# An Aerobic Degradation Model for Landfilled Municipal Solid Waste
## 3 Dynamic Model
### Kinetics of Hydrolysis
The influence factor of temperature can be expressed by Equation (12):  
\[f(T)=\frac{(T-T_{\text{max}})(T-T_{\text{min}})^{2}}{(T_{opt}-T_{\text{min}})[(T_ {opt}-T_{\text{min}})(T-T_{opt})-(T_{opt}-T_{\text{max}})(T_{opt}+T_{\text{min} }-2T)]} \tag{12}\]  
where \(T_{\text{min}}\), \(T_{opt}\), \(T_{\text{max}}\) represent the minimum, optimal, and maximum temperatures, respectively, which were set at 0, 50, and 65 \({}^{\circ}\)C [23]. When the temperature is less than 10 \({}^{\circ}\)C, the temperature influence factor is less than 0.1; when the temperature is between 10 and 50 \({}^{\circ}\)C, the temperature influence factor increases linearly with the temperature; when the temperature is greater than 50 \({}^{\circ}\)C, the temperature influence factor decreases parabolically until it is 0 (Figure 1). The effect of moisture content is mainly reflected in the ability of microorganisms to use nutrients and its effect on reducing air-permeable space. [PERSON] (1993) proposed the correction formula of water content on degradation rate. [PERSON] et al. (2015) used it to simulate the compost degradation process, and the results showed that the formula had a good fitting effect on moisture content. The water content is between 0 and 0.3, and the water content influence factor changes slowly at less than 0.1; the water content is between 0.3 and 0.6, showing a non-linear and rapid growth; and the water content is greater than 0.6, slowly approaching 1 (Figure 2). \[f(C_{water})=\frac{1}{e^{(-17.684C_{water}+7.0622)}+1} \tag{13}\]  
Figure 1: Temperature influence factor changes with temperature. Generally, if oxygen in the voids of MSW is more than 5% of the total volume, the metabolism by aerobic microorganisms is not seriously inhibited. The adjustment of the oxygen content (volume fraction) to the reaction rate was first proposed by [PERSON] (1993) and can be expressed by Equation (14). [PERSON] (2001) used it to define oxygen uptake reaction kinetics and obtained good fitting results. The oxygen concentration influence factor shows a rapid increase with the oxygen volume fraction at the beginning and then slowly approaches 1 (Figure 3). \[f(C_{\text{O2}})=\frac{C_{\text{O2}}}{C_{\text{O2}}+2\%} \tag{14}\]  
Figure 3: Oxygen concentration influence factor change with oxygen concentration. Figure 2: Water content influence factor changes with water content. The metabolic processes of microorganisms have different carbon and nitrogen requirements when obtaining energy and synthesizing cell material. Carbon is the primary energy source of microorganisms, while nitrogen is mainly used by microorganisms to synthesize protoplasm for cell production. The higher the carbon-to-nitrogen ratio, the more organic matter that microorganisms can use, the stronger the microbial activity, and the higher the digestion efficiency. However, when the nitrogen source exceeds the optimal range, or the carbon source is insufficient, microbial activity decreases, and the degradation process slows. When the carbon-to-nitrogen ratio is too low, the nitrogen becomes excessive. During the reaction, the amount of nitrogen in the microbial decomposition medium exceeds the requirements to synthesize cells. The excess nitrogen will be released in the form of NH\({}_{3}\), which causes nitrogen loss and produces a foil odor. The recommended carbon-nitrogen ratio range for domestic waste processing is 20:1-40:1, and the optimal range is 25:1-30:1 [47]. Here, the Gaussian function was used to fit the experimental data and construct the formula that describes the effect of the initial carbon-nitrogen ratio on the reaction rate. The initial carbon-nitrogen ratio influence factor and initial carbon-nitrogen ratio exhibit normal distribution relationships, reaching a peak when the carbon to nitrogen ratio is 28 (Figure 4). \[f(C_{C/N})=e^{-(C_{C/N}-28)/14)^{2}} \tag{15}\]