A modeling approach for estimating hydrogen sulfide solubility in fifteen different imidazole-based ionic liquids

Abstract

Absorption has always been an attractive process for removing hydrogen sulfide (H₂S). Posing unique properties and promising removal capacity, ionic liquids (ILs) are potential media for H₂S capture. Engineering design of such absorption process needs accurate measurements or reliable estimation of the H₂S solubility in ILs. Since experimental measurements are time-consuming and expensive, this study utilizes machine learning methods to monitor H₂S solubility in fifteen various ILs accurately. Six robust machine learning methods, including adaptive neuro-fuzzy inference system, least-squares support vector machine (LS-SVM), radial basis function, cascade, multilayer perceptron, and generalized regression neural networks, are implemented/compared. A vast experimental databank comprising 792 datasets was utilized. Temperature, pressure, acentric factor, critical pressure, and critical temperature of investigated ILs are the affecting parameters of our models. Sensitivity and statistical error analysis were utilized to assess the performance and accuracy of the proposed models. The calculated solubility data and the derived models were validated using seven statistical criteria. The obtained results showed that the LS-SVM accurately predicts H₂S solubility in ILs and possesses R², RMSE, MSE, RRSE, RAE, MAE, and AARD of 0.99798, 0.01079, 0.00012, 6.35%, 4.35%, 0.0060, and 4.03, respectively. It was found that the H₂S solubility adversely relates to the temperature and directly depends on the pressure. Furthermore, the combination of OMIM⁺ and Tf₂N^-, i.e., [OMIM][Tf₂N] ionic liquid, is the best choice for H₂S capture among the investigated absorbents. The H₂S solubility in this ionic liquid can reach more than 0.8 in terms of mole fraction.

Introduction

In the recent century, the need for fossil fuels has risen due to the high levels of energy required for the rapid industrialization of the world¹. The extraction of oil and gas from underground fields and their combustion for generating heat/energy² has undesired environmental effect³ and is accompanied by the production of large amounts of undesired pollutants^4–6, mainly carbon monoxide (CO)⁷, carbon dioxide (CO₂)^8,9, sulfur dioxide (SO₂)¹⁰, and hydrogen sulfide (H₂S)^11,12. The most widely used method for removing these gases is absorption¹³. The absorption processes can help humans meet environmental standards and attenuate the global warming issue¹⁴. Nowadays, the absorption process using alkanolamine- based solvent is one of the most developed and industrially interesting approaches¹⁵. However, the loss of mono-ethanolamine, diethanolamine, N-methyl-diethanolamine, and di-isopropanol amine creates environmental problems, and they have also produced some highly corrosive byproducts^16,17. As an alternative and promising approach, scientists have investigated ionic liquids (ILs)^18–20. Ionic liquids are comprised of cations and anions and have an asymmetric organic cation structure, which results in being liquid at room temperature²¹. Ionic liquids possess outstanding thermal stability and a superior ability to solve organic and non-organic problems^18,19. These features are highly attributed to their cation and anion particles. Cations and anions of ionic liquids can be easily modified to make them suitable for many specific applications¹⁹. Furthermore, having an insignificant vapor pressure, ILs have been considered promising candidates for sweetening processes with the minimum environmental effect and solvent loss²². A central factor that must be appraised in gas sweetening processes is the solubility of gases in liquids under various dominated operational conditions^23,24. Although many references in the literature have calculated or experimentally obtained the CO₂ solubility in ILs^{16–18,25–27}, authentic data representing the H₂S solubility in ILs is scarce. Therefore, developing robust predictive models is crucial for precisely and expeditiously estimation of H₂S solubility data in various operational conditions. In recent years, numerous investigations have been performed to evaluate gas solubility in different ILs. Shariati and Peters²⁸ implemented the Peng–Robinson (PR) equation of state to obtain the solubility of CHF₃ in [C₂mim][PF₆] under various pressures and temperatures. Kroon et al.²⁹ estimated the solubility of CO₂ in different ILs at high pressures less than 100 MPa. Wang et al.³⁰ used the square-well chain fluid equation of state (EoS) to assess several gases’ solubility in ILs. Researchers have used numerous EoSs and methods to evaluate both H₂S and CO₂ solubility in ILs^25,31,32. Nevertheless, none of the above-mentioned approaches could be generalized to different systems. As a result, a range of more general approaches must be applied to forecast gas solubility in ILs. Recently, many intelligent methods, such as artificial neural networks (ANNs)³³ have been applied for predicting various properties in chemical engineering, including crstallinity^34,35, thermal conductivity^36,37, viscosity³⁸, heat capacity³⁹, and solubility of different gases in solutions^40,41.

Predication of CO₂ solubility in ILs is not an exception, and many soft computing methods have been used for this purpose^42,43. In contrast, we found that limited investigations have been done regarding the estimation of H₂S solubility in ILs using artificial intelligence (AI) models, and further research activities are needed. In the present work, we implement different intelligence models, including multilayer perceptron neural network (MLPNN), adaptive neuro-fuzzy inference system (ANFIS), least-squares support vector machine (LS-SVM), radial basis function neural network (RBFNN), cascade feedforward neural network (CFFNN), and generalized regression neural network (GRNN) for accurate estimation of the H₂S solubility in ILs. For this aim, fifteen ILs under different pressure and temperature conditions are investigated. In addition, the preciseness and reliability of the best method have been compared with the UNIFAC EoS. Several graphical and statistical techniques are used to evaluate the performance of the developed models, and the relevancy factor investigates the effect of each input parameter. Moreover, the trend analysis is carried out to assess the capability of the proposed models in detecting the physical trend between the H₂S solubility and different temperatures and pressures. At last, the Leverage approach is made to check the validity of the data and feasible region of the best-proposed model.

