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.