchinson and colleagues showed that a shear mimic plus laboratory centrifugation predicted industrial behavior that settling calculations alone did not capture [4]. Treat shear, lysis and downstream filterability as separate gates.

Nomenclature

Keep geometry, machine correction and operating margin visible

On small screens, swipe horizontally to compare symbol, unit and evidence.

SymbolDefinition and SI basisRequired evidence

QVolumetric feed rate, m³ h⁻¹Calibrated flow and steady-state sample window

Σ_thDocumented theoretical equivalent settling area, m²Reproducible geometry equation or OEM value identified explicitly as uncorrected

cMachine-specific non-ideality correction factor, 0 < c ≤ 1Relevant pilot, scale-down or supplier basis; never apply c again to an already effective Sigma

q_ΣQ/(cΣ), m h⁻¹ or L m⁻² h⁻¹Reference condition that actually met the declared separation endpoint

f_opSelected fraction of the matched candidate flow, 0–1Named operating margin kept separate from c

C_s, η_m, M_usableFeed solids, kg m⁻³; mass-capture fraction; usable bowl solids, kgAll three must use the same dry or wet mass basis

d_eq, κStokes-equivalent diameter diagnostic, µm, and explicit convention factorViscosity and density difference; κ = 1 or 2 per the chosen relationship; not measured d50

Qualified reference basis

Anchor the calculation to a feed-specific separation endpoint

1 · Define the response

State whether the run qualified on centrate turbidity, solids remaining, particle-size distribution, product recovery, viable-cell removal, impurity release or downstream filterability. Record sample timing and analytical uncertainty. A reference flow without its response is not a scale-up basis.

2 · Make machines comparable

Use the same Sigma convention and document bowl speed, disc geometry or OEM rating and correction factor for each machine. Published conventions can differ by angle definition, disc count and a factor of two. Never mix geometric and effective Sigma silently.

3 · Preserve feed state

Record cell line, culture age, viability, cell and debris distribution, solids concentration, viscosity, temperature, hold time and any flocculation. Joseph et al. linked centrifugation conditions to subsequent depth and sterile filtration, showing why the harvest result belongs to a connected process model [3].

03 · Interactive engineering calculator

Scale qualified Q/(cΣ), then close the solids and schedule screen

SI units · local browser calculation · no data upload

Qualified reference Reference flow, Q_refm³/h Reference Sigma basis Reference Sigma input, Σ_refm² Reference correction, c_ref0–1For effective Sigma, set c = 1 so the area is not corrected twice.

Candidate machine Candidate Sigma basis Candidate Sigma input, Σ_candm² Candidate correction, c_cand0–1For effective Sigma, set c = 1 so the area is not corrected twice. Operating fraction, f_op0–1

Harvest and solids Harvest volume, V_batchm³ Feed solids, C_skg/m³ Mass-capture fraction, η_m0–1

Bowl and discharge Rated solids capacitykg Usable capacity fraction0–1 Downtime per mid-run dischargemin

Stokes diagnostic only Dynamic viscosity, μmPa·s Density difference, Δρkg/m³ Stokes convention, κ1 or 2

Qualified reference qΣ0.375 L/m²·hQ_ref / (c_refΣ_ref) Matched candidate flow4.50 m³/hSame qualified Q/(cΣ) Selected operating flow3.60 m³/h80.0% of matched flow Operating qΣ0.300 L/m²·hAfter explicit operating fraction Harvest feed time5.00 hExcludes startup flush and transfer Captured solids138.24 kgV_batch × C_s × η_m Usable bowl capacity120.0 kgRated capacity × usable fraction Captured-solids rate27.65 kg/hAt selected operating flow and η_m Discharge interval4.34 hAt constant feed and mass capture Mid-run discharges12 bowl loads total Total elapsed time5.25 hFeed time plus entered discharge downtime Stokes-equivalent diagnostic2.14 µmκ = 1; not measured d50 or cut size

04 · Calculation method

Transfer the qualified challenge before applying an explicit operating fraction

Qualified separation challenge

q_Σ,ref = Q_ref / (c_refΣ_ref)

Use only a reference condition that met the stated clarification endpoint.

Matched and operating flow

Q_match = q_Σ,refc_candΣ_cand

Q_op = f_opQ_match

Keep machine correction and chosen operating margin separate.

Harvest and solids

t_process = V_batch / Q_op

m_solids = V_batchC_s

Then calculate usable bowl capacity, loads and discharge downtime.

Diagnostic idealization

d_eq = √[18μ(q_Σ,op/3600)/(κΔρg)]

State κ explicitly. Stokes-equivalent only; not a real particle-size cut or predicted clarity.