Theoretical background

Multilayer perceptron neural network

A feedforward MLPNN has three layers of input, interiors, and output^44–46. MLPNN benefits from a unique training approach known as the backpropagation, and the utilized activation functions in this method are non-linear⁴⁷. The three most common types of activation functions are specified as follows^37,48:

$$\text{Linear:}f\left(x\right)=x$$ 1

$$\text{Logarithm sigmoid:}f\left(x\right)=\frac{1}{1+e^{-x}}$$ 2

$$\text{Tangent sigmoid:}f\left(x\right)=\frac{e^x-e^{-x}}{e^x+e^{-x}}.$$ 3

Adaptive neuro-fuzzy inference system

A combination of ANN with fuzzy logic will result in the emergence of ANFIS systems. Typically, two common structures for FIS approaches exist: (a) Mamdani et al. and (b) Takagi–Sugeno^36,49. What is specific about Mamdani et al. method⁵⁰ is that a list of if–then rules must be defined for the fuzzy inference system, while the fussy interface proposed by Takagi–Sugeno creates its own rules based on the intrinsic features of the provided experimental data to the modeling endeavor. If the output data is nonlinearly dependent on the input data, Takagi–Sugeno ANFIS method will be more useful. Five distinct layers are a typical architecture for the ANFIS structure⁵¹. The Fuzzification layer is the first layer in which the conversion of inputted data into linguistic data occurs. The fuzzification process will be done utilizing the defined membership functions. The second layer is used for the model validation by computing a range of parameters known as the firing strengths. The estimated firing strengths are normalized in the next layer, and the fourth layer is responsible for representing outputs’ linguistic terms. Ultimately, all rules attributed to any individual output are combined in the fifth layer⁵⁰.

Least square-support vector machine

As a robust method for pattern recognition⁵² and regression⁵³, the LS-SVM is a widely-used and well-developed method. The SVM formulates the function as is given in Eq. (4).

$$f\left(x\right)=w^T\left(x\right)\phi \left(x\right)+b$$ 4

where the output layer’s transposed vector is denoted by w^T, the kernel function and bias are given as φ(x) and b, respectively^54,55. The size of the input data set and the output ensemble are the determining factors for the SVM’s dimension. The parameters of w and b are then determined by the cost function, given in Eq. (5)⁵⁴.

$$Costfunction=\frac{1}{2}{w}^T+c\sum _{k=1}^N\left({\xi }_k-{\xi }_k^{\ast }\right)$$ 5

The reliable results are possible to achieve by minimizing the cost function considering the following constraints⁵⁴:

$$\left\{\begin{array}{c}{y}_k-w^T\phi \left(x_k\right)-b\le \epsilon +{\mathrm{\xi }}_{\text{k}},k=1,2,\dots ,N\\ w^T\phi \left(x_k\right)+b-y_k\le \epsilon +{\mathrm{\xi }}_k^{\ast },k=1,2,\dots ,N\\ {\xi }_k,{\xi }_k^{\ast }\ge 0\\ \end{array}\right)$$ 6

where the k_th inputted data and its corresponding output are shown by x_k and y_k, respectively. In this formulation, ε stands for the accurateness of the function results, and the maximum acceptable errors are given by ${\xi }_k$ and ${\xi }_k^{\ast }$ Indicates the slack variable. The deviations from ε are determined by c values.

Radial basis function neural network

RBF neural networks are robust predicting methods that use a simpler structure in comparison to MLP networks, the learning step in them is much faster than the MLP’s learning procedure⁵⁶. Like major artificial neural networks, RBF has three layers: the input layer, the interior layers, and the result layer. The radial basis function is applied to the nodes of hidden layers. Using a linear optimization mechanism, the RBFNN will return precise results when the least mean square error is achieved. Despite all existing similarities between MLPNN and the RBFNN structures, RBFNN utilizes a complex RBF function for hidden layers³⁶.

Cascade fee-forward neural network

The implemented CFFNN in this study could be contemplated as a type of feedforward neural network where the input neurons are connected to all neurons located in the following layers^57,58. A range of various learning algorithms is applied to CFFNN models. As one of the most general formulations, the gradient descent algorithm with the momentum is introduced as follows⁵⁹:

$$\mathrm{\Delta }w\left(i+1\right)=-\alpha \frac{\partial E_P}{\partial w}+\mu \mathrm{\Delta }w\left(i\right)+\gamma e_s,\mathrm{\Delta }b\left(i+1\right)=-\alpha \frac{\partial E_P}{\partial b}+\mu \mathrm{\Delta }b\left(i\right)+\gamma e_s$$ 7

where the weight of neurons is denoted by $w$, the learning pace is shown by α, and bias and the number of training steps are given by b and i, respectively. In these formulations, the momentum parameter is presented by $\mu$, and the deviation of outputs from the modeling target is represented by $\gamma$⁶⁰. Although the updating algorithm for weight factors (given in Eq. 4) is precise, it is just applicable to a small ensemble of data. The weights updating formulation with two terms (i.e., the formulation without $\gamma e_s$) is a better choice for modeling of large-scale databanks. Equation (8) shows that the cost function is defined by summation of the square error.

$$SSE=\sum _{p=1}^n{E}_p=\sum _{p=1}^n{\left(t_p-o_p\right)}^2$$ 8

where the target and the output patterns are shown by t_p and o_p. The training procedure will not stop unless a pre-defined desirable sum of square errors is obtained⁶¹.

