Engineering Guide · Vendor-Neutral

Constant P/V vs Constant kLa Scale-Up: Which Bioreactor Criterion Preserves What

Published September 8, 2026 · 12 min read
Constant P/V vs constant kLa scale-up side-by-side comparison torque P / V = constant Constant P/V Preserves energy dissipation Shear · mixing · heat transfer VS air DO kLa = constant Constant kLa Preserves oxygen transfer capacity OTR · DO setpoint · OUR margin kLa ∝ (P/V)^0.4 · vs^0.5 (Van't Riet)
Figure 1: Constant P/V holds the total energy dissipation constant across scale, targeting shear and mixing. Constant kLa holds oxygen transfer capacity constant, targeting OTR-limited processes. Both criteria are linked by the Van't Riet correlation but diverge whenever sparging conditions change.
Quick Verdict

For sparged stirred tanks with matched superficial gas velocity, constant P/V and constant kLa give nearly identical impeller speeds because kLa scales as (P/V) to the 0.4 power in the Van't Riet correlation. They diverge when superficial gas velocity, sparger geometry, or broth viscosity change with scale. Choose constant kLa when oxygen transfer is your rate-limiting step; choose constant P/V as the default for shear and mixing.

Key differences at a glance

Side-by-side comparison

Factor Constant P/V Constant kLa
What it preserves Energy dissipation per m³ (turbulence, shear, mixing intensity, heat transfer) Volumetric oxygen mass transfer coefficient (OTR capacity at fixed driving force)
Governing formula N_large = N_small · (D_small / D_large)2/3 Match kLa = 0.026 · (P/V)0.4 · vs0.5
Primary use case Shear-driven or mixing-driven processes; default when no clear bottleneck OTR-limited processes (high-density microbial, dense mammalian perfusion)
Typical target range 1–5 W/m³ mammalian, 3–10 kW/m³ microbial 5–40 h−¹ mammalian, 100–500 h−¹ microbial
Sensitivity to sparger design Low — depends only on impeller power High — micro versus macro sparger changes gas hold-up and bubble size
Sensitivity to gas hold-up Moderate (via gassed-to-ungassed power correction) High (vs term directly controls kLa)
Viscous / non-Newtonian broths Predictable via Metzner-Otto method for apparent viscosity Van't Riet vs exponent unreliable above µ = 0.05 Pa·s
Ease of measurement across scales Torque or motor power — measured continuously Dynamic gassing-out or off-gas mass balance — periodic characterisation
Industry adoption Default across most CHO and moderate microbial scale-ups Chosen for OTR-critical processes and cross-platform SUB transfers

Values reflect typical published ranges for stirred tanks with radial and axial impellers. Vendor datasheets and your own characterisation runs take precedence.

Constant P/V in detail

Constant power per unit volume — commonly written P/V or Pg/V — scales impeller speed so the ungassed power dissipated per cubic metre of working volume matches the source scale. Because ungassed impeller power is P = Np · ρ · N³ · D5 and working volume is proportional to D³ for geometrically similar vessels, holding P/V constant reduces to the well-known impeller-speed rule.

Constant P/V impeller speed rule N_large = N_small · (D_small / D_large)2/3

How it works

The rule assumes the same impeller type at both scales (same Np), turbulent flow (Re > 10,000, which is almost always true for the target vessel), and geometrically similar spargers. P/V is easy to measure — every modern bench-scale bioreactor (Applikon my|Control, Sartorius Biostat, Eppendorf BioFlo) reports shaft torque and motor power continuously, so verifying that the target vessel delivers the specified P/V is a routine PAT check rather than a special study.

The physical rationale is that at the same energy dissipation rate, the Kolmogorov microscale (the smallest eddy size before viscous dissipation takes over) is similar across scale. For turbulent stirred tanks this means similar shear on hydrodynamic-scale objects (cells, aggregates, droplets), similar bubble breakup, and roughly similar mixing intensity per unit volume. Nienow (2006) argues P/V is really a stand-in for εT, the local turbulent energy dissipation rate, which is the more fundamental quantity for cell damage and mass transfer.

When constant P/V wins

Choose constant P/V when the process is shear-sensitive, mixing-limited, or when a general default is needed at the start of scale-up development. It is the industry default for CHO fed-batch below 2,000 L, for microbial processes at moderate cell density, and for any greenfield vessel qualification where the specific rate-limiting step has not yet been identified. It also wins when comparing vessels of similar geometry, because the (D_ratio)2/3 rule needs no vessel-specific kLa data.

