Abstract
Flue gas desulfurization (FGD) is a critical process for reducing sulfur dioxide (SO_{2}) emissions from industrial sources, particularly power plants. This research uses calcium silicate absorbent in combination with machine learning (ML) to predict SO_{2} concentration within an FGD process. The collected dataset encompasses four input parameters, specifically relative humidity, absorbent weight, temperature, and time, and incorporates one output parameter, which pertains to the concentration of SO_{2}. Six ML models were developed to estimate the output parameters. Statistical metrics such as the coefficient of determination (R^{2}) and mean squared error (MSE) were employed to identify the most suitable model and assess its fitting effectiveness. The random forest (RF) model emerged as the topperforming model, boasting an R^{2} of 0.9902 and an MSE of 0.0008. The model's predictions aligned closely with experimental results, confirming its high accuracy. The most suitable hyperparameter values for RF model were found to be 74 for n_estimators, 41 for max_depth, false for bootstrap, sqrt for max_features, 1 for min_samples_leaf, absolute_error for criterion, and 3 for min_samples_split. Threedimensional surface plots were generated to explore the impact of input variables on SO_{2} concentration. Global sensitivity analysis (GSA) revealed absorbent weight and time significantly influence SO_{2} concentration. The integration of ML into FGD modeling offers a novel approach to optimizing the efficiency and effectiveness of this environmentally crucial process.
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Introduction
Sulfur dioxide (SO_{2}) is a prominent atmospheric contaminant that plays a substantial role in the degradation of air quality. This pollutant notably influences the natural environment and the global climate system^{1,2}. Industrial processes, especially those involving fossil fuel combustion, are recognized as significant sources of SO_{2} emissions^{3}. Power plants and industries contribute to over 70% of the total anthropogenic SO_{2} emissions, making them the primary contributors to this environmental concern^{4}. Various technologies have been developed to mitigate SO_{2} emissions, such as fuel switching^{5}, catalytic converters^{6}, coal preparation^{7}, lowsulfur fuels^{8}, boiler modernization^{9}, fluidized bed combustion^{10}, and flue gas desulfurization^{11}. When choosing a method for removing or lowering the emission of SO_{2} from flue gases, it is necessary to consider a range of criteria. The ideal approach should encompass safety, environmental sustainability, and costeffectiveness while minimizing potential losses and eliminating the issue of fouling^{12}.
Flue gas desulfurization (FGD) is one of the most effective emission control technologies used in power plants, and it plays a pivotal role in reducing SO_{2} emissions^{13}. Several FGD systems have been developed, and the selection process involves considering technical factors and making an economic decision. Notable concerns encompass the extent of desulfurization achievable by the technique and its adaptability. The majority of FGD systems employ an alkali sorbent, such as limestone (calcium carbonate), quicklime (calcium oxide), hydrated lime (calcium hydroxide), or occasionally sodium and magnesium carbonate and ammonia, to trap the acidic sulfur compounds present in the flue gas. Regardless of the circumstances, the alkalis chemically interact with SO_{2} in the presence of water (such as a mist of slurry containing the sorbent) to generate a combination of sulfite and sulfate salts. This reaction might occur either inside the entire solution or on the moistened surface of the solid alkali particles^{14}. FGD technologies are frequently categorized into wet, semidry, or dry processes^{15}.
The ADVACATE process was created as an alternative way to clean the flue gas in coalfired power plants by duct injection. It offers a smaller physical size and lower initial cost than wet desulfurization systems, making it a practical option for upgrading existing plants to meet stricter flue gas cleaning standards^{16}. The ADVACATE process involves the introduction of ADVACATE solids into the coolside duct to mitigate the presence of SO_{2}, NO_{x}, and several other pollutants within the flue gas. The removal process occurs in the gas duct, and the bag filter particle control device exhibits greater significance. Solid ADVACATE materials are formed through the chemical reaction between hydrated lime and recycled fly ash derived from power plants. The chemical as mentioned above process results in the formation of a calcium silicate hydrate solid with a significant degree of porosity, enabling it to retain a considerable quantity of water (~ 50 wt.%) while maintaining the handling characteristics of a powder, as shown by Eqs. 1–3. A substantial quantity of water and alkalinity facilitates the elimination of acid gases and the efficient conversion of solids^{17}.
Figure 1 depicts the stages of preparation. Depending on the size of the starting silica particles, the first step is grinding. The silica undergoes a hightemperature reaction with lime and other additions in an aqueous medium. After the sludge has been dewatered and dried, it can be sent to the source sites. The gas–solid contact can be achieved using a ductinjection/baghouse filter configuration. The gas can also be utilized as a filter medium in a fixed bed medium.
