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. 2021 Oct 8;6(41):26919–26931. doi: 10.1021/acsomega.1c03214

AdaBoost Metalearning Methodology for Modeling the Incipient Dissociation Conditions of Clathrate Hydrates

Sepehr Keshvari †, Saeid Abedi Farizhendi ‡, Mohammad M Ghiasi §,*, Amir H Mohammadi §,*
PMCID: PMC8529602  PMID: 34693113

Abstract

graphic file with name ao1c03214_0010.jpg

This paper proposes the AdaBoost metalearning methodology to combine the outcomes of tree-based models of classification and the regression tree (CART) algorithm for estimating the equilibrium dissociation temperature of clathrate hydrates. In addition to the AdaBoost-CART models, models based on the adaptive neuro-fuzzy inference system (ANFIS) and artificial neural network (ANN) approaches were also developed. Training and testing of the models were done utilizing a gathered database of more than 3500 experimental data on incipient dissociation conditions of CO2 and other hydrate systems. With the average absolute relative deviation percent (AARD%) between 0.03 and 0.07, 0.04 and 1.09, and 0.09 and 1.01, which were obtained by the presented AdaBoost-CART, ANFIS, and ANN models, respectively, the targets were reproduced with satisfactory accuracy. However, for all of the studied clathrate hydrate systems, the proposed AdaBoost-CART models provide more reliable results. Indeed, the obtained AARD% values for tree-based models are lower than those of other models.

1. Introduction

Gas molecules of proper sizes in the presence of adequate water may form and stabilize icelike structures known as clathrate hydrates or gas hydrates. Low temperature and high pressure accelerate hydrate formation. There are three main crystalline structures of clathrate hydrates, structure/type I (sI), structure II (sII), and structure H (sH), depending on the size of hydrate formers. In most cases of natural gases, there is a low possibility of structure H hydrate formers such as methylcyclopentane.1 As a result, only structure I representing small molecules such as methane and structure II representing large molecules such as propane can be formed.

In the petroleum industry, formation of hydrates is a serious problem as it can block pipelines and equipment and cause damage to the production and processing facilities. There are different methods that can be used to avoid hydrate formation. One of the most commonly used methods is the injection of inhibitors. Typically, there are two main categories of inhibitors: low-dosage hydrate inhibitors (LDHIs) and thermodynamic inhibitors.2 Thermodynamic inhibitors can shift the stability zone of the hydrate to low temperatures and/or high pressures. LDHIs can be classified into three categories, namely, antiagglomerants, kinetic inhibitors, and dual-purpose inhibitors.3 It is believed that they are more economical than thermodynamic inhibitors.4

On the other hand, there are other research areas of hydrate formation. These areas include the usage of hydrates in desalination of seawater,5,6 storage and transportation of natural gases,7 separation processes,8 energy resources,9,10 carbon dioxide capture and sequestration,11,12 and usage of the hydrate technology in the food processing industry.13,14

Several methods and models have been presented in the literature for calculating/predicting the equilibrium conditions of dissociation of gas hydrates. Katz and colleagues15−17 established a set of vapor–solid coefficients to calculate the hydrate stability conditions. For some pure gases, several empirical models have been developed.18,19 Chart-based models of Katz20 and Baillie and Wichert21 are other types of tools of interest for application. A more sophisticated method based on statistical thermodynamics was presented by van der Waals and Platteeuw.22 A semitheoretical approach is another suggested tool for hydrate formation calculations.23 In addition, several studies investigated the applicability of machine learning algorithms, such as ANN, ANFIS, LSSVM, and Extra Trees, for calculating/estimating the phase equilibria of clathrate hydrates.24−29

This study aims at modeling the incipient dissociation conditions of clathrate hydrates of various gases in pure water or aqueous solutions of alcohol(s) and/or electrolyte(s) by employing a machine learning-based methodology known as AdaBoost-CART. The AdaBoost algorithm is one of the most employed methodologies in boosting models in the area of data mining/machine learning.30 With a solid foundation and theoretical basis, AdaBoost provides great outcomes in practical applications. Boosting is an ensemble learning method. Ensemble methods are based on the idea that the final decisions of a larger group of people are typically better than that of an individual expert. For example, instead of having a single CART model, combining several CART models for the target of interest provides better results. Therefore, the individual models of ensemble methods are known as weak learners. As a boosting algorithm, AdaBoost can develop a robust regression model by combining several weak regressors. In this study, the CART algorithm was used as a weak learner.

