T3 · Kinetics / RateBioprocessing & Microbial FermentationGATE 2027 Syllabus (XE5)

Monod Specific Microbial Growth Rate for PHA Biopolymers

Empirical model relating the specific microbial growth rate mu of polymer-producing bacteria (such as Cupriavidus necator producing PHB) to limiting carbon substrate concentration [S].

Mathematical Canonical Relation

\mu = \mu_{max} \frac{[S]}{K_s + [S]}

Plain Formula: mu = mu_max * ( [S] / (Ks + [S]) )

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Max Growth Rate mu_max (h⁻¹) (h⁻¹)

mu_max_val

Range: 0.1 – 1 h⁻¹

Substrate Concentration [S] (g/L) (g/L)

S_conc

Range: 0.1 – 50 g/L

Monod Constant Ks (g/L) (g/L)

Ks_val

Range: 0.05 – 5 g/L

Calculated Engineering Result:

mu = 0.324 h⁻¹

Engineering Verification:Biomass doubling time td = 2.1 hours.

When & Where to Apply

Use when modeling fed-batch bioreactors for bacterial polyhydroxyalkanoate (PHA/PHB) biosynthesis to optimize biomass growth before nutrient-limiting polymer accumulation phase.

Common Engineering Pitfalls

Confusing biomass growth phase with PHA polymer accumulation phase (PHA accumulates when nitrogen/phosphorus is limited)
Forgetting that at high substrate concentrations [S] >> Ks, the growth rate reaches asymptotic mu_max

Physical Assumptions & Validity Limits

Single growth-limiting substrate (e.g., glucose, glycerol, or volatile fatty acids)
No substrate inhibition or toxic metabolite accumulation
Homogeneous mixing throughout the agitated bioreactor vessel

Variable Symbols, SI Units & Dimensional Specifications

Symbol	Variable Meaning	SI Unit	Typical Industrial Range
\mu	Specific microbial growth rate	h⁻¹	—
\mu_{max}	Maximum specific growth rate	h⁻¹	0.35
[S]	Limiting substrate concentration (glucose)	g/L	5.0
K_s	Substrate affinity constant (half-velocity constant)	g/L	0.50

Worked Industrial Numerical Example: Cupriavidus necator Growth Rate for PHB Production

Problem Statement:

In a 10,000 L fermenter producing polyhydroxybutyrate (PHB), Cupriavidus necator has mu_max = 0.36 h^-1 and Ks = 0.40 g/L for glucose. Calculate specific growth rate mu when glucose concentration is maintained at [S] = 3.6 g/L.

Step-by-Step Numerical Solution:

Apply Monod equation: mu = mu_max * [S] / (Ks + [S])
Substitute values: mu = 0.36 * 3.6 / (0.40 + 3.6)
Evaluate denominator: 0.40 + 3.6 = 4.0 g/L
Calculate mu: 0.36 * (3.6 / 4.0) = 0.36 * 0.90 = 0.324 h^-1

Final Answer: mu = 0.324 h⁻¹ (Doubling time t_d = 2.14 hours)

💡 Engineering Interpretation: At 3.6 g/L glucose, the culture grows at 90% of its maximum potential rate, rapidly building bacterial cell density before nutrient starvation triggers intracellular PHB granule accumulation.

Authoritative Technical Reference & Source Provenance

Bioprocess Engineering: Basic Concepts by Michael L. Shuler & Fikret Kargi (Prentice Hall, 2002), 2nd Edition, Page 160-168.

Connected Curriculum Lesson

Microbial Fermentation Kinetics & PHA Production

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