Method: the five checks

Scope

The engine answers one question: when a fermentation that works at bench scale is moved into a geometrically similar stirred tank of 150 L or 1,000 L, which physical limit is crossed first, and at what volume. It uses mass and energy balances and published correlations only. It is deterministic: the same inputs always give the same result, and every number on screen can be traced to an equation below.

It does not predict titre, yield or product quality, and it has no model of cell physiology beyond the rates you give it.

Status. Prototype engine using published correlations and illustrative literature values. Not validated. The example reactors and processes on the home page are illustrative literature values, not measurements.

Inputs and confidence

Every process input carries a confidence level, and every check inherits the worst confidence among the inputs it uses.

Level	Meaning
A	Measured by the customer on this process.
B	Measured at another scale or condition. A measured reactor kLa or mixing time counts as B.
C	Estimate or literature value. Correlation-based kLa and mixing time count as C.

Symbol	Input	Unit
X	Target biomass concentration	g DW/L
qO2	Specific oxygen uptake rate	mmol/(g h)
qS	Specific substrate uptake rate	g/(g h)
S_res	Residual substrate in the bulk during feeding	g/L
DO_crit	Critical dissolved oxygen, fraction of saturation	-
T	Broth temperature	°C
tip_max, (P/V)_max	Tolerated impeller tip speed and power per volume	m/s, W/m³
y_H2, y_O2	Gas feed mole fractions (gas fermentation only)	-

Internally everything is SI. Two conversions are worth stating: OUR = qO₂·X gives mmol/(L h), numerically equal to mol/(m³ h); and a gas flow in vvm becomes a superficial velocity v_s = (vvm·V/60)/(πT²/4).

Reactor and scale-up rules

Each scale is a geometrically similar stirred tank: tank diameter T, liquid height H, impeller diameter D, n identical impellers. With H/T fixed, T grows as V^1/3. Turbulent, ungassed power draw:

P = n_{imp} N_{p} ρ N^{3} D^{5}

[POWER-NP]

N_p ≈ 5 for a Rushton turbine, ρ = 1000 kg/m³, N in 1/s, D in m. Power is summed over impellers, which assumes they do not interact.

The base scale (where the process is known to work) runs at its maximum stirrer speed and gas flow unless a measured operating point is given. The stirrer speed at the target scale follows one of four rules, then is capped at that vessel's maximum:

Constant P/V: same power per volume as the base.
Constant tip speed: same πND as the base.
Constant kLa: P/V solved from the kLa correlation so kLa equals the base value.
Max capacity: each vessel at its own maximum stirrer speed and gas flow.

Gas flow is held at the base vvm, capped at each vessel's maximum, so superficial velocity grows with scale.

1. Oxygen transfer

Volumetric mass transfer coefficient, van 't Riet (1979), SI units (kLa in 1/s, P/V in W/m³, v_s in m/s):

k_{L} a = 0.026 {(P / V)}^{0.4} v_{s}^{0.5}

coalescing [KLA-VANTRIET]

k_{L} a = 0.002 {(P / V)}^{0.7} v_{s}^{0.2}

non-coalescing [KLA-VANTRIET]

If the reactor's kLa was measured, the correlation is calibrated by the constant factor measured/predicted at the measurement point. Oxygen solubility from Henry's law, pure water, head pressure only:

C^{*} = H_{cp} (T) p_{head} y_{O_{2}}, H_{cp} = 1.3 \times 10^{- 5} e^{1500 (1 / T - 1 / 298.15)}

[HENRY-O2]

H_cp in mol/(m³ Pa), T in K. Check value: 37 °C, air, 1.01325 bar gives 0.227 mol/m³ (about 7.3 mg/L).

Maximum transfer rate while holding dissolved oxygen above the critical level, the oxygen supplied with the gas, and the ratio:

{OTR}_{\max} = k_{L} a C^{*} (1 - {DO}_{crit}) r_{O_{2}} = \max (\frac{OUR}{{OTR}_{\max}}, \frac{OUR}{O_{2, fed}})

check 1

2. Substrate gradients

Regime analysis (Sweere et al. 1987; Oosterhuis 1984) compares the time needed to mix the vessel with the time the cells need to consume the substrate present in the bulk. If mixing is slower, the feed zone and the rest of the tank see different substrate concentrations.

