# Introducing Temperature as Variable Parameter into Kinetic Models for Anaerobic Fermentation of Coffee Husk, Pulp and Mucilage

^{1}

^{2}

^{*}

## Abstract

**:**

^{−1}volatile solids(VS), respectively, for a fermentation temperature of 37 °C; 156.8, 234.8 and 287.1 L kg

^{−1}VS, respectively, for 30 °C; and 139.9, 196.2 and 255.9 L kg

^{−1}VS, respectively, for 21°C. Two kinetic models, namely, the modified Logistic model (LOG) and the modified Gompertz model (GOM), were applied to fit experimental data and the respective kinetic constants were generated. Both models exhibited a very good fit to the measured data points (R

^{2}> 0.987). The relationship of kinetic constants of substrates with fermentation temperatures was established and inserted into the LOG and GOM models; thus, generalized LOG and GOM models were obtained to predict SMY of the substrates at any temperature between 21 °C and 37 °C.

## 1. Introduction

^{−1}) is harmful to methanogens, especially for the thermophile consortia [27]. Several models are applied to describe the effect of temperature on the rate of anaerobic fermentation. The Arrhenius model, for instance, was applied to estimate bacterial growth and product formation rate. However, the limitation of the model was the indefinite increase of the growth rate with increasing temperatures. The Ratkowsky models, however, predict that the bacterial growth rate increases with increasing temperatures from the initial to the optimum temperature and decreases when the temperature increases further [28]. Ying Jin, et al. [29] reported that the effect of temperature on the methane yield was insignificant at lower temperatures.

## 2. Materials and Methods

#### 2.1. Raw Materials and Inoculum

#### 2.2. Chemical Analysis

#### 2.3. Anaerobic Batch Digestion Tests

_{4}). The fermentation temperature and ambient air pressure were measured while taking the gas reading and used to normalize the gas yield at standard temperature and pressure (273.15 K, 101.325 kPa). The specific methane yield (L CH

_{4}kg

^{−1}VS) was calculated from the ratio of the net normalized methane yield to the amount of organic dry matter of substrate added in the digester. The biodegradability of substrates was calculated according to the mass ratio of the produced gas to the amount of VS loaded in the digester.

#### 2.4. Data Fitting to Models

#### 2.4.1. Modified Gompertz (GOM) Model

_{t}is cumulative methane (L kg

^{−1}VS) produced at time t (d), S is the ultimate methane production potential (L kg

^{−1}VS), R

_{m}is maximum daily methane yield (L kg

^{−1}VS d

^{−1}), λ is the lag time (d), which is the minimum time necessary to produce methane, and e is 2.7183. The parameters S, R

_{m}and λ are constants that could be determined by applying non-linear regression equations.

#### 2.4.2. Modified Logistic (LOG) Model

_{m}and λ respectively. Afterwards the kinetic constants were plotted against the temperature for each substrate. Curve fitting was executed on the kinetic constants to establish a mathematical relationship to the temperature. Consequently, the kinetic parameters were inserted to the LOG and GOM models to estimate the methane yield of husks, pulp and mucilage at any temperature ranging from 21 °C through 37 °C.

#### 2.5. Statistical Analysis

^{2}) according to Equations (3)–(5).

## 3. Results and Discussion

#### 3.1. Model Fitting for Different Fermentation Temperatures

_{m}) and lag time (λ).

_{m}) and higher lag time (λ) than samples evaluated at the other operating temperatures. The pulp exhibited longer lag phases at the beginning of the digestion period. The highest methane yield was predicted from mucilage followed by pulp and husk.

_{m}) and lag time (λ) higher than that of the GOM model. However, the SMY of digesters operated at 37 °C was underestimated by the LOG model. There was a slight difference in the correlation coefficients (R

^{2}) between the experimental dataset and the SMY predicted by the models.

#### 3.2. Model Fitting for Fermentation Temperature as an Input Variable

_{m}) fits with the exponential function. The coefficients c

_{1}–c

_{7}are shown in Table 3.

_{m}) increased with increasing temperatures for all substrates and there was only a small difference between the substrates, regardless of the model type (Figure 2B). The fitting equation indicated strong dependence of the methane yield rate on temperature. The highest yield rate from each fermentation temperature was recorded from mucilage followed by pulp and husks. The Ratkowsky, Hinshelwood and Schoolfield growth models estimate that the growth yield increases from a minimum temperature to the optimum temperature and then starts to decrease again [29]. In this study, the fermentation temperature did not exceed the optimum temperature of 37 °C and hence, did not show a decline. Pham, et al. [42] reported that the maximum daily methane yield rate rose exponentially within a range from low to optimal fermentation temperatures, in full-scale continuous feed biogas digesters fed with pig and cow manure.