The FGD process has shown promising potential for efficient SO_{2} removal. Dzhonova et al.^{18} studied the WellmanLord method for removing SO_{2} from flue gases in combustion systems. The method uses sodium sulfite to absorb SO_{2} and produce sodium bisulfite. The regenerated solution can be reused in the absorber. The authors found the method more costeffective than other FGD methods and suggested techniques to enhance it. They introduced a new technology with lower steam consumption, heat utilization for heating district heating water, and lower capital costs. The study by Özyuğuran and Meriçboyu^{19} compared the desulfurization efficiencies of hydrated lime and dolomite absorbents from flue gases. They subjected them to sulfation at 338K and measured their weight increase during the SO_{2} reaction. The researchers found that the total sulfation capacities increased with increased surface areas and decreased mean pore radius, indicating that the physical properties of absorbents significantly influence their sulfation properties. A study developed by Xu et al.^{20} integrated the FGDCABR system to remove NO_{x} and SO_{2} from flue gas, achieving 100% removal efficiency. The primary sulfur compound was sulfide, with the spray scrubber partially facilitating NO_{x} removal through sulfideoxidizing and nitratereducing bacteria enrichment. Most NO_{x} was converted into harmless N_{2} in the expanded granular sludge bed reactor. StaniendaPilecki^{21} explored the use of limestone sorbents with increased magnesium content in FGD processes in power stations. Triassic limestones in Poland, consisting of low magnesium calcite, high magnesium calcite, dolomite, and huntite, have various magnesium contents. The increased magnesium content in the sorbent positively impacted the dry method of desulfurization, especially when using fluidized bed reactors. Because magnesium ions are unstable, they made it easier to remove carbon from carbonate phases at temperatures similar to those used to remove carbon from dolomite. This results in a faster and more effective desulfurization process.
Over the past few years, numerous methods have been proposed to predict SO_{2} and other emissions from power plants. Among these approaches, mathematical models, and machine learning (ML) models have generated significant scientific interest. However, accurately modeling the concentration of SO_{2} is a challenging task mathematically. Some studies simplify this system by incorporating assumptions, leading to errors in predictions. Furthermore, the calculations utilized in these mathematical models require substantial computing resources^{22}. ML approaches are extensively considered due to their accuracy, fast speed, and capability to do nonlinear calculations, diagnosis, and learning. Additionally, recent advancements in predictive modeling techniques, such as adaptive sampling based surrogate modeling, have gained popularity ^{23}. So far, extensive studies have been carried out in the field of FGD by ML approach. Zhu et al. ^{24} developed a highly effective ML approach for estimating SO_{2} absorption capacity in deep eutectic solvents (DESs). Based on critical parameters like molecular weight, water content, pressure, and temperature, the model was the most accurate in forecasting 480 DESSO_{2} phase equilibria, ensuring its dependability and generalizability. Grimaccia et al.’s ^{25} study aimed to create a model for a proprietary SO_{2} removal technology at the Eni oil and gas treatment plant in southern Italy. The goal was to develop an ML algorithm for unit description, independent of the licensor and more flexible. The model used ANNs to predict three targets: SO_{2} flow rate to the Claus unit, SO_{2} emissions, and steam flow rate to the regenerator reboiler. The datadriven technique accurately predicted targets, allowing optimal control strategies and plant productivity maximization. Xie et al.^{26} introduced a long shortterm memory (LSTM) neural network to improve the WFGD process in thermal power plants. The model achieved a high prediction accuracy of 97.7%, surpassing other models. The modified LSTM model was rigorously tested and validated, demonstrating good prediction effect and high stability. Yu et al.^{27} developed a dynamic model to predict SO_{2}NO_{x} emission concentration in fluidized bed units, aiming to meet emission standards and create an environmentally friendly pollutant removal mode. The model used Pearson coefficients, an extreme learning machine, and a quantum genetic algorithm to optimize connection weights, accurately imitating actual data trends. Yin et al.^{28} developed a hybrid deep learning model integrating a convolutional neural network (CNN) and LSTM to improve the accuracy of predicting SO_{2} emissions and removal in limestonegypsum WFGD systems. The model captures local and global dynamics and temporal characteristics and introduces an attention mechanism (AM) to allocate weights to the outlet SO_{2} sequence at different time points. The model outperforms alternative methodologies in predictive accuracy. Makomere et al.'s^{29} research examined the effectiveness of ANN in modeling desulfurization reactions using Bayesian regularization and Levenberg–Marquardt training algorithms. The shrinking core model was used, revealing the chemical reaction as the ratecontrolling step. Bayesian regularization was preferred due to its flexibility and overfitting minimization capabilities. The hyperbolic tangent activation function showed the best forecasting ability. An investigation by Uddin et al.^{30} on the limestoneforced oxidation (LSFO) FGD system in a supercritical coalfired power plant. Monte Carlo experiments showed that optimal operation could reduce SO_{2} emissions by 35% at initial concentrations of 1500 mg/m^{3} and 24% at initial 1800 mg/m^{3} concentrations. These findings were crucial for reducing emissions in coal power plants and developing effective operational strategies for the LSFO FGD system. Fedorchenko et al.^{22} presented an optimization strategy for FGD using data mining. A modified genetic method based on ANNs was developed, allowing for better prediction of time series characteristics and efficiency. The method used adaptive mutation, allowing less important genes to mutate more likely than high suitability genes. Comparing this method with other methods, the new method showed the smallest predictive error and reduced prediction time, thereby increasing efficiency and reducing SO_{2} emissions. Adams et al.^{31} developed a deep neural network (DNN) and least squares support vector machine (LSSVM) to predict SO_{x} and NO_{x} emissions from coal conversion in energy production. The models were trained on commercial plant data and examined the impact of dynamic coal and limestone properties on prediction accuracy. The results show that training without assumptions improved testing accuracy by 10% and 40%, respectively. Interactive and pairwise correlation features reduced computational time by 46.67% for NO_{x} emission prediction. A summary of the studies conducted in the field of ML for FGD and their results are given in Table 1.
