Physics-Constrained Neural Network Framework for Predicting Terminal Settling Velocity

Predicting the terminal settling velocity of particles in non-Newtonian fluids remains a persistent challenge in multiphase flow modeling, with applications spanning chemical processing, mineral processing, and geophysical fluid dynamics. Traditional approaches—empirical drag correlations and fully resolved numerical simulations—trade off generalizability, physical fidelity, and computational cost, while purely data-driven machine learning models are efficient but unconstrained by fundamental physical principles such as the force balance governing a settling particle, which can yield predictions that are locally accurate yet inconsistent with the underlying governing mechanics. This work introduces a Physics-Constrained Neural Network (PCNN) that embeds the terminal force-balance equation into the training loss through an independent drag closure—a generalized Reynolds number and sphere drag correlation corrected for particle sphericity and Herschel–Bulkley rheology—combined with a data-driven loss term. The framework was trained and evaluated on an experimental dataset spanning a range of particle shapes, densities, and fluid rheological parameters, with the physics weighting and network architecture selected through a sensitivity analysis, and benchmarked against Random Forest, Decision Tree, and XGBoost over repeated runs with multiple random seeds. The best PCNN configuration (λ = 0.3) achieved R2 = 0.943 ± 0.011, a more stable and accurate result than its unconstrained (λ = 0) counterpart across every evaluation metric, though it did not surpass the strongest data-driven baselines (XGBoost: R2 = 0.970 ± 0.004; Random Forest: R2 = 0.963 ± 0.008). A complementary modified Z-score diagnostic showed that, despite a wider overall residual spread, the PCNN produced 41–54% fewer extreme-residual predictions than XGBoost across all thresholds tested, indicating a lower proportion of predictions that deviate sharply from the model’s own typical error behavior. These results suggest that embedding a physically motivated force-balance constraint, even from a general-purpose drag correlation not calibrated to this dataset, can improve the consistency of a model’s error behavior without matching the raw accuracy of a well-tuned data-driven ensemble. The PCNN framework is thus positioned as a step toward hybrid models whose value lies in more predictable error behavior rather than outright accuracy gains, with the mismatch between the drag correlation’s calibration domain and the present dataset identified as a key target for future refinement.