Generalized regression neural network

In utilizing the GRNN predictive method, there is no need for an iterative training process¹³. Instead, between the output and input vectors, any possible arbitrary functions are approximated. In addition to that, this approach is consistent because as larger datasets are fed to the model, the model return more precise results⁶². Such as the problems solved by the standard regression methods, the GRNN model is also suitable for predicting variables that are intrinsically continuous⁶². According to the definition of this method, the best and most accurate result for a dependent variable (y) will be obtained when an independent variable x and the training dataset are given, and the model commences minimizing the mean-squared error for the given x data points⁶².

Experimental data acquisition and preliminary analysis

Data gathering

In the current study, a collection of 792 datasets regarding to H₂S solubility in fifteen different ionic liquids, including [OMIM][Tf₂N], [OMIM][PF₆], [HMIM][PF₆], [BMIM][Tf₂N], [HMIM][Tf₂N], [EMIM][Tf₂N], [HOeMIM][Tf₂N], [BMIM][BF₄], [BMIM][PF₆], [EMIM][PF₆], [EMIM][eFAP], [HOeMIM][OTF], [HOeMIM][PF₆], [HEMIM][BF₄], [EMIM][EtSO₄] were assembled (the full form of these ionic liquids are introduced in Table 1). The range of operating conditions, i.e., pressure (P) and temperature (T), ionic liquid inherent characteristics, i.e., critical pressure (P_c), critical temperature (T_c), and the acentric factor (ω) are listed in Tables 1 and 2. The range of absorbed hydrogen sulfide by different ionic liquids as the dependent variable is also reported in Table 1. Indeed, these variables are enough to derive a global model for determining the amount of captured H₂S in ILs. The gathered data points were divided into two main subsets, including training (85% of the datasets) and testing (15% of the datasets). These groups have been used in a systematic trial-and-error procedure to find the optimal configuration of the model structures and evaluate their performances.

Table 1.

The range of operating conditions during absorbing H₂S molecules by different ionic liquids.

Ionic liquid (Full name)	Abbreviation	Temperature (K)	Pressure (bar)	Solubility (Mole fraction)	Number of data	References
1-Ethyl-3-methylimidazolium ethylsulfate	[EMIM][EtSO4]	303.15–353.15	1.14–12.70	0.012–0.118	36	17
1-Ethyl-3-methylimidazolium hexafluorophosphate	[EMIM][PF6]	333.15–363.15	1.45–19.33	0.032–0.359	40	25
1-Ethyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide	[EMIM][Tf2N]	303.15–353.15	1.08–16.86	0.049–0.609	42	25
1-Ethyl-3-methylimidazolium tris(pentafluoroethyl) trifluorophosphate	[EMIM][eFAP]	303.15–353.15	0.58–19.42	0.022–0.592	79	26
1-Hexyl-3-methylimidazolium bis(trifluoromethanesulfonyl)imide	[HMIM][Tf2N]	303.15–353.15	0.69–20.17	0.029–0.701	87	16,18
1-Hexyl-3-methylilmidazolium hexafluorophosphate	[HMIM][PF6]	303.15–343.15	1.11–11.00	0.050–0.499	67	16
1-Butyl-3-methylimidazolium tetrafluoroborate	[BMIM][BF4]	303.15–343.15	0.61–8.36	0.030–0.354	42	63
1-Butyl-3-methylimidazolium hexafluorophosphate	[BMIM][PF6]	298.15–403.15	0.69–96.30	0.016–0.875	81	63,64
1-Butyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide	[BMIM][Tf2N]	303.15–343.15	0.94–9.16	0.051–0.510	44	63
1-Octyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide	[OMIM][Tf2N]	303.15–353.15	0.94–19.12	0.063–0.735	47	18
1-n-Octyl-3-methylimidazolium hexafluorophosphate	[OMIM][PF6]	303.15–353.15	0.85–19.58	0.046–0.697	48	31
1-(2-Hydroxyethyl)-3-methylimidazolium bis(trifluoromethylsulfonyl)imide	[HOeMIM][Tf2N]	303.15–353.15	1.56–18.32	0.057–0.572	41	27
1-(2-Hydroxyethyl)-3-methylimidazolium trifluoromethanesulfonate	[HOeMIM][OTf]	303.15–353.15	1.06–18.39	0.035–0.548	41	27
1-(2-Hydroxyethyl)-3-methylimidazolium hexafluorophosphate	[HOeMIM][PF6]	303.15–353.15	1.34–16.85	0.034–0.462	47	27
1-(2-Hydroxyethyl)-3-methylimidazolium tetrafluoroborate	[HEMIM][BF4]	303.15–353.15	1.21–10.66	0.020–0.247	50	23

Table 2.