Because P/V correlates with kLa via the Van't Riet correlation (see below), a constant-P/V scale-up in a sparged stirred tank with matched vs also holds kLa within roughly 10–20 percent — enough for most moderate-OUR processes. This dual coverage is why P/V has become the de facto starting point for scale-up conversations.

Constant kLa in detail

The volumetric oxygen mass transfer coefficient kLa (units h−¹) sets the ceiling on oxygen transfer rate: OTR = kLa · (C* − CL). When peak oxygen uptake rate (OUR) approaches the OTR ceiling, holding kLa constant across scale is the only way to preserve the dissolved oxygen setpoint under the same driving force. The workhorse correlation for stirred tanks is Van't Riet's (1979):

Van't Riet correlation (coalescing broths) kLa = 0.026 · (Pg/V)0.4 · vs0.5
where Pg/V is gassed power per unit volume (W/m³) and vs is superficial gas velocity (m/s)

How it works

The 0.4 exponent on P/V says less than half the mass transfer response comes from mechanical energy; the 0.5 exponent on vs says gas hold-up matters as much as agitation. That is why holding P/V constant is not the same as holding kLa constant — if VVM drops from 1.0 to 0.5 vvm at scale (common because gas hold-up rises and foaming becomes worse), vs0.5 alone can pull kLa down 30 percent even at unchanged P/V.

Measuring kLa across scales requires dedicated characterisation runs: dynamic gassing-out (deoxygenate with nitrogen, re-aerate, fit ln(C* − CL) versus time to a first-order model), the sulfite oxidation method, or continuous off-gas O2/CO2 analysis for OUR-based mass balance. Hamilton VisiFerm and Mettler Toledo InPro optical DO probes are the standard measurement device; PreSens non-invasive sensor spots enable the same characterisation in shake flasks and microtiter plates where invasive probes are impractical.

When constant kLa wins

Constant kLa wins when OTR is the rate-limiting step or when the scale-up crosses a technology boundary that changes the P/V-to-kLa relationship. Concrete examples: high-cell-density E. coli fed-batch (peak OUR 150–250 mmol/L/h), dense CHO perfusion above 50 million viable cells/mL, yeast fermentation on methanol (Pichia pastoris qO2 4–12 mmol/g/h), transfer between a Rushton-turbine glass vessel and an axial-impeller SUB, or transfer between a macrosparger pilot and a microsparger production vessel.

Constant kLa also wins for cross-platform transfers. When a process moves between vendors — from an Applikon bench glass vessel to a Cytiva Xcellerex XDR or Thermo HyPerforma DynaDrive SUB — the impeller geometries, power numbers, and sparger designs all differ. Matching kLa on measured data is more reliable than matching P/V on a correlation the target vessel may not obey.

Pros and cons

Constant P/V

Advantages

  • Simple, closed-form calculation using only vessel geometry and impeller speed — no empirical constants required.
  • Continuously measurable via shaft torque or motor power on every modern bioreactor.
  • Preserves shear, mixing intensity, and heat transfer simultaneously — three physical quantities for the price of one criterion.
  • Correlates with kLa within 10–20 percent when superficial gas velocity is matched — giving reasonable OTR performance as a byproduct.

Disadvantages

  • Does not preserve kLa when superficial gas velocity, sparger design, or gas hold-up changes with scale.
  • Correlations built on Rushton turbine data may not apply to axial impellers common in single-use bioreactors.
  • For viscous or non-Newtonian broths, the P/V relationship shifts and Metzner-Otto corrections are needed.
  • Says nothing about oxygen transfer directly — risks silent OTR shortfall for high-OUR processes at scale.

Constant kLa

Advantages

  • Directly targets the oxygen transfer ceiling — the most common rate-limiting mechanism in high-cell-density processes.
  • Robust across vessel platforms because it relies on measured kLa data rather than a specific power correlation.
  • Preserves the dissolved oxygen setpoint under a fixed driving force, so cascade tuning transfers with less rework.
  • Handles technology transitions cleanly (glass to SUB, Rushton to hydrofoil, macro to microsparger).