_{m}and λ) were estimated for fermentation temperatures of 21 °C, 30 °C and 37 °C to generalize the models (Table 4). The performance of the generalized models was validated with the measured SMY and showed high goodness of fit (R

^{2}> 0.959).

^{2}> 0.98, MAE 4.32–10.4); therefore, the models are applicable to estimate the SMY of husks, pulp and mucilage at any fermentation temperature between 21 °C and 37 °C.

#### 3.3. Model Validation

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclatures

Symbol | Definition |

SMY | specific methane yield |

BMP | bio-chemical methane potential |

GOM | modified Gompertz model |

LOG | modified logistic model |

gLOG | generalized logistic model |

gGOM | generalized Gompertz model |

AIC | Akaike information criterion |

ha | hectare (=10,000 m^{2}) |

a.s.l. | above sea level |

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**Figure 1.**Measured specific methane yield (SMY) of coffee husks, pulp and mucilage in the course of batch residence time (t) at different fermentation temperatures and curves fitted with the modified Gompertz model (GOM) and modified Logistic model (LOG); error bars show standard deviation, n = 6.

**Figure 2.**Lag period λ (A), maximum daily methane yield R

_{m}(B) and ultimate methane yield S (C), from the modified Logistic model (LOG) and modified Gompertz model (GOM) vs. fermentation temperature T.

**Figure 3.**Measured specific methane yield (SMY) of coffee husks, pulp and mucilage in the course of batch residence time (t) at different fermentation temperatures and curves fitted with the generalized modified Logistic model (gLOG) and generalized modified Gompertz model (gGOM); error bars show standard deviation, n = 6.

**Figure 4.**Residual specific methane yield (SMYR) of husks, pulp and mucilage in the course of batch assay residence time (t) at different fermentation temperatures as compared to the prediction by generalized modified Logistic model (gLOG) and generalized modified Gompertz model (gGOM) vs residence time (t).

**Figure 5.**Comparison of the generalized modified Logistic model (gLOG) and generalized modified Gompertz (gGOM) model on fitting experimental datasets with the Akaike information criterion (AIC) for husks, pulp and mucilage.

**Figure 6.**Daily specific methane yield (SMYd) projections by the generalized modified Gompertz model (gGOM) for husk, pulp and mucilage at fermentation temperatures 21 °C–37 °C vs. residence time (t).

**Table 1.**Temperature data for some of coffee growing areas in Ethiopia with elevation (H), mean annual temperature (T mean), mean annual maximum temperature (T max) and mean annual minimum temperature (T min) (adapted from [34]).

Stations | H (m a.s.l.) | T min (°C) | T max (°C) | T mean (°C) |
---|---|---|---|---|

Gambella | 480 | 20.4 | 34.8 | 27.6 |

Arbaminch | 1290 | 17.9 | 29.6 | 23.8 |

Jinka | 1430 | 15.3 | 26.9 | 21.1 |

Agaro | 1560 | 12.3 | 28.2 | 20.3 |

Bonga | 1725 | 11.6 | 27.4 | 19.5 |

Harer | 1856 | 13.8 | 24.4 | 19.1 |

Gimbi | 1870 | 13.2 | 26.5 | 19.9 |

Yirga chefe | 1925 | 10.3 | 24.7 | 17.5 |

Gore | 2002 | 13.7 | 23.4 | 18.6 |

Nekemte | 2080 | 11.9 | 24.0 | 18.0 |

**Table 2.**Measured specific methane yield (SMY) of coffee husks, pulp and mucilage at different fermentation temperatures together with ultimate methane yield (S), maximum daily methane yield (R

_{m}) and lag time (λ) as estimated by the modified Logistic model (LOG) and modified Gompertz model (GOM); mean ± SD. MAE: mean absolute error.