The critical temperature, pressure, and acentric factors of ILs used in this study.

Abbreviation	Tc (K)	Pc (bar)	ω	References
[EMIM][EtSO4]	1061.1	40.4	0.3368	65
[EMIM][PF6]	663.5	19.5	0.6708	65
[EMIM][Tf2N]	1244.9	32.6	0.1818	65
[EMIM][eFAP]	830. 7	100.3	1.5099	66
[HMIM][Tf2N]	876.2	22.2	1.3270	65
[HMIM][PF6]	754.3	15.5	0.8352	65
[BMIM][BF4]	632.3	20.4	0.8489	65
[BMIM][PF6]	708.9	17.3	0.7553	65
[BMIM][Tf2N]	1265	27.6	0.2656	65
[OMIM][PF6]	800.1	14.0	0.9069	19
[OMIM][Tf2N]	923.0	18.7	1.3310	65
[HOeMIM][Tf2N]	1297	33.1	0.5171	27
[HOeMIM][OTf]	1059.1	36.7	0.6526	27
[HOeMIM][PF6]	766.9	20.2	1.0367	27
[HEMIM][BF4]	691.9	24.7	1.1643	67

Outlier detection

Outliers are typically an inevitable part of every dataset; therefore, eliminating outliers is extremely important for good quality and reliable modeling. Outliers can drastically plummet the model’s accuracy and robustness. The current study reaps the outstanding rewards of utilizing a combination of Leverage and the Hat matrix methods according to the below equation⁶⁸:

$$H=X{\left(X^{T}X\right)}^{-1}{X}^T$$ 9

where X is the matrix of independent variables in the [n × m] shape, i.e., numbers of features × numbers of measurements. The process of outlier detection is done using a William plot. Calculation, normalization, and illustration of residual values with respect to the hat value are performed by developing this plot. Simultaneously, a warning leverage value (H*) is calculated using the following expression⁶⁹:

$$H^{\ast }=\frac{3\left(n+1\right)}{m}.$$ 10

Statistical criteria for model assessment

Once the models are developed, the accuracy can be evaluated by various statistical approaches to determine their robustness. In the current investigation, the following criteria were utilized for assessing models’ accuracy^70,71:

$$\text{Root mean square error}\left(\text{RMSE}\right)={\left(\frac{{\sum }_{i=1}^N{(y_{i,pred.}-y_{i,exp.})}^2}{N}\right)}^{1/2}$$ 11

$$\text{Coefficient of determination}\left({\text{R}}^2\right)=1-\frac{{\sum }_{i=1}^n{(y_{i,pred.}-y_{i,exp.})}^2}{{\sum }_{i=1}^n{(y_{i,pred.}-{\overline{y}}_{i,exp.})}^2}$$ 12

$$\text{Absolute average relative deviation}\left(\text{AARD\%}\right)=\frac{100}{N}\sum _{i=1}^N\frac{\left|y_{i,pred.}-y_{i,exp.}\right|}{y_{i,exp.}}$$ 13

$$\text{Mean absolute error}\left(\text{MAE}\right)=\frac{{\sum }_{i=1}^N\left|y_{i,pred.}-y_{i,exp.}\right|}{N}$$ 14

$$\text{Relative absolute error}\left(\text{RAE\%}\right)=\frac{100\times {\sum }_{i=1}^N\left|y_{i,pred.}-y_{i,exp.}\right|}{{\sum }_{i=1}^N\left|\overline{y}-y_{i,exp}\right|}$$ 15

$$\text{Root relative squared error}\left(\text{RRSE\%}\right)=100\times \sqrt{\frac{{\sum }_{i=1}^N{\left(y_{i,pred.}-y_{i,exp.}\right)}^2}{{\sum }_{i=1}^N{\left(\overline{y}-y_{i,exp.}\right)}^2}}$$ 16

$$\text{Mean square errors}\left(\text{MSE}\right)=\frac{1}{N}\sum _{i=1}^N{\left(y_{i,pred.}-y_{i,exp.}\right)}^2$$ 17

In these equations, employed for statistical evaluation of the results, $y_{i,exp.}$ and $y_{i,pred.}$ show the experimentally measured and the predicted H₂S solubilities, respectively. The notation $\overline{y}$ is the average value of $y_{i,exp.}$, and N stands for the number of data points.

Result and discussion

Development phase

All considered machine learning mthods^72–74 have some parameters that need to be tuned using historical data of a given problem and an optimization algorithm⁷⁵. This research utilizes 792 experimental data of H₂S solubility in fifteen ILs versus pressure, temperature, acentric factor, critical pressure, and temperature. The collected databank was randomly split into 673 training and 119 testing datasets.

In the training stage, a machine learning method receives the numerical values of independent as well as dependent variables, while its parameters are unknown^76–78. The intelligent model estimates the H₂S solubilities from the available independent variables. The deviation between these estimated values and actual H₂S solubilities are then needed to be minimized by an optimization algorithm. Indeed, the optimization algorithm continuously updates the parameters of a machine learning method to converge to this minimum value.

In the testing stage, a trained machine learning method receives the independent variables only and calculates the H₂S solubility helping the adjusted parameters. The independent variables and machine learning parameters are known in the testing stage, while the dependent variables are unknown.

