Bioreactor Scale-Up Calculator: Constant P/V, kLa & Tip Speed

Constant P/V (power input per unit volume) is the most widely used scale-up criterion for aerobic fermentations. It preserves similar oxygen transfer rates (kLa) and bulk mixing intensity across scales, since kLa correlates strongly with volumetric power input. This calculator uses the relationship P = N_p × ρ × N³ × D₁⁵ (where N_p is the power number, ρ is fluid density, N is impeller speed, and D_i is impeller diameter) to match P/V between your source and target vessels. For highly aerobic processes like E. coli high-cell-density fermentation, constant P/V at 2-5 kW/m³ is the industry standard starting point.

P/V is calculated as the ungassed power input divided by the working volume. Ungassed power P = N_p × ρ × N³ × D_i⁵, where N_p is the impeller power number (5.0 for a standard Rushton turbine, 1.7 for a pitched blade, 0.35 for a marine propeller), ρ is fluid density (typically ~1000 kg/m³), N is impeller speed in rev/s, and D_i is impeller diameter in metres. This calculator applies this formula for both source and target vessels. Gassed power is typically 50-70% of ungassed power and depends on aeration rate, but ungassed P/V is the standard design basis for scale-up comparisons.

Use constant tip speed (π × N × D_i) when shear sensitivity is your primary concern, such as with mammalian cell cultures, filamentous fungi, or shear-sensitive microbial strains. Tip speed directly relates to the maximum shear rate near the impeller blade. Use constant P/V when oxygen transfer is the limiting factor, as in high-cell-density bacterial fermentation. Note that keeping constant tip speed during scale-up results in a significant decrease in P/V (and therefore kLa), since P/V scales as D_i^(2/3) at constant tip speed. Conversely, constant P/V increases tip speed at larger scales. The choice depends on whether oxygen supply or cell damage is your bigger risk.

kLa generally decreases with increasing bioreactor scale when operating at the same P/V, because large vessels have longer mixing times and less uniform energy dissipation. At constant P/V and aeration rate (vvm), kLa may drop by 20-40% from bench (5-10 L) to production scale (5000-10,000 L). This is because large reactors have lower surface-area-to-volume ratios for gas-liquid contact and less effective gas dispersion. This calculator accounts for these scale effects by computing kLa from the Van't Riet correlation (kLa = C × (P/V)^a × v_s^b) at both scales, so you can see the predicted kLa change before committing to a production run.

The most common scale-up challenges are oxygen transfer limitations, mixing time heterogeneity, and gradient formation. At production scale, mixing times increase from seconds (bench) to 30-120 seconds (10,000 L), creating spatial gradients in pH, dissolved oxygen, substrate, and temperature. Cells circulating through these zones experience fluctuating environments that can trigger stress responses, reduce productivity, or alter metabolism. CO2 stripping also becomes problematic -- dissolved CO2 accumulates at large scale due to hydrostatic pressure and reduced surface-area-to-volume ratio. This calculator helps you anticipate these issues by comparing key parameters (P/V, tip speed, Re, mixing time, kLa) across scales simultaneously.

Impeller tip speed is calculated as v_tip = π × N × D_i, where N is the rotational speed in rev/s and D_i is the impeller diameter in metres. For a standard Rushton turbine in microbial fermentation, typical tip speeds range from 3-7 m/s. For mammalian cell culture, tip speeds are kept below 1.5-2.0 m/s to avoid hydrodynamic damage. This calculator computes tip speed automatically for both source and target vessels. When scaling up at constant tip speed, the required RPM at the larger scale is N₂ = N₁ × (D_i1 / D_i2). Note that Rushton turbines generate higher local shear than axial-flow impellers at the same tip speed.

The power number N_p is dimensionless and roughly constant in the turbulent regime (Re > ~10⁴). Typical turbulent values are: Rushton 6-blade disc turbine ~5.0-6.0; pitched-blade turbine (45°) ~1.3-1.7; marine propeller ~0.35; hydrofoil / elephant-ear impellers (e.g. A310) ~0.3-1.0; and anchor ~0.35. N_p feeds the power draw directly through P = N_p × ρ × N³ × D⁵, so a Rushton turbine draws more than ten times the power of a marine propeller at the same speed and diameter.

Mixed stacks are common: a radial Rushton at the bottom to break up the sparged gas, with axial pitched-blade turbines above it for bulk circulation. Tick Mixed stack under the impeller inputs on either vessel and you can set the type at each position independently, bottom to top. The calculator then computes the effective power number as the sum over the stack, N_p,eff = ΣN_p, instead of assuming one type repeated. A bottom Rushton (N_p 5.0) with two pitched-blade turbines (N_p 1.7 each) gives N_p,eff = 5.0 + 1.7 + 1.7 = 8.4, against 15.0 for three Rushtons, so the mixed stack draws about 44% less power at the same speed and diameter.

