Bioprocess Engineering Formulas: μmax, Kinetics & Cheat Sheet

In bioprocess, μ_max (the maximum specific growth rate) is the highest rate at which biomass can grow per unit biomass when no nutrient is limiting, with units of inverse time (h⁻¹). It is the asymptotic ceiling of the Monod model μ = μ_max · S / (K_s + S) as substrate concentration S becomes large. Typical μ_max values are E. coli 0.8–2.0 h⁻¹ in rich medium (LB, K-12 vs BL21) and 0.7–0.8 h⁻¹ in minimal glucose medium; Pichia pastoris 0.15–0.30 h⁻¹ on glycerol; Saccharomyces cerevisiae 0.45 h⁻¹ on glucose; CHO 0.025–0.04 h⁻¹; HEK293 0.025–0.035 h⁻¹. μ_max is measured in batch culture from the slope of ln(X) vs t during early exponential phase, where substrate is in clear excess. It anchors fed-batch design: the set growth rate is chosen below μ_max to avoid overflow metabolism (acetate in E. coli, lactate in CHO).

Bioprocess kinetics is the quantitative description of cell growth, substrate uptake, and product formation rates inside a bioreactor. The three core equations are: (1) growth: dX/dt = μ · X, with the Monod substrate dependence μ = μ_max · S / (K_s + S); (2) substrate uptake: dS/dt = −μ/Y_X/S · X − m_s · X, where Y_X/S is the biomass yield on substrate and m_s is the maintenance coefficient; (3) product formation: dP/dt = q_P · X, where q_P (g product / g cell / h) follows growth-associated (Luedeking–Piret α · μ), non-growth-associated (β), or mixed forms. These rate equations are integrated against the mass balance for batch, fed-batch, or continuous (chemostat) reactors. At steady state in a chemostat, μ equals the dilution rate D by definition.

The Monod equation describes how specific growth rate depends on the concentration of a single limiting substrate: μ = μ_max · S / (K_s + S). Here μ_max is the maximum growth rate, S is the limiting substrate concentration (g/L), and K_s is the half-saturation constant (the S at which μ = μ_max/2). At S much greater than K_s, μ approaches μ_max; at S much less than K_s, the relationship is approximately first-order in S. Typical K_s values are very low for glucose: E. coli 4 mg/L, S. cerevisiae 25 mg/L; Pichia on glycerol 10–30 mg/L; CHO on glucose 50–100 mg/L. The Monod model is the foundation for chemostat steady-state design and fed-batch substrate-limited feeding strategies. See the Monod kinetics deep dive for derivations and worked examples.

k_La (the volumetric oxygen mass transfer coefficient, units h⁻¹) is most commonly estimated using the Van’t Riet correlation: k_La = a · (P/V)^b · (v_s)^c, where P/V is gassed power per unit volume (W/m³), v_s is superficial gas velocity (m/s), and a, b, c are empirical constants. Typical exponents are b = 0.4–0.7 and c = 0.4–0.5; for coalescing media (water) a is around 0.026 with b = 0.4 and c = 0.5, while for non-coalescing media (salt solutions, broth) a is around 0.002 with b = 0.7 and c = 0.2. k_La can also be measured experimentally with the dynamic gassing-out method: degas the broth with N₂, switch to air, fit dC/dt = k_La · (C* − C). The full k_La guide covers derivations, correlations, and three measurement methods.

Power per unit volume (P/V, W/m³) is the standard scale-up criterion for geometrically similar bioreactors: P/V = (N_p · N³ · D_i⁵ · ρ) / V, where N_p is the impeller power number (dimensionless), N is rotational speed (rev/s), D_i is impeller diameter (m), ρ is broth density (kg/m³), and V is liquid volume (m³). Power numbers depend on impeller geometry in the turbulent regime: Rushton 6-bladed disc 5.0–5.5, pitched-blade (4-blade, 45°) 1.2–1.7, hydrofoil (elephant-ear / SC-3) 0.3–0.8. P/V scales as N³ · D_i² / V at constant geometry — useful for scale-up because doubling the impeller diameter requires reducing speed by a factor of 2^2/3 to hold P/V constant. Typical industrial values are 1–5 kW/m³ for microbial fermentations and 30–100 W/m³ for mammalian cell culture to control shear.

