Introduction

Airlift Contactors (ALC) are a modified form of conventional bubble column with a draft tube placed in the column. Airlift contactors are widely used in chemical and bioprocesses such as low waste conversion of ethylene and chlorine to dichloroethane, biological wastewater treatment, production of single cell proteins1,2,3,4,5. Based on the recirculation airlift contactors are classified as internal-loop airlift Contactors and External loop airlift contactors1,3,5. A typical airlift loop contactor consists of two zones namely, riser and downcomer sections. These sections are either separated by an internal draft tube in an internal airlift contactor or connected to one another through an external conduit1,3. A typical internal loop airlift contactor (ILAC) is shown in Fig. 1. The section where the gas is sparged is called the riser and the section where the liquid flows down is termed as downcomer. The gas is either sparged through the draft tube or through the annulus region to develop liquid circulation in ILAC. Sparging is done with the help of spargers that are usually placed at the bottom of the ILAC column. Circulation of the liquid is induced due to the density difference between the fluids in riser and downcomer sections. The liquid circulation ensures homogeneous flow behaviour and thereby also increases heat and mass transfer rates. ILAC’s are particularly used for processes that need rapid and uniform distribution of reaction components and for multiphase operations requiring high heat and mass transfer rates. For instance, in the wastewater treatment process, the sparging of air results in a pressure gradient in the airlift column. This pressure gradient helps in circulation of wastewater and aids to suspend flocs. ILAC is also used when long contact times between gas-liquid operations are desired or where increased turbulence is required6,7,8. Another interesting application of airlift reactor technology is the use of airlift contactor for growing microalgae3. Recent works5 demonstrated the use of modified airlift contactor or also termed as photobioreactor as a promising reactor for dark fermentation of spent wash using microalgae. Microalgae serves as bioenergy fuel stock and hence is looked upon as a promising candidate to reduce the dependency on fossil fuels. Airlift photobioreactor is found to be effective for microalgae cultivation because of the absence of moving parts, which helps in growth of microalgae and in higher mass transport (oxygen transport rates) in such reactors5. Thus, a deep insight on the hydrodynamics of ILAC’s serves as a prerequisite to ensure these performance aspects in processes involving ILAC. High throughput can be achieved in ILAC owing to their efficient mixing ability1,3,5.

Fig. 1
figure 1

Schematic representation of a typical ILAC.

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The hydrodynamic behavior of equipment is influenced by the system dimensions, fluid properties and the operating conditions of the system. The major hydrodynamic parameters that affect the design, operation and performance of ILACs are: (1) liquid circulation velocity, (2) gas holdup, (3) mass transfer coefficient. Among these parameters, the gas holdup and the mass transfer coefficient are considered as the most important parameters and are of much research interest1,3,5. The mass transfer coefficient is based on the resistance offered by the liquid side to mass transfer in gas–liquid mass transfer applications, the knowledge of which helps in achieving desired performance of ILAC. The gas holdup affects the liquid circulation velocity and the mixing behaviour. Gas holdup is crucial for the design and operation of any multiphase contactor. Holdup determines the interfacial area per unit volume available for phase and energy transport across the phase boundary. Thus, knowledge over gas holdup characteristics enables us to design an efficient ILAC with good transport characteristics.

Gas holdup in an ILAC plays a crucial role in determining its overall performance. Accurate prediction of gas holdup is essential for the design and optimization of airlift contactors. A few correlations are available for the prediction of gas holdup in riser of an ILAC9,10,11. The correlations reported by Korpijarvi et al.9 and Kilonzo et al.11 for riser gas holdup were more generic and predicted the gas holdup considering only the superficial gas velocity in the riser section. Blazej et al.10 developed an empirical correlation for gas holdup using experimental data collected from three different column geometries of ILAC. All these correlations are based on data from air-water experimental system.

