Bioreactor Scale-Up: P/V, Tip Speed and Mixing Time

The geometry is a cylindrical working volume with H = T, one centrally mounted impeller, and four baffles. Let $N$ be revolutions per second, $D$ impeller diameter and $T$ tank diameter. The length scale factor is $\lambda=(V_2/V_1)^{1/3}$ and $D_2=\lambda D_1$. Density, viscosity, impeller geometry and power number stay equal.

Power and kinematics follow $P=N_p\rho N^3D^5$, $u_{\rm tip}=\pi ND$ and $Re=\rho ND^2/\mu$. Power is the ungassed shaft power delivered to the liquid, excluding motor and transmission losses. The default $N_p=5.47$ is the measured Rushton example in Loo et al. (2019), attached PDF page 5, Power Number results; replace it with a value for the actual impeller.

For turbulent single-phase blending, the Grenville–Nienow correlation at H/T = 1 is

Rodgers et al. (2011), Eq. 1, reproduces the general correlation and explains its aspect-ratio limit. Here this reduces to the same dimensionless blend time at both scales.

Aeration is specified at vessel conditions. $Q_g=\beta V$, $u_g=Q_g/(\pi T^2/4)$ and $P_g=f_gP$. For constant vvm, $u_{g,2}/u_{g,1}=\lambda$; for constant superficial gas velocity this ratio is one.

Estimated kLa uses van’t Riet’s coalescent, nonviscous-liquid model, reproduced in DECHEMA (2020), attached PDF page 17, Eqs. 7–11:

Use P_g/V in W/m³ and u_g in m/s to obtain kLa in s⁻¹, then convert to h⁻¹. The normalized calculation steps preserve these empirical SI conventions.

Required speed ratios are $\lambda^{-2/3}$ for constant P/V, $\lambda^{-1}$ for constant tip speed, and 1 for constant mixing time. For constant estimated kLa the speed ratio is $\lambda^{-2/3}(u_{g,2}/u_{g,1})^{-0.5/(3\times0.4)}$.

What each choice changes: Constant P/V increases tip speed and lengthens mixing time. Constant tip speed reduces P/V and lengthens mixing time; kLa can rise or fall depending on the gas rule. Constant mixing time raises P/V as $\lambda^2$ and tip speed as $\lambda$. Constant estimated kLa changes power density, mixing time and tip speed according to the aeration rule. The table gives the numerical trade-offs for the current inputs.

• Geometrically similar baffled tanks, cylindrical liquid volume, H/T = 1, one impeller, and unchanged relative clearance and baffle dimensions.
• Incompressible Newtonian liquid; density, viscosity and the turbulent ungassed power number remain unchanged.
• The entered gassed/ungassed power fraction is an assumption applied equally at both scales. The default 0.7 is an illustrative estimate, not a measured value for the default tank.
• kLa uses coalescent, water-like gas–liquid behavior with the same liquid composition, gas, sparger characteristics and pressure regime. Salts, antifoam and surfactants can invalidate that estimate.
• The equations for geometric scaling and the four speed ratios are derived here from the cited power, kinematic, mixing and mass-transfer relationships.
• The animation is a schematic top view of a six-blade impeller at each tank’s own drawing scale. Angular motion uses physical seconds and calculated rpm. It does not simulate the liquid flow.

Use for preliminary scale-up comparison, with Re ≥ 10,000 and approximately 0.25 ≤ D/T ≤ 0.50. The Reynolds warning considers every row of the comparison table, including criteria that are not currently selected. The 5 mPa·s viscosity reminder is a conservative screening flag for the nonviscous kLa model, not a published sharp boundary.

DECHEMA reports application of the cited van’t Riet models in conventional sparged stirred vessels from 50 L to 4 m³ (PDF page 17). Outside this range, the scale-up equations remain useful but absolute kLa is an extrapolation. No accuracy guarantee is implied within that range either: confirm mixing time, gassed power and kLa experimentally.

The 95% blend time is a turbulent single-phase bulk-mixing estimate. Gas loading, feeding location, zoning, non-Newtonian rheology, solids, multiple impellers and tall tanks require validated extensions. Tip speed is a comparative kinematic quantity, not a universal cell-damage threshold; Nienow (2006) discusses why hydrodynamic and biological constraints must be assessed together.

Neither matching kLa nor P/V establishes oxygen sufficiency or equivalent cell growth. Oxygen demand, saturation concentration, CO₂ stripping, flooding, foam, heat transfer, motor sizing and mechanical design are outside the model. Gas flows must be evaluated at vessel conditions; standard volumetric flow needs pressure and temperature correction.