Estimating Dense-Packed Zone Height in Liquid-Liquid Separation: A Physics-Informed Neural Network Approach

TL;DR

This work tackles the challenge of estimating the dense-packed zone height $h_{ m DP}$ in a liquid–liquid gravity settler using a physics-informed neural network (PINN) framework that leverages scarce experimental data and abundant synthetic data from a low-fidelity mechanistic model. A two-stage training strategy—pretraining on mechanistic-model data and fine-tuning with experiments—paired with an EKF-inspired state estimator enables real-time tracking of $h_{ m HP}$, $h_{ m DP}$, and outlet flows from inexpensive volume-flow measurements. Ensemble predictions and a simple outlet-height NN demonstrate enhanced accuracy over purely data-driven or purely mechanistic approaches, with the PINN ensemble outperforming alternatives in interpolation and extrapolation scenarios. The results underscore the practical potential of PINN-based soft sensing for flooding risk management in LLS, while acknowledging limitations such as the band-shaped DPZ assumption and data gaps that motivate future extensions to nonuniform DPZ geometries and richer experimental data.

Abstract

Separating liquid-liquid dispersions in gravity settlers is critical in chemical, pharmaceutical, and recycling processes. The dense-packed zone height is an important performance and safety indicator but it is often expensive and impractical to measure due to optical limitations. We propose to estimate phase heights using only inexpensive volume flow measurements. To this end, a physics-informed neural network (PINN) is first pretrained on synthetic data and physics equations derived from a low-fidelity (approximate) mechanistic model to reduce the need for extensive experimental data. While the mechanistic model is used to generate synthetic training data, only volume balance equations are used in the PINN, since the integration of submodels describing droplet coalescence and sedimentation into the PINN would be computationally prohibitive. The pretrained PINN is then fine-tuned with scarce experimental data to capture the actual dynamics of the separator. We then employ the differentiable PINN as a predictive model in an Extended Kalman Filter inspired state estimation framework, enabling the phase heights to be tracked and updated from flow-rate measurements. We first test the two-stage trained PINN by forward simulation from a known initial state against the mechanistic model and a non-pretrained PINN. We then evaluate phase height estimation performance with the filter, comparing the two-stage trained PINN with a two-stage trained purely data-driven neural network. All model types are trained and evaluated using ensembles to account for model parameter uncertainty. In all evaluations, the two-stage trained PINN yields the most accurate phase-height estimates.

Figures (13)

• Figure 1: Piping and instrumentation diagram of the experimental setup with DN 200 separator in the RWTH Aachen Fluid Process Engineering (AVT.FVT) lab. Blue, orange, and green colors mark the aqueous, organic, and dispersion phases (DPZ). Reproduced from Zhai.2025.
• Figure 2: Detection of DPZ heights along the separator with effective length of 1m. Images refer to detections in QIR03 (left) and QIR04 (right). In each image, four heights are detected for same widths. The distances between inlet and first height detection, width of QIR03, width between QIR03 and QIR04, and width of QIR04 is $a=21cm$, $b=36cm$, $c=10cm$, $d=33cm$, respectively.
• Figure 3: Pre-processed inlet volume flow rate and phase height measurements for the training trajectory, measured at eight different positions at QIR03 and QIR04 (see \ref{['fig:Separator_CamPositions']}).
• Figure 4: Post-processed experimental trajectories obtained from the gravity settler. The quantities $\hw$ and $\hp$ denote the average heavy-phase height and DPZ height, respectively, computed over the eight detection positions. $\Qin$ denotes the inlet volume flow rate, and $\Qb$ and $\Qt$ denote the bottom and top outlet volume flow rates, respectively.
• Figure 5: Mechanistic model schematic of the pilot-scale liquid–liquid separator. Adapted from VELIOGLU2025108899.