**3. Development of an efficiency evaluation method for dark fermentation hydrogen production**

To develop a model for evaluating the efficiency of dark fermentation, the system must first be defined and limited as a process. This process is to be worked out as a PN. The results of the modeled biomass conversion will then be compared with experimental data, and the efficiency of the process will be summarized based on an efficiency factor.

#### **3.1 Definition of the system under consideration**

Before a system can be considered mathematically, its accounting limits must first be defined. The system of dark fermentation in a biogas plant must be represented mathematically. In particular, changes in the process flow leading to hydrogen production are relevant. The balance space must contain all desired processes and structures [33]. This work focuses solely on the biodegradation of biomass without technically required secondary components (**Figure 4**).

The input is the already prepared biomass, and the output is the end products of the reaction pathways during fermentation, mainly hydrogen and methane formation. The material balance is formed based on the effluents described. Here, all material flows have to be considered, which influence the considered target substances, in this case, the hydrogen formation [33].

**Figure 5** shows the transit-adjusted distribution of the chemical reactions considered. It is a highly simplified representation of the relevant reactants and products. This figure focuses on classifying the products and intermediates to the degradation steps of anaerobic digestion. It should be noted that the degradation steps are not strictly sequential. To simplify matters, overlaps have not been considered in this figure.

Each reaction is divided into "contributes positively" (green), "negatively" (red), "not directly" (blue), or "not at all" (black) to hydrogen formation. In the red

*Optimizability of Biogenic Hydrogen Production DOI: http://dx.doi.org/10.5772/intechopen.111853*

#### **Figure 4.**

*Schematic illustration of the accounting process.*

#### **Figure 5.**

*Schematic illustration of the four steps of biomass degradation to methane —Hydrolysis, acidogenesis, acetogenesis, and methanogenesis—Their intermediates and their interactions (transit-adjusted balance area; green: Contributes positively to H2 formation; red: Contributes negatively to H2 formation; blue: Does not contribute directly to H2 formation; and black: Does not contribute at all to H2 formation).*

reactions, H2 is consumed, whereas H2 is formed in the green reactions. Undesired byproducts, which are formed independently of the desired H2, but do not negatively influence the yield of the main product, are shown in orange.

There are two transit streams: the inert biomass parts, e.g., lignin, and the inert AA. These are hardly or not at all degraded. The desired main product is H2; the undesired by-products are CH4, CO2, and NH4. Ammonia is formed during AA degradation. Carbon dioxide is produced during sugar fermentation and acetoclastic methanogenesis.

It is striking that the sugar fermentation shows red and green reactions, while the AA fermentation and fat degradation show only green reactions. In particular, the lactate and ethanol pathways are H2-consuming, whereas in the other reaction pathways, H2 is produced during the further metabolism of OAs.

By definition, methanogenesis is no longer part of dark fermentation. Since it is the primary H2 consumer, it is nevertheless considered.

#### **3.2 Development of a Petri net**

PNs are mathematical structures based on graph theory. They function as a formal modeling language with strict syntax [34]. PNs are directed bipartite multigraphs. They are constructed from two types of nodes (bipartite): places and transitions. Places describe states, which can be, e.g., molecular species and physical parameters like temperature. They are graphically represented as circles. Transitions are state changes and represent, e.g., chemical reactions, changes of location, or an interaction. They are graphically represented as squares. So-called tokens mark places. The terminology used for the developed model is shown in **Figure 6**.

With PNs, causal processes can be represented particularly well, although they can also run independently. Thus, PNs are well suited for describing complex biochemical systems, such as gene regulation or metabolism. The exact syntax of PNs prevents mathematically nonsensical models, and the graphical representation allows an intuitive application that hardly requires mathematical or computer science knowledge.

The balance boundary of the considered process was drawn in the last section. The boundary encompasses all anaerobic digestion, including methanogenesis. The PN was developed based on **Figure 5**, which summarizes the results of the literature review on biomass degradation. Accordingly, the structure of the PN (**Figure 7**), created in SNOOPY (version 1.22), is similar to that of the schematic illustration of the degradation process (see **Figure 5**).

