## 3 Results and Discussion
### Estimation of Kinetic Parameters during Anaerobic Co-Digestion  
Apart from the cumulative biogas yield, the decomposition rate, lag phase, and biogas production rate are also important factors in assessing the efficiency of anaerobic co-digestion (AcoD) [39; 40]. To evaluate these kinetic parameters, first-order, modified Gompertz, logistic, and Cone models were applied. The specific parameters are summarized in Table 1.  
The biogas production potential (\(P_{0}\)) estimated for the four models increased with the growing proportion of FW, ranging from 284.78 to 964.00 mL/g VS at various mixing ratios of FW and MA; these results were in line with the result of [PERSON] et al. [41]. However, unlike [PERSON] et al. [41] who reported a P0 range of 88.00-670.60 mL/g VS and the highest biogas production occurring in the FW/MA = 0.8.0.2 group, our experiment showed the highest biogas production at a ratio of 3:1. This discrepancy is primarily due to differences in the types of microalgae used. The observation was further verified in the previous study as a large amount of organic matter derived from FW promoted an increase in biogas production [32].  
The decomposition rate (\(k\)) is another key indicator reflecting digestion efficiency and substrate biodegradability. The k values ranged from 0.22 to 0.45/d in this study, which is consistent with the k values for anaerobic digestion of FW reported in previous studies [40] ([PERSON] & [PERSON], 2013). Interestingly, k values of co-digestion systems were better than those for mono-digestion systems, particularly at the FW/MA mixing ratios of 1:1 and 1:3 (Figure 2a); this indicated that adding MA detritus to FW accelerated the total decomposition rate. Similarly, the maximum biogas production rate (\(R_{m}\)) was obtained at the FW/MA ratio of 1:1, indicating that appropriate mixing ratios could promote biogas production (Figure 2b). These results show that co-digestion of FW and MA had an impact on the organic matter biodegradability and conversion rate. In all cases, the estimated lag phase (\(\lambda\)) disappeared, which could be attributed to the readily available biodegradation components in the FW and MA [30]. This suggests that the large, easily biodegradable proportion of FW and MA was directly subjected to acidification and methanogenesis without the need for first undergoing solubilization and hydrolysis [41].  
Figure 1: The effects of mixing ratios on biogas yields: (**a**) daily biogas yield and (**b**) cumulative biogas yield. EBY represents cumulative experimental biogas yield, and CBY represents calculated biogas yield according to the proportions of FW and MA applied in the co-substrates and their separate biogas yield during the mono-digestion process.  
The reliability and accuracy of the four applied models were evaluated by comparing and calculating the correlation coefficients (R\({}^{2}\)), the residual sums of squares (RSS), and the differences between experimental and model-predicted data (Dem). The Cone model outperformed the other models, with the highest R\({}^{2}\) in various mixing ratios (above 0.99); R\({}^{2}\) values of the first-order, modified Gompertz, and logistic models ranged from 0.96 to 0.98, 0.91 to 0.96, and 0.88 to 0.94, respectively. Additionally, the Cone model gave the lowest RSS and Dem, which further confirmed the agreement of predicted results with experimental data. Overall, the Cone model fit best in this study, which was highly consistent with the previous study [41; 42], which examined co-digestion of FW and MA (_Scenedesmus_ sp., _Chlorella_ sp., _Spirulina platensis_).