# Mathematical Analysis and Update of ADM1 Model for Biomethane Production by Anaerobic Digestion

^{1}

^{2}

^{*}

## Abstract

**:**

^{®}and Simulink Toolbox, we discovered that the model fails to predict the microbiological behavior of the system. The structure of the ADM1 model was then modified by adding substrate consumption yields in equations describing microbial growth, to better reflect the consortium behavior. The updated model was tested by modifying several parameters: the coefficient of decomposition was identified to increase biomethane production. Approaching mathematical models from a microbiological point of view can lead to further improvement of the models themselves. Furthermore, this work represents additional evidence of the importance of informatics tools, such as bioprocess simulations to foster biomethane role in bioeconomy.

## 1. Introduction

_{4}), is a prominent target to be substituted with renewable alternatives, as it is still widely used in the EU for industrial, domestic, and transport sectors. Its direct replacement is biomethane, which is the final product of anaerobic fermentation of organic matter, as part of the gaseous mixture called biogas. Biogas is in fact a blend of mainly CH

_{4}and CO

_{2}: it can be deployed directly as a source of energy, but the presence of the already fully oxidized CO

_{2}reduces the overall calorific value. For this reason, several “upgrading” processes exist, aimed at purifying biomethane from biogas, to expand its use also to domestic and transport sectors via injection into the conventional pipelines [3,4].

_{2}, volatile fatty acids (VFAs, e.g., propionate, butyrate, acetate), and H

_{2}(stage II—acidogenesis) [8]. These molecules are then converted into acetic acid (stage III—acetogenesis) and sequentially transformed into biomethane (CH

_{4}) (stage IV—methanogenesis) (Figure 1) [8]. These reactions are carried out by different species of microbes, whose symbiosis and syntropy in the consortium are the key for the anaerobic digestor to function. In a simplified model of these reactions, described in Figure 1, stage I is carried out by glucose-fermenting acidogens, able to both hydrolyze the fibers and carbohydrates contained in the substrate and transform glucose into VFAs. Propionate and butyrate-degrading acetogens accumulate acetic acid from corresponding VFAs, whereas acetoclastic methanogens complete the reaction to biomethane [9,10,11].

## 2. Materials and Methods

#### 2.1. Mathematical Model of Anaerobic Digestion

#### 2.2. Variables and Constant Description

#### 2.2.1. Variables

_{0}(g/L) is the concentration of soluble organic compounds, measured as volatile solids, S

_{NH}

_{4}

^{+}is the concentration of ammonia, S

_{1}(g/L) is the concentration of glucose, S

_{2}(g/L) is the concentration of propionate, S

_{3}(g/L) is the concentration of butyrate, S

_{4}(g/L) is the concentration of acetate, X

_{1}(g/L) is concentration of glucose-fermenting acidogens, X

_{2}(g/L) is concentration of propionate-degrading acetogens, X

_{3}(g/L) is concentration of butyrate-degrading acetogens, X

_{4}(g/L) is concentration of acetoclastic methanogens, and µ

_{1}, µ

_{2}, µ

_{3}, and µ

_{4}(d

^{−1}) refer to specific growth rates, described as follows:

#### 2.2.2. Constants

^{−1}) = 0.1072—dilution rate; β (d

^{−1}) = 0.31—hydrolytic rate; K

_{i}

_{,o}(g/L) = 0.23—inhibition constant, reflecting the decrease of hydrolytic rate due to VFAs accumulation; K

_{i}

_{,NH4}

^{+}(g/L) = 0.5—inhibition constant reflecting the decrease of acetoclastic methanogenesis rate due to ammonia accumulation; K

_{i}

_{,acet/prop}(g/L) = 0.96—product inhibition constant, reflecting the decrease of propionate degradation rate due to acetate accumulation; K

_{i}

_{,acet/but}(g/L) = 0.72—product inhibition constant, reflecting the decrease of butyrate degradation rate due to acetate accumulation; Y

_{e}= 0.55—coefficient of decomposition, counting what part of insoluble organic compounds are transformed to soluble compounds; S

_{oi}= 30.6 g/L—concentration of insoluble organic compounds, measured as total solids; S

_{i}

_{1}, S

_{i}

_{2}, S

_{i}

_{3}, and S

_{i}

_{4}(g/L) are the concentrations of the corresponding substrates in the influent; S

_{i}

_{1}= 5.1, S

_{i}

_{2}= 1.6, S

_{i}

_{3}= 0.1, and S

_{i}

_{4}= 3.1.