Moreover, several ANN and ANFIS models were developed as a basis of comparison. To this end, previously published data on the phase equilibria of clathrate hydrates were gathered. Information regarding the collected databases is given in Section 2. Section 3 presents the employed methods’ background and the procedure for model development. The obtained results are discussed in Section 4 using statistical parameters and graphical evaluation. Finally, key findings of the work are summarized in Section 5.

2. Experimental Data

In this study, the previously published experimental data for hydrate systems of methane, ethane, propane, i-butane, hydrogen sulfide, nitrogen, and gas mixture were gathered from the available literature. This collection contains a total number of 3514 equilibrium data representing the equilibrium dissociation conditions of the hydrate systems as mentioned earlier. The gathered data points have been reported by Mohammadi and Richon,31−34 Mohammadi et al.,35−37 Haghighi et al.,38−41 Najibi et al.,42 Tohidi et al.,43,44 Chapoy and Tohidi,45 Nixdorf and Oellrich,46 Kharrat and Dalmazzone,47 Masoudi et al.,48,49 Jager et al.,50 Jager and Sloan,51 Maekawa,52,53 Verma,54 McLeod and Campbell,55 Verma et al.,56 Thakore and Holder,57 de Roo et al.,58 Adisasmito and Sloan,59 Adisasmito et al.,60 Song and Kobayashi,61 Ghavipour et al.,62 Lafond et al.,63 Ng et al.,64,65 Ng and Robinson,66−70 Jhaveri and Robinson,71 Robinson and Mehta,72 Robinson and Hutton,73 Robinson and Ng,74 Kang et al.,75 Atik et al.,76 Nasab et al.,77 Bishnoi and Dholabhai,78 Ross and Toczylkin,79 Dholabhai et al.,80,81 Dholabhai and Bishnoi,82 Roberts et al.,83 Ma et al.,84 Reamer et al.,85 Galloway et al.,86 Falabella,87 Deaton and Frost,88 John and Holder,89 Holder and Grigoriou,90 Holder and Hand,91 Holder and Godbole,92 Godbole,93 Avlonitis,94 Kamath and Holder,73 Kubota et al.,95 Miller and Strong,96 Kobayashi et al.,97 Englezos and Ngan,98 Breland and Englezos,99 Patil,100 Vlahakis et al.,101 Schneider and Farrar,102 Rouher and Barduhn,103 Larson,104 Selleck et al.,105 Unruh and Katz,16 Miller and Smythe,106 Bond and Russell,107 Carroll and Mather,108 Marshall et al.,109 van Cleeff and Diepen,110 Hemmingsen et al.,111 Mei et al.,112 Kamari and Oyarhossein,113 Wu et al.,114 Wilcox et al.,115 Schroeter et al.,116 Lapin and Cinnamon,73 and Paranjpe et al.117

Table 1 gives the existing additives in hydrate systems of C1, C2, C3, i-C4, H2S, N2, and gas mixture. As can be seen, various alcohols and/or salts including NaCl, KCl, CaCl2, MgCl2, methanol (MeOH), ethylene glycol (EG), diethylene glycol (DEG), triethylene glycol (TEG), 1-propanol, and 2-propanol are available in the aqueous phase.

Table 1. Available Additives in the Aqueous Phase of the Studied Hydrate Systems.

  maximum concentration (wt %)
gaseous/vapor system NaCl KCl CaCl2 MgCl2 MeOH EG DEG TEG 1-prop. 2-prop.
C1 22.03 15.00 25.74 15.00 85.00 70.00 50.00 50.00 20.00 20.00
C2 20.00 10.00 15.00 no 50.00 no no 40.00 no no
C3 20.03 20.00 15.20 no 50.00 no no no no no
i-C4 no no no no no no no no no no
H2S 26.40 no 36.00 no 50.00 15.00 no no no no
N2 no no no no no no no no no no
gas mix. 20.20 15.01 33.00 10.00 60.00 70.00 no no no no
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Pressure and temperature ranges for the gathered database are summarized in Table 2.

