⚡ electrical engineering

Machine Learning Modeling of the Volumetric Overall Mass Transfer Coefficient in a Novel L-Shaped Pulsed Sieve Plate Extraction Column

This study demonstrates that an Artificial Neural Network (ANN) model significantly outperforms empirical correlations and other machine learning algorithms in accurately predicting the volumetric overall mass transfer coefficient for a novel L-shaped pulsed sieve plate extraction column, while identifying dispersed phase holdup and drop size as the dominant influencing factors.

Original authors: vahid rafiei, sadegh moradi

Published 2026-08-12

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: vahid rafiei, sadegh moradi

Original paper licensed under CC BY 4.0 (https://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 trying to mix two liquids that hate each other, like oil and water, but you need to swap a specific ingredient between them. This is the job of liquid-liquid extraction, a process used everywhere from making medicine to cleaning up nuclear waste. To make this swap happen fast, engineers use tall, shaking towers filled with metal plates. Inside, the liquids break into tiny droplets, creating a massive surface area where the ingredient can jump from one liquid to the other. The speed of this jump is measured by something called the mass transfer coefficient. Think of it as the "efficiency score" of the tower: a higher score means the tower is smaller, cheaper, and faster at doing its job.

For decades, scientists have tried to write simple math formulas to predict this efficiency score. They wanted a "one-size-fits-all" rule that would work for any tower shape or liquid mixture. But here's the problem: these old formulas are like trying to use a map of a city to navigate a jungle. They work great for standard, straight-up towers, but they fall apart when the tower has a weird shape or the liquids behave differently. They often guess wrong by huge margins, making it risky to build new, better machines. This is where machine learning steps in. Instead of forcing the data into a rigid, pre-written formula, machine learning acts like a super-smart detective that looks at thousands of clues to figure out the hidden patterns on its own. It doesn't need to know the rules of physics beforehand; it just learns from the data what actually happens.

This paper tackles a very specific, tricky puzzle: a brand-new type of extraction tower shaped like the letter L. Unlike the usual straight towers, this one has a horizontal section where the liquids slide sideways, followed by a vertical section where they settle down. The researchers found that this L-shape creates two completely different worlds inside the same machine. In the horizontal part, the liquids churn up into a chaotic, high-energy "dispersion" regime with lots of tiny droplets. In the vertical part, they calm down into a "mixer-settler" mode with fewer, larger droplets. Because the physics are so different in each half, the old "one-size-fits-all" formulas completely failed to predict how well the tower worked, missing the mark by huge amounts.

To solve this, the team from Arak University didn't try to force a new math equation onto the problem. Instead, they fed experimental data from two different chemical mixtures (toluene-water and n-butyl acetate-water) into three different machine learning "brains": an Artificial Neural Network (ANN), a Random Forest (RF), and a Support Vector Regression (SVR) model. They taught these models to predict the efficiency score (the mass transfer coefficient) based on real measurements of how fast the liquids were flowing, how hard the tower was pulsing, and exactly how big the droplets were.

The results were a clear victory for the Artificial Neural Network (ANN). While the other models did okay, the ANN became a crystal ball, predicting the tower's performance with stunning accuracy. It got the right answer 98% of the time (a test R² value above 0.98) and was off by less than 6% on average. In contrast, the old formulas from textbooks were off by anywhere from 33% to 74%, essentially useless for this new L-shaped design. The machine learning model learned that the most important clues were the amount of liquid trapped in the tower (holdup) and the size of the droplets. It figured out that in the horizontal section, shaking the tower harder made the droplets smaller and the efficiency skyrocket. But in the vertical section, shaking it harder actually made the efficiency drop because the droplets didn't have enough time to settle.

The paper also showed that the "best" chemical mixture depended on which part of the tower you were looking at. The toluene system worked better in the horizontal section, while the n-butyl acetate system was superior in the vertical section. The ANN model naturally figured out this complex switch without being explicitly told the rules, proving it had truly learned the underlying physics. To help engineers who can't use a computer model, the team also created a new, simpler math formula based on their findings, which was much better than the old ones (though still not as perfect as the AI).

Ultimately, this study proves that for weird, new machine shapes like the L-shaped tower, you can't rely on the old rulebooks. The "one-size-fits-all" formulas are broken for these geometries. Instead, using machine learning to learn directly from the data is the key to designing faster, more efficient chemical plants. The AI didn't just guess; it learned the secret language of the L-shaped tower, showing that when you have a novel design, you need a novel way of thinking to make it work.

Technical Summary: Machine Learning Modeling of the Volumetric Overall Mass Transfer Coefficient in a Novel L-Shaped Pulsed Sieve Plate Extraction Column

1. Problem Statement

The volumetric overall mass transfer coefficient ( $K_{oca}$ ) is a critical design parameter for liquid-liquid extraction columns. While empirical correlations exist for conventional vertical pulsed sieve plate columns (PSPCs), they suffer from significant limitations when applied to novel geometries. Specifically, correlations developed for vertical configurations cannot be reliably extrapolated to the L-shaped pulsed sieve plate extraction column (LPSPC). The LPSPC features a unique two-regime operation: a horizontal section operating in the dispersion regime with high dispersed phase holdup (16–27%) and a vertical section operating in the mixer-settler regime with low holdup (1–3%). Existing literature correlations, when applied to this geometry, yield prediction errors (Average Absolute Relative Error, AARE) ranging from 33% to over 74%, rendering them unsuitable for quantitative column design. Furthermore, no data-driven model had previously been reported for predicting $K_{oca}$ in the LPSPC.