Disadvantages

  • Requires empirical kLa versus power curves for each vessel and impeller type — needs characterisation runs.
  • Often demands higher power draw at scale, raising local shear and threatening shear-sensitive cells.
  • Says nothing about mixing time — large-scale kLa-matched vessels can develop pH and substrate gradients.
  • Van't Riet correlation exponents shift for viscous broths, coalescing versus non-coalescing media, and dual-impeller systems.

Which should you choose?

Match the criterion to the rate-limiting physics of the process. Below are the four situations that account for the majority of stirred-tank scale-up decisions.

CHO fed-batch at 10–2,000 L, moderate density

Peak OUR under 5 mmol/L/h; agitation-driven shear is the main risk. P/V holds shear and mixing at reasonable values while kLa remains within acceptable band.

Choose Constant P/V

High-cell-density E. coli or yeast fermentation

Peak OUR above 100 mmol/L/h; DO setpoint below 30 percent. Match kLa to OUR peak plus 30–50 percent safety margin; verify shear cap separately.

Choose Constant kLa

Cross-platform transfer (SUB to SUB or glass to SUB)

Different impeller geometry, power number, and sparger design. Correlation-based P/V may not apply; measured kLa transfers more reliably.

Choose Constant kLa

Dense CHO perfusion above 50 M cells/mL

Both shear and OTR are active constraints. Use constant kLa but hard-cap tip speed at 2 m/s and close any OTR gap with dual sparger or O2 enrichment.

Choose Constant kLa + tip speed cap

Real-world use cases

Typical setups where the choice of criterion has produced the standard scale-up path.

CHO mAb, 10 L to 2,000 L
Constant P/V at 30–50 W/m³

Rushton or hydrofoil impellers at the source scale hold P/V near 40 W/m³. Scaling with the (D_ratio)^0.667 rule gives 88 rpm at 2,000 L, tip speed 1.6 m/s, kLa around 12 h−¹. Comfortably above the 4–6 mmol/L/h peak OUR.

E. coli high-density fed-batch
Constant kLa at 400–500 h−¹

Peak OUR of 180–220 mmol/L/h at 60 gDCW/L demands kLa near 500 h−¹ to hold DO at 30 percent. At 1,000 L, constant P/V alone would leave a 30 percent OTR shortfall; the fix is dual Rushton plus 1.5 vvm air with O2 trim.

AAV HEK293 transient transfection
Constant P/V with vendor kLa cross-check

Transfection window forces low shear (tip speed below 1.5 m/s) so P/V dominates the sizing at 2–10 W/m³. Cross-check the vendor's Sartorius Biostat STR or Cytiva Xcellerex kLa curve to confirm 5–10 h−¹ delivered.

Shake flask to 2 L STR
Constant kLa from RAMOS characterisation

Flask kLa via a Kuhner RAMOS or PreSens sensor spot gives 30–80 h−¹. Match with a bench-scale sulfite study on the STR — typically 300–400 rpm at 0.5 vvm hits the same band and gives a clean flask-to-STR transfer.

Not sure which criterion fits your target vessel?

The Scale-Up Calculator applies all five criteria (P/V, tip speed, kLa, Reynolds, mixing time) side-by-side to your source and target geometry, and flags out-of-range warnings. Compare N, tip speed, P/V and estimated kLa in one view.

Open the Scale-Up Calculator

Operational cost and lifecycle considerations

The cost of the criterion is really the cost of getting it wrong

Choosing constant P/V for a process that is actually OTR-limited produces a batch that hits DO zero at mid-log and either drops titer 20–40 percent or triggers off-gassing safety interlocks. Choosing constant kLa for a shear-sensitive process without a tip speed cap can lyse cells and drop viability into the single digits by day 10.

Neither criterion carries a direct capital cost — both are engineering practices, not products. The real cost comes from the instrumentation you need to verify them across scale, the characterisation runs to build the correlations, and the failed batches when the wrong criterion has been chosen for the process at hand.

The bench-to-clinical process-development timeline for a mAb typically includes 3–8 scale-up runs across three vessel sizes. A single failed engineering batch at 200 L costs $50k–$150k in media, labor and lost calendar time; a failed engineering batch at 2,000 L costs $250k–$800k. Choosing the wrong criterion once at 2,000 L therefore pays for a lot of upfront kLa characterisation at bench.