Considering the prevailing research landscape focused on traditional modeling approaches in the realm of FGD, this study strategically addresses critical research gaps. Specifically, our work pioneers the application of ML techniques to model and predict the performance of calcium silicate absorbents within the context of a sand bed reactor. Additionally, using ML in sand bed reactors in FGD is a new idea that goes against traditional ways of doing things and shows how advanced modeling techniques can be used to get the best results in this reactor. This study, therefore, endeavors to fill existing research gaps and advance the state of knowledge in the field. The study used data from experiments on FGD with a calcium silicate absorbent in a sand bed reactor as both input and output for the ML method. This research aims to utilize ML models to estimate the concentration of SO_{2} accurately and quickly in flue gas. For implementing the proposed models, 323 experimental data points collected from this work were considered. A statistical evaluation and comparison of the accuracy of the constructed ML models was conducted based on the coefficient of determination (R^{2}) and mean squared error (MSE), and the best model was chosen. The results of this study can be used in power plants, environmental regulations, engineering and design, research, and development in the future.
Theoretical background
Setup description
The reaction between SO_{2} and solid absorbents was studied in Arthur's sand bed reactor system^{17} and shown in Fig. 2. Compressed SO_{2}/N_{2} (~ 0.5%) was diluted with either nitrogen or air, depending on the desired oxygen content, to create a simplified flue gas. The flow rates of all gases were controlled using mass flow meters and a controller box. Water was supplied to a helical Pyrex evaporator through an Infusion Pump, which humidified the flue gas. The temperature in the furnace was regulated using a voltage controller. The flow rate of water from the syringe pump was measured by monitoring the weight of the water output over time. The sand bed reactor used in the experiment was made of glass and had dimensions of 7.5 inches in length and 1.5 inches in diameter. A 2mm coarse glass frit was placed at the bottom of the reactor to support the mixture of sand and absorbent. The reactor was sealed using a ground glass fitting secured with a metal clamp and rubber bands. It was positioned upright in a water bath, which was temperaturecontrolled using a dedicated controller. The concentration of SO_{2} was measured using an SO_{2} analyzer, and the output from the analyzer was automatically collected using a digitizer and PC for data analysis. A bypass line was incorporated within the temperaturecontrolled water bath to establish a stable operational state for the synthesized flue gas and the analytical system before the onset of the chemical reaction. The flue gas, characterized by concentrations spanning from 0 to 2000 parts per million (ppm), underwent substantial dilution with ambient air from the facility to attain concentrations within the 0 to 50 ppm range, a requisite for the analyzer. This dilution process concurrently addressed issues related to gas condensation within the analytical system by reducing the relative humidity of the gas. The predominant portion of the effluent gas stream was directed through a sodium hydroxide (NaOH) scrubbing system, which typically operated under a pH level of 13. A small vacuum pump integrated into the SO_{2} analyzer extracted a small portion of the gas.
Data collection
Since the concentration of SO_{2} can be affected by different operating conditions, there is a need to investigate the relationship between the outlet concentration and the parameters affecting the outlet concentration. Relative humidity, absorbent weight, temperature, and time play an essential role in the concentration of SO_{2}. Therefore, relative humidity, absorbent weight, temperature, and time were included among the input variables. The SO_{2} concentration was also considered as output. Hence, this study incorporates the input variables of maximum level (max), minimum level (min), average level (mean), and standard deviation (STD), as presented in Table 2. The training and testing data for the models were acquired from Arthur^{17}, yielding a dataset comprising 323 data points. The Pearson correlation coefficient matrix is the covariance of the two mentioned features and the product of their standard deviation. The correlation among the selected variables is analyzed and presented in the heatmap in Fig. 3.
Model selection
In this study, all ML analyses were conducted using the Python programming language. Various ML methods and models are available to solve clustering, classification, and regression problems. However, the challenge lies in determining which model and combination of hyperparameters would work best for a specific dataset. The optimization algorithm in this case, involves multiple learning algorithms (models) and hyperparameters. It is necessary to explore numerous combinations to maximize predictive accuracy and find the optimal set of hyperparameters. In this study, six models are used: artificial neural network (ANN), multilayer perceptron (MLP), radial basis function neural network (RBFNN), random forest (RF), extra trees regression (ETR), and support vector regression (SVR). The procedure to reach the best ML model is shown in Fig. 4.