Synthetic worked example

From a 0.300 m³/h reference run to a 3.60 m³/h operating screen

Qualified reference

A synthetic reference machine processed 0.300 m³ h⁻¹ at 1,600 m² theoretical Sigma and c = 0.50, giving 800 m² effective Sigma and q_Σ,ref = 0.375 L m⁻² h⁻¹. This fixture assumes the reference condition met its declared separation target; it is not literature-derived guidance.

Candidate throughput

A candidate with 30,000 m² theoretical Sigma and c = 0.40 has 12,000 m² effective Sigma, matching the challenge at 4.50 m³ h⁻¹. Ignoring both correction factors would instead predict 5.625 m³ h⁻¹—25% above the corrected match. Applying f_op = 0.80 gives 3.60 m³ h⁻¹ and a 5.00 h feed time.

Solids and elapsed time

At 8 kg m⁻³, the batch presents 144 kg feed solids. An explicitly synthetic 0.96 mass-capture fraction estimates 138.24 kg captured solids on the same mass basis. A 150 kg rated capacity at 0.80 usable fraction holds 120 kg, so the screen requires two bowl loads and one mid-run discharge. Adding 15 minutes gives 5.25 h total elapsed time.

Download the synthetic disc-stack centrifuge scale-up ledger CSV

Worked calculation ledger

Each result retains the exact convention and capacity basis

On small screens, swipe horizontally to inspect every substitution.

QuantitySubstitutionResult

Qualified q_Σ0.300 / (0.50 × 1,600)0.000375 m/h = 0.375 L m⁻² h⁻¹

Matched candidate flow0.000375 × 0.40 × 30,0004.50 m³/h

Operating flow0.80 × 4.503.60 m³/h

Harvest feed time18 / 3.605.00 h

Usable bowl capacity150 × 0.80120 kg

Captured solids and bowl loads18 × 8 × 0.96 = 138.24; ceil(138.24 / 120)2 loads; 1 mid-run discharge

Total elapsed time5.00 + 1 × 15/605.25 h

Stokes-equivalent diagnosticμ = 1.5 mPa·s; Δρ = 50 kg/m³; κ = 1; operating q_Σ2.14 µm; κ = 2 would give 1.51 µm; diagnostic only

Scale-up gate matrix

A Q/Σ match passes only the settling gate

GateEvidenceDecision

Comparable feed and endpointViability, solids, PSD, viscosity, temperature, hold and the same analytical responseReject transfer when the separation challenge changed materially.

Sigma conventionSpeed, geometry/OEM value, disc count, angle definition and correction-factor provenanceReject mixed geometric, effective or factor-of-two conventions.

Solids and scheduleUsable solids capacity, discharge trigger, discharge loss and downtime, startup and flushConfirm the batch fits the available manufacturing window.

Shear and lysisRepresentative feed-zone exposure, LDH or justified damage marker, PSD and product recoveryDo not infer equivalent damage from Q/Σ.

Downstream consequenceCentrate turbidity, impurity release, depth-filter capacity and sterile-filter performanceVerify the complete clarification train, not centrifuge yield alone.

Method rejection and limitations

Reject a neat hydraulic answer when the physical basis is not transferable

Failure modeWhy the result failsRequired control

Reference run lacks an endpointQ/Σ has no qualified relationship to clarity, recovery or filterability.Attach raw run conditions, response data, acceptance criterion and uncertainty.

Correction factors are copiedNon-ideal feed acceleration and flow distribution are machine and method dependent.Use scale-down, pilot or OEM evidence for both machines and declare the convention.

Stokes diagnostic called d50Real feeds are polydisperse; particles may be non-spherical and settling may be hindered.Measure PSD and clarification response. Keep d_eq labelled as an idealized diagnostic.

Constant Q/Σ implies equal shearFeed-zone energy dissipation and residence time can change independently of settling challenge.Run a relevant shear mimic or pilot comparison and evaluate lysis and downstream consequences.

Total bowl volume used as solids capacityWorking capacity, discharge logic and solids rheology determine breakthrough and interruptions.Use qualified usable solids mass and verify discharge timing and recovery.

Operating margin is hidden inside cCalibration and conservatism cannot be audited or updated independently.Report c and f_op separately and prevent double derating.

Screen used for equipment purchaseMechanical limits, containment, CIP/SIP, utilities, controls, footprint and vendor guarantees are outside the equations.Complete vendor design review, representative trials, hazard assessment and site qualification.

Engineering checklist

Minimum evidence before approving a centrifuge scale-up basis

Feed and response

Cell line and harvest age; viability; total and viable cell density; solids mass; PSD; viscosity; density difference; temperature; hold history; flocculation; target turbidity or solids; recovery; impurity release; filterability; analytical precision; representative batch range.