Table 2. Operating Ranges of the Collected Experimental Data for the Studied Hydrate Systems.

system T (K) P (kPa)
C1 194.6–326.8 760.0–1000 000.0
C2 200.8–304.6 8.3–160 600.0
C3 229.3–279.0 48.2–44 850.0
i-C4 241.4–275.1 17.6–169.0
H2S 250.5–302.7 34.0–55 160.0
N2 272.0–305.5 14480.0–328 890.0
gas mix. 214.8–305.0 46.9–111 798.0
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3. Modeling Procedure

3.1. General Procedure

With the objective of developing AdaBoost-CART, ANN, and ANFIS models for representing/predicting the hydrate dissociation temperatures (HDTs) where pure water or aqueous solutions of alcohol(s) and/or electrolyte(s) are present, the HDT is assumed to be a function of gas composition (Zi), concentration of inhibitor(s) (Cadditive), and equilibrium pressure of the hydrate system (P)

3.1. 1

For pure gases, eq 2 represents the relationship between the dependent and independent variables

3.1. 2

As a general step for developing the AdaBoost-CART, ANN, and ANFIS models, the collected data bank for each hydrate system was randomly divided into two sub-data-sets. The majority of the data (90% of the whole data bank) was employed as the training data set to train the model. The remainder of the data bank was used to evaluate the capability of the presented models in estimating the unseen targets. It should be noted that the Python programming language was used to develop AdaBoost-CART models. In the case of ANN and ANFIS, MATLAB was employed.

3.2. AdaBoost-CART Modeling

Ensemble methods like boosting algorithms are categorized as supervised learning methodologies. Using ensembles contributes to obtaining better results. This is owing to the fact that the ensemble reduces the risk of selecting a poor-performance regressor/classifier by combining the outputs of a group of classifiers (for classification problem) or regressors (for regression analysis).

Among all of the algorithms developed as boosting methods, the most successful methodology is known to be AdaBoost. Indeed, this method turned the boosting from a mere conjecture into application and reality. In addition to the fact that the AdaBoost is an efficient learning algorithm, it also illuminates the development of several other learning algorithms.118 In data mining, it is believed that this algorithm is one of the top 10 methods.119 This study implements the AdaBoost (Adaptive Boosting) meta-algorithm to develop several ensemble models for representing/estimating the equilibrium HDT of various systems.

Since the modeling of interest is a regression analysis, AdaBoost.R2,120 which is developed for regression problems, was used. In this boosting algorithm, an initial weight, wi, is assigned to each training observation. Employing the average loss function, defined by eq 3, the algorithm adjusts the weights so that the average loss becomes below the defined criterion.120

3.2. 3

where L̅ and Li are the average loss function and the loss function (linear, square, or exponential), respectively. pi represents the probability that data point i is in the training observation. This parameter can be expressed using eq 4(120)

3.2. 4

More information regarding the algorithm of AdaBoost.R2 can be found elsewhere.120,121

The CART was selected as the weak regressor of the AdaBoost method. The CART method was presented by Breiman et al.122 for generating tree-based classification or regression models. Our previous work provides details regarding application of CART for modeling.121,123 Specifications of the presented AdaBoost-CART models for estimating the HDT of the investigated hydrate systems are summarized in Table 3. The digraph of the proposed AdaBoost-CART models for the studied hydrate systems is available upon request to the authors.

Table 3. Specifications of the Presented AdaBoost Models for the Studied Hydrate Systems.

system number of trees maximum depth
C1 7 45
C2 3 40
C3 4 45
i-C4 2 25
H2S 6 40
N2 2 25
gas mix. 8 55
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Digraphs of the developed Adaboost-CART models can be found in Appendix A.

3.3. ANFIS Modeling

With a combination of ANN and fuzzy inference system (FIS), Jang124 presented a rule-based method, namely, ANFIS. The rules of this method are developed during the training process. In addition to the input and output layers, ANFIS has other layers (as hidden layers) including the fuzzification layer, the rule layer, normalization layer, and the defuzzification layer.

In this study, the Gaussian MF is employed. Furthermore, the ANFIS was trained using the hybrid learning algorithm of least squares and backpropagation. Specifications of the presented ANFIS models are summarized in Table 4.

Table 4. Specifications of the Presented ANFIS Models for the Studied Hydrate Systems.

  system
parameter C1 C2 C3 i-C4 H2S N2 gas mix.
cluster center’s range of influence 0.41 0.11 0.21 0.35 0.13 0.18 0.26
number of inputs 11 6 5 1 5 1 23
number of fuzzy rules 2 3 4 2 13 3 4
maximum epoch number 700 500 100 200 200 180 200
initial step size 0.10 0.10 0.10 0.10 0.05 0.05 0.05
step size decrease rate 0.95 0.95 0.90 0.90 0.90 0.85 0.95
step size increase rate 1.10 1.05 1.05 1.05 1.05 1.15 1.15
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Structures of the constructed ANFIS models for estimation of the dissociation conditions of C1, C2, C3, i-C4, H2S, N2, and gas mixture are shown in Figure 1.