The impeller Reynolds number, used to assess mixing regime in a stirred-tank bioreactor, is Re = (ρ · N · D_i²) / μ, where ρ is broth density (kg/m³), N is impeller rotational speed (rev/s), D_i is impeller diameter (m), and μ is dynamic viscosity (Pa·s). Regimes for stirred tanks: laminar Re < 10, transitional 10 < Re < 1×10⁴, and turbulent Re > 1×10⁴. Most industrial bioreactors operate firmly in the turbulent regime (Re ≈ 10⁵–10⁶), which is required for the Van’t Riet k_La correlation and N_p-table power numbers to apply. Mycelial broths and high-viscosity cultures can drop into the transitional regime — in which case N_p rises sharply and mixing time gets dramatically longer.

The most-used equation in bioprocess design is the Monod equation, μ = μ_max · S / (K_s + S), which links specific growth rate to limiting substrate concentration and underpins batch, fed-batch, and chemostat models. Runners-up by frequency are the volumetric oxygen transfer relation OTR = k_La · (C* − C_L) for aeration sizing, the exponential feed rule F(t) = (μ · X₀ · V₀) / (Y_X/S · S_f) · e^μt for fed-batch dosing, and the impeller power draw P = N_p · ρ · N³ · D_i⁵ for scale-up. Together with the yield-coefficient definitions (Y_X/S, Y_P/S), these four sit inside almost every upstream process model. Published bioreactor textbooks (Doran 2013; Shuler and Kargi 2017) build their entire upstream chapters around them, and every calculator on this cheat sheet reduces to one of these forms.

In a stirred bioreactor, k_La is calculated either from an empirical correlation or from a direct dynamic experiment. The Van’t Riet correlation k_La = a · (P/V)^b · v_s^c uses gassed power per unit volume P/V (W/m³) and superficial gas velocity v_s (m/s); for water-like coalescing broths a is around 0.026 with b = 0.4 and c = 0.5, and for non-coalescing salt-rich media a is around 0.002 with b = 0.7 and c = 0.2. For direct measurement, the dynamic gassing-out method degases the broth with N₂, switches to air, and fits dC/dt = k_La · (C* − C) to the recorded DO trace; the negative slope of ln(C* − C) versus time gives k_La in h⁻¹. Typical microbial values reach 200–800 h⁻¹ at industrial scale; mammalian bioreactors sit at 5–50 h⁻¹ because impellers must run at lower tip speed to protect cells. See the OTR/k_La estimator to run both routes side by side.

Y_X/S (yield of biomass on substrate, g biomass / g substrate) and Y_P/S (yield of product on substrate, g product / g substrate) both quantify how efficiently a limiting substrate is converted, but they track different sinks. Y_X/S measures how much cell mass forms per gram of substrate consumed: typical values are 0.4–0.5 g/g for E. coli on glucose, 0.3–0.5 g/g for S. cerevisiae, and 0.02–0.06 g dry cell weight per gram glucose for CHO. Y_P/S measures how much target product (protein, metabolite, or organic acid) forms per gram of substrate and is set by pathway stoichiometry plus flux distribution; ethanol from S. cerevisiae reaches 0.48 g/g (near the 0.51 g/g theoretical maximum), while a recombinant protein might sit at 0.05–0.20 g/g. Overall carbon closure requires Y_X/S + Y_CO2/S + Y_P/S + Y_residual/S = 1.0 on a carbon-mole basis, which is how off-gas RQ measurements validate a fed-batch run.