## 4 Results and Discussion
Different machine learning algorithms were trained and tested for a comparative analysis with the ANN model, including DT [44], RFR [45], XGBOOST [46], KNN [47], and SVM [48]. The results of this evaluation are summarized in Table 7, which shows the average values and standard deviations for six models obtained through K-fold cross-validation. The findings indicate that the ANN model exhibits high accuracy and relative stability. Based on these results, the ANN model is used for further research. Using advanced models like SVR could obtain higher accuracy; however, considering the broader context of our research and its specific objectives, the ANN model is used for the following reasons: (1) Consistency with initial research design: this study was initially conceptualized and executed with an ANN-based approach and was structured around this methodology. (2) Balance between accuracy and complexity: while the SVR model showed marginally better performance, the ANN model still provides high accuracy. Additionally, ANN offers a balance between predictive power and model interpretability, which aligns with our research aims. (3) Scope of study: comprehensive revisions to include a different AI model would require substantial changes in the analysis and the manuscript's narrative, which might shift the focus away from the original scope of our research. The calculation method proposed by [PERSON] et al. [34], known as the connection weights algorithm, is demonstrated in Equation (5). It involves summing the products of weights associated with the connections among input, hidden, and output layers in ANN model. \[RI_{x}=\sum_{y=1}^{M}w_{xy}w_{yz} \tag{5}\]  
where \(RI_{x}\) is the relative importance of input variable \(x\), \(M\) is the total number of hidden neurons, \(w_{xy}\) is the weight of the connection between input and hidden layers, and \(w_{yz}\) represents the weight of the connection between hidden and output layers. Figure 6 displays the relative importance of different input variables on particle settling velocity. The consistency index (x\({}_{6}\)) has the greatest impact, followed by the particle diameter (x\({}_{2}\)), fluid density (x\({}_{5}\)), settling area ratio (x\({}_{4}\)), particle density (x\({}_{1}\)), flow behavior index (x\({}_{7}\)), and sphericity (x\({}_{3}\)). The absolute values determine the importance, whereas the sign (positive or negative) indicates the direction of the effect of each input on the output. As depicted in Figure 6, an increase in the consistency index, settling area ratio, and fluid density results in a decrease in particle settling velocity. Conversely, an increase in particle diameter, particle density, flow behavior index, and sphericity lead to an increase in particle settling velocity. An increase in particle density (x\({}_{1}\)) results in \begin{table}
\begin{tabular}{c c c} \hline \hline
**Models** & \
\begin{tabular}{c} **Average R\({}^{2}\) Value from K-Fold** \\ **Cross-Validation** \\ \end{tabular} &
\begin{tabular}{c} **Standard Deviation of R\({}^{2}\) Values from** \\ **K-Fold Cross-Validation** \\ \end{tabular} \\ \hline DT & 0.8016 & 0.0580 \\ RFR & 0.9374 & 0.0112 \\ XGBOOST & 0.9679 & 0.0093 \\ KNN & 0.8922 & 0.0204 \\ SVM & 0.9083 & 0.0287 \\ ANN (one-hidden-layer) & 0.9706 & 0.0085 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Model accuracy compared with other models. a higher gravitational force, leading to an increased settling velocity. A larger particle diameter (x\({}_{2}\)) also increases gravitational force, enhancing the settling velocity. The cubic relationship between diameter and volume implies a stronger positive influence of diameter compared to density. As sphericity (x\({}_{3}\)) increases, particles become more sphere-like, thus increasing their settling velocity. However, this effect is less pronounced due to minor variations in sphericity within our experimental data range. A larger settling area ratio (x\({}_{4}\)) increases the cross-sectional area facing resistance, thus significantly impacting the settling velocity.