Machine learning and response surface analysis of mean drop size and dispersed phase holdup in an L-shaped pulsed sieve plate column

Abstract

This study explores the hydrodynamic behavior of four distinct nitric acid-TBP/kerosene systems in an L-shaped pulsed sieve-plate column (LPSPC), with a focus on predicting Sauter mean drop diameter (SMDD) and dispersed phase holdup (DPH). The complex interactions between operating parameters, pulsation intensity, flow rate, and hydrodynamic performance were modeled using response surface methodology (RSM) and artificial neural networks (ANN). This is the first side-by-side assessment of ANN versus RSM for LPSPC. The model was trained on experimental dataset, with 70% for training, 15% for validation, and 15% for testing, respectively. A compact two-hidden-layer topology with 10 and 5 neurons with trainbr was selected after a systematic search over 1–4 layers and three training algorithms. The ANN model achieved R² values of 0.9950, 0.9970, 0.9900, and 0.9822 with MSE values of 0.000314, 0.000451, 0.000019, and 0.000001 for SMDD-horizontal, SMDD-vertical, DPH-horizontal, and DPH-vertical, respectively. At the same time, RSM was slightly better for SMDD-vertical. Complementing these data-driven models, new dimensionless semi-empirical correlations derived via Buckingham’s π-theorem agreed well with experiments, yielding AAREs of 6.95% and 8.29% for SMDD, and 6.56% and 8.68% for DPH in horizontal. Together, these tools provide accurate and usable surrogates for analyzing and optimizing extraction systems relevant to the purification of yellow cake.

Introduction

The liquid-liquid extraction process is one of the separation procedures in which the extraction of the soluble component is transferred between aqueous and organic phases¹. Identifying the main concepts in separation processes, utilizing appropriate extraction equipment, and interpreting experimental results from a semi-industrial to an industrial scale are among the issues that the development of research in each area makes the solvent extraction method more practical² Today, solvent extraction is widely recognized as an effective separation method, and hundreds of extractors have been applied on semi-industrial and industrial scales. One critical liquid-liquid contactor equipment used in solvent extraction is pulsed columns (PCs). The advantages of PCs over other extraction columns include high separation efficiency, less space required, and no internal moving parts, which require less maintenance and repair than other columns equipped with a mechanical stirrer (especially when working with corrosive and radioactive materials). These advantages have made this type of column commonly utilized in the oil, nuclear, and biochemical industries⁶. In PCs, the applied pulse provides mechanical energy, acting like an agitator to generate smaller droplets and enhance efficiency. PCs are generally built in both horizontal and vertical configurations⁷. Horizontal pulsed columns (HPCs) present several advantages over vertical pulsed columns (VPCs). HPCs do not require high-ceiling buildings, which reduces radiation shielding costs when handling radioactive materials. They are also easier to repair, maintain, and assemble in high-activity environments. Furthermore, HPCs offer improved process control and reduced equipment contamination. Despite these benefits, HPCs also have drawbacks, such as limited phase separation due to the lack of gravity assistance and lower flooding points compared with VPCs. Experience with both VPCs and HPCs has shown that each design has unique advantages and drawbacks. Combining HPC and VPC does not simply add their disadvantages; rather, their drawbacks are mitigated by one another. The LPSPC, with its horizontal-vertical design, represents an innovative extractor. By including a horizontal section, the column height is reduced, while capacity is improved due to suction forces at the connection of the two sections. Research on LPSPC is still limited, and studies have generally focused on standard systems such as water-toluene, water-butanol, and water-butyl acetate^8,9,10,11,12. Akhgar et al. were the first to investigate process parameters such as pulsation intensity (A_f), continuous and dispersed phase flow rates, and physical properties of the characteristic velocity in LPSPC at a semi-industrial scale³. Pulsation intensity (Af) is defined as the relative movement of the fluid induced by the pulsation mechanism. It is typically expressed as:

$$\:{A}_{f}=Amplitude\left(A\right)\times\:frequency\left(f\right)$$

(1)

Where f is the pulsation frequency (s^− 1), A is the pulsation amplitude (m). Pulsation intensity (A_f (m/s)) reflects the level of agitation caused by the pulsation in the column³. They independently examined the effect of parameters in this research. Results showed that in the horizontal section, characteristic velocity increases with pulsation intensity and with higher phase flow rates. In the vertical section, velocity increases with dispersed phase flow rate but decreases with continuous phase flow and pulsation intensity. Amani et al. later investigated droplet size behavior and distribution in an LPSPC for three liquid systems: toluene/water, butyl acetate/water, and normal butanol/water⁴. They reported an inverse relationship between pulsation intensity and droplet size, with higher pulsation intensity producing smaller droplets. However, droplet size increases directly with flow rate. Another influential factor was the interfacial tension, which was obtained in this way: as the interfacial tension increases, the droplets will be significantly larger. Fig. 1 presents a schematic of an LPSPC.

Fig. 1

Schematic of the structure and layout of an LPSPC.

In extraction columns, droplet diameter plays a critical role in hydrodynamic behavior and mass transfer. The amount of dispersion phase holdup (DPH) depends on droplet size, which also influences flooding, interfacial area, and mass transfer coefficients⁴. Therefore, droplet diameter can be named the main hydrodynamic parameter in extraction columns, and other hydrodynamic and mass transfer parameters are somehow related to droplet size. In PCs, the size distribution of drops is not uniform throughout the column’s length, so the droplet’s mean diameter is considered the droplet’s mean characteristic diameter, which can be defined as SMDD. By Eq. (2), the SMDD (d₃₂) has been determined:

$$\:{\text{d}}_{32}=\frac{\sum\:_{\text{i}=1}^{\text{n}}{\text{n}}_{\text{i}}{\text{d}}_{\text{i}}^{3}}{\sum\:_{\text{i}=1}^{\text{n}}{\text{n}}_{\text{i}}{\text{d}}_{\text{i}}^{2}}$$

(2)

Where n_i indicates the quantity of droplets having a diameter of d_i^10,12,13. Table 1 shows the relationships provided by different researchers to calculate SMDD (d₃₂) in PCs.

