## 3 Results and Discussion
### Stage 2
#### 3.2.3 Correlations and Empirical Models  
Strong correlations were noted between the SCO\({}_{2}\) dose and the production of biogas/CH\({}_{4}\). In the variants V1-V4, the determination coefficients reached R\({}^{2}\) = 0.8755 for biogas and R\({}^{2}\) = 0.8881 for CH\({}_{4}\) (Figure 5a). In the variants V4-V6, their values reached R\({}^{2}\) = 0.8459 for biogas and R\({}^{2}\) = 0.8567 for CH\({}_{4}\) (Figure 5a). In V1-V4 and V4-V6, strong correlations between the pH value of the AGS and the biogas volume were found, as indicated by the determination coefficients reaching R\({}^{2}\) = 0.9409 and R\({}^{2}\) = 0.9991, respectively (Figure 5b). Similar correlations were noticed in the case of CH\({}_{4}\), with determination coefficients of R\({}^{2}\) = 0.9558 and R\({}^{2}\) = 0.9998, respectively (Figure 5b). The increase in the biogas and CH\({}_{4}\) yields observed in V1-V4 was due to the improvement of the AGS biodegradability as a result of its conditioning with SCO\({}_{2}\). The substrate availability and biodegradability in V4-V6 were similar due to the concentrations of the compounds determined in the dissolved phase. However, the high SCO\({}_{2}\) does applied resulted in a pH decrease, which, in turn, had an inhibiting effect on the  
Figure 4: pH values after SCO\({}_{2}\) pre-treatment and AD. The results obtained in V1-V4 demonstrated a correlated surface effect of the COD concentration and pH value of the supernatant on the yields of biogas (Figure 6a) and CH4 (Figure 6b). A mathematical model was developed that enabled the estimation of biogas production. Because a linear correlation was found in the variants V1-V4, they were considered in the estimation. It was found that dependent variables such as the COD concentration and pH value of the supernatant had a statistically significant effect on the volume of biogas produced. The developed model of biogas production (1) reflects ca. 88.41% of the changes in the process of biogas acquisition (determination coefficient R\({}^{2}\) = 0.8841) and is characterized by an estimation error of \(\pm\)38.987. Figure 5: Correlations between (**a**) SCO\({}_{2}\) dose, (**b**) pH of AGS, and biogas and methane volumes. Figure 6: Correlation between pH, dissolved COD concentration, and the (**a**) biogas and (**b**) CH4 yields. \[\text{BIOGAS}=0.4921\text{COD}-0.3065\text{pH}+230.33 \tag{1}\]  
BIOGAS--the amount of produced biogas (cm\({}^{3}\)/gVS),  
COD--COD concentration in the dissolved phase, mgO\({}_{2}\)/dm\({}^{3}\),  
pH--pH value (-)  
The monitoring of the organic compound concentrations in the supernatant is one of the ways to assess the effectiveness of the sludge preparation processes of a given substrate subjected to disintegration [42; 43]. The possibility to state reliable mathematical models to estimate the AD course based on the organic compound content in the supernatant was proven [44]. Hence, such an assessment is essential from the practical standpoint, as it allows for the omission of highly advanced measurements that enable an estimation of the effectiveness of the applied disintegration methods. As in any technology, in the case of the use of the AGS pre-treatment with SCO\({}_{2}\), weaknesses can also be identified. Certainly, a limitation is the low reconnaissance and low technological readiness level (TRL) based only on laboratory-scale work. The lack of data from installations operated on a pilot or fractional technical scale limits the possibility of a reliable and credible estimation of both technological efficiency and economic profitability [21; 29]. The production of SCO\({}_{2}\) in the biogas upgrading process is also not well known, which may favor, and significantly support, the profitability of the entire technological system. Obtaining SCO\({}_{2}\) from installations dedicated to its production may eliminate this method of sewage sludge conditioning due to the investment and operating costs incurred [26].