Despite the simplicity of ILAC designs, existing empirical correlations for gas holdup prediction often fail under varying geometric and operational conditions. Empirical models for ILAC have used very few parameters in correlations limiting the generalized applicability. motivating the need for robust Machine Learning (ML) models. ML models employ a data driven approach and can handle multiple operating parameters and capture nonlinear correlations among the operating variables. ML models have been successfully employed in various multiphase reactor modelling problems. Recent ML-based studies in multiphase reactors include employing artificial neural networks in prediction of water holdup in two phase flow, prediction of slug liquid holdup and Prediction of solid holdup in a gas-solid circulating fluidized bed riser12,13,14. Zu et al.15 provide a comprehensive review of ML for hydrodynamics, transport, and reactions in multiphase flows and reactors. Another recent review16 deals extensively with multiscale Modelling and artificial Intelligence for multiphase flow. In our present study, we utilize Random Forest (RF) Regression, a robust black-box model, to predict gas holdup.

Methods

Data set is curated from the data collected from seven different sources9,10,17,18,19,20. Experimental data on riser gas holdup in an ILAC are collected from reported previous studies. We have used air-water and air- Carboxymethyl Cellulose (CMC) systems with different dilutions. Supplementary Table S1 of the supporting information gives the summary of data collected on gas holdup in the riser section of ILAC from different experimental studies on ILAC. Also, distribution of variables is described in Supplementary Note. Outlier detection methods did not show any outliers in the curated dataset. The data includes details such as column geometry, range of superficial gas velocity and the number of data points in the reported work. GetData Graph Digitizer software was used to extract the experimental gas holdup data from seven sources as mentioned. A total of 324 data points are consolidated. The collected data on gas-holdup in the riser section of ILAC is made available at https://doi.org/10.7910/DVN/YHJV2H. The collected data on the airlift column contains ten features which are related to geometry, fluid properties and operating conditions. The ten attributes are (1) Superficial gas velocity, (2) Draft tube diameter, (3) Downcomer diameter, (4) Draft tube height, (5) Density of gas, (6) Viscosity of gas, (7) Downcomer height, (8) Viscosity of liquid, (9) Density of liquid and (10) Sparger pore size. The riser holdup is the output parameter which we intend to predict using ML methods. Data points were chosen such that (1) there are a minimum of 25 data points in a given source (2) Sources together represent experiments covering a wide range of governing input parameters. The features together affect the gas-liquid mass transfer efficiency, flow circulation patterns and mixing characteristics. The importance of these features and the interaction between them depend upon the specific combination of these for a given instance in the dataset. Data-driven approaches with explainability can bring out these characteristics.

Explainable data driven prediction of gas holdup

Algorithm employed

In this work, we employ the RF algorithm as the black box model for predicting gas holdup in internal draft airlift loop contactors. To optimize its performance, a Genetic Algorithm (GA) is employed to tune the hyperparameters of the RF model. Furthermore, to enhance its interpretability SHapley Additive exPlanations (SHAP) methodology is employed. In our study, we leverage different Python libraries to implement various components of our research methodology. Specifically, we utilize the following libraries:

  • Scikit-learn (sklearn): We employ the sklearn library for implementing the RF model. Scikit-learn21 is a popular library for ML and consists of a plethora of tools and algorithms for various tasks, including classification, regression, and clustering.

  • Pymoo: We utilize the Pymoo library for implementing GA for hyperparameter tuning. Pymoo22 is a versatile optimization library that offers a collection of multi-objective optimization algorithms and tools.

  • SHAP: We make use of the SHAP Python package to incorporate SHAP values and generate explanations for our black-box model. SHAP23 is a powerful interpretability technique that allows us to attribute feature contributions to individual predictions.