Table 1 Equations for SMDD in PCs.

Another key design variable is DPH (φ), the ratio of the dispersed phase volume to the total phase volume²³:

$$\:\phi\:=\frac{{v}_{d}}{{v}_{d}+{v}_{c}}$$

(3)

Where, $\:{v}_{c}$ and $\:{v}_{d}$ indicate the volume of the continuous and dispersed phases, respectively^1,9. In Table 2, the empirical relationships presented by other researchers to predict DPH in PCs are presented:

Table 2 Empirical formulas for DPH in PCs.

Recent studies underscore the breadth of ANN and metaheuristic tools for nonlinear process modeling and optimization across chemical and energy systems^26,27. Predicting droplet behavior and DPH in LPSPC can be effectively modeled using an artificial neural network (ANN) and response surface methods (RSM). These methodologies enable the integration of various operational parameters and their effects on hydrodynamic behavior, particularly in liquid-liquid extraction systems. RSM has been used to model droplet size and DPH in rotating disc contactors²⁸, while ANN has shown high predictive power in pulsed columns²⁹. Ghaemi et al. modeled liquid-liquid systems in an Oldshue–Rushton column, achieving R² values of 0.9975 for RSM and 0.9905 for ANN. The optimal ANN structure consisted of four layers with 15, 20, 10, and 2 neurons, respectively³⁰. Hemmati et al. also demonstrated ANN and RSM accuracy in pulsed disc-and-doughnut columns³¹. RSM and ANN models demonstrated acceptable predictive accuracy, with maximum R² values of 0.996 and 0.987, respectively. However, such models have mainly been applied to conventional extraction columns, and no prior studies have developed ANN and RSM models for LPSPCs. Recently, Ravandeh and Khooshechin investigated SMDD and drop size distribution in an L-shaped pulsed packed column with ANN and semi-empirical correlations, reporting R² of 0.986 for mean drop size and 0.981 for drop size distribution. However, all these studies were limited to standard aqueous–organic systems and did not involve industrially relevant chemical systems³². In addition, Sinha and Vincent investigated hydrodynamic parameters such as slip velocity and characteristic velocity in pulsed disc-and-doughnut columns using ANN modeling, achieving high predictive accuracy (R² > 0.98) in standard aqueous–organic systems. However, all these studies were limited to standard systems and did not investigate industrially relevant chemical systems³³. In light of the above, the present study addresses this research gap by examining droplet size (SMDD) and DPH in an LPSPC using four new chemical systems critical in yellow cake purification³⁴: 17% nitric acid with 5, 15, and 30% (v/v) TBP/kerosene mixtures. While earlier studies on LPSPCs focused only on standard systems and examined process parameters independently, this work is the first to apply and directly compare both RSM and ANN modeling for LPSPCs under industrially relevant conditions. The novelty lies in comparing these two approaches to develop accurate predictive models for SMDD and DPH in complex systems, providing both a deeper understanding of LPSPC hydrodynamics and reliable tools for semi-industrial and industrial process design and optimization.

Materials and methods

Two-phase liquid system

In this study, four chemical systems were investigated: water-kerosene, 17% nitric acid-5% (v/v) TBP/kerosene, 17% nitric acid-15% (v/v) TBP/kerosene, and 17% nitric acid-30% (v/v) TBP/kerosene. Kerosene, TBP, and nitric acid were obtained from the Isfahan Uranium Conversion Facility (UCF) Plant. The dispersed (organic) phase was Kerosene with different percentages of TBP, and the continuous (aqueous) phase is a 17% nitric acid solution and water. Total tests were performed below the flooding point⁴. Volume ratios of each phase were measured, mixed, and then shaken to reach equilibrium and saturation. Then, two phases were separated, and their physical characteristics were measured. Each experiment was performed in triplicate to ensure the accuracy and reproducibility of the physical property measurements. Table 3 indicates the physical properties of these systems. The viscosities and densities of each phase were determined using a DVI Prime viscometer and pycnometer. The interfacial tension of the liquids was measured using a Krüss tensiometer based on the Du Noüy ring method^5,10. In this technique, a platinum ring is carefully immersed in the liquid and then slowly pulled upward through the liquid surface. The maximum force required to detach the ring from the surface is recorded, which corresponds to the surface tension. All measurements were conducted at ambient temperature and repeated three times to ensure accuracy and reproducibility.

Table 3 Physical specifications of the liquid-liquid systems (20 °C).

Description of equipment

The main part of a column containing plates is called the active area. The horizontal and vertical sections are each 146 cm long with an inner diameter of 6.2 cm. For better fluid mixing, twenty-four pairs of semi-perforated plates are utilized in the horizontal part, and 29 fully perforated plates are used in the vertical part. Table 4 shows the exact dimensions of the used plates.

Table 4 Structural features of LPSPC.

A sample image of the plates used in the horizontal part of the column can also be shown in Fig. 1. The pulse maker part of the column includes an air regulator, a compressor, and a pair of solenoid valves. The compressor forces air into the regulator, reducing pressure. After the regulator, two solenoid valves are arranged in parallel, which alternately enter the circuit. The valve in the circuit opens and closes at regular intervals depending on the frequency setting. Changing the air regulator pressure adjusts the pulse amplitude, while the frequency is adjusted using a regulator located in the path of the solenoid valves’ on/off switch. It should be noted that the frequency setting also affects the amplitude. To observe the pulsation range, a glass tube is mounted parallel to the column, connected at the bottom to the settling section of the column and at the top to the inlet of the compressed air line. The pulsation range is measured by observing the fluctuation in this tube and its relation to the fluctuation in the active part of the column. The pulsation intensity is calculated by multiplying the oscillation amplitude by the frequency applied to the valves in the airflow path. Furthermore, two pumps are used to feed the dispersed and continuous phases into the column. The constant phase enters the column from the vertical part (top) and the dispersed phase from the horizontal part (bottom). The two-phase flow counter-currently flows within the column. Although the pumps were graded and the volumetric flow rate could be observed, two glass rotameters were installed at the pump outlets to adjust and measure the feed and solvent flow rates accurately. These rotameters were calibrated and adjusted before use. The column also has an upper and a lower settler: the lower settler separates the aqueous phase, and the upper settler separates the dispersed phase. An optical sensor is installed at the upper settler to manage the interface between the two phases. When the interface location changes, the sensor sends a signal to a solenoid valve in the continuous phase flow outlet. When the valve opens, the aqueous phase exits the LPSPC, maintaining the interface at a constant position. Finally, the LPSPC has four similar tanks, two for storing feed and solvent, and two for collecting products at the top and bottom of the column.

