## 3 Results
### Dynamic Biogas Process Modeling
#### 3.4.2 Gray Box Biogas Process Models
The photovoltaic system was basically used to load a battery system to allow base load at night. Thermal behavior and thermal energy consumption of the AD constitute another research focus and lead to different types of awareness. Using a kinetic model of AD, a methane production rate with one kinetic parameter based on Hill [73], Reference [103] investigated the behavior of a 45 m\({}^{3}\) swine manure digester, placed below ground, to simulate the effect of heating the digestate in order to observe higher methane production rates. Thus, the methane production rate depends on an ultimate methane yield as a function of the digestate temperature in addition to hydraulic retention time and an experimentally evaluated kinetic parameter. The heater is modeled as a solar collector on the roof of the digester. A comprehensive dynamic energy equation as thermal model of the digester and biogas is presented, supplemental to the AD kinetic model, including heat flow from the solar collector, heat transfer between the boundary layer of gas and digestate surface, heat losses to the ground and convective heat flows of the input manure mass flow. The authors could show by using the coupled biochemical and thermal model that heating leads to higher methane production and verified the simulation with 240 h of experimental data of days in September. A hybrid energy system model was developed by [104] to optimize the biogas production of a biogas digester. The MATLAB(r) Simulink model contained a solar-assisted biogas reactor with a dynamic model of a water tank, electrical gas compressor, gas storage system and a simplified AD model. The methane production rate as an AD model was based on a production rate as a function of hydraulic retention time, a volatile solid concentration of the feedstock with its maximum methane yield and the temperature inside the digester. Energy balance equations considered heat losses to the environment, energy transfer by solar collectors as a function of daytime solar radiation and energy consumption of a gas compressor. The gas storage system contained an overall dynamic mass balance equation. An optimized methane production rate as balance equation containing hydraulic retention time, methane yield of feedstock, and a temperature pending kinetic parameter was predicted for cloudy days using a solar-based heating system with a water tank as a heat buffer. Additional investment costs due to solar collectors, pumps, heat exchanger, valves, pipes and several other components would need 5.5 years to return the investment as result of higher biogas production rates. The study in [105] covered a similar research field. The authors presented a thermal model of the digester, calculating Nusselt numbers for heat transfer coefficients to simulate heat losses and transfer of a heating system integrated into a digester. A simplified ADM1 was integrated into the model due to the coupling of the model developed with more complex components, such as thermal storage tanks, thermal solar collectors or biogas upgrading units. The model was examined from the plausibility of the AD process: Increasing the input flow rate of the feedstock increased the biogas production rate proportionally and reduced the digester temperature without using a heating system. Increasing the feedstock input flow by 21% (626 to 756 kg/h) led to a proportionally increased biogas flow (96 to 115 kg/h) and a temperature reduction of 1 Kelvin, while increasing the temperature by about 1 Kelvin increased the biogas production by about 0.2% (95.54 to 95.75 kg/h). Gray box models are also designed for modeling the influence of process parameters on biogas plant operation. As a detailed process simulation model, [PERSON] [9; 106; 107] designed an operator training simulator of full-scale biogas plants as a tool for bioprocess engineering education. The AD process inside the simulation was modeled using AMOCO [72].