Random forest regressor

Bagging, essentially consisting of bootstrap aggregation, is a widely utilized ensemble classification algorithm. RF basically consists of an ensemble24 of decision tree models which are unpruned and developed for classification problems. RF, a very useful and performance enhancing improvement over simple bootstrap aggregation, has many desirable properties. For regression problems the final output is obtained by mean aggregation of outputs of individual decision trees. RF inherently induces two sets of randomness for performance enhancement: (A) For each tree instances are sampled employing bootstrap sampling with replacement. (B) Each node in every tree considers distinct random subset of features for splitting of nodes while keeping a previously defined subset size constant across nodes in every tree. Sampling via bootstrapping utilizes two thirds of the data instances for each tree. The examples not utilized are the out of bag examples (OOB). OOB examples arise because for each tree in the forest, random sampling of the training data is selected with replacement. The sample size is typically equal to the original dataset size (though some observations will be duplicated while others are left out). On an average, each bootstrap sample contains about 63.2% of unique observations from the original dataset. The remaining 36.8% of observations not selected become the “Out-of-Bag” (OOB) sample, which can be used for validation. While these OOB samples could theoretically be used for model validation, in our study we opted for a 5-fold cross-validation approach on the training data, which provides a more robust assessment of model performance by evaluating it across multiple train-validation splits.

Here are some important hyperparameters (notation used per sklearn).

  • Number of trees (n_estimators): This parameter represents the quantity of decision trees to be included in the RF. Increasing n_estimators can improve performance but can increase algorithmic complexity.

  • Maximum number of features (max_features): This parameter dictates the maximum number of attributes randomly selected for node splitting.

  • Maximum depth of trees (max_depth): This parameter is responsible for setting the maximum depth of each tree. Trees with high depth value can reveal intricate relationships. Reducing the depth of trees can lead to better generalization and reduction in overfitting.

  • Minimum samples for splitting (min_sample_split): This parameter sets the number of minimum samples that need to be employed for node splitting.

  • Minimum samples in leaf (min_sample_leaf): Least number of instances required to be at a leaf node are governed by this parameter.

RF models are often considered “black box” models because its internal workings can be less interpretable and very complex compared to other simpler models. RF consists of multiple decision trees and combination of these trees makes the overall model more accurate, but at the same time increases the complexity and opacity. RF consists of multiple hierarchical structures of nodes and splits. Understanding how RF combines these trees for decision making becomes more challenging. Especially with an increase in the number of trees resulting in complexity increase, because RF does not provide any functional relationships to reveal the algorithm behavior.

RF was selected as the modelling approach for two key reasons: First, gas holdup exhibits non-linear relationships with input parameters, making linear models suboptimal. Second, RF’s ensemble nature provides robustness against overfitting. We have also included results with linear regression model (which could be considered as one of the baselines). As the number of attributes and data points are not very large, deep learning approach was not considered.

Optimization of model hyper-parameters using GA

GA starts by creating an initial set of potential solutions. Every trial solution is a unique combination of five hyperparameters for our model. The fitness function typically corresponds to a performance metric such as mean square error or R2 score. R2 score indicates the variance proportion in the designated target variable (gas holdup) which can be explained by the input features of RF. For higher R2 score values, the regression model will be better fit for the input data. The R2 score equation is given as,

$${\text{R}}^{2} = 1 - \left( {{\text{SSR}}/{\text{SST}}} \right)$$

where SSR is the sum of the squared differences between the model predictions and the actual values for gas holdup. And SST represents the total sum of squared differences between the actual values and the mean of the actual values of gas holdup. In this work we have used the R2 score as the fitness function. Each solution’s fitness is determined by training the model using the specific set of hyperparameters on a training dataset and assessing its performance on a separate validation dataset. Each solution is a real valued vector. Each element of the vector corresponds to a specific hyper-parameter. As all the hyper-parameters take only integer values, the real values are rounded to the nearest integer before fitness evaluation.

Choice of the genetic operations

  • Selection: Several selection operators are available in GA literature, we chose tournament selection25 as it is a common selection operator employed to select solutions from the set for reproduction based on their fitness values. It is a probabilistic selection method that mimics the concept of a tournament, where individuals compete against each other to determine their chances of being selected.

  • Crossover: Simulated Binary Crossover (SBX) is widely employed genetic operators in real-coded GA for problems where the solutions are represented as real-valued vectors2627. SBX is a crossover operator designed to mimic the crossover process in binary-coded GA but adapted for real-valued representations. It has a parameter called distribution index (η) which controls the variance of the crossover simulations. It influences the probability of generating offspring close to the parents or exploring further away in the problem landscape. Increase in η tends to generate offsprings in the neighbourhood of the parents, while a lower value promotes exploration of distant regions.