_{m}= 1.3—coefficient in the Contois growth rate model for µ

_{4}, reflecting the decrease of acetoclastic methanogenesis rate due to biomass accumulation; K

_{S}

_{1}(g/L) = 4.8—saturation constant for glucose-fermenting acidogens; K

_{S}

_{2}(g/L) = 0.93—saturation constant for propionate-degrading acetogens; K

_{S}

_{3}(g/L) = 0.176—saturation constant for butyrate-degrading acetogens

_{max}

_{1}(d

^{−1}) = 0.7—maximum specific growth rate of glucose-fermenting acidogens at 34 °C; µ

_{max}

_{2}(d

^{−1}) = 0.54—maximum specific growth rate of propionate-degrading acetogens at 34 °C; µ

_{max}

_{3}(d

^{−1}) = 0.68—maximum specific growth rate of butyrate-degrading acetogens at 34 °C; µ

_{max}

_{4}(d

^{−1}) = 0.45—maximum specific growth rate of acetoclastic methanogens at 34 °C; b

_{i}(i = 1, …, 4)—mortality rates for each of the four microbial populations (it was supposed that b

_{i}= 0.05µ

_{maxi}). It was assumed that a part of the dead cells is transformed into soluble organics with recycling conversion factor λ (λ > 0 and λ < b

_{i}).

_{glu/X}

_{1}= 12.9 g/g biomass, Y

_{acet/X}

_{1}= 20 g/g biomass, Y

_{prop/X}

_{1}= 2.94 g/g biomass, Y

_{prop/X}

_{2}= 10.2 g/g biomass, Y

_{but/X}

_{1}= 3.08 g/g biomass, Y

_{but/X}

_{3}= 11.9 g/g biomass, Y

_{acet/X}

_{2}= 8 g/g biomass, Y

_{acet/X}

_{3}= 1.54 g/g biomass, Y

_{acet/X}

_{4}= 16 g/g biomass, and Y

_{CH}

_{4/X4}= 4 L/g biomass.

#### 2.3. Balancing the Reaction Equations of Anaerobic Digestion Mathematical Model

_{r}G’°) and equilibrium constant (K’

_{eq}) were calculated. The values of pH, pMg, and ionic strength were kept as the default ones (7.5, 3.0, and 0.25 M, respectively).

#### 2.4. Simulation Studies

^{®}(R2019b version, MathWorks, Natick, MA, USA). To improve the model, different simulation scenarios were run to validate model behavior. Based on our observations, the differential equations of the model were modified on Simulink. Simulations were then run again to verify appropriate behavior. To identify significant process parameters, the values for some parameters were modified. Thus, we can foresee the effect of such modifications on the production of biomethane. Excel (Office 365, Microsoft, Albuquerque, NM, USA) was then used to calculate equations of the relationship between single parameters (D, Y

_{acet/X}

_{1}, S

_{1i}, S

_{0i}, Y

_{e}) and biomethane production (Q), described in Section 3.4.

## 3. Results and Discussion

#### 3.1. Biochemical Description of the Model

^{+}becomes the electron sink of the reaction (Reaction 3, Table 1), whereas H

_{2}is the receiver of the four electrons (Reaction 1, Table 1) when propionate-degrading acetogens are involved. Accordingly, butyrate as mid-step involves the use of both H

_{2}and NAD

^{+}as electron sinks of the fermentation (Reaction 2, Table 1). Considering kinetic parameters such as the estimated Gibbs free energy (Δ

_{r}G’°) and equilibrium constant (K’

_{eq}), it was clear that the direct fermentation of glucose into acetate is thermodynamically the most favorable, suggesting that the microorganisms involved in such metabolism are pivotal for the accumulation of biomethane. In agreement with this observation, bioaugmentation of acetate-type fermentation species have been proposed to ameliorate anaerobic digestion of residual biomasses into biomethane [23].