Experiments

It is required to saturate the organic phase (kerosene with different percentages of TBP) with the aqueous phase (17% nitric acid solution) before performing the experiments to prevent mutual solubility of the phases. For this purpose, the two phases were thoroughly mixed and shaken, and then maintained in a tank for several hours to reach equilibrium, after which they were easily separated. The LPSPC was set up in a systematic sequence to ensure proper operation. First, all tanks and the column were thoroughly cleaned with detergents and rinsed with distilled water. Next, the drain valves of the collection tanks and the column were securely closed, and the feed and solvent tanks were supplied with pre-prepared saturated phases. The compressor was then turned on to initiate pulsing within the column. The aqueous phase pump was activated and set to its maximum capacity until the height of the continuous phase reached the optical sensors located in the upper settling section, at which point the flow was stopped. Subsequently, the pumps for both the constant and dispersed phases were turned on, and the flow rate of each phase was adjusted to the desired values. The frequency and pulse amplitude were also set according to the experimental requirements. After completing these adjustments, the column required 60–90 min to reach a steady state, depending on the chemical system and phase flow rates. The primary parameters investigated during each experiment were droplet diameter and DPH. The column was cleaned and replenished with fresh materials after each experiment to minimize errors. The drop size in this study was determined using the digital photography method. Fig. 2 shows an example of the images of the rising drops in LPSPC.

Fig. 2

An example of images of dispersed phase droplets in LPSPC (a), (b) Horizontal, and (c) Vertical sections.

The size of the drops has been determined with the aid of the gap between the plates and the plate thickness scale. In fact, the diameter of the drops in the recorded photographs was measured according to the distance between the two plates as a base. Then, according to the ratio of the distance between two plates in the column (actual) and the photo, a coordinate grid was defined, enabling the measurement of the diameter of drops easily. In other words, the distance between the two plates and the drops measured have the exact location and conditions; as a result, the plate and droplet positions in the captured images are also similar. Therefore, when the droplet diameter is measured relative to the plate’s distance, the effect of the camera lens and the impact of glass curvature can be practically ignored, because both the actual droplet and sieve plate sizes have the same effect on the captured images. In addition, the margin of error in measuring droplets in extraction columns can also depend on factors such as the resolution and quality of images, lighting, and the movement and deformation of droplets. We used a very high-quality camera to reduce the error. Taking all things into account, the possible error is very small. Photography was done at six points of the active area of the column, three points in the horizontal part, and three points in the vertical part of the column. The photos captured have been analyzed to calculate the dimensions of the drops. More than 1500 drops have been measured for each test to be statistically verified. Finally, by Eq. (4), the d₃₂ has been determined for each test. In other words, Drops are generally spherical in shape; however, they can sometimes appear ellipsoidal. Ellipsoidal drop size is measured in the following way:

$$\:{d}_{i}=\sqrt[3]{{d}_{H,i}^{2}{d}_{L,i}}$$

(4)

Where, d_H is major axis, and d_L is the minor axis^12,13. This research aimed to measure the diameter of nearly all the drops present in the images. The smallest diameter of the drops is reported to be 0.5 mm. It should be noted that according to the selected operating ranges, the working flow regime is dispersed, and therefore, the droplet sizes are not expected to be reduced significantly below 0.5 mm⁷. But a few drops are very small, which are not visible in the photo and cannot be recorded or measured. This research used the interfacial displacement method to calculate the total DPH within the column. After stabilizing the system, the position of the interface between the two phases was determined in the upper settler. Then, all column outlets and inlets were shut off simultaneously. After a while, the position of the phases interface in the column is re-evaluated. Under these circumstances, by measuring the variation in the dispersed phase height, using the primary and secondary positions, the amount of dispersed phase was calculated. As mentioned earlier, the amount of DPH is the ratio of the volume of the dispersed phase to the total volume of the continuous and dispersed phases inside the column. Also, to calculate the amount of dispersed phase in the horizontal segment of the column, the Melnik method has been used, which involves calculating the length of the arc of the dispersed phase (Fig. 3).

Fig. 3

Cross-section of the horizontal part of LPSPC³⁵.

In this method, by measuring S₀ and calculating L_i (Eqs. (5) and (6)), it is possible to calculate the quantity of DPH:

$$\:{L}_{i}=2[{r}_{i}^{2}-{r}_{0}^{2}{sin}^{2}\left(\frac{\pi\:}{2}-\frac{{S}_{0}}{2{r}_{0}}\:\right)]$$

(5)

$$\:\phi\:=\frac{1}{2}-\frac{1}{\pi\:}{{cos}}^{-1}\left(\frac{{L}_{i}}{2{r}_{i}}\right)-\frac{{L}_{i}}{2\pi\:{r}_{i}^{2}}\sqrt{{r}_{i}^{2}-\frac{{L}_{i}^{2}}{4}}\:\:\:\:\:\:\:$$

(6)

Where r_i, S₀, and L_i represent the radius of the horizontal segment of LPSPC, wet circumference, and the interfacial chord length of the two phases, respectively³⁵. Finally, the DPH amount of the vertical section was calculated from the difference between the DPH of the entire column and its horizontal segment. The fitted parameters were determined by minimizing the average absolute relative error (AARE), which was employed as the objective function^1,2,3,4,5:

$$\:AARE=\frac{1}{N}\sum\:_{i=1}^{N}\frac{\left|Experimental\:value-Calculated\:value\right|}{Experimental\:value}\times\:100\%$$