Figure 1.

Figure 1

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ANFIS structure for (a) methane, (b) ethane, (c) propane, (d) i-butane, (e) hydrogen sulfide, (f) nitrogen, and (g) gas mixture hydrate systems. The blue circle indicates “and”.

3.4. ANN Modeling

The artificial neuron receives the input signals (xn) employing weights (wmn) and bias (bm) terms. Consequently, the linear combiner output (rm) is calculated by eq 5(125)

3.4. 5

where n and m denote the number of input signals and the neuron’s number, respectively. By applying an activation function, the output signal of the neuron (ym) is computed

3.4. 6

where f is the activation function.

Among the available architectures for ANN, this study employs the widely used ANN, namely, the multilayer perceptron (MLP) ANN. To design a MLP-ANN, several parameters need to be determined: the number of neurons in the input, hidden, and output layers and the values of the weights and biases of the neurons. The number of input neurons (I) and output neurons (O) is equal to the number of independent variables and dependent variables, respectively. However, there is no universal rule to obtain the optimum number of hidden neurons (H). To develop ANN models for estimating the HDT of the investigated hydrate systems, the number of hidden neurons was changed from 5 to 15. Subsequently, the performance of the created ANN model was evaluated. Furthermore, the MLP-ANN models were trained by the optimization algorithm of Levenberg–Marquardt.126,127 It should be noted that all of the ANN models utilized the transfer function of the hyperbolic tangent sigmoid type. Table 5 gives the topology of the best developed ANN model for each system. For all of the developed ANN models, the hyperbolic tangent sigmoid transfer function was used.

Table 5. Topology of the Presented ANN Models for the Studied Hydrate Systems.

system topology
C1 11-9-1
C2 6-8-1
C3 5-10-1
i-C4 1-7-1
H2S 5-8-1
N2 1-9-1
gas mix. 23-12-1
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4. Results and Discussion

4.1. Model Assessment Criteria

With the aim of the assessment of the accuracy of the developed models, statistical parameters including coefficient of determination (R2), average relative deviation percent (ARD%), and average absolute relative deviation percent (AARD%) were utilized. ARD% defines the distribution of errors between negative and positive values. AARD% is a measure of the accuracy of the model. The R2 value mathematically describes the goodness of fit. The higher the R2 value, the more the data points fitted to the model.

4.2. Model Assessment Results

Results of the performance evaluation of the developed AdaBoost-CART, ANFIS, and ANN models for estimating the equilibrium dissociation temperature of C1, C2, C3, i-C4, H2S, N2, and gas mixture hydrate systems are summarized in Tables 67–8.

4.2.
4.2.
4.2.

Table 6. R2 Values for the Developed AdaBoost-CART, ANFIS, and ANN Models.

  AdaBoost
 ANFIS
 ANN
system train test overall train test overall train test overall
C1 0.9996 0.9994 0.9996 0.9759 0.9865 0.9773 0.9483 0.7617 0.9452
C2 0.9977 0.9977 0.9977 0.8890 0.9197 0.8925 0.9208 0.9827 0.9231
C3 0.9988 0.9982 0.9982 0.9700 0.9881 0.9724 0.8851 0.9480 0.8862
i-C4 0.9928 0.9978 0.9967 0.9954 0.9983 0.9950 0.9972 0.9880 0.9970
H2S 0.9997 0.9934 0.9994 0.9742 0.9638 0.9739 0.9644 0.9785 0.9649
N2 0.9982 0.9993 0.9992 0.9997 0.9998 0.9997 0.9774 0.9940 0.9783
gas mix. 0.9985 0.9980 0.9980 0.8704 0.8812 0.8711 0.8420 0.8551 0.8526
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Table 7. ARD% Values for the Developed AdaBoost-CART, ANFIS, and ANN Models.

  AdaBoost
 ANFIS
 ANN
system train test overall train test overall train test overall
C1 0.00 0.01 0.00 0.01 0.07 0.01 –0.20 –0.11 –0.19
C2 0.01 0.02 0.01 0.03 0.20 0.05 –0.06 0.07 –0.06
C3 –0.01 –0.04 –0.01 0.01 0.30 0.04 0.06 0.01 0.05
i-C4 0.01 0.23 0.03 0.00 0.11 0.01 –0.01 –0.10 –0.01
H2S –0.06 0.01 0.00 0.01 0.26 0.03 –0.05 0.14 –0.03
N2 0.00 0.02 0.00 0.00 –0.04 0.00 –0.06 –0.21 –0.08
gas mix. 0.00 0.00 0.00 0.02 0.02 0.02 0.04 0.22 0.06
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Table 8. AARD% Values for the Developed AdaBoost-CART, ANFIS, and ANN Models.

