Residual energy evaluation in vortex structures: On the application of machine learning models

Vortex structures are widely employed for energy dissipation in urban surface water conveyance systems. When transporting wastewater through these networks, a substantial amount of water energy is dissipated. The effectiveness of these structures is usually evaluated by their efficiency in dissipating energy. Recent literature reviews on vortex structures have emphasized that, despite numerous experimental studies aimed at assessing their hydraulic performance, a reliable mathematical model to predict the residual energy head ratio remains elusive. In this study, resilient numerical models employing Artificial Intelligence (AI) methodologies (such as non-parametric regression, decision trees, and ensemble learning) have been structured by reliable experimental tests. By analyzing the experiments, three primary factors, referred to as flow Froude number (Fr), the ratio of sump height (Hs) to shaft diameter (D), and the ratio of drop total height (L) to shaft diameter (D) were determined to estimate the residual energy head ratio. Through experimental study, the residual energy head ratio is computed as a ratio of downstream flow energy (E2) to upstream flow energy (E1) at vortex structure. During the training and testing phases of AI models, the results of statistical tests, serving as quantitative evaluations, have shown that ensemble learning models namely Adaptive Boosting (AdaBoost) and Categorical Boosting (CatBoost) models had higher level of efficiency in the E2/E1 predictions and followed by Model Tree (MT), K-Nearest Neighbor (KNN), Support Vector Machine (SVM), Extreme Gradient Boosting (XGBoost) and Multivariate Adaptive Regression Spline (MARS). Additionally, the second-order regression-based equation was obtained from Fully Factorial Method (FFM) which had lower level of precision (R = 0.8275, RMSE = 0.1156, and MAE = 0.0846) in the residual energy head ratio predictions when compared with all predictive AI models. Variations of three effective factors (i.e., Fr, L/D, Hs/D) versus the predicted E2/E1 ratios were in well agreement with observational tests. Moreover, the results of Sobol's index indicated that Fr number was determined as the most effective parameter in the evaluation of residual energy head ratio in the vortex structure.