(7)

RSM theory

The experimental design method is a scientific approach to understanding the relationships between input variables and their impact on process outputs. This method enables process modeling, cost optimization, and the determination of the correlation coefficient between input and output variables³⁶. One of the statistical modeling techniques is RSM. This approach utilizes the experimental design method to develop regression equations based on quantitative data obtained from experiments³⁷. This method models the relationships between dependent and independent variables³⁸. The primary objectives of the model are to establish an approximate relationship between Y (the response variable) and the control variables, evaluate the significance of factors represented by x₁, x₂,., x_k through hypothesis testing, and identify the optimal settings for achieving maximum or minimum response within a specified region³⁹. Eq. (8) represents the quadratic model used in RSM.

$$\:Y={\beta\:}_{0}+\sum\:_{j=1}^{k}{\beta\:}_{j}{x}_{j}+\sum\:_{j=1}^{k}{\beta\:}_{jj}{{x}_{j}}^{2}+\sum\:_{i}\sum\:_{j>=2}^{k}{\beta\:}_{ij}{{x}_{i}x}_{j}+{e}_{i}$$

(8)

In this model, Y represents the predicted response, while x_i and x_j denote the independent variables. The model intercept coefficient is indicated by β₀. The coefficients β_j, β_jj, and β_ij correspond to the linear, quadratic, and interaction terms. The parameter k signifies the number of independent variables, and e_i accounts for the error term⁴⁰. In the RSM method, the experimental data fit the quadratic model described in Eq. (8), and the model’s coefficients are determined. The accuracy of the resulting model is assessed through an analysis of variance (ANOVA), the correlation coefficient (R²), and the model’s p-value. Tables 5 and 6 summarize the values of the independent parameters and responses, respectively, organized according to their symbol assignment, response, and trial range.

Table 5 Information on the independent variables used in the experimental design.

The selection of factor levels and their corresponding ranges was based on preliminary experimental observations and published literature on liquid-liquid extraction in pulsed sieve-plate columns. The goal was to define a range broad enough to capture process variability within safe and controllable operational limits to avoid flooding, droplet coalescence, or oscillation instabilities. These ranges were validated through pre-screening tests to ensure that all operating points generated measurable and consistent holdup and droplet data suitable for response surface modeling.

Table 6 Summary of response variables.

The RSM-CCD approach evaluated the influence of various factors on droplet size across five levels. Initially, the two phases were saturated with each other, and the droplet size was measured under steady-state conditions using a photographic technique involving a Nikon D5000 digital camera. In pulse columns, the movement of the dispersed phase upward within the column alters the droplet size distribution, primarily due to the processes of droplet breakage and coalescence.

ANN theory

The ANN method has gained significant popularity over the past two decades due to its ability to model computational networks based on the structure and function of the human brain and nervous system⁴¹. Initially developed by mathematicians and computer scientists, ANNs aim to mimic real-world neural networks. These models utilize a connectionist approach to process information, providing numerous benefits such as rapid processing speed, strong input-output data relationships, compatibility with various networks, resilience to noisy data, parallel processing ability, fault tolerance, and adaptability through learning mechanisms⁴². ANNs are adaptive systems that learn from available data, enabling them to map input parameters to output parameters without requiring explicit knowledge of the underlying complex relationships⁴³. This adaptability is achieved by modifying the network structure in response to external or internal information obtained during the learning process. Contemporary statistical models often incorporate non-linear neural networks to model intricate input-output relationships and uncover hidden data patterns^44,45. An ANN generally consists of three layers: input, hidden, and output. Each layer is made up of interconnected neurons, where the network’s structure is characterized by weight parameters and activation functions. Throughout the learning phase, a training algorithm fine-tunes the weights and biases to reduce the error between the actual and predicted values of the parameters^36,46. The fundamental processing units of ANNs are neurons. Each neuron gathers information from multiple inputs (x_i​), multiplies them by corresponding weights (w_i​), sums the results, and adds a bias term (b). This process can be mathematically expressed, as shown in Eq. (9)⁴⁷.

$$\:net=\left(\sum\:_{i=1}^{n}{w}_{i}{x}_{i}\right)+b$$

(9)

This process culminates in feeding the results into a transfer function (f), which generates the output values (y) based on Eq. (10) ⁴⁸.

$$\:y=f\left(net\right)$$

(10)

To ensure reliable results from the neural network, data normalization is essential. For this purpose, all data were scaled within the range of -1 to 1 using Eq. (11):

$${X_{norm}}=2\frac{{(X - {X_{\hbox{min} }})}}{{({X_{\hbox{max} }} - {X_{\hbox{min} }})}} - 1$$

(11)

Here, X_norm​ represents the normalized data, X is the input variable, and X_max and X_min ​ denote the maximum and minimum values of the data, respectively⁴⁹. To determine the network parameters, the network error should be minimized by reducing the MSE at each step during the training iterations⁵⁰. MSE and R² are used to assess the model’s performance against the validation dataset. A value of R² close to 1 and an MSE approaching zero indicate high network accuracy. The calculations for MSE and R² are as follows^51,52:

$$\:MSE=\frac{1}{n}\sum\:_{i=1}^{n}{\left({Y}_{predicted}-{Y}_{actual}\right)}^{2}\:\:\:\:\:\:$$

(12)

$$\:{R}^{2}=\frac{\sum\:_{i=1}^{n}{\left({Y}_{predicted}-{Y}_{actual}\right)}^{2}}{\sum\:_{i=1}^{n}{\left({Y}_{predicted}-{Y}_{mean}\right)}^{2}}$$

(13)

Y_predicted, Y_actual, Y_mean, and N denote the predicted Y value obtained from the neural network, the actual Y value, the average Y value, and the total number of data points, respectively.

