🤖 machine learning

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

This paper proposes a two-stage training framework for a Physics-Informed Neural Network (PINN) that, when integrated into a state estimation filter, enables the accurate estimation of liquid-liquid phase heights using only available volume flow measurements.

Original authors: Mehmet Velioglu, Song Zhai, Alexander Mitsos, Adel Mhamdi, Andreas Jupke, Manuel Dahmen

Published 2026-04-28

📖 3 min read☕ Coffee break read

CC BY 4.0

Original authors: Mehmet Velioglu, Song Zhai, Alexander Mitsos, Adel Mhamdi, Andreas Jupke, Manuel Dahmen

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are running a massive industrial kitchen that makes giant batches of salad dressing. To keep the dressing perfect, you need to make sure the oil and vinegar separate exactly the right way. If the "middle layer" (the cloudy part where they are mixing) gets too thick and rises too high, it spills over the sides, ruins the batch, and creates a huge, expensive mess.

In the industrial world, this is called Liquid-Liquid Separation. The "middle layer" is called the Dense-Packed Zone (DPZ).

The Problem: The "Blindfolded Chef"

The problem is that these separation tanks are huge, dark, and oily. Trying to see exactly how high that middle layer is using a camera is like trying to measure the height of a layer of fog in a dark forest using a flashlight from a mile away. The cameras are blurry, the lighting is bad, and sometimes they just miss the mark entirely.

If you can't see the layer, you can't stop the "spill" (called flooding) before it happens.

The Solution: The "Smart Ghost" (The PINN)

The researchers decided to stop relying solely on "eyes" (cameras) and instead build a "Smart Ghost"—a digital brain called a Physics-Informed Neural Network (PINN).

Think of it this way:

The Textbook (Pre-training): Before the "Smart Ghost" ever sees the real kitchen, it spends weeks reading physics textbooks. It learns the "rules of the universe"—like how gravity pulls heavy things down and how liquids flow. This is the Mechanistic Model.
The Real-World Internship (Fine-tuning): Now, the Ghost enters the actual kitchen. It watches the real, messy, blurry camera footage. It says, "The textbook says the layer should be here, but the camera shows it's actually a bit higher. I'll adjust my brain to match reality."
The Super-Senses (The EKF): Finally, they give the Ghost a special tool called a Kalman Filter. Imagine the Ghost is wearing a blindfold, but it can "feel" the vibrations of the pipes. Even if it can't see the layer, it can feel how much liquid is flowing in and out of the pipes. By combining its "physics brain" with these "pipe vibrations," it can calculate exactly how high the layer is without ever actually seeing it.

Why is this a big deal?

Usually, computers are either "Purely Logical" (they follow math rules but fail when things get messy) or "Purely Observational" (they learn from data but fail when they see something new).

This paper combines both. It’s like a student who has both a perfect memory of their textbook and the common sense to know when a real-world situation is slightly different from the book.

The Result:
The researchers proved that this "Smart Ghost" could predict the height of that messy middle layer incredibly accurately, even when the flow of liquid changed in ways the computer had never seen before. It’s like being able to tell exactly how much salt is in a soup just by listening to the bubbles, even if you're blindfolded!

In short:

They created a digital brain that uses the laws of physics and flow measurements to "see" through the darkness of industrial tanks, preventing expensive accidents before they happen.

Technical Summary: Estimating Dense-Packed Zone Height in Liquid-Liquid Separation via PINNs

1. Problem Statement

In industrial liquid-liquid separation (e.g., pharmaceutical, chemical, and recycling processes), the accumulation of a dense-packed zone (DPZ)—a layer of droplets that coalesce slower than they sediment—is a critical performance and safety concern. If the DPZ height exceeds a critical level, it causes "flooding," which disrupts separation efficiency and downstream operations.