The accuracy of all machine learning methods in the training and testing stages has been monitored using different statistical indices [i.e., Eqs. (11–17)]. Then, it is possible to find the best model using the ranking analysis. Figure 1 represents a general flowchart for model development in the present study.

Figure 1.

General sketch for development of the proposed models.

Table 3 shows the complete information about the applied trial-and-error procedure during model development. This table shows the numbers of hidden neurons for the MLPNN, CFFNN, and RBFNN, spread factor for the RBFNN and GRNN, cluster radius for the ANFIS, and kernel type for the LS-SVM model is the deciding features in the trial-and-error analyses. Table 3 also presents the cumulative numbers of the developed model for each machine learning class. Generally, 740 models are developed in this study.

Table 3.

General information about the model development phase.

Model	Decision features	Range of decision features	Numbers of iteration	Numbers of developed models
ANFIS	Cluster radius training algorithm	0.5–1 (10 values) Backpropagation and hybrid	10 per cluster radius	200
LS-SVM	Kernel types	Linear, polynomial, Gaussian	30 per kernel function	90
MLPNN	Numbers of hidden neurons	1–9 (9 values)	10 per hidden neuron	90
CFFNN	Numbers of hidden neurons	1–8 (8 values)	10 per hidden neuron	80
RBFNN	Numbers of hidden neurons Spread values	1–9 (9 values) 10–6-10 (20 values)	20 per hidden neuron	180
GRNN	Spread values	10–6-10 (100 values)	One per spread factor	100

Assessment phase

Statistical analyses

After the model development phase, monitoring their accuracy in the training and testing stages employing various statistical criteria is necessary. In this way, it is possible to find the most accurate model in each class using the ranking analysis. The prediction uncertainty of the most precise model in each category in terms of seven statistical criteria is summarized in Table 4. This table states that the cluster radius of 0.5 and Gaussian kernel are the best features for the ANFIS and LS-SVM paradigms. Furthermore, nine, six, and nine hidden neurons are the best topologies of the MLPNN, CFNN, and RBFNN models. The best spread factor for the RBFNN and GRNN models are 3.1579 and 0.00210, respectively. As this table shows, almost all intelligent models are sufficiently robust for estimating hydrogen sulfide solubility in various ionic liquid media. All models show the R² values greater than 0.99, apart from RBFNN.

Table 4.

Statistical evaluation of the best selected model in each class.

Model	Key feature	Stage	AARD%	MAE	RAE%	RRSE%	MSE	RMSE	R2
ANFIS	Cluster radius = 0.5	Training	5.20	0.0075	5.34	6.84	0.00014	0.01178	0.997657
		Testing	4.99	0.0075	5.93	7.62	0.00014	0.01180	0.997105
		Total	5.17	0.0075	5.42	6.94	0.00014	0.01178	0.997587
LS-SVM	Gaussian kernel	Training	3.89	0.0058	4.17	6.21	0.00011	0.01063	0.998073
		Testing	4.78	0.0070	5.45	7.26	0.00014	0.01165	0.997427
		Total	4.03	0.0060	4.35	6.35	0.00012	0.01079	0.997980
MLPNN	9 hidden neurons	Training	4.76	0.0067	4.85	6.60	0.00013	0.01131	0.997819
		Testing	3.55	0.0067	5.01	7.38	0.00014	0.01189	0.997363
		Total	4.58	0.0067	4.87	6.72	0.00013	0.01140	0.997746
CFFNN	6 hidden neurons	Training	4.95	0.0074	5.47	7.20	0.00014	0.01202	0.997408
		Testing	5.71	0.0069	4.33	7.26	0.00018	0.01340	0.997475
		Total	5.06	0.0073	5.27	7.21	0.00015	0.01224	0.997400
GRNN	Spread = 0.00210	Training	0.59	0.0011	0.81	2.66	0.00002	0.00460	0.999646
		Testing	25.72	0.0478	40.10	39.23	0.00344	0.05864	0.926456
		Total	4.36	0.0082	5.89	13.62	0.00053	0.02312	0.990788
RBFNN	9 hidden neurons, spread = 3.1579	Training	33.62	0.0469	33.62	35.99	0.00380	0.06161	0.932988
		Testing	26.51	0.0405	30.99	33.19	0.00285	0.05340	0.943624
		Total	32.55	0.0460	33.22	35.61	0.00365	0.06045	0.934448

Graphical inspection

Different graphical inspections, such as cross-plot and distribution of residual errors, were performed to illustrate the efficiency of the developed models and compare their performances. Figure 2 shows the cross plots of all the implemented approaches and confirms an excellent agreement between experimental and predicted mole fractions of H₂S in ILs due to the concentrated accumulation of the training and testing data around the unit slope line. In addition, the relative deviation of the investigated models from experimental data is depicted in Fig. 3. The error distribution provides a suitable visual comparison between the models’ performances. In this figure, LS-SVM, MLPNN, CFFNN, and ANFIS have small scattering to anticipate H₂S solubility in various ILs, while the relative deviation of the training and testing data points for GRNN and RBFNN models exceed 40%. These findings confirm the obtained results in Table 4.

Figure 2.

Cross plots of the best model in each class.

Figure 3.