#### 3.2. Microbiological Analysis of ADM1 Model

_{1}, X

_{2}, and X

_{3}) on the production of biomethane. We singularly simulated the nullification of acetate production yields by X

_{1}, X

_{2}, and X

_{3}(Y

_{acet/X}

_{1}, Y

_{acet/X}

_{2}, and Y

_{acet/X}

_{3}, respectively) to assess their impact on the final biomethane yield over time (Q): the lower this value when a single parameter is set to zero, the higher the importance of such element in the system. When Y

_{acet/X}

_{2}and Y

_{acet/X}

_{3}were set to zero, Q did not decrease significantly from the original one (Q = 0.45 L/d, Figure 2A), whereas Y

_{acet/X}

_{1}= 0 resulted in Q = 0.18 L/d, witnessing the importance of the metabolism of X

_{1}on the overall process over X

_{2}and X

_{3}(in accordance with the thermodynamical analysis). We then expanded the simulation by setting to zero the yield of glucose consumption by X

_{1}(Y

_{glu/X}

_{1}= 0) to completely eliminate the action of this species earlier in the process, expecting a strong reduction of Q. Surprisingly, the simulation resulted in an infinite production of biomethane in the first days of fermentation (Figure 2B), underlying some inaccuracy in the model itself. In the model X

_{1}, both hydrolyzed fibers and fermented glucose into VFAs; therefore, the subsequent production of biomethane is strongly dependent on its activity. Since Y

_{glu/X}

_{1}represents the ability of X

_{1}to consume glucose and, therefore, to grow and produce VFAs, the fact that its simulated nullification was so beneficial for the production of biomethane was at least suspicious from a microbiological point of view. Therefore, we analyzed the description of the equations to understand the reasons of this unexpected finding and how to solve it, for better representing the microbiological reality of anaerobic digestion and for further improving the model itself.

#### 3.3. Update to ADM1 Model Structure

_{glu/X}

_{1}appears in Equation (3) as a negative contributor to the titer of glucose in the digestor. The nullification of Y

_{glu/X}

_{1}, therefore, increased the amount of glucose, which in turn increased µ

_{1}(Equation (11)) and the titer of X

_{1}(Equation (1)), and subsequently, the titer of the microbial species as well. The paradox laid on the fact that although X

_{1}was simulated of not being able to consume glucose, it was still growing and producing VFAs, while microbiologically the opposite should occur. In order to meet this fundamental requirement, we modified the equations describing microbial growth (Equations (2), (4), (6) and (8)) by adding the corresponding yields of substrate consumption (Y

_{glu/X1}, Y

_{prop/X2}, Y

_{but/X3}, Y

_{acet/X4}) as follows:

_{glu/X1}reduced the value to Q = 0.1104 L/d, in accordance with the modifications proposed for the model (Figure 3). The value of Q was not zero because of the acetate present in the influent (S

_{i}

_{4}), transformed by X

_{4}into biomethane according to Equations (9) and (10). Furthermore, the value of Q obtained with this new version of the model was higher compared to the original [11], although it should be noticed that the constant values for updated model need to be validated experimentally. Nevertheless, the goal of this work was to provide an updated version of the model that could be a closer description of the microbial consortium responsible for the anaerobic digestion. Indeed, despite that the model analyzed here does not involve hydrogenotrophic methanogens, since their minor contribution in biomethane production compared to acetoclastic methanogens [10], the proposed modifications can be considered valid for the original ADM1 model as well.

#### 3.4. Identifying Significant Process Parameters

_{1}, X

_{2}, X

_{3}, X

_{4}). As a starting point, the dilution rate (D) was modified according to the maximum value available at the plant of an industrial partner of the project (D = 0.14 d

^{−1}), with 5000 L as the operative volume. The updated set of values resulted in Q = 1.198 L/d, displayed in Figure 4 together with other parameters, such as substrate, glucose and VFAs titer, microbial titer, and specific growth rate. The stoppage time was set to 60 d, since the maximum Q was already reached.

^{2}+ 8.8125D + 0.124, R

^{2}= 0.99), underlying the limit of the dilution that can cause wash out. With the polynomial above, it was also possible to calculate the dilution rate that provided the highest production of biomethane (D = 0.4458 d

^{−1}, Q = 2.062 L/d). The ability to foresee this value is clearly an advantage of the use of mathematical modeling.

_{1}metabolism in anaerobic digestion, in particular for the production of acetate, Y

_{acet/X}

_{1}was modified to assess its impact on Q. Similarly, values for the initial concentration of glucose in the influent (S

_{1i}) were considered, to simulate the effect of the addition of another residual biomass containing glucose. Finally, the concentration of insoluble organic compounds (S

_{oi}), and the coefficient of decomposition (Y

_{e}), counting which part of the insoluble organic compounds are transformed to soluble compounds, were considered as well. Table 2 shows that none of the modified parameters caused the same increase in Q and that the more the value was increased the lower was the effect on the corresponding ratio. In addition, implementation of S

_{1i}had only minor effects, showing that additional glucose would not greatly impact Q. On the other hand, S

_{oi}and Y

_{e}showed to be crucial for a relevant implementation of Q; unsurprisingly, the same increment of S

_{oi}and Y

_{e}produced the same values of Q, since they are both factors of Equation (1).