  AdaBoost
 ANFIS
 ANN
system train test overall train test overall train test overall
C1 0.03 0.04 0.03 0.51 0.55 0.51 1.03 0.86 1.01
C2 0.07 0.05 0.07 1.10 1.08 1.09 0.88 0.31 0.83
C3 0.05 0.04 0.05 0.33 0.48 0.35 0.60 0.23 0.56
i-C4 0.03 0.23 0.05 0.13 0.14 0.13 0.08 0.12 0.09
H2S 0.02 0.16 0.04 0.51 0.46 0.50 0.64 0.33 0.61
N2 0.03 0.09 0.04 0.04 0.05 0.04 0.21 0.21 0.21
gas mix. 0.03 0.03 0.03 0.91 0.91 0.91 0.92 1.37 0.96
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For all of the investigated systems, the values of R2 for the presented AdaBoost-CART models are higher than 0.99. Hence, in terms of the R2 parameter, the presented AdaBoost-CART models have better performances in estimating/representing the target values. Furthermore, for the hydrate systems of C1, C3, H2S, N2, and gas mixture, the developed ANFIS models have higher R2 values as compared to the ANN models. For the remaining systems including C2 and i-C4, the developed ANN models show better R2 values.

According to the obtained values for ARD% that are tabulated in Table 7, it can be concluded that the errors arising from the presented AdaBoost-CART models for C1, H2S, N2, and gas mixture hydrate systems are equally distributed between negative and positive values. On the other hand, since the value of ARD% of the developed ANN model for C1 hydrate is equal to −0.19, this model considerably underestimates the targets. For other developed models, the values of the ARD% parameter are approximately close to zero.

As a degree of scatter, the values of AARD% reveal the excellent performance of the proposed AdaBoost-CART models to estimate the dissociation temperature of the investigated hydrates. For all of the systems, the presented AdaBoost-CART models represent the target values with AARD% between 0.03 and 0.07 (Table 8). Relative deviations of the outcomes of the developed models for C1, C2, C3, i-C4, H2S, N2, and gas mixture hydrate systems are demonstrated in Figures 2–45678, respectively. As can be seen from Figure 2, relative errors of the AdaBoost-CART model for C1 hydrate are distributed between −1.5 and 1.0. On the other hand, the relative errors of the ANFIS and ANN models range from −6.0 to 4.0. According to Figure 3, the relative deviations of the outputs of the AdaBoost-CART model for C2 hydrate have values between −2.0 and 1.0. For the C2 hydrate system, the error ranges of the ANFIS and ANN models are [−16.0, 4] and [−15, 4], respectively. For all other hydrate systems, except for i-C4 hydrate, the error domains of the developed AdaBoost-CART models are more limited than the error domains of the ANFIS and ANN models. In the case of the i-C4 hydrate system, the error range of the AdaBoost-CART model is from −1.2 to 0.6. [−0.5, 1.0] and [−0.4, 1.0] are the ranges for the ANFIS and ANN models, respectively.

Figure 2.

Figure 2

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Relative deviations of the outcomes of the developed models (a) AdaBoost-CART, (b) ANFIS, (c) ANN, and (d) LSSVM for the methane hydrate system.

Figure 4.

Figure 4

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Relative deviations of the outcomes of the developed models (a) AdaBoost-CART, (b) ANFIS, (c) ANN, and (d) LSSVM for the propane hydrate system.

Figure 5.

Figure 5

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Relative deviations of the outcomes of the developed models (a) AdaBoost-CART, (b) ANFIS, (c) ANN, and (d) LSSVM for the i-butane hydrate system.

Figure 6.

Figure 6

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Relative deviations of the outcomes of the developed models (a) AdaBoost-CART, (b) ANFIS, (c) ANN, and (d) LSSVM for the hydrogen sulfide hydrate system.

Figure 7.

Figure 7

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Relative deviations of the outcomes of the developed models (a) AdaBoost-CART, (b) ANFIS, (c) ANN, and (d) LSSVM for the nitrogen hydrate system.

Figure 8.

Figure 8

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Relative deviations of the outcomes of the developed models (a) AdaBoost-CART, (b) ANFIS, (c) ANN, and (d) LSSVM for the gas mixture hydrate system.

