Scaling a stirred-tank bioreactor from bench to production at fixed geometric ratios (D/T = 0.35, H = T) does not preserve every flow variable at once — turning up the impeller enough to keep mixing fast enough at large scale drives local energy dissipation, and therefore shear, far higher than it was at bench scale.
Power draw: P = Np · ρ · N³ · D⁵ (Np ≈ 5, Rushton turbine)
Mean dissipation: ε = P / (ρ·V)
Kolmogorov scale: η = (ν³ / ε)^(1/4) ν = μ/ρ
Pumping / mixing: Q = Nq · N · D³, θm ≈ 4·(V/Q) (Nq ≈ 0.75)
η is the size of the smallest turbulent eddies — the scale at which kinetic energy is finally dissipated as heat. When η is much larger than a suspended cell (≈15 µm for CHO cells), the cell simply rides the eddy. Once N and D push η down toward the cell diameter, the eddy itself is small enough to tear at the cell membrane, and viability drops sharply — modelled here as
viability ≈ 100 · η⁴ / (η⁴ + d_cell⁴)
- Impeller speed — raises N³ into the power draw, shrinking η roughly as N^(-3/4).
- Vessel scale — larger V (at fixed D/T) needs a disproportionately bigger impeller diameter D to hit the same mixing time, and P grows with D⁵, so η at production scale is usually far smaller than at bench scale even at "the same" tip speed — this is the classic scale-up shear trap.
- Viscosity — raises ν and hence η directly, cushioning cells against shear but also slowing oxygen transfer (not modelled here).
- Inject tracer dye — releases a dye pulse into the circulating flow field; the readout θm is the time for ~4 impeller circulations to fully redistribute it, the standard rule-of-thumb blend-time estimate.
Real bioprocess engineers walk this exact trade-off when scaling CHO or hybridoma cultures: enough power for good mixing and oxygen transfer, without enough local dissipation to shear-kill the very cells being grown.