Biotechnology Letters

, Volume 29, Issue 5, pp 737–741

# Fermentation kinetics including product and substrate inhibitions plus biomass death: a mathematical analysis

• Mathieu Bouville
Original Research Paper

## Abstract

We propose an analytical solution of the kinetic equations describing fermentations. Equations are solved in phase space, i.e. the biomass concentration is written explicitly as a function of the substrate concentration. These results hold even when cell death and an arbitrary number of substrate/product inhibitions are accounted for. Moreover, constant yield needs not be assumed.

## Keywords

Biomass yield Fermentation Growth kinetics Inhibition Model

## Nomenclature

Ki

Inhibition coefficient (g/l)

ki

De-dimensionalized K i , k i  = K i /S 0 (−)

KD

Cell death coefficient (g/l h)

m

Maintenance coefficient (h−1)

nD

Number of biomass death terms (–)

nI

Number of inhibitions (–)

Pi

Product concentration (g/l)

pi

De-dimensionalized P i , p = P i /(α i S 0 ) (–)

S

Substrate concentration (g/l)

s

Normalized substrate, s = 1 − S/S 0 (–)

t

time (h)

X

Viable biomass concentration (g/l)

x

Normalized biomass, x = (X − X 0 )(1 + ρ)/(YS 0) (–)

Y

Biomass yield (–)

αi

Stoichiometric coefficient for product P i (–)

κ

De-dimensionalized K D , κ = mYK D /(αS 0) (–)

μ

Maximum specific growth rate (h−1)

μ′

Change in μ due to inhibition (–)

ρ

Ratio of time constants, ρ = m Y/μ (–)

σi

a root of $$\mu '\left( {1 - s} \right) + \rho \,k_1 \prod\nolimits_j {\left[ {1 + \left( {1 - s} \right)/k_j } \right]}$$ (–)

φi

Defined in Eq. 11 (–)

ψ

Distance to constant yield, ψ = k 1 ρ/(1 + ρ) (–)

0

Initial value

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