The relative deviations of the selected models for estimating the H₂S solubility.

Ranking analysis

The six models were selected before, and their accuracy in the training and testing stages and over the whole of the database was monitored using seven statistical matrices. It is hard to most accurate one through visual inspection. Therefore, the ranking analysis is employed to do so⁴⁰. Figure 4 provides the results of model ranking in each stage based on the average values of the seven statistical criteria reported in Table 4. The GRNN model in the learning step is the best model; nevertheless, it shows the worst performance in the testing stage. This sharp contrast between the learning ability and the testing results indicates the overfitting of the GRNN model. On the other hand, the LS-SVM with the second-ranking in the training stage and the first ranks for the testing stage and over the whole database is the best model for predicting H₂S solubility in ionic liquid media.

Figure 4.

Developed models ranking based on seven different statistical indices.

According to the results of Fig. 4, the developed predictive models can be summarily ranked in terms of their accuracy as follows: LS-SVM > MLPNN > ANFIS > CFFNN > GRNN > RBFNN.

Figure 5 depicts the performance of different models to predict the experimental data of H₂S solubility in [BMIM][Tf₂N] ionic liquid versus pressure at 343.15 K. The LS-SVM is the most precise model for estimating the H₂S solubility in ionic liquid media.

Figure 5.

The prediction ability of the developed models for H₂S solubility in [BMIM][Tf₂N] (T = 343.15 K).

As mentioned earlier, the LS-SVM approach was selected as the best model. In order to better describe the excellent performance of the LS-SVM model, its residual errors (RE) versus the frequency are plotted in Fig. 6. Histograms related to the training, testing, and all data points reveal that the maximum frequency could be seen around residual errors of zero, and virtually all data points are predicted with − 0.05 < RE <  + 0.05.

Figure 6.

Histograms of the residual errors (RE) of the LS-SVM for predicting H₂S solubility in different IL media (a) training group, (b) testing group, and (c) all datasets.

Equations of state

The prediction uncertainty (in terms of AARD) of the LS-SVM and UNIFAC EoS⁷⁹ to estimate the 792 collected H₂S solubility in various IL media are compared in Fig. 7. It can be seen that the LS-SVM (AARD ~ 14%) and UNIFAC (AARD ~ 30%) show the maximum uncertainty for the H₂S solubility in the [HMIM][PF₆] ionic liquid. Although the UNIFAC has its second-highest AARD of 27% for [HOeMIM][Tf₂N], the obtained value by the LS-SVM is about thirteen times lower (AARD ~ 2%). Indeed, the results predicted by the LS-SVM method are remarkably better than the UNIFAC results. The overall AARD% of UNIFAC for H₂S solubility in all IL media is ~ 14%, while it is about 4% for the LS-SVM model. Excluding hydrogen sulfide solubility in the [HMIM][PF₆] and [HMIM][TF₂N] ionic liquids, the LS-SVM predicts all other systems with the AARD of lower than 5%. It confirms the LS-SVM excellent capability for a wide range of IL/H₂S systems.

Figure 7.

Comparison between LS-SVM and UNIFAC uncertainties⁷⁹ to estimate the H₂S solubility in various ILs.

Table 5 presents the accuracy of the proposed LS-SVM model and other approaches reported in the literature based on AARD%. As can be observed, the implemented model in this work shows precise performance for estimating H₂S solubility in ILs with AARD% less than that for other EoS models and intelligent algorithms. In addition, other reports have developed for a smaller number of ILs and data points compared to the present study.

Table 5.

Comparison between the developed LS-SVM model and the available approaches in the literature for estimating H₂S solubility in ILs based on AARD%.

Model	No. of data points	No. of ILs	AARD (%)	Ref
Peng-Robinson (kij = 0)	465	11	38.95	80,81
Soave–Redlich–Kwong (kij = 0)	465	11	36.43	80,81
Peng-Robinson	465	11	4.90	80,81
Soave–Redlich–Kwong	465	11	4.87	80,81
Peng-Robinson (kij = 0)	664	14	196.76	66
Peng-Robinson	664	14	8.35	66
RETM-CPA	317	5	4.41	82
SAFT-VR	225	6	5.44	83
Peng-Robinson-Two State	636	12	3.40	84
Peng-Robinson (kij = 0)	664	14	25.13	85
Peng-Robinson	664	14	3.67	85
FC-ELM	722	16	2.33	86
Gene expression programming	465	11	4.38	80,81
MLPNN	496	12	1.94	87
MLPNN	664	14	2.07	66
MLPNN	664	13	9.08	88
RBFNN	664	13	26.15	88
ANFIS	664	13	38.44	88
LS-SVM	792	15	4.02	This work