Machine and operation

Machine identifier; bowl and feed-zone design; speed; OEM or calculated Sigma; convention; c factor and source; steady-state sampling; startup; flow calibration; pressure; backpressure; usable solids capacity; discharge trigger, loss and downtime; flush; containment; CIP/SIP.

Scale-up and governance

Reference acceptance decision; matched flow; explicit operating fraction; sensitivity cases; shear mimic or pilot result; downstream filter response; batch-window fit; utilities; mechanical and safety review; model version; source ownership; reviewer; vendor confirmation; qualification plan.

08 · Primary technical sources

References

Ambler, C. M. “The Theory of Scaling Up Laboratory Data for the Sedimentation Type Centrifuge.” Biotechnology and Bioengineering 1(2), 185–205 (1959). doi:10.1002/jbmte.390010206.
Maybury, J. P.; Hoare, M.; Dunnill, P. “The Use of Laboratory Centrifugation Studies to Predict Performance of Industrial Machines: Studies of Shear-Insensitive and Shear-Sensitive Materials.” Biotechnology and Bioengineering 67(3), 265–273 (2000). doi:10.1002/(SICI)1097-0290(20000205)67:3<265::AID-BIT2>3.0.CO;2-J.
Joseph, A.; Kenty, B.; Mollet, M.; Hwang, K.; Rose, S.; Goldrick, S.; Bender, J.; Farid, S. S.; Titchener-Hooker, N. “A Scale-Down Mimic for Mapping the Process Performance of Centrifugation, Depth and Sterile Filtration.” Biotechnology and Bioengineering 113, 1934–1941 (2016). doi:10.1002/bit.25967.
Hutchinson, N.; Bingham, N.; Murrell, N.; Farid, S.; Hoare, M. “Shear Stress Analysis of Mammalian Cell Suspensions for Prediction of Industrial Centrifugation and Its Verification.” Biotechnology and Bioengineering 95(3), 483–491 (2006). doi:10.1002/bit.21029.
Zaman, F.; Allan, C. M.; Ho, S. V. “Ultra Scale-Down Approaches for Clarification of Mammalian Cell Culture Broths in Disc-Stack Centrifuges.” Biotechnology Progress 25(6), 1709–1716 (2009). doi:10.1002/btpr.275.
Shekhawat, L. K.; Sarkar, J.; Gupta, R.; Hadpe, S.; Rathore, A. S. “Application of CFD in Bioprocessing: Separation of Mammalian Cells Using Disc Stack Centrifuge During Production of Biotherapeutics.” Journal of Biotechnology 267, 1–11 (2018). doi:10.1016/j.jbiotec.2017.12.016.
Dölle, M.; Hervé, M.; Königsson, S.; Thorwid, P.; Rockberg, J.; Chotteau, V. “Efficiency and Scalability in Harvesting Mammalian Cell Cultures: A Scale-Down Approach to Continuous Centrifugation.” Journal of Biotechnology 411, 89–101 (2026). doi:10.1016/j.jbiotec.2026.01.012.
U.S. Patent US7157276B2. “Use of Depth Filtration in Series with Continuous Centrifugation to Clarify Mammalian Cell Cultures.” Equations 12–15 and stated Sigma-theory limitations. Official patent record.

FAQ

Technical FAQ

Questions engineers ask about Sigma factor and centrifuge scale-up

What is the Sigma factor of a disc-stack centrifuge?

The Sigma factor is a theoretical equivalent settling area, expressed in square metres. It combines centrifuge geometry and rotational speed so separation challenge can be compared across scales under declared assumptions and one documented convention.

How do you scale up a disc-stack centrifuge?

Establish a Q/(cΣ) value that meets the clarification target at reference scale, document machine-specific correction factors, and use that qualified challenge to calculate candidate flow. Then check solids capacity, discharge time, shear-sensitive responses and downstream filterability separately.

Can bench centrifuge data predict continuous centrifuge throughput?

Bench data can support a prediction when laboratory Sigma, spin time, acceleration and deceleration are treated consistently and the production-machine correction is calibrated. A shear mimic or pilot comparison may also be necessary for fragile mammalian cells.

Does constant Q/Σ guarantee the same centrate quality?

No. Constant Q/Σ preserves an idealized settling challenge; it does not guarantee equivalent feed-zone shear, cell lysis, particle-size distribution, non-uniform flow, hindered settling or solids-discharge behavior.

Should I calculate Sigma or use the manufacturer’s value?

Use a manufacturer-supplied value when its bowl speed and convention are documented. Calculate Sigma from geometry only when the required radii, disc count and angle definition are reliable, and do not mix geometric and effective values.

How do I estimate centrifuge harvest time?

Divide harvest volume by the selected operating flow, then add startup, steady-state establishment, each mid-run discharge, flush and transfer time. Report feed-on processing time and total elapsed manufacturing time separately.