Table II.

Maximum OUR and k_La values predicted by the single eigenmode solution of the reaction–diffusion equation model for the dynamic gassing-out procedure , and values predicted by solution of the model for steady state fermentation conditions .

	Steady state solution		Dynamic gassing-out solution
Edge case	OURmax|ss	kLa|ss	OURmax|dg	kLa|dg
Perfect mixing	kLa|dgC*	kLa|dg	kLa|dgC*	kLa|dg (measured)
Diffusion only; homogenous OUR; homogenous cell distribution
Diffusion only; point source OUR (at bioreactor bottom); all cells on bioreactor bottom
Well mixed; point source OUR (at bioreactor bottom); all cells on bioreactor bottom	kLa|dgC*	kLa|dg	kLa|dgC*	kLa|dg (measured)

L refers to the characteristic length for diffusion in the liquid phase (typically the height of media in the bioreactor chamber), D to the diffusivity of oxygen in the medium, and C* to the liquid phase concentration of oxygen at equilibrium with the gas phase concentration. Well-mixed systems satisfy the assumptions of the 1st order ODE model presented as Equation (1), and thus all values can be calculated directly from the k_La value measured during the gassing-out procedure. Establishing the similarity between the two k_La values for the diffusion bioreactor with homogenous cell distribution/OUR is important, as it confirms the validity of using k_La measured in these devices for predicting their OTR performance, and validity of their use when comparing OTRs across volume scales. The scalar factor between the two values is ∼1.23. With the 3rd case, the diffusion bioreactor with heterogeneous cell distribution/OUR, steady state OTR performance is predicted to be less than half (∼0.41) of that expected from the measured k_La value. This strikes a note of caution, both when attempting scale up, and designing devices for cell types that are more prone to sedimentation—particularly if stagnant zones are created by cellular fouling. Details of this analysis are given in the Supplementary Information Section.