Multilayer perceptron (MLP)

The MLP is a neural network composed of several layers of basic sigmoid processing units, also known as neurons, which have two possible states. These neurons are linked through weighted connections. The structure of the MLP includes an input layer at the base, one or more hidden layers, and an output layer at the top. Each layer’s neurons are connected to neurons in the neighboring layers, with no intra-layer connections. The weights influence the strength of the interactions between the activity levels of the neurons⁵³. MLP utilizes the backpropagation algorithm as part of its feed-forward architecture. This method involves propagating errors backward through the network to fine-tune the weights, enabling the model to generate accurate outputs. Unlike the traditional linear perceptron, the MLP is designed to manage non-linearly separable input data effectively⁵⁴. The weights and bias values are adjusted according to the selected training algorithm to achieve their optimal values during each epoch. Determining the number of hidden layers and neurons in the MLP network is often performed through trial and error. By analyzing the MSE, the ideal network configuration can be identified by modifying the number of neurons and the dimensions of the hidden layers⁵⁵. The mathematical basis of the MLP network is represented by Eq. (14).

$$\:y={f}_{2}\:\left(\sum\:_{j=1}^{N}{w}_{j}{f}_{1}\left(\sum\:_{i=1}^{n}{h}_{ij}{x}_{i}+{b}_{j}\right)+{b}_{0}\right)$$

(14)

The hidden layer’s weight matrix, bias vector, and activation function are represented as h_ij, b_j, and f₁, respectively. Similarly, the output layer’s weight matrix, bias vector, and activation function are denoted as w_j, b₀, and f₂⁵⁶. Fig. 4 illustrates the structure of an MLP with two hidden layers. During the training process, the goal is to minimize the MSE by generating an error signal based on the activation function. This error signal is then propagated backward through the layers, and the weights are adjusted accordingly. Weights that result in a lower MSE are considered more desirable. Table 7 contrasts the three optimizers we evaluated for training the MLP.

Table 7 Training algorithms evaluated for the MLP^57,58.

ANN structure design

The architecture of the ANN developed in this study is illustrated in Fig. 4. The input layer consisted of five independent operating parameters: Q_c, Q_d, A_f, σ, and TBP. The model was designed to predict four critical process responses: horizontal DPH, vertical DPH, horizontal SMDD, and vertical SMDD. Prior to training, the input and output data were normalized to the range [− 1,1]. The dataset was then randomly partitioned into three subsets: 70% for training, 15% for validation, and 15% for testing. The optimal network architecture, determined through systematic evaluation of different hidden layer configurations and training algorithms, consisted of two hidden layers with 10 and 5 neurons, respectively. The first and second hidden layers employed tansig and logsig activation functions to capture nonlinear interactions of varying complexity, while the output layer used a linear (purelin) activation function to preserve the physical scaling of the continuous targets. This compact two-layer topology achieved the best balance between flexibility and variance control, as demonstrated in the performance comparisons presented in Figs. S1–S2 in the supplementary. The ANN was trained using the trainbr. This algorithm minimizes a penalized objective function that simultaneously reduces data error and weight magnitudes, effectively incorporating built-in L₂​ regularization⁵⁸. Consequently, the approach mitigates overfitting without relying exclusively on validation-based stopping rules. To ensure strict separation between optimization and evaluation, the network divider was set to divide train, restricting the optimizer to training data only, while validation and test subsets were retained for post-training diagnostics. The MSE was employed as the loss function. Following training, network outputs were denormalized back to their physical units using the inverse of the normalization transformation. Model performance was assessed on all three data subsets, and both the R² and MSE were reported for each response. Empirical results demonstrated that the two-hidden-layer (10–5) ANN trained with trainbr achieved the highest R² values and the lowest MSE values among all tested configurations (1–4 hidden layers and three training functions). Importantly, this configuration also avoided the instability and tendency toward overfitting observed in deeper networks when trained on the available dataset. These findings, summarized in Fig. S1 (R²) and Fig. S2 (MSE), strongly support the adoption of the two-layer ANN as the final architecture.

Fig. 4

Schematic diagram of the MLP network with two hidden layers.

Results and discussions

RSM result

This section examines various fitting models and their effectiveness in aligning with data. The quadratic model, was chosen as the most accurate model based on its optimal R² value. Additionally, the influence of variable parameters on SMDD and DPH as well as their optimization using RSM, are also discussed.

The results of the ANOVA analysis are presented in Table 8. The F-value evaluates the overall significance of the model, while the P-value represents the probability associated with the ANOVA analysis. Parameters with P-values below 0.05 are considered statistically significant, whereas P-values exceeding 0.1 indicate that the parameters are not statistically significant⁵⁹. The findings of this research reveal that the models for d₃₂ h, d₃₂ v, holdup h, and holdup v demonstrated significant F-values of 52.48, 184.81, 45.28, and 25.8, respectively. The p-values for the model terms (< 0.0001) confirm that these models are both significant and reliable. The significance of each parameter is detailed in Tables 8 and 9.

Table 8 ANOVA results for the two studied responses (d₃₂ h, d₃₂ v).

Table 9 ANOVA results for the two studied responses (holdup h and holdup v).

The models’ Adeq Precision signal-to-noise ratio values are well above 4, signifying that the developed models are suitable for industrial process design. Table 10 provides key fit statistic parameters, including R², predicted R², and adjusted R², for the respective responses.

Table 10 The statistical parameters of the quadratic model are employed to model responses.