The physics behind the sum is that for impellers spaced at least one impeller diameter apart the flow fields do not interact strongly, so the total ungassed power is additive. At closer spacing the upper impellers draw less than their standalone N_p and the sum overestimates the true power, so treat the additive result as an upper bound on tightly spaced stacks.

One useful consequence: every criterion on this page resolves to a ratio of source to target power number. Constant tip speed, constant Reynolds and constant mixing time do not contain N_p at all. Constant P/V, constant kLa and constant OTR contain it only as (ΣN_p)_source / (ΣN_p)_target. So if your bench and production vessels run the same stack composition, that ratio cancels and the target RPM is identical whether you model the stack as mixed or uniform. What the mixed-stack input changes is the absolute power draw and P/V readouts, and the target RPM in the case that actually matters: when the composition differs between scales.

Constant power-per-volume (P/V) preserves turbulence and oxygen transfer and is the most common criterion for aerobic microbial scale-up, with typical targets around 2-5 kW/m³. Constant tip speed (π × N × D_i) preserves the maximum impeller shear and suits shear-sensitive mammalian cells. Constant kLa directly preserves oxygen-transfer capacity. These parameters cannot all be held simultaneously across scales, so pick the one that controls your process and let the others move.

Scale-up from 5 L to 2000 L is a 400-fold volume jump, and it is normally done in two or three steps (5 L to 50 L to 500 L to 2000 L) rather than a single leap so you can retune fluid dynamics and control loops at each stage. Fix your source-vessel operating point (RPM, aeration in vvm, temperature, DO setpoint) and then hold one dimensionless criterion constant across each transition. For microbial fermentation the default is constant P/V at 3-5 W/kg (up to 20 W/kg for high-cell-density E. coli). For mammalian cell culture, use constant tip speed at 1.5-2 m/s or constant kLa. Enter your 5 L vessel as the source and the 2000 L vessel as the target above. This calculator computes the required target RPM at each criterion and shows how the other parameters (kLa, Re, mixing time, OTR, power) drift so you can choose the trade-off before the tank is running.

For CHO cell culture, constant tip speed at 1.5-2.0 m/s is the safest first pass because CHO lines are shear-sensitive and hydrodynamic damage at the impeller blade is a top failure mode at production scale. Constant P/V is used when oxygen supply drives the process (typical CHO fed-batch targets are 20-50 W/m³, well below microbial values), but at constant P/V the tip speed grows with impeller diameter and can push cells past their shear tolerance. Constant kLa is the tightest option for oxygen-limited high-density perfusion or intensified fed-batch. In practice, most CHO scale-up protocols hold tip speed constant, let P/V drop, and then recover kLa through sparger geometry or a small bump in vvm. Run the calculator at both constant tip speed and constant kLa to see the gap and decide where to compensate.

Constant power per volume (P/V) preserves the total energy dissipation rate in the tank, which controls bulk mixing, oxygen transfer (kLa) and the average shear field. Constant tip speed (v_tip = π × N × D_i) preserves the peak shear rate right at the impeller blade tip, which is the local hydrodynamic environment that actually damages fragile cells. The two criteria diverge quickly with scale: at constant P/V the tip speed rises with impeller diameter to roughly the two-thirds power, so a 200x scale-up can nearly double tip speed. Conversely, holding tip speed constant drops P/V (and kLa) by roughly 1/D_i^(4/3). Pick P/V when oxygen or mixing is the limit; pick tip speed when cell damage is the limit. This calculator shows both simultaneously so you can see the trade-off in one glance.

The parameters that govern performance are power per unit volume (P/V), impeller tip speed, the volumetric oxygen transfer coefficient k_La, the Reynolds number, and bulk mixing time. The honest answer is that you cannot maintain all of them: geometric similarity fixes the relationships between them, so pinning one forces the rest to move. In practice you hold the single parameter that controls your process — k_La or P/V when oxygen transfer limits an aerobic culture, tip speed when hydrodynamic shear limits a mammalian line, mixing time when feed or pH gradients drive the failure — and then check that the others stay inside acceptable absolute bounds rather than at their bench values. Product quality follows the same logic: what must be preserved is the cells’ environment (dissolved oxygen, shear exposure, gradient duration), not the dimensionless number itself. Use the Scale journey panel above to pick a criterion and watch the other four diverge across your actual volume range, with the shear and power ceilings for your organism class checked at every intermediate scale.

Because the parameters are not independent — they are all functions of impeller speed N and impeller diameter D_i, and scale-up gives you only one free variable (N) once the vessel geometry is chosen. P/V scales as N³D_i² under geometric similarity, tip speed as ND_i, Reynolds as ND_i², and mixing time as 1/N. Fixing any one of those fixes N, which then determines all the others. Concretely, for a 100-fold volume increase at geometric similarity, holding P/V constant raises tip speed by about 1.7×; holding tip speed constant instead drops P/V to roughly a fifth of its bench value, taking k_La down with it. There is no setting of N that satisfies both. That is why scale-up is a choice about which risk you are willing to carry, and why the trade-off is worth seeing plotted rather than read off a table.