  • Mutation: Polynomial Mutation28 is a popular mutation operator used in real-coded GA to introduce random changes to the gene values within a solution. Polynomial mutation operator modifies values of genes based on a polynomial distribution. A parameter known as the distribution index (µ) controls the polynomial distribution shape which is used to perturb the gene values. It affects the magnitude of the mutation. Larger values of µ generate a larger perturbation, while a lower value produces smaller perturbations.

Both mutation and crossover operations are applied elementwise, individually to each element (gene) of the parents to produce offspring. A random number is uniformly generated between the range of 0 and 1 for every element. If this number is below a specified threshold, that element of the offspring is generated by performing genetic operations on the corresponding elements of the parents. For crossover, if the ith element is selected, then the ith element of the offspring is generated via crossover operation on the corresponding elements of the parents, else if it is not selected, the ith element of any of the parent is directly used as the ith element of the offspring. Similarly in the case of mutation, if the ith element is selected, the mutation operator perturbs the ith element of the single parent, else its ith element is directly used as the ith element of the offspring. The thresholds are known as probability of mutation (pm) and probability of crossover (pc). For crossover, it is usually set to a high value and for mutation, it is set as 1/ (total number of elements). We set the values of parameters of these operations to the values which have been commonly used to solve various problems27. The parameter values are set as follows: for SBX crossover, the probability of crossover (pc) is 0.9 and the distribution index (η) is 2. And for mutation, the probability of mutation (pm) is 1/5 and the distribution index (µ) is 50.

Model explanation with SHAP

SHAP based on cooperative game theory utilizes the principle of Shapley values, which has its roots in cooperative game theory. The Shapley values provide an unbiased method for distribution of rewards distribution between participating entities. To determine a feature’s (participant) contribution, the computation involves analysing the model’s output by considering various combinations of features. Shapley values essentially quantify the impact of including or excluding a particular feature on the overall model performance. At first glance, it may seem unclear how this could be applicable, as a feature cannot be absent from the input when feeding into a ML model. However, let assume for now that the model has the capability to accept subsets of features as input.

Let \(\:x\) be a test instance in \(\:{R}^{n}\) for which we seek an explanation of the model prediction \(\:f\left(x\right)\). Given that the model can compute the outcome f(xS) when only a subset\(\:S\subset\:F\) features are present. The feature importance assigned to attribute xi for explanation of the prediction f(x) is denoted by \(\:{\varphi\:}_{i}\). These important values uniquely satisfy specific desirable properties as represented by the equation below.

$$\phi _{i} = \mathop \sum \limits_{{S \subset F\backslash \left\{ i \right\}}} \frac{{\left| S \right|!\left( {\left| F \right| - \left| S \right| - 1} \right)!}}{{\left| F \right|!}}f\left( {x_{{S \cup \left\{ i \right\}}} } \right) - f\left( {x_{S} } \right)$$

The calculation for SHAP value for attribute, i can be given as follows:

  1. 1)

    For every i consider all possible combinations (coalitions) of subsets (S) of other attributes which do not include i.

  2. 2)

    For every S, calculate the change in model prediction when i is added:

$$f\left( {S \cup \left\{ i \right\}} \right) - f\left( S \right)$$
  1. 3)

    Find the weight of each marginal contribution using the probability of encountering subset S. The weight formula accounting for number of ways S can be formed to ensure fairness can be calculated as:

$$\frac{{\left| S \right|!\left( {\left| f \right| - \left| S \right| - 1} \right)!}}{{\left| f \right|!}}$$
  1. 4)

    Sum over the weighted marginal contributions across all possible S to get the SHAP value for i.

  2. 5)

    Final prediction for data instance is sum of SHAP values for all attributes plus baseline prediction:

$$f\left( x \right) = \phi _{0} + \mathop \sum \limits_{{i = 1}}^{n} \phi _{i}$$

where \(\:{\varphi\:}_{0}\) denotes the baseline prediction and \(\:{\varphi\:}_{i}\) denotes the attribute contribution of the ith attribute. This is also known popularly as SHAP value.