The following regression Eq.s (15–18) are fitted models derived from Eq. (8), with coefficients estimated based on experimental data. These models effectively demonstrated the influence and interactions of the input variables on the output responses.

$$\begin{aligned} d_{32}h & = 1.08958 + 0.000502059*A + 0.253033*B + - 0.204276*C + 0.178042*D \\ & + - 0.0767463*E + 0.0143065*AB + 0.000267976*AC + 0.00783051*AD + 0.000339406*AE \\ & + - 0.0362977*BC + 0.0339788*BD + - 0.00575666*BE + - 0.00299706*CD + \\ & 0.0296921*CE + - 0.0481958*DE + 0.0336548*A^{2} + - 0.0192*B^{2} + - 0.0462276*C^{2} + 0*D^{2} + 0*E^{2} \\ \end{aligned}$$

(15)

$$\begin{aligned} d_{32}v & = 1.81688 + 0.016836*A + 0.236107*B + - 0.409959*C + 0.358187*D \\ & + - 0.0660112*E + 0.0369466*AB + - 0.0624406*AC + 0.0177112*AD \\ & + 0.0138001*AE + 0.0716404*BC + 0.0160343*BD + - 0.0542108*BE + - 0.0691172*CD \\ & + - 0.00391862*CE + - 0.0592057*DE + 0.0318239*A^{2} + 0.00882479*B^{2} + 0.0492425*C^{2} + 0*D^{2} + 0*E^{2} \\ \end{aligned}$$

(16)

$$\begin{aligned} Holdup~h & = 0.158205 + 0.00547505*A + 0.0291438*B + - 0.0433697*C + 0.0252527*D \\ & + - 0.0111798*E + 7.82086e - 05*AB + - 0.00299829*AC + 0.000576624*AD + - 0.00059045*AE \\ & + - 0.0181171*BC + 0.00808136*BD + 0.00145129*BE + - 0.00935833*CD + 0.00482977*CE \\ & + 9.28771e - 06*DE + 0.0109592*A^{2} + 0.0164381*B^{2} + - 0.0110492*C^{2} + 0*D^{2} + 0*E^{2} \\ \end{aligned}$$

(17)

$$\begin{aligned} Holdup~v & = 0.0302711 + 0.000324622*A + 0.00649152*B + - 0.00677681*C + 0.013307*D \\ & + 0.0167848*E + 0.000962659*AB + - 0.00100408*AC + 0.000184424*AD + 0.000576391*AE \\ & + - 0.00255443*BC + - 0.00142882*BD + 0.000470878*BE + - 0.000416396*CD + - 0.00172471*CE \\ & + 0.00990748*DE + 0.00165846*A^{2} + 0.000643722*B^{2} + - 0.00174831*C^{2} + 0*D^{2} + 0*E^{2} \\ \end{aligned}$$

(18)

The perturbation graphs illustrate the influence of all parameters on response performance, using the operating midpoint range as the reference point. These graphs compare the effects of all factors on holdup h, holdup v, $\:{\text{d}}_{32}$h and $\:{\text{d}}_{32}$v at the center point, providing a clear visualization of each parameter’s impact. The perturbation scheme for evaluating all operating parameters is presented in Fig. 5(a-d).

Fig. 5

Perturbation graphs for (a) d₃₂ h, (b) d₃₂ v, (c) holdup h, and (d) holdup v.

The Pearson correlation coefficient matrix represents the covariance ratio between each pair of variables and the product of their standard deviations. It provides insights into the linear relationships between the variables. Fig. 6 illustrates the Pearson correlation coefficient matrix.

Fig. 6

The Pearson correlation matrix between any two variables based on the total database.

The optimization process was conducted to evaluate and optimize the experimental parameters for the liquid-liquid extraction process in the LPSPC, as shown in Table 11. In the optimization process, goals were identified as “in range,” “maximize,” or “minimize” based on their role in the performance of the liquid-liquid extraction system and their influence on the desired outcomes. Each goal was defined to ensure the process operated within feasible boundaries while achieving optimal efficiency and effectiveness. For the independent variables, such as the Q_c, Q_d, A_f,, σ, and TBP, the goal was “in range”. This ensures these variables remain within their practical and operational limits during optimization. Specific objectives were determined based on the desired process performance for the response variables. The droplet sizes, d₃₂ h and d₃₂ v, were set to “minimize” because smaller droplets enhance the interfacial area between the dispersed and continuous phases, promoting mass transfer and improving extraction efficiency. Reducing droplet size also ensures better dispersion and uniform flow within the column. Conversely, the holdup values, holdup h and holdup v, were set to “maximize” because higher holdup levels indicate a greater proportion of the dispersed phase in the system, which improves phase interaction and mass transfer. Maximizing holdup enhances the overall efficiency of the extraction process by increasing the residence time of the dispersed phase in contact with the continuous phase. By setting these goals appropriately, the optimization process ensures that the independent variables are controlled within practical ranges while achieving the best possible combination of minimized droplet sizes and maximized holdup values, thereby improving the performance of the LPSPC.

Table 11 Optimize the experimental parameters for LPSPC.

ANN results

A comprehensive hyperparameter study was conducted to identify the optimal ANN for predicting the hydraulic responses. We evaluated 1–4 hidden layers in combination with three training algorithms, such as trainbr, trainlm, and trainscg, and, for each configuration, computed R² and MSE on training, validation, and test sets. Comparative summaries in Figs. S1-S2 show that the two-hidden-layer architecture consistently delivered the strongest performance across all four outputs, yielding the highest R² and lowest MSE⁴¹. The three-layer model exhibited slightly degraded validation scores, indicating the onset of overfitting. In contrast, the four-layer model showed an evident decline in both validation and test sets, confirming reduced generalizability with the available data. Among training algorithms, trainbr outperformed trainlm and trainscg, providing more stable training and superior predictive accuracy; accordingly, it was adopted for the final model. The ANN model demonstrated excellent predictive performance for all four output variables, as summarized in Table 12 and Fig. S6 (a-d) in the supplementary. For the particle size diameter in the d32 h​, the network achieved an R² of 0.9950 and an MSE of 0.000314, indicating strong agreement with experimental data. Similarly, d32 v​ was predicted with R² of 0.9970 and MSE of 0.000451. The holdup h​ achieved R² of 0.9900 and MSE of 0.000019, whereas the holdup v​ reached R² of 0.9822 and MSE of 0.000001. These results indicate that the ANN model captured the nonlinear relationships within the dataset with high accuracy.

Table 12 Training performance metrics for the developed model.