Figure 3.

Figure 3

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Relative deviations of the outcomes of the developed models (a) AdaBoost-CART, (b) ANFIS, (c) ANN, and (d) LSSVM for the ethane hydrate system.

4.3. Sample Results

The capability of the developed AdaBoost-CART model in estimating the experimental data of Haghighi et al.38 for the methane hydrate + ethylene glycol (EG) + water system is compared to that of the presented ANFIS and ANN models in Table 9. As can be observed from Table 9, using the proposed AdaBoost-CART model, all of the reported data are reproduced without error. Employing the ANFIS and ANN models results in errors from 0.03 to 2.62 and 0.29 to 4.76 K, respectively.

Table 9. Results of the Presented Models in Comparison with the Experimental Data Reported in ref (38) for Methane Hydrate.

    T (K) error (K)
EG (wt %) P (kPa) exp. AdaBoost ANFIS ANN AdaBoost ANFIS ANN
10 6379 279.40 279.40 279.37 278.66 0.00 0.03 0.74
17 600 288.25 288.25 288.79 283.49 0.00 0.54 4.76
37 448 293.95 293.95 295.63 291.11 0.00 1.68 2.84
20 7159 277.75 277.75 276.81 275.79 0.00 0.94 1.96
17 779 284.90 284.90 284.68 280.51 0.00 0.22 4.39
29 917 289.25 289.25 289.45 285.52 0.00 0.20 3.73
30 6862 273.35 273.35 271.73 271.94 0.00 1.62 1.41
18 586 281.15 281.15 280.18 277.34 0.00 0.97 3.81
31 690 284.80 284.80 285.06 282.91 0.00 0.26 1.89
40 5055 264.95 264.95 263.89 266.10 0.00 1.06 1.15
15 255 274.10 274.10 272.39 271.02 0.00 1.71 3.08
23 166 277.05 277.05 276.90 274.66 0.00 0.15 2.39
31 386 279.05 279.05 279.88 278.25 0.00 0.83 0.80
50 12 621 265.35 265.35 263.19 262.66 0.00 2.16 2.69
21 724 269.65 269.65 269.87 267.06 0.00 0.22 2.59
30 910 271.55 271.55 274.17 271.26 0.00 2.62 0.29
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Results of the presented models in comparison with the experimental data by Ross and Toczylkin79 for ethane hydrate in the aqueous solution of triethylene glycol (TEG) are summarized in Table 10. Except for two data points, the AdaBoost-CART model regenerated all other targets without error. On the other hand, the ANFIS and ANN models reproduced the data with an average error of 2.9 and 0.93 K, respectively.

Table 10. Results of the Presented Models in Comparison with the Experimental Data Reported in ref (79) for Ethane Hydrate.

    T (K) error (K)
TEG (wt %) P (kPa) exp. AdaBoost ANFIS ANN AdaBoost ANFIS ANN
40 1970 275.0 275.0 277.01 275.65 0.00 2.01 0.65
2300 275.8 275.8 277.05 276.16 0.00 1.25 0.36
3300 277.9 277.9 277.19 277.36 0.00 0.71 0.54
20 770 281.7 283.0 279.61 280.70 1.30 2.09 1.00
33 570 283.0 283.0 281.39 283.24 0.00 1.61 0.24
20 790 273.7 273.7 281.25 272.84 0.00 7.55 0.86
1290 276.5 278.0 281.32 277.35 1.50 4.82 0.85
1540 278.0 278.0 281.36 278.45 0.00 3.36 0.45
2630 283.0 283.0 281.51 280.43 0.00 1.49 2.57
9720 285.5 285.5 282.49 287.05 0.00 3.01 1.55
28 270 288.0 288.0 285.07 287.08 0.00 2.93 0.92
36 270 289.0 289.0 286.18 288.80 0.00 2.82 0.20
10 1000 277.0 277.0 283.49 277.13 0.00 6.49 0.13
1800 282.0 282.0 283.60 281.65 0.00 1.60 0.35
3720 286.3 286.3 283.87 283.07 0.00 2.43 3.23
23 270 289.0 289.0 286.58 287.94 0.00 2.42 1.06
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Table 11 gives the outputs of the presented models versus the experimental data of Ng and Robinson66 for propane hydrate in methanol (MeOH) solution. The maximum error obtained using the AdaBoost-CART model is equal to 0.93 K. The worst estimations of the ANFIS and ANN models have errors of 2.23 and 7.68 K, respectively. Table 12 shows the outcomes of the developed tools in comparison with the experimental data reported by Holder and Godbole92 for i-butane hydrate in pure water. For a pressure of 35.1 kPa, the estimation of the AdaBoost-CART model has a 2.8 K deviation from the experimental HDT. Under other conditions, the results of the AdaBoost-CART model are better than the estimations of the ANFIS and ANN models.