Artificial neural network
An ANN is a computational model inspired by the workings of the human brain. It comprises many individual units, like artificial neurons, which are connected by coefficients known as weights. These weights together form the network structure and enable it to process information. Each of these processing units, often called processing elements (PE), has inputs with different weights, a transfer function, and produces a single output. Think of PE as an equation that balances its inputs and outputs. ANNs are often called connectionist models because the connection weights effectively serve as the network's memory^{32}. While a single neuron can handle simple informationprocessing tasks, the true power of neural computation comes to light when these neurons are interconnected within a network. Whether ANNs possess accurate intelligence remains a topic of debate. Notably, ANNs typically consist of only a few hundred to a few thousand PEs, whereas the human brain contains about 100 billion neurons. So, artificial networks with the complexity of the human brain are still far beyond our current computational capabilities. The human brain is much more intricate, and many intellectual functions remain unknown. However, ANNs excel at processing large amounts of data and can make surprisingly accurate predictions. Nonetheless, they do not possess the kind of intelligence that humans do. Therefore, it might be more appropriate to refer to them as examples of computer intelligence. In the field of neural networks, various types of networks have been developed over time, and new ones continue to emerge regularly. However, they can all be categorized based on the functions of their neurons, the rules they use to learn, and the formulas governing their connections^{33}.
Multilayer perceptron
The perceptron algorithm, initially proposed by Rosenblatt in the late 1950s, has gained significant recognition as a prevalent and regularly utilized model in supervised ML^{34}. Compared to more intricate models, the MLP offers higher model quality, simplicity of implementation, and shorter training duration^{35}. In the MLP network, the input layer receives information and transmits it to the output layer, reflecting the final findings. Meanwhile, the hidden layers within the network do the initial processing of the received data. The hidden layers of the neural network receive the weights and biases and subsequently propagate the values to the output layer through the utilization of activation functions^{36}. Figure 5 illustrates the primary architecture of the MLP. The Eq. (4) comes from the MLP feature approach. In this equation, the output vector is denoted as g, the weight vector of factors is given by w, x_{i}^{k} indicates the reference vector, and θ denotes the threshold limit^{37}.
The output of the MLP neural network can be derived in the following manner:
where γ_{jk} stands for the influence exerted by neuron j in layer k, while β_{jk} signifies the bias weight associated with neuron j within layer k. The term F_{k} denotes the nonlinear activation transfer function about layer k, and w_{ij} represents the connection weights.
Radial basis function neural network
The RBFNN possess a robust mathematical basis deeply based on regularization theory, which is employed to address illconditioned problems^{38}. The RBFNN model's versatility stems from its outstanding efficiency, simplicity, and speed, making it suitable for various applications^{39}. An RBFNN is structured with three distinct layers: the input, hidden, and output layers. Each layer is assigned distinct tasks^{40}. The transfer function within RBFNN exhibits nonlinearity when mapping inputs to hidden layers, but it demonstrates linearity when mapping hidden layers to output layers^{41}. Equation (6) displays the Gaussian transfer function used by the RBFNN for processing inputs^{42}.
where the input variable is denoted as x, the center point is represented by c_{i}, the bias is symbolized as b, and the spread of the Gaussian function is indicated by σ_{i}. Figure 6 illustrates an essential schematic representation of the RBFNN.
Random forest
The RF algorithm is widely recognized in the field of ML for its ability to construct predictive models, and it was initially proposed by Breiman^{43} in 2001. This supervised learning technique is a composite model consisting of several tree predictors. Each tree predictor is constructed based on the values of an independent random vector, and all vectors are created with the same configuration. This method is applicable for solving classification and regression issues^{44,45}. The functioning of the RF model is depicted in Fig. 7. Each regression tree’s output was added together to get the result shown in Eq. (7) below^{46}:
where T_{i}(x), x, and K represent an individual regression tree that is constructed using a subset of input variables and bootstrapped samples, a vector input variable, and the number of trees, respectively.
RF can assess the significance of input features, improving model's performance when dealing with datasets with many dimensions. The process entails quantifying the average reduction in predictive accuracy resulting from altering a single input variable while holding all other variables constant. This process entails assigning a score that represents the relative relevance of each variable, which then aids in selecting the most impactful features for the ultimate model^{47}.
Extra trees regression
