Application of mathematical models to ethanol fermentation in biofilm reactor with carob extract


Mathematical models not only ensure information about kinetic-metabolic nature of fermentations, but also facilitate their control and optimization. In the study, flexible ten models were evaluated and employed to describe the ethanol fermentation in a biofilm reactor with a carob extract medium (CEM). Findings indicated that W model well fitted the experimental data of cell growth (root mean square error (RMSE) = 0.289 g/L, mean absolute error (MAE) = 0.237 g/L, regression coefficient (R2) = 0.9944, bias factor (BF) = 1.021, and accuracy factor (AF) = 1.047), ethanol production (RMSE = 1.609 g/L, MAE = 1.277 g/L, R2 = 0.9774, BF = 1.178, and AF = 1.283), and substrate consumption (RMSE = 2.493 g/L, MAE = 1.546 g/L, R2 = 0.9931, BF = 1.001 and AF = 1.053). In the prediction of kinetic parameters, W model also gave better and well-directed results compared with the other mathematical models used in the study. When an independent set of experimental data was used in the validation of mathematical models, similar validation results were obtained and W model was also successful. Consequently, W model could be used for more progress of fermentation process in biofilm reactor with CEM, which can serve as a universal equation.

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Fig. 1
Fig. 2


μ max :

Maximum specific growth rate (1/h)

AF :

Accuracy factor (−)

A m :

Upper asymptote (g/L)

A o :

Lower asymptote (g/L)

A t :

The predicted biomass, ethanol, and sugar concentration at time “t” (g/L)

BF :

Bias factor (−)

DF :

Degree of freedom (−)

e :

Euler number, 2.718 (−)

h o :

A parameter calculating the initial physiology state of the cells (g/L)

k :

A parameter governing the rate at which the response variable approaches its potential maximum (−)

m :

Slope (−)


Mean absolute error (g/L)


Modified Gompertz model (−)


Modified logistic model (−)


Morgan-Mercer-Flodin model (−)


Modified Richards model (−)

n :

Number of observations (−)

P :

Ethanol concentration (g/L)

P max :

Maximum ethanol production (g/L)

P min :

Minimum ethanol production (g/L)

Q :

Maximum production, consumption, and growth rate (g/L)

Q P :

Maximum ethanol production rate (g/L)

Q S :

Maximum sugar consumption rate (g/L)

Q X :

Maximum cell growth rate (g/L)

R 2 :

Regression coefficient (−)


Re-modified Gompertz model (−)


Re-modified logistic model (−)


Re-modified Richards model (−)


Root mean square error (g/L)


Residual sugar concentration (g/L)


Residual sum of squares (g/L)

S max :

Maximum sugar concentration (g/L)

S min :

Minimum sugar concentration (g/L)


Sugar utilization yield (%)

t :

Sampling time (h)


Theoretical ethanol yield (%)

v :

Dimensionless shape parameter (−)

X :

Biomass concentration (g/L)

X max :

Maximum biomass production (g/L)

X min :

Minimum biomass production (g/L)

x t :

Experimental value at time “t” (g/L)

Y P/S :

Ethanol yield (%)

Y P/X :

Product yield per biomass (g ethanol/g cell)

y t :

Predicted value at time “t” (g/L)

Y X/S :

Biomass yield (%)

β :

Growth displacement (−)

δ :

Allometric constant (−)

λ :

Lag time (h)


The relative error sum of squares (−)


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This study was funded by the Akdeniz University Research Foundation (Grant number 2014.02.0121.020).

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Correspondence to Irfan Turhan.

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Germec, M., Cheng, K., Karhan, M. et al. Application of mathematical models to ethanol fermentation in biofilm reactor with carob extract. Biomass Conv. Bioref. 10, 237–252 (2020).

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  • Carob extract
  • Biofilm reactor
  • Ethanol
  • Optimization
  • Predicting with flexible functions