Table 11. Results of the Presented Models in Comparison with the Experimental Data Reported in ref (66) for Propane Hydrate.

    T (K) error (K)
MeOH (wt %) P (kPa) exp. AdaBoost ANFIS ANN AdaBoost ANFIS ANN
5.00 234 272.12 272.12 272.12 271.77 0.00 0.00 0.35
259 272.58 272.58 272.71 271.99 0.00 0.13 0.59
316 273.28 273.28 273.60 272.50 0.00 0.32 0.78
405 274.18 274.18 274.25 273.27 0.00 0.07 0.91
468 274.79 274.79 274.44 273.79 0.00 0.35 1.00
794 275.02 274.97 274.60 276.33 0.05 0.42 1.31
1720 275.09 274.97 277.16 282.77 0.12 2.07 7.68
6340 274.97 274.97 274.71 274.16 0.00 0.26 0.81
10.39 185 268.30 269.23 268.76 271.17 0.93 0.46 2.87
228 269.23 269.23 269.17 271.23 0.00 0.06 2.00
352 271.07 271.07 269.69 271.40 0.00 1.38 0.33
360 270.93 271.07 269.71 271.41 0.14 1.22 0.48
415 271.59 271.59 269.78 271.48 0.00 1.81 0.11
434 271.82 271.59 269.80 271.51 0.23 2.02 0.31
737 272.07 272.07 269.87 271.89 0.00 2.20 0.18
984 272.10 272.07 269.87 272.18 0.03 2.23 0.08
6510 272.08 272.08 269.98 272.45 0.00 2.10 0.37
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Table 12. Results of the Presented Models in Comparison with the Experimental Data Reported in ref (92) for i-Butane Hydrate.

  T (K) error (K)
P (kPa) exp. AdaBoost ANFIS ANN AdaBoost ANFIS ANN
17.6 241.4 241.4 242.26 241.37 0.00 0.86 0.03
20.2 243.4 243.4 243.90 243.38 0.00 0.50 0.02
26.4 248.4 248.4 247.60 246.10 0.00 0.80 2.30
35.1 253.7 256.5 252.25 253.69 2.80 1.45 0.01
42.8 256.5 256.5 255.89 256.50 0.00 0.61 0.00
53.5 259.7 259.7 260.25 259.69 0.00 0.55 0.01
66.4 263.3 263.3 264.58 263.20 0.00 1.28 0.10
85.5 268.1 268.1 269.37 267.94 0.00 1.27 0.16
89.7 269.4 269.4 270.18 268.85 0.00 0.78 0.55
91.3 269.5 269.5 270.47 269.43 0.00 0.97 0.07
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The reported data by Mohammadi and Richon34 for hydrogen sulfide hydrate in a solution of salts and/or alcohol and the results of the developed models are reported in Table 13. As can be seen, the best results are obtained from AdaBoost for all of the reported thermodynamic conditions. The experimental data by Nixdorf and Oellrich46 for the nitrogen hydrate + water system versus the outputs of the presented AdaBoost-CART, ANFIS, and ANN models are given in Table 14. At a pressure of 17 668.0 kPa, the ANFIS model provides better estimation as compared to the AdaBoost-CART model. However, the overall performance of the AdaBoost-CART model is better than that of both the ANFIS and ANN models. Table 15 gives the estimations of the presented models in comparison with the experimental data published by Kamari and Oyarhossein113 for natural gas hydrate in pure water. The results prove the ability of AdaBoost-CART to estimate the HDT of the studied hydrate system of natural gas.

