Bioreactor Oxygen Transfer: kLa, OTR and Max Cell Density

The liquid oxygen balance is $dC/dt=k_La(C^*-C)-OUR$, with $OUR=q_{O2}X$.

Solubility. Freshwater air saturation follows the Benson–Krause polynomial in USGS Technical Memorandum 2011.03, attached file page 6, equation 7. The same page, equation 11, gives water vapor pressure. The tool scales by the dry-gas pressure ratio $(p-p_v)/(101.325-p_v)$, oxygen mole fraction relative to 0.2095, and the user-entered broth/water solubility ratio. This uses ideal Henry-law pressure scaling and omits the small near-atmospheric virial correction in USGS equation 9.

Transfer. Van 't Riet (1979), DOI 10.1021/i260071a001, gives the customary coalescing $k_La=0.026(P_g/V)^{0.4}u_g^{0.5}$ and non-coalescing $k_La=0.002(P_g/V)^{0.7}u_g^{0.2}$ forms, with SI numerical inputs W/m³ and m/s and output s⁻¹. The coefficients are also documented in the attached DECHEMA 2020 recommendations, page 17, equations 9–10. Normalization quantities in the steps make those dimensional empirical coefficients explicit. No automatic viscosity or temperature correction is applied to kLa.

The alternative cell-free gassing-out estimate is the two-point logarithmic slope $k_La=\ln[(1-f_1)/(1-f_2)]/\Delta t$. Fractions refer to the final measured equilibrium in that test. The default 0.2 and 0.8 endpoints span the usual 20–80% range. It is not a multipoint statistical regression. The independent reference cases use measured endpoints from DECHEMA 2020, attached page 55, and compare with published regression slopes on page 56; this simplified estimate is not an implementation of the complete DECHEMA measurement protocol.

Capacity and heat. $OTR=k_La(C^*-C_{min})$ and $X_{O2}=OTR/q_{O2}$. Metabolic heat is oxygen uptake times the positive oxycaloric equivalent. The 460 kJ/mol default is an estimate; Luong and Volesky (1980), DOI 10.1002/cjce.5450580412, report 465 kJ/mol for rates and 461 kJ/mol for totals. Required jacket duty adds gassed stirring power. Available heat removal is $UA\Delta T_{mean}$. Cooling biomass capacity uses the heat removal remaining after stirring, clamped to zero. The combined ceiling is the lower of the oxygen and cooling ceilings.

The transient is the closed-form, constant-OUR solution, clipped at zero oxygen. Its trace spans five transfer time constants, capped at two hours. Bubble motion illustrates aeration and is not a predicted bubble trajectory.

• Uniform, well-mixed liquid; constant working volume, temperature, effective pressure, effective dry-gas composition, kLa and specific uptake.
• Biomass is dry mass per liquid volume, not cells/mL. Enter strain-specific uptake per gram dry mass; no cell mass conversion is assumed.
• The minimum DO is a fraction of equilibrium under the selected operating conditions. A sensor calibrated to air saturation may require conversion before use.
• The broth/water solubility ratio and default uptake are illustrative user assumptions. Non-coalescing kLa does not itself correct oxygen solubility for salts.
• All gassed mechanical power becomes broth heat. Feed, evaporation, wall losses and other heat sources are omitted.
• Jacket UA and mean driving temperature difference are supplied by the user. Coolant flow and warming are represented only through these values.

Use for screening nonviscous, sparged stirred tanks and teaching the coupled oxygen/heat balances. Freshwater solubility is restricted to 0–40 °C. USGS documents near-atmospheric solubility methods at about 0.5–1.1 atm; extension to 3 bar absolute and oxygen enrichment uses ideal partial-pressure scaling, not a validated high-pressure broth model.

Van 't Riet correlations describe conventional low-viscosity gas–liquid systems; real broths, antifoam, impellers, spargers, flooding, shear limits and scale can change kLa substantially. The 500–10,000 W/m³ advisory band is a screening guard, not proof that a vessel is within a correlation's domain. Prefer measurements in the intended broth and operating range.

Gassing-out requires no oxygen-consuming cells or reactions, constant operating conditions, a stable equilibrium reading and negligible probe delay. A two-point fit has no confidence interval or diagnostic residuals.

The reported ceilings assume sufficient feed-gas oxygen and a representative gas composition. Gas-phase depletion, oxygen supply flow limits, hydrostatic gradients, non-Newtonian mixing, substrate limitations and biological inhibition are not solved. Do not infer a growth forecast or guaranteed achievable density. The constant-OUR transient ceases to describe cell physiology after oxygen depletion; the zero plateau only marks that boundary. Fermentative or mixed metabolism can invalidate the heat/oxygen estimate.