Biotechnology Letters

, Volume 29, Issue 5, pp 737–741 | Cite as

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

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 Ki, ki = Ki/S0 (−)

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 Pi, p= Pi/(αiS0) (–)

S

Substrate concentration (g/l)

s

Normalized substrate, s = 1 − S/S0 (–)

t

time (h)

X

Viable biomass concentration (g/l)

x

Normalized biomass, x = (X − X0)(1 + ρ)/(YS0) (–)

Y

Biomass yield (–)

αi

Stoichiometric coefficient for product Pi (–)

κ

De-dimensionalized KD, κ = mYKD/(αS0) (–)

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 = k1ρ/(1 + ρ) (–)

Superscript

0

Initial value

References

  1. Bailey JE, Ollis DF (1977) Biochemical engineering fundamentals. McGraw-Hill, New YorkGoogle Scholar
  2. Bazua CD, Wilke CR (1977) Ethanol effects on the kinetics of a continuous fermentation with Saccharomyces cerevisiae. Biotechnol Bioeng Symp 7:105–118PubMedGoogle Scholar
  3. Caro I, Pérez L, Cantaro D (1991) Development of a kinetic model for the alcoholic fermentation of must. Biotechnol Bioeng 38:742–748CrossRefGoogle Scholar
  4. Colombié S, Malherbe S, Sablayrolles JM (2005) Modeling alcoholic fermentation in enological conditions: feasibility and interest. Am J Enol Vitic 56:238–245Google Scholar
  5. Cramer AC, Vlassides S, Block DE (2001) Kinetic model for nitrogen-limited wine fermentations. Biotechnol Bioeng 77:49–60CrossRefGoogle Scholar
  6. Ghose TK, Tyagi RD (1979) Rapid ethanol fermentation of cellulose hydrolysate. II. product and substrate inhibition and optimization of fermenter design. Biotechnol Bioeng 21:1401–1420CrossRefGoogle Scholar
  7. Hill GA, Robinson CW (1990) A modified Ghose model for batch cultures of Saccharomyces cerevisiae at high ethanol concentrations. Chem Eng J 44:B69–80CrossRefGoogle Scholar
  8. Hoppe GK, Hansford GS (1982) Ethanol inhibition of continuous anaerobic yeast growth. Biotechnol Lett 4:39–44CrossRefGoogle Scholar
  9. Maiorella BL, Blanch HW, Wilke CR (1983) By-product inhibition effects on ethanolic fermentation by Saccharomyces cerevisiae. Biotechnol Bioeng 25:103–121CrossRefGoogle Scholar
  10. Marín R (1999) Alcoholic fermentation modelling: current state and perspectives. Am J Enol Vitic 50:166–178Google Scholar
  11. Mitchell DA, von Meien OF, Krieger N, Dalsenter FDH (2004) A review of recent developments in modeling of microbial growth kinetics and intraparticle phenomena in solid-state fermentation. Biochem Eng J 17:15–26CrossRefGoogle Scholar
  12. Monod J (1941) Recherche sur la croissance des cultures bactériennes. HermannGoogle Scholar
  13. Ough CS, Amerine MA (1963) Use of grape concentrate to produce sweet table wines. Am J Enol Vitic 14:194–204Google Scholar
  14. Thatipamala R, Rohani S, Hill GA (1992) Effects of high product and substrate inhibitions on the kinetics and biomass and product yields during ethanol batch fermentation. Biotechnol Bioeng 40:289–297CrossRefGoogle Scholar
  15. Topiwala HH, Sinclair CG (1971) Temperature relationship in continuous culture. Biotechnol Bioeng 13:795–813PubMedCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.Institute of Materials Research and EngineeringSingaporeSingapore

Personalised recommendations