Cost component Constant P/V Constant kLa
Instrumentation requiredShaft torque / motor power (standard)DO probes plus off-gas or sulfite study
Characterisation runs per scaleTypically 0 — derived from geometry2–4 dynamic gassing-out runs per vessel type
Development time to defineHours (calculator run)1–3 weeks per vessel platform
Risk cost of a wrong choiceOTR shortfall — batch failure at scaleExcess shear — catchable in bench trials

Vendor and software landscape

Neither criterion is a product, but the tools that implement each one differ. Below are the platforms most commonly used to verify or optimise each criterion in industry.

Constant P/V implementations

Constant kLa implementations

Frequently asked questions

Are constant P/V and constant kLa the same thing?
No, but for sparged stirred tanks with matched superficial gas velocity they are close. The Van't Riet correlation gives kLa proportional to (P/V) to the 0.4 power times superficial gas velocity to the 0.5 power, so holding P/V constant keeps kLa within about 15 percent as long as the gas velocity, sparger geometry, and broth viscosity do not change with scale. When any of those change, the two criteria diverge and choosing between them matters.
Why does constant P/V not preserve kLa exactly?
kLa depends on two things: energy dissipation (captured by P/V) and gas hold-up (captured by superficial gas velocity vs). The exponents in the Van't Riet correlation are 0.4 on P/V and 0.5 on vs, so power dissipation alone determines less than half the mass transfer response. At scale, spargers, aeration rate (VVM), and gas hold-up all change, and the exponents themselves shift for viscous or non-Newtonian broths. Constant P/V therefore preserves shear and mixing but only approximates kLa.
When should I use constant kLa instead of constant P/V?
Use constant kLa when oxygen transfer is the rate-limiting step. Typical signals: high-cell-density E. coli or yeast fermentations with peak oxygen uptake above 100 mmol per litre per hour, dense CHO perfusion above 50 million viable cells per mL, or any process that already runs at less than 30 percent dissolved oxygen setpoint at the source scale. Also switch to constant kLa when the target vessel uses a different sparger technology (microsparger versus macrosparger) or has different gas hold-up characteristics, because P/V no longer tracks kLa in those cases.
What are typical P/V values for CHO versus E. coli scale-up?
For mammalian cell culture the industry range is 1–5 W/m³ for large stirred tanks (in some CHO fed-batch programs holding 20–60 W/m³ at 2,000 L). Microbial fermentation runs 3–10 kW/m³, with high-cell-density E. coli commonly at 2–5 kW/m³. The mammalian value is set by shear tolerance; the microbial value is set by oxygen demand.
How do I calculate impeller speed at scale using constant P/V?
Because ungassed power scales as P = Np × ρ × N³ × D5, holding power per unit volume constant means Nlarge = Nsmall × (Dsmall / Dlarge)2/3. For example, scaling from a 10 L bioreactor with impeller diameter 0.06 m at 300 rpm to a 2,000 L bioreactor with impeller diameter 0.35 m gives Nlarge = 300 × (0.06/0.35)0.667 = 88 rpm. The equation assumes the same impeller type (same power number Np) and geometrically similar vessels.
Can I use constant P/V for microbial fermentation?
Yes, and it is the default for most low- and moderate-cell-density microbial processes. Where it breaks down is at very high cell densities where OUR peaks approach OTR ceilings. There, keeping P/V constant produces a large-scale vessel with kLa 20–40 percent below the small-scale value once you also account for reduced VVM and different sparger performance. For those runs, size on constant kLa or on peak OTR with a safety margin, then verify the resulting P/V is within the vessel and cell-line envelope.
Does constant kLa work for shear-sensitive cells?
Only with caution. Matching kLa across scale usually means raising power draw at the large scale, which raises Kolmogorov-scale eddy energy and can damage shear-sensitive cells. For shear-limited processes (primary cells, iPSC, high-density CHO perfusion) the preferred approach is constant kLa but with a hard tip speed cap (typically 2 m/s or lower), then plugging any residual OTR gap with dual spargers, oxygen enrichment, or higher backpressure rather than more agitation.
How do single-use bioreactors change P/V and kLa scale-up?
Single-use bioreactors typically use axial or hybrid impellers (elephant ear, marine, hydrofoil) with lower power numbers than Rushton turbines. The classic P/V-versus-kLa correlations were derived for radial-flow Rushton turbines and may not apply. Vendor-supplied kLa versus power curves for each specific vessel and impeller combination are essential, and rely on measured kLa data rather than deriving from a P/V correlation. This is why constant kLa scale-up is often more robust than constant P/V across SUB platforms with different impeller geometries.

Resources and references