Geurts et al.^{48} proposed the ETR method, a developed method derived from the RF model. This approach is a recent advancement in ML, an enlargement of the wellknown RF algorithm. It was made to prevent overfitting. Training each base estimator with a random subset of features is fundamental to the ETR algorithm's success, just as in the RF^{47}. ETR uses the whole training dataset to train each regression tree. On the other hand, RF uses a bootstrap replica to train the model^{49}.
Support vector machine
Previously, supervised learning approaches, specifically SVM, were mainly utilized for classification purposes. However, contemporary research has also demonstrated successful adaptations of these techniques for regression problems^{50}. Furthermore, kernel functions are employed in SVM to transform the training data, thereby mapping it to a space with higher dimensions where the data can be effectively segregated^{51}. SVM models were built using consistent input descriptors and training/testing datasets. Equation (8) within the SVM model is the prediction or approximation function^{52}.
SVM helps minimize systemic risk, diminishing overfitting, lowering prediction errors, and enhancing generalization. SVM does not rely on a predefined structure since it assesses the significance of training samples to determine their contributions. "Support vectors" are only established for models based on specific data samples^{53}. In this research, SVM regression was conducted using the support vector regression (SVR) class available in the scikitlearn API's SVM module. As illustrated in Fig. 8, a model is crafted, and the data is transformed into a chosen dimension.
Error metric
The models are evaluated based on several metrics, including mean absolute error (MAE), mean squared error (MSE), root mean squared error (RMSE), and regression coefficient (R^{2}), to choose the optimal model. The MAE is calculated as the average of the absolute values of the errors. The metric is defined as the arithmetic mean of the absolute differences between the actual values and the corresponding predicted values. The term "MAE" is commonly used to denote a loss function. The primary objective in utilizing this loss function is to minimize it. The definition of MAE is as follows^{54}:
where Y_{predicted} indicates the predicted value, and Y_{actual} represents the actual value of the model.
The MSE denotes the average value of the squared error, as illustrated in Eq. (10). MSE is seen as a loss function that requires minimization. One of the primary rationales for the extensive utilization of MSE in practical ML applications stems from its inherent characteristic of assigning more penalties to more significant errors compared to MAE when employed as the objective function^{54}.
The RMSE is mathematically defined as the square root of the MSE, as demonstrated in the equation below. The RMSE is widely utilized as a loss function due to its interpretative capacity^{54}.
The coefficient of determination (R^{2}) is a way to measure how well the model fits the scientifically reliable results. The better the estimates are based on the experimental data, the closer the R^{2} is to 1. The calculation for R^{2} is as follows^{55}:
where Y_{mean} refers to the average value.
Results and discussion
In this study, Kaggle's CPU session was employed, offering an environment equipped with 4 CPUs. The specifications of these CPUs include an Intel(R) Xeon(R) CPU @ 2.20 GHz with a total of 4 CPU cores, supporting both 32bit and 64bit operations. Dedicating 1 CPU to each trial facilitated the concurrent execution of 4 processes, streamlining the exploration of hyperparameter space for each model. The duration of hyperparameter tuning for individual models spanned from 2 to 3 h, reflecting variations influenced by the intricacies of different models and the extent of the hyperparameter search space. During the hyperparameter tuning and model training phases, approximately 34 GB of RAM was employed. This allocation proved sufficient to manage the computational load throughout these processes.
Hyperparameters optimization
In the ML domain, the crucial role of hyperparameter optimization in developing efficient and precise models is undeniable. The main objective is to finetune each model, ensuring optimal performance across diverse datasets. A cohesive strategy for hyperparameter tuning was adopted, utilizing Ray Tune and various schedulers. The primary focus was to strike a balance between a model's complexity and its predictive accuracy, achieved through meticulous exploration and validation processes. This approach aimed to prevent overfitting and maintain the model's generalization ability. In the tuning process, practices like Kfold cross validation, early stopping, and L2 regularization played a pivotal role, especially for models such as ANN, MLP, and RBFNN. These practices effectively validated the model's performance and mitigated overfitting risks. Ray Tune's ASHAScheduler dynamically adjusted hyperparameters during training across various models, including ANN, RBFNN, RF, ETR, and SVR. The HyperBandScheduler was particularly effective for the MLP model, accelerating the tuning process and ensuring swift convergence to the best hyperparameter configuration. It is worth noting that other methodologies such as multiobjective optimization in neural architecture search (NAS) with algorithms like NSGAII and the utilization of surrogate models for SVR are recognized as valuable tools that complement and enhance optimization strategies^{56,57,58}.
ANN
After considering various factors such as the number of layers, neurons per layer, batch_size, learning_rate, weight_decay, activation_function, optimizer, and epochs, a thorough analysis was conducted to determine the best configuration for the ANN network architecture. The main goal of this analysis was to achieve the most favorable results on the test data. The optimal hyperparameters for the ANN network can be summarized as follows: units_layer1 = 128, units_layer2 = 128, units_layer3 = 32, batch_size = 16, learning_rate = 0.0005, weight_decay = 0.00002, activation_function = Relu, optimizer = Adam, and epochs = 216.