SHAP offers several fundamental advantages over other explainability models like LIME29, particularly in addressing the critical issue of local linearity assumptions and hyperparameter tuning. While LIME relies on the assumption that a model behaves linearly within a defined neighborhood, this assumption frequently proves problematic for complex models where nonlinear interactions are common. The challenge is compounded by the need to choose an appropriate “locality radius” - a hyperparameter that significantly impacts LIME’s explanations but lacks a principled method for selection. If this radius is too small, LIME might miss important feature interactions; if too large, it violates its own core assumption of local linearity. The handling of nonlinear relationships further distinguishes SHAP from LIME. While LIME attempts to approximate nonlinear behavior through local linear models, SHAP naturally captures nonlinear effects and feature interactions through its game-theoretic framework.

Explanation scope

Local explanations focus on providing explanations for model predictions for any given instance in the dataset. SHAP produces additive attribute contributions that indicate influence of each attribute on any given data instance. This helps us in understanding how each attribute contributes to the predictions. The explanations at global level aim to explain the overall behaviour of the black box model. SHAP takes a set of instances and prepares a set of local explanations for each instance. These consist of contribution values of each attribute. To gain insights into the global behaviours, an analysis is performed on the contribution of each attribute over all data instances. This investigation helps understand how different attributes individually and as a group contribute to the prediction ability of the model. Such analysis also provides insights about the global contribution of every attribute in the dataset.

Results

The dataset is divided into an 80:20 train-test split, consisting of 259 instances for training and 65 instances reserved for testing. The model’s hyperparameters are tuned using the training dataset. After determining optimal hyperparameters, the model was retrained on the entire training set, followed by an evaluation on the reserved test dataset.

Hyper-parameter tuning with genetic algorithm

The detailed description of the GA configuration used in our experiment is implemented using the Pymoo package. The GA was initialized with a population size of 40 individuals using integer random sampling. For genetic operators, we employed Simulated Binary Crossover (SBX) with a crossover probability of 0.9 and distribution index η = 2, along with Polynomial Mutation (PM) having a mutation probability of 1/n_feat (where n_feat represents the number of features) and distribution index η = 50. Both genetic operators utilized float variable types with rounding repair to maintain integer solutions. Additionally, duplicate elimination was enabled to maintain diversity in the population. These parameter settings were carefully chosen to balance exploration and exploitation in the search space while ensuring reproducibility of the experimental results.

Hyperparameters of the RF model was optimized using GA. The fitness function used was the mean cross-validated R2 score calculated over 5 folds on the training dataset. The acceptable ranges for each hyperparameter are provided in Table 1, including default values from the sklearn implementation. The optimal values obtained through GA are also listed in Table 1. In the ML literature it is customary to compare the tuned hyperparameters with default values of hyperparameters, for this reason we chose default as a baseline. RF hyperparameters significantly affect the algorithm performance. For example, Higher values of max_depth can provide deeper trees and comprehend finer details, while lower values may lead to underfitting; similarly controlling the values of max_features provides search diversity. For best performance with minimum overfitting, the cross-validation metric is normally used to tune the RF hyperparameters.

Table 1 Hyper-parameter tuned using GA.
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The hyperparameter bounds were selected through comparison with scikit-learn’s defaults, focusing on parameters most likely to impact model performance given our small dataset of 324 points. While scikit-learn defaults to 100 estimators, we explored a range of 50–250 trees to assess the impact of ensemble size. The max_depth parameter, which defaults to None (unlimited) in scikit-learn, was constrained between 2 and 10 to prevent overfitting while maintaining sufficient complexity. For max_features, we explored a range from 4 to all 10 features, compared to scikit-learn’s regression default of using all features, to investigate if random feature subset selection during splitting could improve performance through increased tree diversity. The min_sample_leaf and min_sample_split parameters retained scikit-learn’s default values (1 and 2 respectively), validating these defaults for our specific case.