95% mixing time, turbulent regime. One impeller, Grenville; several Rushton impellers with H/T > 1, Cooke et al. (Rodgers et al. 2011), with N_p the summed power number:

N t_{95} = 5.2 N_{p}^{- 1 / 3} {(T / D)}^{2} {(H / T)}^{0.5}

[MIX-GRENVILLE]

N t_{95} = 3.3 N_{p}^{- 1 / 3} {(H / D)}^{2.43}

[MIX-COOKE]

t_{S} = \frac{S_{res}}{q_{S} X} r_{mix} = \frac{t_{95}}{t_{S}}

check 2 [REGIME]

A measured mixing time replaces the correlation and is scaled as t₉₅ ∝ 1/N. For a gas fermentation with no liquid-fed carbon source, q_S and S_res are empty and the check reports "cannot evaluate": dissolved H₂ consumption would be the analogue, and H₂ transfer limitation is not modelled.

3. Heat removal

Metabolic heat is proportional to oxygen consumed; all shaft power ends as heat. Removal is through the jacket only.

Q_{met} = \frac{OUR}{3600} Δ H_{O_{2}} V, Δ H_{O_{2}} = 460 kJ/mol

[HEAT-O2]

r_{heat} = \frac{Q_{met} + P}{U A Δ T_{cool}}

check 3

Luong and Volesky (1980) report 0.461 MJ per mol O₂ across A. niger, E. coli and Candida. Evaporation and sensible heat of the gas are ignored. Wetted-wall area grows as V^2/3 while heat grows as V, which is why this ratio rises with scale.

4. Shear

u_{tip} = π N D r_{shear} = \max (\frac{u_{tip}}{u_{tip, \max}}, \frac{P / V}{{(P / V)}_{\max}})

check 4

Tip speed and P/V are proxies for the hydrodynamic stress cells feel. The tolerated values are process inputs; the presets use 6 m/s as an assumption for bacteria and yeast, with no P/V limit.

5. H₂ and off-gas flammability

Only for gas fermentations that feed hydrogen with oxygen. A mixture is flagged as flammable when both the fuel and the oxidant exceed their limits: hydrogen above its lower flammability limit (4%) and oxygen above the limiting oxygen concentration (5% with N₂ as diluent).

r_{gas} = \min (\frac{y_{H_{2}}}{0.04}, \frac{y_{O_{2}}}{0.05})

check 5 [H2-LIMITS] [H2-LOC]

This is a screening rule, not a ternary flammability diagram, and it applies the limits with no safety margin. Headspace composition drifts as gas is consumed. It flags; it does not certify safety. Keeping oxygen below the limit (Ishizaki et al. 2001 use 6.9% v/v) lowers C* and so tightens check 1.

Thresholds and sensitivity

Ratio	Status
< 0.8	Within margin
0.8 to 1.0	Warning
> 1.0	Limit crossed

These thresholds are product choices, not values from the cited papers; regime analysis in particular compares orders of magnitude, so the boundary is indicative. The first limit is the failing check with the largest ratio at the smallest failing scale.

Sensitivity: every input with confidence B or C is multiplied by 0.7 and by 1.3 and the assessment is rerun. A C-level input whose change alters the verdict or the first limit is listed as something to measure at bench scale. Temperature is excluded, since 30% of a Celsius value has no physical meaning.

Known limits

Ungassed power. Gassed power is lower, so P, and with it kLa, is overestimated. Non-conservative.
Pure-water solubility, no gas depletion. Medium salts lower C*, and O₂ is depleted along the bubble path in tall tanks. Both non-conservative. Hydrostatic pressure is ignored, which is conservative.
Oxygen gets easier with scale. At constant vvm the superficial gas velocity grows with tank size, so the correlation's kLa can rise with volume. Real large tanks often show the opposite locally.
Whole-vessel balances. Each check is an independent ratio over the whole vessel. Local oxygen limitation inside a glucose-rich feed zone, seen in large E. coli fed-batches, is a coupling the engine cannot represent.
Validity range. The van 't Riet fit's original range of volume, P/V and v_s was not checked; at 1,000 L the example vessel reaches v_s of about 0.03 m/s.
No magnitude. The engine says which limit and where. It does not predict how much yield is lost.

Sources

Status tags: verified citation and number checked; secondary number taken from a named secondary source; assumption no source found, treated as confidence C.