Relevancy analysis

As stated earlier, the best model was determined LS-SVM with the input parameters, including pressure, temperature, acentric factor, and critical pressure and temperature. In order to study the influence of input parameters on the dissolved mole fraction of H₂S in ionic liquids, the relevancy factor was utilized⁸⁹. This relevancy factor (${\text{r}}_{\text{i}}$) is defined for all independent variables (i) as follows⁹⁰:

$$r_i=\frac{{\sum }_{k=1}^n\left(M_{i,k}-{\overline{M}}_i\right)(N_k-\overline{N})}{\sqrt{{\sum }_{k=1}^n{\left(M_{i,k}-{\overline{M}}_i\right)}^2{\sum }_{k=1}^n{\left(N_k-\overline{N}\right)}^2}}\left(i=1,\dots ,5\right)$$ 18

where $M_{i,k}$,$\overline{M}$,$n$, $N_k$, and $\overline{N}$ represent input parameters, an average of inputs, number of the data points, output parameter, and average of output, respectively. The value of $r_i$ is located within  − 1 to 1, and the large values correspond to the strong correlation. Also, the increasing or decreasing of output parameter with variations in $M_i$ attribute to a positive or negative sign, respectively. Two main techniques, namely Spearman and Pearson⁹¹, relevancy factors were calculated to ascertain the reliability of the interrelation of the considered independent variables with the H₂S solubility as the model's output. According to the results of both methods (Fig. 8), pressure and temperature have the most significant roles in this process, while the acentric factor has the lowest effect. Moreover, it was found that the H₂S solubility adversely relates to the temperature and critical pressure of the ionic liquids. Generally, by increasing the pressure, critical temperature, and the acentric factor of ionic liquids, more H₂S is expected to be captured.

Figure 8.

Dependence of H₂S solubility in ILs on its influential parameters.

Trend analysis of the LS-SVM

Besides being precise, the developed LS-SVM approach should be able to detect the physical trend of the simulated phenomenon. For doing so, the LS-SVM predictions for H₂S solubility in ionic liquids in various temperatures and pressures were compared to the experimentally measured data.

The effect of temperature and pressure

Figure 9 illustrates the solubility of hydrogen sulfide in 1-Butyl-3-methylimidazolium hexafluorophosphate ([BMIM][PF₆]) in terms of operating pressure at various temperatures. The H₂S solubility in [BMIM] [PF₆] was investigated between T = 298.15 K and T = 403.15 K and pressure up to 100 bar. As expected, more H₂S molecules dissolved in the [BMIM][PF₆] by increasing the operating pressure. However, the H₂S solubility in the ionic liquid dramatically decreases as temperature increases. The enhancing effect of pressure is related to the fact that the pressure pushes the H₂S molecules into the liquid phase⁹². Furthermore, this enhancement is more significant in lower pressure. Increasing the kinetic and internal energy of the hydrogen sulfide molecules by increasing the temperature may be responsible for this observation⁹². Furthermore, dissolving H₂S in liquid is an exothermic process. When this gas dissolves in ILs, its molecules interact with ionic liquid molecules and release heat within attractive interaction. Conversely, increasing temperature by adding heat to the solution provides thermal energy that overcomes the attractive forces between the gas and the liquid molecules, thereby decreasing the solubility of the gas.

Figure 9.

The effect of pressure on the H₂S solubility in [BMIM][PF₆] ionic liquid in a wide range of temperatures.

The outstanding performance of LS-SVM for predicting the profile and all distinct data points can be concluded from this figure. Indeed, the proposed LS-SVM model successfully understands the influence of operating pressure and temperature on the hydrogen sulfide solubility in the ionic liquid.

The effect of cation and anion type

Assessing the effects of both anions and cations leads to a deep perception of the behavior of H₂S solubility in ILs. It is generally found that anions have more influence on the solubility of H₂S gas than cations⁹³ As illustrated in Figs. 10a and b, higher H₂S solubility was obtained for ILs with the same [TF₂N]⁻ anion but higher alkyl chain length ([C₈MIM]⁺ > [C₆MIM]⁺  > [C₄MIM]⁺  > [C₂MIM]⁺  > [C₂OHMIM]⁺). This originates because the longer alkyl chain provides more free volume available in ILs. Aki et al.⁹⁴ ascribed this behavior to entropic rather than enthalpic reasons, where the molar density of the ILs decreases as the length of the cation alkyl chain gets larger⁹⁵. As the molar density of the IL decreases, the free volume of the IL enhances the absorption of H₂S to occur through a space-filling mechanism⁹⁶. Consequently, larger free volumes increase H₂S solubility by stronger Van der Waals interactions and more H₂S molecules absorbed in the solvent²³. The above-mentioned trend is an outcome of the variations in molecular interactions of H₂S with ionic liquids, which arise from the differences in the chemical constituents, shapes, and sizes of ILs. In addition, H₂S solubility for ILs with similar [EMIM]⁺ but different types of anions was investigated. It was found that higher H₂S solubility obtains in anions containing more fluorine content ([TF₂N]⁻  > [eFAP]⁻  > [PF₆]⁻  > [EtSO₄]⁻), which is in accordance with the other reports in the literature³¹. Moreover, CO₂ and H₂S could be strongly attracted in ILs containing [Tf₂N]⁻ in compassion with [PF₆]⁻⁹⁷. As can be seen, the LS-SVM model as a non-linear approach possesses the exceptional capability for estimating the H₂S solubility behavior in terms of anion and cation types to obtain reliable quantitative results.

Figure 10.

(a) The effect of cation type on the H₂S solubility in [Tf₂N]-contained ionic liquids (T = 343.15 K) (b) The influence of anion type on the hydrogen sulfide capture ability of the [EMIM]-contained ionic liquids (T = 353.15 K).