The algorithm was run with a population size of 40 with 100 maximum generations (default Pymoo package) chosen to balance between solution quality and computational efficiency. This fixed iteration limit provided sufficient generations for the optimization process while keeping the demonstration of GA’s effectiveness computationally practical while the convergence is determined if no further significant change of the R² score is observed over five successive generations.

In Table 2, a comparison is presented between the RF model utilizing optimal hyperparameters obtained from GA and the default parameter set in terms of performance prediction. Models are evaluated using R2 score and Mean Absolute Error (MAE). It is evident from the table that there is a noticeable improvement in performance when utilizing the optimal hyperparameters. Before conducting RF simulations, we established a base line using linear regression simulations which gave us a test R2 score of 0.7194 and MAE of 0.0144. The default RF demonstrated a significant increase in R² score by 20.74% (0.9268 vs. 0.7194) over linear regression, while reducing the Mean Absolute Error by 45.83% (0.0078 vs. 0.0144). These improvements were further enhanced with the tuned RF, achieving most optimal value of R² of 0.9542 (23.48% increase from baseline) and MAE of 0.0060 (58.33% reduction from baseline). Beyond the numerical improvements, RF offers crucial advantages including the ability to capture non-linear relationships, automatic modeling of feature interactions, robustness to outliers, and no requirement for feature scaling. The combination of these quantitative improvements and theoretical benefits strongly validates the choice of RF, with the subsequent performance gains through hyperparameter tuning further confirming its suitability for this prediction task.

Table 2 Model evaluation before and after hyper-parameter tuning.
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The evaluation of the tuned model on the test set indicated an impressive R2 score of 0.95, and MAE of 0.006, demonstrating its capability in accurately predicting Gas Holdup in Internal Draft Airlift Loop Contactors. Figure 2 displays parity plots to visualize the model’s prediction accuracy. SHAP framework then is employed to get insight into local and global model behaviour.

Fig. 2
figure 2

Parity plots for RF model with default set of hyper-parameters. (a–d) Parity plots for RF model with tuned hyper parameters set.

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Statistical significance of improved R-squared metric

We conducted a comparative analysis of two RF Regression models - a baseline model with default parameters and a tuned model with optimized hyperparameters (n_estimators = 85, max_depth = 12, max_features = 4, min_samples_split = 2, min_samples_leaf = 1). The analysis utilized the entire dataset. A robust 10-fold cross-validation methodology was implemented to evaluate model performance, with consistent random states ensuring reproducibility.

The evaluation revealed statistically significant improvements in the tuned model’s performance compared to the baseline, as evidenced by a p-value of 0.0281 from a paired t-test. The performance was measured using R-squared scores across all folds, calculating both mean performance and standard deviation, along with absolute and relative improvements. The paired t-test specifically accounted for fold-to-fold variations, and the significant p-value (p < 0.05) strongly suggests that the tuned model’s superior performance was not due to random chance, validating the effectiveness of the hyperparameter optimization.

SHAP explanations

SHAP has the ability of providing local and global level explanations in order to comprehend how a ML model operates. Local explanations focus on interpreting individual instance predictions generated by the ML model. Additive feature contributions (SHAP values) for each prediction can be generated by SHAP, indicating the influence of each attribute on that specific instance prediction. Additivity implies the contributions along with the mean prediction of the model over the training data adds up to the predicted values. To explain the model behaviour on a particular data instance, the SHAP values are calculated and represented using a waterfall plot. It depicts the contribution of every attribute to the prediction output. Understanding the local level explanations, one can gain insights regarding why the model made a specific prediction for a particular input. The waterfall plot consists of horizontal bars representing different attributes, the bars are vertically stacked, and each bar length represents the impact of that attribute on that instance prediction. The waterfall plots in Fig. 3 illustrate how different features influence gas holdup predictions across four different test instances. Each plot starts from a base value of E[f(x)] = 0.052 (mean of gas holdup of all instances in the dataset) and shows how individual features push the prediction higher or lower. For instance, in Fig. 3a, draft tube height and downcomer height contribute positive SHAP values, indicating these geometric parameters at their given values promote gas holdup higher than the base value. Conversely, in Fig. 3b, superficial gas velocity shows a substantial negative contribution (-0.02), suggesting that its current value (0.063) leads to gas holdup lesser than the base value.