MLP
The optimal configuration of the MLP network architecture was determined by considering several factors, including the number of layers, the number of neurons for each layer, dropout, weight_decay, learning_rate, batch_size, test_size, activation_function, optimizer, and the number of epochs. This comprehensive analysis aimed to produce the most favorable outcomes on the test data. The ideal hyperparameters for the MLP network are summed up as follows: units_input = 256, units_hidden = 32, num_layers = 5, dropout = 0.0491, weight_decay = 0.00008, learning_rate = 0.0003, batch_size = 32, test_size = 0.2, activation_function = Relu, optimizer = Adam, and epochs = 100.
RBFNN
The training of the RBFNN involves optimizing many network characteristics, including the number of epochs, hidden_features, weight_decay, learning_rate, activation_function, and optimizer to attain optimal performance on the test data. The optimized hyperparameters include the following values: the number of epochs = 1500, the hidden_features = 50, the weight_decay = 0.00000001, learning_rate = 0.1, activation_function = Relu, and optimizer = Adam. Figure 9 illustrates the learning curve according to the most influential architecture of the MLP, ANN, and RBFNN.
RF
To enhance the performance of the RF algorithm, it is necessary to select appropriate hyperparameters carefully. The hyperparameters typically considered for optimization include n_estimators, max_depth, bootstrap, max_features, min_samples_leaf, criterion, and min_samples_split. For the specific case at hand, the ideal values for these hyperparameters are determined to be 74, 41, false, sqrt, 1, absolute_error, and 3 respectively, for n_estimators, max_depth, bootstrap, max_features, min_samples_leaf, criterion, and min_samples_split.
ETR
To optimize ETR, these hyperparameters are assessed: (n_estimators, max_depth, min_samples_leaf, bootstrap, max_features, min_samples_leaf, criterion, and min_samples_split), the ideal values are n_estimators = 70, max_depth = 12, bootstrap = false, max_features = log 2, min_samples_leaf = 1, criterion = poisson, and min_samples_split = 5.
SVR
The hyperparameters typically considered during the optimization of SVR include kernel, C, degree, gamma, coef0, epsilon, shrinking, and tol. In this case, the ideal values for these hyperparameters are as follows: kernel = RBF, C = 99.5403, degree = 3, gamma = scale, coef0 = 0.8938, epsilon = 0.0589, shrinking = true, and tol = 0.0014.
Comparison predictions
The models were retrained using the specified hyperparameters on training (70%), validation (20%), and testing (10%) datasets for each case. Following guidelines like those described in^{59}, we constructed the testing dataset to ensure uniform coverage across the entire operational domain. This was achieved by systematically sampling points across the full range of each variable, including relative humidity, absorbent weight, temperature, time, and SO_{2} concentration. The graph in Fig. 10 compares the estimated SO_{2} concentration with the experimental values of the test groups. The performance of the models was evaluated using analytical criteria, namely the MAE, MSE, RMSE, and R^{2}, as indicated in the previous equations. The outcomes are presented in Table 3. The high R^{2} value of 0.9902 and low MSE value of 0.0008 indicate that the RF model is suitable for estimating SO_{2} absorption by calcium silicate based on operational and absorption conditions. The model’s performance over the uniformly sampled testing dataset, which encapsulates the entire domain of FGD conditions, yielded a consistent accuracy, demonstrating its robustness and reliability in various operational scenarios. This precise ML model can predict the SO_{2} concentration under different operational conditions for new absorbents. The ML models developed in this study can reduce the time and cost associated with experimental screening tests for various absorbents used in different scenarios, thereby promoting costeffective and environmentally friendly generation for sustainability. Figure 10 demonstrates a high level of accuracy in the relationship between the RF model outputs and the SO_{2} concentration data. The RF model achieves the most accurate results, accurately estimating the experimental data.
A random selection of five test data points was made from the set of considered data to assess the validity of the acquired models. The data shown in Table 4 provides information on the experimental concentration of SO_{2}. The calculated value is determined based on the specific operating conditions for each model. Furthermore, the RF model had the highest level of accuracy in predicting SO_{2} concentration across most cases, surpassing all other models. Figure 11 shows a radar chart to compare the R^{2} value of the models. Based on the data given, it can be concluded that the RF algorithm has superior performance in predicting experimental data about SO_{2} concentration. The training algorithm of the network aims to minimize the average error. Therefore, the RF model was employed to generate threedimensional graphs that illustrate the correlation between input parameters or operational circumstances and the concentration of SO_{2}. Figure 12 depicts the threedimensional curves of the RF forecasting model. The collection of data on the curves was conducted to enhance comprehension of the impact of relative humidity, absorbent weight, temperature, and time on the concentration of SO_{2}. The values of the constant parameters are determined by averaging the remaining inputs. A generalized optimal RF model to provide SO_{2} concentration performance for analyzing the influence of (a) relative humidity and absorbent weight; (b) relative humidity and temperature; (c) relative humidity and time; (d) absorbent weight and temperature; (e) absorbent weight and time; and (f) temperature and time, while other parameters are kept constant at 44% relative humidity, 