^{2}= 0.91), showing that an improvement in the ability of the microbial consortium to hydrolyze fibers is not proportionally beneficial in the production of biomethane (Figure S1). These observations from the simulation of anaerobic digestion of sludge may pave the way for further implementation of the real industrial process, and in turn, to the validation of the modified equations from the original ADM1.

## 4. Practical Applications of This Work and Future Research

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Simplified biochemical model of anaerobic digestion of biomass by the consortium of glucose-fermenting acidogens (X

_{1}), propionate-degrading acetogens (X

_{2}), butyrate-degrading acetogens (X

_{3}), and acetoclastic methanogens (X

_{4}).

**Figure 2.**Simulation of biomethane yield (Q) during time using parameters from [11] (panel

**A**) and with Y

_{glu/X}

_{1}= 0 (panel

**B**).

**Figure 3.**Simulation of biomethane yield (Q) over time with updated equations using parameters from [11] (panel

**A**) and with Y

_{glu/X}

_{1}= 0 (panel

**B**).

**Figure 4.**Simulation of biomethane yield over time (panel

**A**), compound titers (panel

**B**), microbial titers (panel

**C**), and microbial specific growth rates (panel

**D**) over time with updated equations, and D = 0.14 d

^{−1}.

**Table 1.**Stoichiometric description of the reactions occurring during anaerobic digestion from glucose to acetate, via propionate (1) or butyrate (2) production, or direct fermentation to acetate (3), with kinetic parameters of such reactions.

Reaction | Estimated Δ_{r}G’° (KJ/Mol) | K’_{eq} |
---|---|---|

(1) Glucose + 2 H_{2}O <=> 4 CO_{2} + 2 CH_{4} + 4 H_{2} | −148.6 ± 35.0 | 1.1 × 10^{26} |

(2) Glucose + 2 NAD^{+} + 2 H_{2}O <=> 4 CO_{2} + 2 CH_{4} + 2 H_{2} + 2 NADH | −217.7 ± 28.6 | 1.5 × 10^{38} |

(3) Glucose + 4 NAD^{+} + 2 H_{2}O <=> 4 CO_{2} + 2 CH_{4} + 4 NADH | −286.9 ± 26.3 | 1.9 × 10^{50} |

**Table 2.**Effect of the increment of the value of some parameters of the anaerobic digestion model on the value of biomethane production (Q).

Parameter | Value | Ratio | Q Value | Q Ratio | Q Ratio/Parameter Value Ratio |
---|---|---|---|---|---|

Y_{acet/X}_{1} | 20 | 1.198 | |||

40 | 2 | 1.933 | 1.62 | 0.81 | |

60 | 3 | 2.548 | 2.13 | 0.71 | |

S_{1i} (g/L) | 5.1 | 1.198 | |||

10.2 | 2 | 1.504 | 1.26 | 0.63 | |

15.3 | 3 | 1.807 | 1.51 | 0.50 | |

S_{0i} (g/L) | 30.6 | 1.198 | |||

61.2 | 2 | 2.001 | 1.67 | 0.84 | |

91.8 | 3 | 2.796 | 2.33 | 0.78 | |

Y_{e} | 0.55 | 1.198 | |||

1.1 | 2 | 2.001 | 1.67 | 0.84 | |

1.65 | 3 | 2.796 | 2.33 | 0.78 |

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**MDPI and ACS Style**

Bertacchi, S.; Ruusunen, M.; Sorsa, A.; Sirviö, A.; Branduardi, P.
Mathematical Analysis and Update of ADM1 Model for Biomethane Production by Anaerobic Digestion. *Fermentation* **2021**, *7*, 237.
https://doi.org/10.3390/fermentation7040237

**AMA Style**

Bertacchi S, Ruusunen M, Sorsa A, Sirviö A, Branduardi P.
Mathematical Analysis and Update of ADM1 Model for Biomethane Production by Anaerobic Digestion. *Fermentation*. 2021; 7(4):237.
https://doi.org/10.3390/fermentation7040237

**Chicago/Turabian Style**

Bertacchi, Stefano, Mika Ruusunen, Aki Sorsa, Anu Sirviö, and Paola Branduardi.
2021. "Mathematical Analysis and Update of ADM1 Model for Biomethane Production by Anaerobic Digestion" *Fermentation* 7, no. 4: 237.
https://doi.org/10.3390/fermentation7040237