While monitoring DPZ height is essential, it is practically difficult because:

Optical Limitations: Industrial separators are often opaque or have limited viewing capacity (gauge glasses).
Data Scarcity: Experimental data for dynamic transitions is expensive and time-consuming to collect.
Model Inaccuracy: Existing mechanistic models (steady-state or lumped) often lack the accuracy (frequently $<25\%$ ) required to predict flooding points under dynamic conditions.

2. Methodology

The authors propose a hybrid framework that combines Physics-Informed Neural Networks (PINNs) with an Extended Kalman Filter (EKF) to estimate phase heights using only readily available volume flow-rate measurements.

A. Two-Stage PINN Training Strategy:
To overcome the lack of high-fidelity experimental data, the authors employ a transfer learning approach:

Pre-training (Physics-Driven): The PINN is first trained on synthetic data generated by a low-fidelity mechanistic model. This stage uses the model's volume balance equations as "soft constraints" (physics regularization) to establish a baseline understanding of the system dynamics.
Fine-tuning (Data-Driven): The pre-trained model is then fine-tuned using scarce, noisy experimental data (phase heights and flow rates). During this stage, the loss function is adjusted to prioritize experimental accuracy while maintaining physical consistency.

B. Model Architecture & Physics Integration:

Inputs: PINN time ( $t$ ), initial phase heights, and inlet volume flow rate ( $Q_{in}$ ).
Outputs: Predicted phase heights ( $h_{DP}, h_{HP}$ ), outlet flows ( $Q_{top}, Q_{bot}$ ), and immeasurable internal flows (coalescence $Q_c$ and sedimentation $Q_s$ ).
Physics Constraints: The network is regularized using macroscopic volume balance equations (algebraic and differential residuals) via automatic differentiation.

C. State Estimation (Deployment):
Since phase heights are not measurable during real-time operation, the trained PINN is embedded into an EKF-inspired framework.

The PINN acts as the state-transition model (predicting the next state) and the measurement model (predicting outlet flows from states).
The filter uses the discrepancy between predicted and measured outlet flow rates to update the estimated phase heights.
Uncertainty Quantification: An ensemble of 40 PINNs is used. The spread of the ensemble is used to adaptively calculate the process noise covariance ( $W_k$ ), allowing the filter to "distrust" the model in regions of high uncertainty.

3. Key Contributions

Hybrid Modeling Framework: Successfully integrates low-fidelity mechanistic models with high-fidelity experimental data through a two-stage PINN training process.
Soft Sensor Development: Demonstrates that phase heights (difficult to measure) can be accurately estimated using only flow-rate measurements (easy to measure).
Uncertainty-Aware Estimation: Uses ensemble learning to provide a robust state estimator that accounts for both model (epistemic) and measurement noise.
Geometric Compensation: Employs a secondary simple neural network to map the "average" DPZ height (assumed by the model) to the "maximum" DPZ height at the separator outlet, addressing the wedge-shaped reality of high-flow regimes.

4. Results

Prediction Accuracy: The two-stage trained PINN significantly outperformed both the mechanistic model and purely data-driven Vanilla Neural Networks (VNN).
Interpolation vs. Extrapolation: The PINN showed high accuracy in interpolation tests. In extrapolation tests (operating at flow rates unseen during training), the PINN maintained much better tracking of the DPZ height compared to the VNN, which tended to overshoot.
Estimation Performance: The PINN-EKF combination successfully reconstructed phase height trajectories from flow-rate data. Even when initial states were unknown, the ensemble mean converged to the experimental values within approximately 100 seconds.
Error Metrics: The PINN achieved lower Root Mean Square Error (RMSE) and lower Interquartile Range (IQR) in both interpolation and extrapolation scenarios compared to the benchmark models.

5. Significance

This research provides a scalable solution for the real-time monitoring of liquid-liquid separators. By transforming a difficult optical measurement problem into a manageable flow-rate estimation problem, the proposed framework offers a practical "soft sensor" for industrial automation. This enables safer operation by providing early warning signs of flooding, even in environments where direct observation is impossible.