The different components were treated as plates and the chemical reactions as transitions. At the arrows, there are numerical values, which reflect stoichiometry. In the model, a reaction will only occur if all necessary components are available in sufficient quantity and if the reaction rates conditional probability of a reaction is high. Once these conditions are fulfilled, the reaction takes place until the conditions are no longer fulfilled. For example, 4 H2 molecules and 1 CO2 molecule are required to hydrogenotrophically form 1 CH4 molecule (see methanogenesis at the bottom of **Figure 7**).

#### **3.3 Calculation of transition rates**

The transition rates correspond to the stochastic probability in a stochastic PN that a transition switches. This corresponds to the reaction rate of the chemical reaction.

Initially, the biomass is added to the digester in grams per liter. The following values are specific conversion values of maize silage (see **Table 1**).

*Optimizability of Biogenic Hydrogen Production DOI: http://dx.doi.org/10.5772/intechopen.111853*

**Figure 6.**

*Graphical representation of the elements that can be used in a Petri net [35].*

Maize silage was chosen as a substrate for simplicity since the data are available. The transition rates have to be adjusted for each new substrate. First, each substrate contains a specific ratio of carbohydrates, proteins, lipids, and inert substances (those that are not degraded and thus do not contribute to the biogas yield).

In this context, a test report on maize silage from 2021 (commissioned by the Fraunhofer IFF) was used and converted into relative shares of the total dry matter (DM). As an example, this calculation is carried out in Eq. 8. Digestible carbohydrates can be found among nitrogen-free extractives. The proteins are noted as "crude proteins." The inert substances are composed of crude ash and crude fiber. The lipids are indicated as "crude fats."

$$\text{protein content} = \frac{\text{raw protein DM}}{\text{total DM}} = \frac{10,4\%}{92,796} = 10,36\% \text{DM} \tag{7}$$

Up to this point, all values are in g DM/L. One unit corresponds to one gram DM per liter of reactor content. These must be converted into several molecules. The calculation is shown for glucose as an example in Eq. 9.

#### **Figure 7.**

*Petri net model of the four steps of biomass degradation to methane-hydrolysis, acidogenesis, acetogenesis, methanogenesis, their intermediates, and their interactions.*

$$N\_{\text{Glucose}} = \frac{m\_{\text{Glucose}}}{M\_{\text{Glucose}}} \ast N\_A = \frac{1 \text{g}}{180,906 \frac{\text{g}}{\text{mol}}} \ast 6,002 \ast 10^{23} \frac{\text{1}}{\text{mol}} = 3,3 \ast 10^{21} \tag{8}$$

*mGlucose*=Mass of glucose, [g].

*MGlucose*=Molecular weight of glucose, [g/mol].

*NA*=Avogadro constant, [1/mol].

*NGlucoce*=Number of molecules per gram of glucose, [�].

Thus, 1 g glucose corresponds to 3\*10<sup>21</sup> glucose molecules. Finally, the reactor volume must be considered for all molecule numbers to know the total number of molecules in the fermenter.

The other conversion factors in **Table 1** were calculated analogously based on average molecular weights.

#### **3.4 Derivation of the differential equations**

The finished PN can be derived with the calculated reaction rates by SNOOPY to a differential equation system. The corresponding differential equations are formed according to the following scheme:

$$\frac{dX}{dt} = \sum\_{i} p\_i \* c\_{in,m} - \sum\_{j}' t - \mathbf{1} \tag{9}$$

*Optimizability of Biogenic Hydrogen Production DOI: http://dx.doi.org/10.5772/intechopen.111853*


**Table 1.**

*Data used when creating the Petri net.*

*dX dt* =Concentration of component X at time t, [].

*pi* =Stoichiometric coefficient of reaction i, [].

*cin*,*<sup>m</sup>*=Concentration of input component m, [].

SNOOPY created the complete differential equation system of the anaerobic degradation. With the help of this chemical equation system, the efficiency analysis of the degradation can be performed.

The results of the modeling of the concentration course in the reactor for the time of the total of 23 reaction steps are shown in **Figure 8**. By modeling each individual reaction step, it was possible to represent the idealized concentration curve of all reactants, products, and intermediates.