[POWER-NP]Rushton, Costich, Everett (1950). Power characteristics of mixing impellers, Parts I and II. Chemical Engineering Progress 46(8):395-404, 46(9):467-476. N_p ≈ 5 for a Rushton turbine via Grenville (2017), IChemE lecture. citation verified Np secondary
[KLA-VANTRIET]van 't Riet, K. (1979). Review of measuring methods and results in nonviscous gas-liquid mass transfer in stirred vessels. Ind. Eng. Chem. Process Des. Dev. 18(3):357-364. doi:10.1021/i260071a001. citation verified coefficients as widely quoted, not checked against the original
[MIX-GRENVILLE]Grenville, R. K. (1992). PhD thesis, Cranfield Institute of Technology; Grenville and Nienow (2004), Blending of miscible liquids, in Handbook of Industrial Mixing, Wiley, 507-542. Equation as quoted in Bågmark (2024), MSc thesis, Lund University. secondary
[MIX-COOKE]Rodgers, Gangolf, Vannier, Parriaud, Cooke (2011). Mixing times for process vessels with aspect ratios greater than one. Chem. Eng. Sci. 66(13):2935-2944. Equation as quoted in Bågmark (2024). secondary
[REGIME]Sweere, Luyben, Kossen (1987). Regime analysis and scale-down: tools to investigate the performance of bioreactors. Enzyme Microb. Technol. 9:386-398. Oosterhuis, N. M. G. (1984). Scale-up of bioreactors: a scale-down approach. PhD thesis, TU Delft. Thresholds 0.8 and 1.0 are product choices. citation verified thresholds assumed
[HENRY-O2]Sander, R. (2023). Compilation of Henry's law constants (version 5.0.0) for water as solvent. Atmos. Chem. Phys.; henrys-law.org, O₂ entry. H_cp = 1.3e-5 mol/(m³ Pa), d ln H/d(1/T) = 1500 K. verified
[HEAT-O2]Luong and Volesky (1980). Determination of the heat of some aerobic fermentations. Can. J. Chem. Eng. 58(4):497-504. doi:10.1002/cjce.5450580412. 0.461 MJ/mol O₂. Also Cooney, Wang, Mateles (1969), Biotechnol. Bioeng. 11:269. verified
[H2-LIMITS]Zabetakis, M. G. (1965). Flammability characteristics of combustible gases and vapors. US Bureau of Mines Bulletin 627. LFL 4% H₂ in air. secondary
[H2-LOC]Perry's Chemical Engineers' Handbook, 8th ed. (2007), p. 23-9: LOC 5% O₂ (N₂), 5.2% (CO₂). Practice: Ishizaki, Tanaka, Taga (2001), Appl. Microbiol. Biotechnol., doi:10.1007/s002530100775, O₂ below 6.9%. secondary
[ECOLI-QO2]Lin, Mathiszik, Xu, Enfors, Neubauer (2001). Biotechnol. Bioeng. 73(5):347-357. O₂ uptake capacity about 5 mmol/(g h) in glucose-limited fed-batch. verified
[YEAST-QO2-QS]van Hoek, van Dijken, Pronk (1998). Effect of specific growth rate on fermentative capacity of baker's yeast. Appl. Environ. Microbiol. 64(11):4226-4233. qO₂ 3.9 mmol/(g h), q_glucose 0.306 g/(g h) at D = 0.15 1/h. verified
[HOB-X]Repaske and Mayer (1976). Dense autotrophic cultures of Alcaligenes eutrophus. Appl. Environ. Microbiol. 32(4):592-597. 25 g/L dry weight. verified
[REACTOR-ILLUSTRATIVE] [A-*]Assumptions. Example reactor geometry (H/T 2, D/T 1/3, two Rushton turbines), maximum speeds and gas flows, head pressure 1.3 bar, U 250 to 500 W/(m² K), coolant ΔT 15 K; E. coli X 40 g/L and qS 0.3 g/(g h); yeast X 30 g/L; H₂-oxidising bacteria qO₂ 10 mmol/(g h) and the 70/20/10 H₂/O₂/CO₂ feed; residual substrate 0.05 g/L; DO_crit 0.2; tip speed limit 6 m/s; temperatures. assumption
[CHECK]Bylund et al. (1998), Bioprocess Eng. 18:171-180, doi:10.1007/s004490050427; Xu et al. (1999), Appl. Microbiol. Biotechnol., doi:10.1007/s002530051433; Larsson et al. (1996), Bioprocess Eng. 14:281-289, doi:10.1007/BF00369471, read via Arulrajah et al. (2025), Bioprocess Biosyst. Eng., doi:10.1007/s00449-025-03182-w. abstracts and review only

Scope

Inputs and confidence

Reactor and scale-up rules

1. Oxygen transfer

2. Substrate gradients

3. Heat removal

4. Shear

5. H2 and off-gas flammability

Thresholds and sensitivity

Known limits

Sources

5. H₂ and off-gas flammability