№82-2

Hybrid monotonic modelling of hydrocyclone efficiency for setpoint control of cut size

A.Ablets ¹,  https://orcid.org/0009-0005-9692-9846

V.Tron ¹     https://orcid.org/0000-0002-6149-5794

¹Kryvyi Rih National University, Kryvyi Rih, UkraineColl.res.pap.nat.min.univ. 2025, 82:16–28

Full text (PDF)

https://doi.org/10.33271/crpnmu/82.016

ABSTRACT

Purpose.To develop and validate a hybrid monotonic model of the hydrocyclone partition curve that combines a physics‐based logistic component, a data-driven residual correction, and an enforced monotonic output - for interpretable prediction of the cut size (d₅₀) and selection of operating setpoints to achieve a target d*₅₀.

The methods.The partition curve (reduced partition to underflow (fraction) vs particle size) is represented by a logistic function with interpretable parameters d₅₀ and steepness. The deviations of experimental data from this logistic baseline are learned by an ensemble machine learning method (gradient boosting regression), and the combined prediction is then calibrated using isotonic regression (PAV algorithm) to ensure strictly monotonic increase and correct probability bounds [0,1]. The model was trained and tested on an open experimental dataset of hydrocyclone tests spanning various apex (underflow orifice) / vortex finder (overflow pipe) diameters and feed pressures. Model validation uses 5-fold grouped cross-validation (GroupKFold by cyclone configuration); in addition, 95% bootstrap confidence intervals were computed for the average error, and probability maps were generated to estimate the chance of hitting a target cut size under varying conditions.

Findings. The hybrid monotonic model achieved an average RMSE of ~0.14 (vs ~0.23 for the logistic-only baseline), demonstrating significantly higher accuracy. The model reproduces known pressure and geometry trends: increasing feed pressure or decreasing the apex/vortex diameters leads to a finer separation (lower d₅₀). The hybrid model also provides valid cut-size estimates even in edge cases of incomplete separation (censored data) and yields robust recommended setpoints to attain a desired P(|d₅₀ -d*₅₀ | ≤Δ).

The originality.We propose a hybrid logistic–isotonic model with residual learning that enforces a physically consistent monotonic partition curve within [0,1] and reduces prediction error. The work quantitatively establishes the dependence of  on setpoints (pressure, apex and vortex‑finder diameters) and introduces probability maps  as a practical criterion for setpoint selection.

Practical implementation.The resulting model can serve as a practical engineering tool for hydrocyclone operation: it enables quick estimation of the cut size with confidence intervals, helps select operating parameters (setpoints) to achieve a target cut size with a quantified probability of success, and provides maps of stable operating regimes - all without time-consuming CFD simulations or extensive experimental trials.

Keywords: hydrocyclone, partition curve, cut size d₅₀, isotonic regression, monotonic calibration, machine learning, setpoint.

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