Substrate | Dig. Temp (°C) | SMY, measured (L kg^{−1} VS) | Model | S (L kg^{−1} VS) | R_{m} (L kg^{−1} VS d^{−1}) | λ (d) | SMY, estimated (L kg^{−1} VS) | Goodness of fit | |
---|---|---|---|---|---|---|---|---|---|

R^{2} | MAE | ||||||||

Husk | 37 | 159.4 ± 1.8 | LOG | 155.2 ± 3.9 | 16.1 ± 1.6 | 2.7 ± 0.5 | 155.2 | 0.997 | 2.888 |

GOM | 158.5 ± 2.4 | 14.8 ± 0.8 | 2.1 ± 0.3 | 158.4 | 0.997 | 2.334 | |||

30 | 156.8 ± 2.6 | LOG | 157.3 ± 4.4 | 9.7 ± 0.8 | 4.1 ± 0.7 | 156.9 | 0.976 | 6.289 | |

GOM | 163.0 ± 6.4 | 9.0 ± 0.9 | 3.0 ± 0.9 | 160.0 | 0.999 | 2.585 | |||

21 | 139.9 ± 6.8 | LOG | 143.2 ± 12.2 | 3.4 ± 0.7 | 1.6 ± 4.9 | 138.7 | 0.997 | 4.186 | |

GOM | 151.4 ± 13.7 | 3.3 ± 0.6 | 0.1 ±3.3 | 139.6 | 0.987 | 6.485 | |||

Pulp | 37 | 244.7 ± 6.4 | LOG | 243.8 ± 3.7 | 21.8 ± 0.8 | 4.3 ± 0.2 | 243.8 | 0.996 | 5.909 |

GOM | 248.2 ± 5.3 | 20.7 ± 1.0 | 3.7 ± 0.3 | 247.8 | 0.998 | 3.230 | |||

30 | 234.8 ± 2.9 | LOG | 236.6 ± 7.3 | 12.0 ± 0.8 | 6.1 ± 0.7 | 232.7 | 0.992 | 6.749 | |

GOM | 253.3 ± 3.8 | 10.7 ± 0.3 | 4.6 ± 0.2 | 235.1 | 0.999 | 1.924 | |||

21 | 196.2 ± 7.6 | LOG | 223.2 ± 29.5 | 4.3 ± 0.6 | 7.2 ± 3.7 | 197.3 | 0.995 | 3.607 | |

GOM | 265.0 ± 52.6 | 3.9 ± 0.4 | 3.5 ± 2.7 | 197.9 | 0.987 | 4.660 | |||

Mucilage | 37 | 294.5 ± 9.6 | LOG | 284.6 ± 7.0 | 24.8 ± 1.7 | 3.1 ± 0.4 | 284.5 | 0.998 | 3.317 |

GOM | 294.4 ± 3.9 | 23.2 ± 0.8 | 2.5 ± 0.2 | 293.8 | 1.000 | 1.377 | |||

30 | 287.1 ± 11.1 | LOG | 288.5 ± 6.2 | 15.9 ± 0.8 | 5.3 ± 0.4 | 286.1 | 0.992 | 6.036 | |

GOM | 304.2 ± 10.6 | 13.9 ± 0.9 | 3.8 ± 0.6 | 289.5 | 0.999 | 3.237 | |||

21 | 255.9 ± 16.6 | LOG | 268.3 ± 16.9 | 6.0 ± 0.6 | 6.5 ± 2.3 | 250.8 | 0.996 | 5.411 | |

GOM | 311.7 ± 27.5 | 5.3 ± 0.4 | 3.5 ± 1.6 | 253.9 | 0.995 | 4.112 |

**Table 3.**Coefficients c

_{1}-c

_{7}for husks, pulp and mucilage, estimated for ultimate methane yield (S), maximum daily methane yield (R

_{m}) and lag period (λ) for the modified Logistic model (LOG) and modified Gompertz model (GOM).

Substrates | Model | S | R_{m} | λ | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{c}}_{1}$ | ${\mathit{c}}_{2}$ | R² | ${\mathit{c}}_{3}$ | ${\mathit{c}}_{4}$ | ${\mathit{c}}_{5}$ | R² | ${\mathit{c}}_{6}$ | ${\mathit{c}}_{7}$ | R² | ||

Husk | LOG | 137.0 | 0.5 | 0.640 | 9.43 | 0.01 | 2.40 | 1.000 | 7.17 | 0.1 | 0.905 |

