A stirred-tank bioreactor supplies oxygen to a growing culture through gas–liquid mass transfer, while the cells consume it. Two coupled first-order ODEs, integrated with 4th-order Runge–Kutta:
OTR = kLa·(C* − CL) [O2 supply, mmol/L/h]
OUR = qO2·X [O2 demand, mmol/L/h]
dCL/dt = OTR − OUR
dX/dt = μmax·CL/(CL+Ko2)·X [O2-limited Monod growth]
kLa = kLa_ref·(N/Nref)^1.2·(vvm/vvmref)^0.5
The agitation and aeration exponents (1.2 on N, 0.5 on gas rate) follow the classic Van't Riet-style correlation k_La ∝ (P/V)^0.4·v_s^0.5, since ungassed power draw of a turbine impeller scales P ∝ N³ (so P/V exponent 0.4 becomes N-exponent ≈1.2). C* is the air-saturation dissolved-oxygen concentration (~0.21 mmol/L at 37 °C, 1 atm).
As biomass X grows exponentially, OUR = qO2·X climbs with it while OTR is capped at kLa·C* (its maximum, reached as CL→0) — the fixed oxygen-transfer ceiling set by the current agitation/aeration setting. Once demand crosses that ceiling, dissolved oxygen collapses toward zero, the Monod term CL/(CL+Ko2) collapses with it, and growth stalls even though nutrients are not limiting — exactly the failure mode that forces real fermentations to raise agitation/aeration or accept a lower cell-density set-point.
- Left pane — reactor cross-section: impeller spin rate tracks N, bubble density tracks aeration, broth color runs green (high DO) → amber → red (anoxic).
- Right pane — DO (%), biomass and the OTR-vs-OUR race over time; the shaded band marks the OUR curve poking above the OTR ceiling.