The residual-versus-fitted plots (Fig. S3 in the supplementary) show a random cloud of points around the zero line with no discernible trend, curvature, or heteroscedastic “funneling,” indicating unbiased predictions, no obvious model misspecification, and approximately constant error variance; the experimental–predicted overlays (Fig. S4 in the supplementary) further reveal trajectories that are nearly superposed for all responses, with any deviations small, scattered, and non-systematic, consistent with strong out-of-sample agreement; finally, the error histograms with kernel-density overlays (Fig. S5 in the supplementary) are unimodal and centered near zero with short, broadly symmetric tails, a distributional pattern consistent with homoscedastic, approximately independent residuals and supportive of the adopted error model⁶⁰. The substantial overlap of predicted trajectories with experiment across all subsets indicates accurate fits and good generalization, with only minor deviations at a few extrema (Fig. 7).

Fig. 7

Sample-wise comparison of experimental measurements (black dashed lines) and ANN predictions for (a) d32 h, (b) d32 v​, (c) holdup h​, and (d) holdup v.

3D response surfaces and comparison of RSM and ANN

Effect of model’s variables on SMDD

Fig. 8 illustrates the effect of interfacial tension and pulsation intensity on the SMDD. As observed, increasing pulsation intensity results in smaller droplet sizes in both column sections. This occurs because a higher pulsation intensity enhances the droplet breakage rate, which surpasses the reduction in coalescence rate, ultimately leading to a decrease in the mean droplet size^{4,10] and [61}. Furthermore, systems with higher interfacial tension generate larger droplets than those with lower interfacial tension. Fig. 8 (a) and (b) correspond to ANN methods, and Fig. 8 (c) and (d) correspond to RSM results. Therefore, the SMDD change process results are the same for both RSM and ANN methods. A comparison of the ANN and RSM surface plots indicates that the predictive models exhibit distinct characteristics. The 3D response surface generated by the ANN appears more complex and captures finer details in the relationship between the variables. In contrast, the surface produced by RSM is notably smoother. This difference suggests that the polynomial-based RSM model possesses a lower sensitivity to potential noise in the data, resulting in a more generalized and smoother approximation of the underlying trend.

Fig. 8

3D plots showing the effect of interfacial tension and pulsation intensity on SMDD for horizontal and vertical sections: (a, b) ANN predictions and (c, d) RSM predictions.

Fig. 9 illustrates the influence of the volumetric flow rates of the continuous and dispersed phases on the SMDD. As shown, an increase in Q_d significantly raises the number of dispersed phase droplets, forming larger droplets. In contrast, variations in the continuous phase volumetric flow rate had minimal impact on SMDD. This limited effect is likely due to the increased flow rate of the constant phase only slightly enhancing resistance in the flow path of the dispersed phase. As a result, DPH increased somewhat, resulting in relatively larger droplets. On the other hand, increasing the volumetric flow rates of the continuous phase increased the intensity of the constant phase collision with the dispersed phase droplets, resulting in droplet rupture, so these two factors can neutralize each other’s effects^4,9,10,35.

Fig. 9

Effect of volumetric flow rates of the continuous and dispersed phases on SMDD (ANN, RSM) for (a, c) horizontal and (b, d) vertical sections.

Fig. 10 shows the influence of interfacial tension and percentage of TBP on SMDD. These two parameters have a direct impact on SMDD. As observed in Fig. 10, increasing interfacial tension leads to an increase in the tendency of droplets to coalesce and an increase in droplet size^4,10. In addition, as the percentage of TBP increases, the interfacial tension decreases, which, according to Fig. 6, reduces the interfacial tension and consequently reduces the droplet size. A comparison between the ANN and RSM surface plots reveals apparent differences in their predictive behavior. The ANN-generated 3D surface is more intricate, capturing subtle variations in the relationships among variables. In contrast, the RSM surface is smoother and more generalized, reflecting the polynomial nature of the model, which makes it less sensitive to data fluctuations but less detailed in representing complex.

Fig. 10

Effect of the interfacial tension and percentage of TBP on SMDD (ANN, RSM) for (a, c) horizontal and (b, d) vertical sections.

Effect of model’s variables on DPH

Fig. 11 demonstrates the effect of pulsation intensity and interfacial tension on the DPH. As depicted, increasing the pulsation intensity decreases DPH across both sections of the column. This is because higher pulsation intensity enhances the droplet breakage rate, which outweighs the reduction in coalescence rate, resulting in smaller mean droplet sizes at higher pulsation intensities. In other words, increasing the pulsation intensity and shear forces has caused the droplet size to become smaller. Consequently, the resistance of the plate holes to the movement of the dispersed phase droplets has decreased, which has led to a decrease in the amount of DPH⁹. In addition, as the interfacial tension increased, the size of the droplets increased, and DPH increased. The ANN surface plots capture more detailed and complex variable interactions, whereas the RSM plots appear smoother and more generalized due to the polynomial nature of the model.

Fig. 11

Effect of pulsation intensity and interfacial tension on DPH (ANN, RSM) for (a, c) horizontal and (b, d) vertical sections.

Fig. 12 illustrates the impact of the volumetric flow rates of the continuous and dispersed phases on the DPH. As shown, higher Q_d values lead to a greater number of dispersed phase droplets, along with the formation of larger droplets. Additionally, an increase in Q_c increases the drag forces between the bulk of the continuous phase and the dispersed phase droplets, which in turn leads to a rise in DPH^9,35.

Fig. 12

Effect of the volumetric flow rates of continuous and dispersed phases on DPH (ANN, RSM) for (a, c) horizontal and (b, d) vertical sections.

Fig. 13 represents the influence of interfacial tension and percentage of TBP on DPH. According to Fig. 13, as interfacial tension increases, the size of the droplets becomes larger and DPH increases. In the vertical part of the column, the higher the percentage of TBP, the lower the interfacial tension. Therefore, smaller droplets are formed, and DPH increases. The reason for this is that by creating smaller droplets, the rate of droplet ascent in the column decreases, and DPH increases³⁵. In the horizontal part of the column, DPH is lower for fluids with a higher TBP percentage. In this part of the column, the lower the TBP, the greater the coalescence of the droplets, leading to an increase in their size. Since larger droplets have a higher upward velocity and a lower axial velocity than smaller droplets (higher TBP percentage), the larger droplets in the horizontal part move much faster towards the continuous phase surface and, by mixing with other droplets, lead to the formation of a uniform phase and, ultimately, DPH increases. The ANN surface plots reveal finer and more complex interactions among variables, while the RSM plots appear smoother and less detailed, making such variations less noticeable in the figures.