2. Methodology

Experimental Setup and Data Collection

The study utilized an L-shaped column equipped with sieve plates, operating with two standard chemical systems recommended by the EFCE:

Toluene-Acetone-Water (TAW)
n-Butyl Acetate-Acetone-Water (BAW)

Experiments were conducted across 36 operating conditions (18 per system) varying pulsation intensity ( $A_f$ ), continuous phase flow rate ( $Q_c$ ), and dispersed phase flow rate ( $Q_d$ ).

Hydrodynamic Measurements: Dispersed phase holdup ( $\phi$ ) was measured via the displacement method, and mean drop size ( $d_{32}$ ) was determined using photographic techniques.
Mass Transfer Determination: The volumetric overall mass transfer coefficient ( $K_{oca}$ ) was not measured directly but derived from experimental concentration profiles using the Back Flow Model (BFM) to account for axial mixing.

Machine Learning Framework

Three machine learning algorithms were developed, trained, and compared to predict $K_{oca}$ for both the horizontal and vertical sections:

Artificial Neural Network (ANN): A Multilayer Perceptron (MLP) with two hidden layers (10 and 5 neurons), trained using Bayesian Regularization to prevent overfitting.
Random Forest (RF): An ensemble of 200 regression trees.
Support Vector Regression (SVR): Using a Radial Basis Function (RBF) kernel.

Input Features: The models utilized operating conditions ( $A_f, Q_c, Q_d$ ) alongside directly measured hydrodynamic parameters ( $d_{32}, \phi$ ). The inclusion of hydrodynamic variables was a strategic choice to encode the nonlinear state of the column, rather than relying solely on operating conditions as proxies.

Data Splitting: The dataset was stratified into training (70%), validation (15%), and testing (15%) subsets. Performance was evaluated using the coefficient of determination ( $R^2$ ), Mean Squared Error (MSE), and AARE.

3. Key Contributions

First ML Application to LPSPC: This study presents the first machine learning models specifically trained to predict $K_{oca}$ in the novel L-shaped column geometry, addressing a gap where traditional correlations fail.
Multi-Section Modeling: Unlike previous studies focusing on single column types, this work simultaneously models two hydrodynamically distinct sections (horizontal dispersion vs. vertical mixer-settler) within a single framework.
Hybrid Approach: The study provides both a high-accuracy "black-box" ANN model and a new semi-empirical dimensionless correlation for the Sherwood number ( $Sh_{oc}$ ), offering flexibility for different design stages.
Mechanistic Insight via Feature Importance: The research quantifies the relative importance of input variables, confirming that hydrodynamic parameters (holdup and drop size) are the dominant drivers of mass transfer performance.

4. Results

Model Performance

The ANN model demonstrated superior predictive capability compared to RF and SVR:

Horizontal Section: $R^2_{test} = 0.9856$ , AARE = 4.83%.
Vertical Section: $R^2_{test} = 0.9812$ , AARE = 5.62%.

In contrast, the RF and SVR models showed lower accuracy (AAREs ranging from 7.34% to 11.24%). Residual analysis confirmed the ANN model was free of systematic bias and satisfied assumptions of homoscedasticity.

Comparison with Literature

When five established empirical correlations for vertical PSPCs were applied to the LPSPC data, they failed significantly:

Horizontal Section AARE: 47.3% – 74.1%.
Vertical Section AARE: 33.1% – 52.8%.
The ANN model reduced these errors by a factor of 7 to 15, demonstrating that existing correlations cannot be extrapolated to the L-shaped geometry.

New Dimensionless Correlation

A new semi-empirical correlation for $Sh_{oc}$ was developed:

Horizontal: $Sh_{oc} = 74.35 Re^{1.41} Sc^{0.583} (1-\phi)$
Vertical: $Sh_{oc} = 0.046 Re^{0.478} Sc^{0.763} (1-\phi)$
This correlation achieved AAREs of 9.30% (horizontal) and 10.17% (vertical), significantly better than literature correlations but less accurate than the ANN.

Physical Insights and Response Surfaces

Opposing Effects of Pulsation: The ANN response surfaces revealed that increasing pulsation intensity ( $A_f$ ) increases $K_{oca}$ in the horizontal section (due to drop breakage and increased interfacial area) but decreases $K_{oca}$ in the vertical section (due to holdup suppression in the mixer-settler regime).
Chemical System Reversal: The models correctly captured a performance inversion between the TAW and BAW systems. TAW performed better in the horizontal section, while BAW performed better in the vertical section. This was attributed to the interplay between interfacial tension (affecting drop size) and the distribution coefficient (affecting driving force), which manifest differently in the two hydrodynamic regimes.
Feature Importance: Sensitivity analysis identified dispersed phase holdup ( $\phi$ ) and mean drop size ( $d_{32}$ ) as the most critical predictors, collectively accounting for ~65–70% of the output variance.

5. Significance and Claims

The authors claim that this work provides a practical predictive tool for the design of L-shaped extraction columns, a geometry previously lacking reliable modeling methods. The study demonstrates that:

Machine Learning is Essential for Novel Geometries: Traditional empirical correlations are insufficient for the LPSPC due to its unique hydrodynamic transitions; data-driven models trained on specific geometry data are necessary.
Hydrodynamic Inputs are Critical: Including measured parameters like holdup and drop size significantly improves model accuracy compared to using only operating conditions.
Optimization Capability: The ANN-based response surfaces allow for the identification of optimal operating conditions (e.g., an $A_f$ range of 0.85–1.00 cm/s) that balance mass transfer performance across both sections of the column.

The paper concludes that while the ANN offers the highest accuracy for design optimization, the new dimensionless correlation serves as a valuable, explicit equation-based alternative for preliminary scale-up calculations where measured hydrodynamic data may not yet be available.