0.0625 g absorbent weight, 43.25°C temperature, and 30.5 min time. Depending on the data presented in Fig. 12, maintaining the process at a higher relative humidity leads to a decrease in SO_{2} concentration. While humidity typically promotes the dissolution of SO_{2}, it can also influence its concentration in the gas phase. High relative humidity can lead to increased water content in the flue gas, which, in turn, enhances SO_{2} absorption and decreases its concentration in the gas phase^{60,61}. With the increase in the weight of the absorbent and with the increase of time, the concentration of SO_{2} increases significantly. This depends on various factors. Initially, increasing absorbent weight enhances SO_{2} absorption by providing more surface area for interaction. However, when saturation is reached, excess absorbent can hinder absorption, potentially leading to increased SO_{2} concentration. also, SO_{2} absorption can reach a chemical equilibrium. Adding absorbent weight might shift this equilibrium towards desorption, resulting in higher SO_{2} concentrations, especially when excess absorbent prevents an absorptionfavorable equilibrium. On the other hand, the rate of SO_{2} absorption depends on surface area and chemical reaction kinetics. Increased absorbent weight can alter reaction kinetics, potentially slowing absorption and causing higher SO_{2} concentrations. Over time, absorbed SO_{2} can desorb back into the gas phase, increasing SO_{2} concentration, particularly with prolonged exposure^{62,63,64}. The optimal range of absorbent weight to keep the SO_{2} concentration low is 0.025–0.06 g. As the desulfurization process begins, SO_{2} concentration increases. After the initial rise, around the 5min mark, SO_{2} concentration reaches a minimum. This phase represents efficient SO_{2} removal from the gas phase as the absorbent starts absorbing SO_{2}. Following the minimum concentration, SO_{2} concentration starts to rise again. This is due to factors like absorbent saturation or changes in the equilibrium between gas and absorbent. Towards the end of the time interval, SO_{2} concentration stabilizes and reaches an equilibrium. This equilibrium reflects a balance between continued SO_{2} release and absorption by the absorbent^{65,66,67}. The performance of SO_{2} concentration was insensitive to temperature changes.
Global sensitivity analysis (GSA)
To identify the primary factors influencing the SO_{2} concentration, we conduct global sensitivity analysis (GSA) utilizing the ML models we developed. In this process, we apply the sensitivity equations provided in reference^{68}. The GSA outcomes, specifically the firstorder and totalorder indices, are presented in Fig. 13 for the ANN, MLP, RBFNN, RF, ETR, and SVR models, respectively. The firstorder index gauges the impact of individual environmental parameters on the output in isolation. Conversely, total order indices measure the influence of an environmental parameter, considering its interactions with other environmental factors^{69}. Due to the computational complexity associated with determining higherorder indices individually, the calculation of totalorder indices is commonly carried out. In all GSA simulations, we utilized 256 samples to assess the impact of each input parameter on the output. As depicted in Fig. 13, the output of all six models is most significantly influenced by the quantities of absorbent weight and time. Specifically, in the RBFNN and ETR models, time and absorbent weight respectively exhibit the foremost impact on the SO_{2} concentration. Conversely, in the RF, SVR, MLP, and ANN models, the absorbent weight and time respectively exert the greatest influence on the SO_{2} concentration. It is noteworthy that the impact of relative humidity and temperature on the SO_{2} concentration in all six models is deemed insignificant.
Conclusion
This research studied calcium silicate absorbent to establish an ML prediction for SO_{2} concentration in an FGD process. The experimental data, which included 323 data sets, was defined with four inputs: relative humidity, absorbent weight, temperature, and time, and one output, including SO_{2} concentration. Six models were created to estimate the output parameters, including ANN, MLP, RBFNN, RF, ETR, and SVR. For the models mentioned earlier, statistical values such as the R^{2} and MSE were determined to determine the optimal model and evaluate the fitting effectiveness. The highest performance was provided by the RF model that demonstrated the best estimation with R^{2} of 0.9902 and MSE of 0.0008, and the optimal hyperparameter values were established as follows: n_estimators = 74, max_depth = 41, bootstrap = false, max_features = sqrt, min_samples_leaf = 2, criterion = absolute_error, and min_samples_split = 3. The predicted SO_{2} concentration closely matched the experimental results, demonstrating the accuracy of the modeling. Threedimensional surface plots were reported to investigate the effect of relative humidity, absorbent weight, temperature, and time on SO_{2} concentration. The findings revealed that absorbent weight and time were the most influential factors in SO_{2} concentration among the four parameters investigated. The results of this investigation indicate that ML methods can significantly improve the prediction of SO_{2} concentration within the range of the experiment. Continued research and development in this field and advances in ML techniques hold great potential for achieving cleaner air quality, reduced environmental impact, and more efficient energy production through enhanced FGD processes. We hope this study contributes to the ongoing efforts to address environmental challenges and promote cleaner, more sustainable industrial practices.
Data availability
The datasets used and analyzed during the current study are available from the corresponding author upon reasonable request.
Abbreviations
 \({\alpha }_{i}:\) :