Fig. 13

Effect of the interfacial tension and percentage of TBP on DPH (ANN, RSM) for (a, c) horizontal and (b, d) vertical sections.

The ANN and RSM models were compared based on Table 13 and the corresponding 3D plots (Figs. 8, 9, 10, 11, 12 and 13). As shown in Table 13, compared to the RSM, the ANN model significantly outperformed classical regression approaches. For instance, RSM achieved R² values of 0.9393, 0.9820, 0.9304, and 0.8839 for d32 h​, d32 v​, holdup h​, and holdup v​, respectively, highlighting the superiority of the ANN in modeling complex hydraulic behavior. Moreover, the scatter plots of predicted versus experimental values demonstrated close alignment across training, validation, and test datasets, confirming that the network generalizes well without overfitting. ANN model provides a reliable and robust predictive tool for hydraulic parameter estimation. The combination of carefully optimized architecture, transfer functions, and training algorithms enabled precise predictions, with consistently high R² and low MSE values across all outputs, making it suitable for both process simulation and optimization applications.

Table 13 Comparison of ANN and RSM Performance.

Predictive correlations for SMDD and DPH

In our previous studies, we have proposed empirical correlations for each of the hydrodynamic parameters^35,61. Based on Buckingham’s π-theorem, an empirical model was developed for SMDD and DPH. This model was presented to predict SMDD and DPH in both sections of the column as a function of the flow rates of both phases and the pulsation intensity, together with the physical properties of the systems, using the dimensional analysis technique as follows (Table 14):

Table 14 Empirical model SMDD and DPH.

The predictive capabilities of the proposed correlations are depicted in Figs. 14 and 15. For SMDD, the correlations yield AARE values of approximately 6.95% and 8.29%, indicating excellent agreement between the predicted values and experimental measurements for LPSPC. In the case of DPH, the correlations produce AARE values around 6.56% and 8.68%, further confirming that the proposed models accurately capture the experimental behavior of the system.

Fig. 14

Comparison of experimental and calculated SMDD values using the empirical model for (a) horizontal and (b) vertical sections.

Fig. 15

Comparison of experimental and calculated DPH values using the empirical model for (a) horizontal and (b) vertical sections.

Table 15 presents a comparison between the experimental results and the SMDD and DPH values calculated using previously reported correlations.

Table 15 Comparison of SMDD and DPH on a PC with the experimental results.

As shown in Table 15, none of the correlations reported for other columns are suitable for our column. This discrepancy is attributed to differences in column geometry and the chemical systems.

Conclusion

This study explored the influence of key hydrodynamic parameters, such as pulsation intensity, interfacial tension, and the volumetric flow rates of continuous and dispersed phases on droplet behavior and DPH in an LPSPC. For the first time, ANN and RSM were combined to model and predict the SMDD and DPH in this column type, focusing on four new chemical systems. The findings demonstrated that pulsation intensity significantly reduces droplet size by increasing the droplet breakage rate relative to coalescence, as confirmed by both ANN and RSM predictions. Systems with higher interfacial tension formed larger droplets due to enhanced coalescence, while an increase in the TBP percentage reduced interfacial tension, resulting in smaller droplet sizes. Additionally, higher dispersed phase flow rates (Q_d) increased both droplet size and DPH. In contrast, changes in the continuous phase flow rate (Q_c) had minimal effect on SMDD but increased DPH due to enhanced drag forces. In addition, new semi-empirical correlations for predicting SMDD and DPH were developed based on experimental data, as the existing correlations from the literature were found unsuitable for the studied systems. These correlations showed good agreement with the measured data, with AAREs of 6.95% and 8.29% for SMDD, and 6.56% and 8.68% for DPH in horizontal and vertical sections, respectively. This provides a reliable tool for design and scale-up purposes. Furthermore, the predictive performance of the ANN and RSM models was excellent. A systematic search over architecture (1–4 hidden layers) and optimizers, including trainlm, trainscg, and trainbr, identified a compact two-hidden-layer MLP trained with trainbr as the best performer. The final ANN achieved training R² values of 0.9950, 0.9970, 0.9900, and 0.9822. RSM showed moderate accuracy, and ANN achieved the highest precision, effectively capturing the complex interactions between LPSPC. The integration of ANN and RSM provided a comprehensive and reliable framework for understanding and optimizing hydrodynamic behavior in LPSPCs. Despite the robust predictive performance of both ANN and RSM models, several limitations must be acknowledged. First, the accuracy and generalizability of the ANN model depend on the quality and volume of the training data, which may limit its extrapolation beyond the studied operating range. Second, the droplet measurements relied on high-resolution digital photography; however, the absence of advanced imaging techniques such as TEM or high-speed video may introduce uncertainty in capturing smaller or fast-moving droplets. Third, while the experimental results were obtained under controlled and repeatable conditions, the influence of long-term operational stability, equipment wear, and process variability on model accuracy has not been evaluated. These limitations underscore the need for further studies that incorporate real-time monitoring, larger datasets, and more advanced imaging tools to enhance the reliability and applicability of the developed models in industrial-scale processes. For future work, the developed ANN and RSM-based modeling frameworks can be extended to other multiphase separation systems, such as inclined sieve-plate columns, rotating disc contactors, or packed columns under pulsed flow. Furthermore, scale-up studies should be conducted to evaluate the models’ predictive capabilities in pilot or industrial-scale units. Integrating the modeling approaches with real-time sensor data and adaptive control systems is also recommended to enhance process optimization, robustness, and energy efficiency in continuous operations.