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.