Fig. 3
figure 3

Waterfall plots for RF model predictions on four instances. In subfigures (a), (b) and (c) exhibit model predictions that are higher than the base prediction value. In subfigure (d), model prediction is lower than base prediction value. Notably, the Superficial gas velocity feature demonstrates a substantial contribution across all instances. It is important to observe that as the attribute value decreases, its positive contribution towards the model output also decreases and even become negative. We could also observe a similar trend in the contribution of the feature draft tube diameter.

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Positive SHAP values indicate that a feature’s value for that instance pushes the prediction towards higher gas holdup than the baseline. For example, in Fig. 3a, the large positive SHAP value (+ 0.15) for superficial gas velocity indicates that this feature’s value strongly suggests higher gas holdup in the contactor than average. This aligns with physical expectations, as higher gas velocities typically increase gas holdup. Conversely, negative SHAP values indicate that a feature’s value pushes the prediction towards lower gas holdup than the baseline. This can be seen in instances where geometric parameters like draft tube diameter or downcomer height contribute negative SHAP values, suggesting their values for those cases promote gas holdup lesser than the baseline value in the system. The magnitude of these SHAP values quantifies the strength of each feature’s influence - larger absolute values indicate stronger effects on gas holdup prediction, while values closer to zero suggest minimal impact. This interpretation helps connect the mathematical model’s predictions to the physical behavior of the airlift contactor system.

The plots also reveal consistent patterns across instances. Superficial gas velocity consistently shows strong influence. Geometric parameters like draft tube diameter and downcomer diameter show moderate impacts, while fluid properties (viscosity of gas, density of liquid) generally have smaller contributions. Notably, when superficial gas velocity has higher values (0.268 and 0.183 in bottom plots), it contributes strong positive SHAP values (+ 0.03 and + 0.01 respectively), aligning with the physical expectation that higher gas velocities generally increase holdup. This interpretation helps bridge the gap between the model’s mathematical predictions and the underlying physical behavior of the airlift contactor.

Based on Fig. 3, a significant impact of superficial gas velocity was observed on the model’s predictions and the positive impact of this feature as well as draft tube diameter on the predictions decreased as the feature value decreased. To investigate if this trend holds true more generally, we created a scatter plot using SHAP matrix. This scatter plot visualizes the relationship between the SHAP values and the corresponding feature values taken by the instances (over the entire dataset). Figure 4 shows the scatter plot for features superficial gas velocity and draft tube diameter.

Fig. 4
figure 4

SHAP Scatter plot for the features. (A) Superficial gas velocity and (B) Draft tube diameter. It indicates an increasing trend in the SHAP value with relation to the feature value. It can also be seen that for lower value instances, this influence is negative.

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SHAP values inherently capture feature interactions through their game theoretic foundation, where each feature’s contribution is calculated by considering all possible coalitions (combinations) with other features. To visualize these interactions, particularly between superficial gas velocity and draft tube diameter, we used scatter plots that show SHAP values against feature values, with coloring based on the interacting feature. Figure 5 visualization reveals that the impact of superficial gas velocity on gas holdup predictions (shown by SHAP values) is modulated by draft tube diameter. Specifically, for a given superficial gas velocity, its positive contribution to gas holdup tends to be slightly stronger when the draft tube diameter is smaller.

Fig. 5
figure 5

Measuring the impact of superficial gas velocity in the presence of Draft tube diameter.