GOM | 130.3 | 0.8 | 0.999 | 6.42 | 0.58 | 5.93 | 1.000 | 7.3 | −0.1 | 0.985 | |

Pulp | LOG | 196.6 | 1.3 | 0.987 | −5.68 | 10.00 | 15.9 | 1.000 | 11.2 | −0.2 | 0.954 |

GOM | 286.7 | −1.0 | 0.977 | −2.44 | 6.36 | 12.4 | 0.999 | 3.47 | 0.02 | 0.570 | |

Mucilage | LOG | 248.9 | 1.1 | 0.646 | −14.3 | 19.60 | 24.7 | 1.000 | 11 | -0.2 | 0.937 |

GOM | 334.8 | -1.0 | 0.978 | -50.9 | 56.90 | 55.8 | 1.000 | 4.96 | -0.1 | 0.536 |

**Table 4.**Specific methane yield (SMY) of coffee husks, pulp and mucilage at different fermentation temperatures as estimated by the generalized modified Logistic model (gLOG) and generalized modified Gompertz model (gGOM) together with ultimate methane yield (S), maximum daily methane yield (R

_{m}) and lag time (λ).

Substrate | Dig. Temp (°C) | Model | S (L kg^{−1} VS) | R_{m} (L kg^{−1} VS d^{−1}) | λ (d) | SMY estimated (L kg^{−1} VS) | Goodness of fit | |
---|---|---|---|---|---|---|---|---|

R^{2} | MAE | |||||||

Husk | 37 | gLOG | 155.5 | 17.29 | 3.5 | 154.5 | 0.993 | 4.86 |

gGOM | 159.9 | 15.03 | 3.6 | 159.5 | 1.000 | 1.86 | ||

30 | gLOG | 152.0 | 9.86 | 4.2 | 150.7 | 0.994 | 4.05 | |

gGOM | 154.3 | 9.07 | 4.3 | 151.8 | 0.997 | 4.82 | ||

21 | gLOG | 147.5 | 9.44 | 5.1 | 139.6 | 0.959 | 7.60 | |

gGOM | 147.1 | 7.00 | 5.2 | 133.5 | 0.986 | 10.02 | ||

Pulp | 37 | gLOG | 244.7 | 21.67 | 3.8 | 244.1 | 0.998 | 3.29 |

gGOM | 249.7 | 20.67 | 4.2 | 248.5 | 0.992 | 6.72 | ||

30 | gLOG | 235.6 | 11.93 | 5.2 | 231.4 | 0.995 | 4.31 | |

gGOM | 256.7 | 10.70 | 4.1 | 238.2 | 0.997 | 4.06 | ||

21 | gLOG | 223.9 | 4.32 | 7.0 | 197.0 | 0.987 | 6.48 | |

gGOM | 265.7 | 3.92 | 3.9 | 197.1 | 0.991 | 6.08 | ||

Mucilage | 37 | gLOG | 289.6 | 23.16 | 3.6 | 289.5 | 0.989 | 9.30 |

gGOM | 297.8 | 24.90 | 1.3 | 293.7 | 0.998 | 4.06 | ||

30 | gLOG | 281.9 | 13.92 | 5.0 | 277.7 | 0.991 | 7.34 | |

gGOM | 304.8 | 15.96 | 2.0 | 294.4 | 0.960 | 14.99 | ||

21 | gLOG | 272.0 | 5.30 | 6.8 | 242.6 | 0.980 | 12.15 | |

gGOM | 313.8 | 6.00 | 2.9 | 267.2 | 0.981 | 12.27 |

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**MDPI and ACS Style**

Chala, B.; Oechsner, H.; Müller, J.
Introducing Temperature as Variable Parameter into Kinetic Models for Anaerobic Fermentation of Coffee Husk, Pulp and Mucilage. *Appl. Sci.* **2019**, *9*, 412.
https://doi.org/10.3390/app9030412

**AMA Style**

Chala B, Oechsner H, Müller J.
Introducing Temperature as Variable Parameter into Kinetic Models for Anaerobic Fermentation of Coffee Husk, Pulp and Mucilage. *Applied Sciences*. 2019; 9(3):412.
https://doi.org/10.3390/app9030412

**Chicago/Turabian Style**

Chala, Bilhate, Hans Oechsner, and Joachim Müller.
2019. "Introducing Temperature as Variable Parameter into Kinetic Models for Anaerobic Fermentation of Coffee Husk, Pulp and Mucilage" *Applied Sciences* 9, no. 3: 412.
https://doi.org/10.3390/app9030412