Weight for feature vector
 b:

Bias (–)
 c_{i} :

Center points (–)
 f:

Function
 F_{k} :

Nonlinear activation transfer functions
 g:

Output vector (–)
 G:

Gaussian function
 i:

Subscripts refer to the initial condition
 i:

Number of neurons in the hidden layer
 k:

Position vector
 K:

Kernel function
 K:

Number of trees
 n:

Number of neurons
 N:

Number of datasets for training (–)
 R^{2} :

Coefficient of determination
 t:

Time (min)
 T_{i} :

The result from each tree
 w:

Weight factor (–)
 W:

Absorbent weight
 W_{ij} :

Weight related to each hidden neuron (–)
 x:

Input variable (–)
 x_{i} :

The ith feature vector (–)
 x_{i} ^{k} :

Reference vector (–)
 \({\beta }_{jk}:\) :

Bias weight for neuron j in layer k
 \({\gamma }_{jk}:\) :

Neuron j’s output from k’s layer
 \(\uptheta :\) :

Threshold limit (–)
 ξ:

Slack variable
 σ:

Width of Radial Basis Function Neural Network (RBFNN) kernel (–)
 σ_{i} :

Spread of Gaussian function (–)
 ANN:

Artificial neural network
 CFF:

Cascaded forward neural network
 CNN:

Convolutional neural networks
 DFGD:

Dry flue gas desulfurization
 DNN:

Deep neural network
 ELM:

Extreme machine learning
 ETR:

Extra trees regression
 FGD:

Flue gas desulfurization
 GSA:

Global sensitivity analysis
 LSTM:

Long shortterm memory
 LSSVM:

Least squared support vector machine
 MAE:

Mean absolute error
 MAPE:

Mean absolute percentage error
 ML:

Machine learning
 MLP:

Multilayered perceptron
 MSE:

Mean squared error
 NAS:

Neural architecture search
 RBFNN:

Radial basis function neural network
 RF:

Random forest
 RMSE:

Root mean square error
 RNN:

Recurrent neural network
 STD:

Standard deviation
 SVR:

Support vector regression
 WFGD:

Wet flue gas desulfurization
 Activation function:

The activation function is a mathematical function between the input feeding the current neuron and its output going to the next layer
 Bias:

Bias is a constant that helps the model in a way that can fit best for the given data
 Epoch:

In the training process, the inputs enter each training step and give outputs compared with the target to calculate an error. With this process, weights and biases are calculated and modified in each epoch
 Neurons:

Neurons are the basic units of the large neural network
 Weight:

Represents the importance and strengths of the feature/input to the Neurons
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K.N.: Conceptualization, Methodology, Software, Conceived and designed the experiments, Validation, Formal analysis, Investigation, Resources, Data curation, Writing  original draft, Writing  review & editing. M.S.K.Y.: Software, Conceived and designed the experiments, Validation, Formal analysis, Investigation, Resources, Data curation, Writing  original draft, Writing  review & editing. H.J.: Software, Writing review & editing, Validation, Formal analysis, Data curation. F.B.: Software, Writing  review & editing, Validation, Formal analysis, Data curation. A.G.: Corresponding author: aghaemi@iust.ac.ir Conceptualization, Methodology, Software, Conceived and designed the experiments, Supervision, Funding acquisition, Software, Validation, Formal analysis, Investigation, Resources, Visualization, Project administration, Supervision visualization, Writing  original draft, Writing  review & editing. M.R.M.: Supervision, Funding acquisition, Software, Validation, Formal analysis, Investigation, Resources, Visualization.
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Naderi, K., Kalami Yazdi, M., Jafarabadi, H. et al. Modeling based on machine learning to investigate flue gas desulfurization performance by calcium silicate absorbent in a sand bed reactor. Sci Rep 14, 954 (2024). https://doi.org/10.1038/s41598024515867
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DOI: https://doi.org/10.1038/s41598024515867