The effect of ionic liquid type (cation + anion)

Previous analyses show that the [OMIM]⁺ and [Tf₂N]^- provide the highest H₂S absorption capacity for the ionic liquid. This section aims to investigate whether their combination poses the highest absorption capacity or not. The effect of absorbent type (combination of cation and anion) on the H₂S dissolution in the ionic liquid at the temperature of 333.15 K is depicted in Fig. 11. Generally, an astonishing agreement exists between the calculated hydrogen solubility and the experimentally measured values. As expected, the combination of the [OMIM]⁺ and [Tf₂N]^-, i.e., [OMIM][Tf₂N] ionic liquid, shows the highest tendency for absorbing the H₂S molecules. On the other hand, the [eMIM][EtSO₄] is the worst medium for capturing the hydrogen sulfide molecules.

Figure 11.

Comparing the H₂S capture tendency of various ionic liquids at T = 333.15 K.

Maximizing H₂S solubility in ionic liquid

The previous investigations approved that combining the [OMIM]⁺ and [Tf₂N]^-, i.e., [OMIM][Tf₂N] ionic liquid synthesized the best medium for absorbing the hydrogen sulfide molecules. This section uses the developed LS-SVM approach to graphically determine the operating condition that maximizes hydrogen sulfide absorption capacity of the [OMIM][Tf₂N] ionic liquid. Figure 12 presents pure simulation results for the effect of simultaneous change of pressure and temperature on the H₂S solubility in the [OMIM][Tf₂N] ionic liquid. It can be seen that increasing the pressure and decreasing the temperature gradually increases the H₂S dissolution in the considered ionic liquid. Therefore, the maximum hydrogen solubility of ~ 0.8 is achievable at the highest pressure and lowest temperature.

Figure 12.

Three-dimensional illustration of temperature and pressure coupled effect on the H₂S solubility in [OMIM][Tf₂N].

Applicability of LS-SVM and outlier detection

During the model development, standard residuals were calculated and plotted. Data points with standard residuals in the range of − 3 to + 3 (illustrated on the y axis) and Hat indexes limited to 0. The calculated H* (x-axis) values have been recognized as good data points. William’s plot related to the developed LS-SVM model is shown in Fig. 13. As can be seen in this plot, the significant numbers of the point are located in the good Leverage area (0 ≤ H ≤ 0.022 and − 3 ≤ SR ≤ 3)⁹⁸. Hence, the Leverage approach confirms the validity and reliability of the proposed LS-SVM model for estimating H₂S solubility in ILs. Furthermore, the number of outliers is too small to affect the modeling generalization negatively.

Figure 13.

Outlier/valid data detection by the Leverage method.

Application range of the constructed LS-SVM model

Table 1 shows that the H₂S solubility data utilized to develop the LS-SVM model are only about imidazole-based ionic liquids containing F atoms. Therefore, this intelligent approach is only valid for the utilized ionic liquids in the reported pressure and temperature ranges.

On the other hand, many different non-F functionalized ionic liquids have also been utilized for H₂S absorption. It is possible to collect a databank for H₂S solubility in non-F functionalized ionic liquids (or for both F functionalized and non-F functionalized ionic liquids), develop different machine learning methods, compare their accuracy, and find the most accurate model.

Conclusion

The absorption process is likely the most widely used method for H₂S removal. Untapped potentials and favorable characteristics of ionic liquids have been enticing for scientists to investigate their H₂S removal capacity. However, experimental endeavors are not only costly but time-consuming. On the other hand, since there are many affecting parameters and the interactions between IL and H₂S molecules are complex, accurate results cannot be achieved by the equations of state. Fortunately, AI methods can bypass theoretical equations and solve complicated problems expeditiously and accurately. The current study investigated H₂S solubility in fifteen ILs by implementing six robust AI methods, including MLPNN, LS-SVM, ANFIS, RBFNN, CFFNN, and GRNN. The temperature, pressure, acentric factor, critical pressure, and critical temperature of investigated ILs are influential variables of the current study. The validation of the derived models was approved using seven statistical criteria. It was found that the LS-SVM was the best predictive model having R², RMSE, MSE, RRSE, RAE, MAE, and AARD of 0.99798, 0.01079, 0.00012, 6.35%, 4.35%, 0.0060, and 4.03%, respectively. It was found that temperature and the critical pressure of the liquid are adversely related to the H₂S solubility. However, the pressure, critical temperature, and acentric factor of ionic liquids increase H₂S dissolution in ionic liquids. The outlier detection method justified that a relatively substantial number of data points are valid and have enough quality to be incorporated into the modeling procedure. Finally, the maximum hydrogen solubility of ~ 0.8 is achievable by [OMIM][Tf₂N] ionic liquid at the highest pressure and lowest temperature.

Author contributions

J.A. Writing - Review and Editing, Investigation. M.H. Writing - Review and Editing. SH. EF. Writing - Review and Editing. B.V. Conceptualization, Methodology, Writing - Review and Editing, Resources, Supervision, Validation, Investigation, Resources.

Data availability

A user-friendly and straightforward Matlab-based code has been prepared to use by other research groups (please see Supplementary Information: supplementary_file\Matlab_code). The collected experimental databank has been added to the revised manuscript (please see Supplementary Information: supplementary_file\Database).

Competing interests

The authors declare no competing interests.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-022-08304-y.