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Global level explanations reveal which attributes are important to the black-box model, highlighting their significance. Furthermore, these explanations can provide ideas regarding the latent patterns and trends of attributes across different data instances. This study requires a large dataset that represents the data distribution. To perform study on the global extent, initially a SHAP coefficient matrix was formulated. This is achieved by performing SHAP calls on all the data instances in the entire dataset and storing the corresponding coefficients of attributes in a matrix data structure which has the same number of columns as attributes and rows as data instances. To get the magnitude of attribute importance across all the instances, we computed the mean of the absolute SHAP values of each attribute. This is achieved by considering the column-wise mean of the absolute values from the SHAP coefficient matrix. Bar plot containing the global feature importance is depicted in Fig. 6. It can be seen from this bar plot that superficial velocity is the highest ranked feature. The other top ranked features include draft tube diameter, downcomer diameter and draft tube height. It can also be seen that the mean SHAP value for superficial velocity (denoted by length of the bar) is much higher than the other features. The SHAP ranking aligns quite well with the physical expectations of gas-liquid-systems. The top ranked attribute, superficial velocity directly impacts, bubble dynamics, available gas liquid interfacial area and extent of mixing. Draft tube diameter and downcomer diameter also play a major role in determining gas liquid hydrodynamics and bubble swarm characteristics.

Fig. 6
figure 6

Global feature importance by taking mean of absolute SHAP values across all the examples in the dataset.

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SHAP bar plots can facilitate identification of the best feature subset, utilizing information from the bar plots, we can adopt an iterative approach for feature selection. We begin by training our model using only the most important feature and in each subsequent iteration, we gradually incorporate additional features in descending order of importance. Further to evaluate the performance of the model with each feature subset, we utilized the test dataset. A graphical representation of the results of this evaluation are depicted in Fig. 7. By considering the global feature importance, we can establish a ranking for the features, as evident from the SHAP bar plot. The exact numerical values of the model performance are provided in Table 3. By reducing the number of features, we can simplify the model without significantly sacrificing its performance. The stepwise feature selection shows that 8 features can retain high predictive accuracy. Drop in performance when fewer features are used indicate that although the dropped features have lesser importance, they still possess some information and correlate (although to a lesser extent) with the gas holdup. The minimal impact of sparger pore size may be due to its indirect influence compared to superficial gas velocity.

Fig. 7
figure 7

Impact on the model performance by adding the attributes with the decreasing global importance one by one.

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Table 3 RF regression model evaluation on the test dataset with attribute selection.
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Discussion and conclusion

The RF model tuned with a GA achieved a significant enhancement in predictive accuracy on the test set. Parity plots further validated the model’s reliability. To promote model transparency, SHAP values provided both local and global explanations. At a local level, the waterfall plot utilizes SHAP values to visually explain the prediction of the black-box model for a specific instance. This plot effectively illustrates the contributions of various features towards the model’s prediction, providing valuable insights. At the global level, computing the mean absolute SHAP values enabled a quantitative ranking of feature importance. Guided by these insights, an iterative feature selection process identified an optimal subset of the top 8 most influential features. This reduction in input dimension from 10 to 8, captures most informative features without compromising performance. The R2 score on the test set was quite high even after dropping the bottom 2 features. We also found the feature ‘Superficial gas velocity’ having a significant impact on gas holdup prediction of the RF model. The SHAP scatter plot provides a more comprehensive understanding of how individual features contribute when they vary throughout the entire dataset. These visual representations significantly enhance our understanding of the model’s behavior and the significance of different features.

Physical and phenomenological models along with empirical correlations have long been used for designing separation equipment. Our data driven based RF approach can handle many attributes and build a comprehensive model capable of handling nonlinearity effectively. There are no ML based models for prediction of gas holdup in ILACs. Our model can complement conventional approaches and can be used as a first step in obtaining optimal configurations and geometries of ILACS and multiphase flow equipment. The tuned RF model can be used for designing industrially important ILACS for wastewater treatment and microalgae cultivation. This study demonstrates the efficacy of RF regression optimized by GA for predicting gas holdup in ILACs, achieving an R² of 0.9542. The integration of SHAP-based interpretability provides valuable insights into feature contributions, paving the way for more transparent and efficient reactor design processes. Future work could focus on expanding the dataset to include industrial-scale ILACs with complex geometries and varying fluid properties to generalize the model’s applicability. The current work can also lead to directions towards developing hybrid combinations of phenomenological and data driven modelling approaches.