Table 13. Results of the Presented Models in Comparison with the Experimental Data Reported in ref (34) for Hydrogen Sulfide Hydrate.

concentration (wt %) P (kPa) T (K) error (K)
NaCl CaCl2 MeOH EG   exp. AdaBoost ANFIS ANN AdaBoost ANFIS ANN
5 0 0 15 180 272.7 273.4 274.93 274.48 0.70 2.23 1.78
315 278.3 278.3 277.34 276.28 0.00 0.96 2.02
584 284.4 284.4 282.16 280.39 0.00 2.24 4.01
1082 290.1 290.1 291.07 290.23 0.00 0.97 0.13
5 0 10 0 189 273.4 273.4 275.16 272.06 0.00 1.76 1.34
288 277.2 277.2 277.76 276.03 0.00 0.56 1.17
456 281.6 281.6 280.61 281.54 0.00 0.99 0.06
777 286.8 286.8 284.69 287.84 0.00 2.11 1.04
0 0 30 0 236 267.5 267.5 270.79 266.79 0.00 3.29 0.71
338 271.1 271.1 271.08 270.41 0.00 0.02 0.69
496 274.8 274.8 271.52 274.98 0.00 3.28 0.18
0 0 50 0 201 254.1 254.1 253.89 254.31 0.00 0.21 0.21
328 260.0 260.0 258.52 259.18 0.00 1.48 0.82
464 264.2 264.2 262.93 263.57 0.00 1.27 0.63
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Table 14. Results of the Presented Models in Comparison with the Experimental Data Reported in ref (46) for Nitrogen Hydrate.

  T (K) error (K)
P (kPa) exp. AdaBoost ANFIS ANN AdaBoost ANFIS ANN
16 935 273.67 273.67 273.61 273.13 0.00 0.06 0.54
17 668 274.07 274.20 274.00 273.50 0.13 0.07 0.57
19 521 275.11 275.11 274.96 274.40 0.00 0.15 0.71
20 748 275.77 275.77 275.55 274.97 0.00 0.22 0.80
24 092 277.27 277.27 277.02 276.40 0.00 0.25 0.87
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Table 15. Results of the Presented Models in Comparison with the Experimental Data Reported in ref (113) for Gas Mixture Hydrate (C1 = 81.55%, CO2 = 3.31%, N2 = 0.17%, C2 = 5.37%, i-C4 = 2.23%, n-C4 = 0.51%, i-C5 = 1.00%, n-C5 = 0.52%, C6 = 0.45%, C7 = 0.75%, C8+ = 0.70%, H2S = 1.05%, and C2H4 = 2.39%).

  T (K) error (K)
P (kPa) exp. AdaBoost ANFIS ANN AdaBoost ANFIS ANN
848.0 274.78 274.78 278.73 281.59 0.00 3.95 6.81
1620.2 279.80 279.80 280.53 281.70 0.00 0.73 1.90
2275.2 282.03 282.03 282.05 281.80 0.00 0.02 0.23
3102.6 284.87 284.87 283.96 281.92 0.00 0.91 2.95
4136.8 286.53 286.53 286.26 282.05 0.00 0.27 4.48
5308.9 288.18 288.18 288.61 282.20 0.00 0.43 5.98
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5. Conclusions

A type of ensemble method, AdaBoost algorithm, presented in this work was utilized to combine CART models to present models capable of representing/estimating the HDT of the systems of C1, C2, C3, i-C4, H2S, N2, and gas mixture. This study is the first to investigate the clathrate hydrate systems using the AdaBoost-CART technique. Furthermore, the hydrate systems mentioned earlier were modeled by employing the ANFIS and ANN approaches. To achieve the objective of this study, more than 3500 experimental data representing solid–ice–vapor/gas and solid–liquid–vapor/gas equilibrium conditions of the C1, C2, C3, i-C4, H2S, N2, and gas mixture hydrate systems were collected from the literature.

The obtained results reveal that the developed models on the basis of the AdaBoost-CART method accurately estimate/represent the equilibrium HDT of the investigated hydrate systems with AARD% between 0.03 and 0.07. To evaluate the estimation capability of the proposed AdaBoost-CART models, ANFIS and ANN methods were selected as the basis of comparison. It was found that the ANFIS and ANN models cannot rival the AdaBoost-CART models in terms of accuracy and reliability.

The machine learning approach can be further studied by its comparison with other methods such as molecular dynamics simulation, Monte Carlo simulation, and statistical thermodynamics.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.1c03214.

  • Digraphs of the created Adaboost-CART models (PDF)

Author Contributions

Conceptualization, S.K. and M.M.G.; methodology, S.K. and M.M.G.; validation, S.K., S.A.F., and M.M.G.; formal analysis, S.K., S.A.F., and M.M.G.; investigation, S.K., S.A.F., and M.M.G.; resources, S.K., S.A.F., and M.M.G.; data curation, S.K., S.A.F., and M.M.G.; writing: original draft preparation, S.K., S.A.F., and M.M.G.; writing: review and editing, A.H.M.; visualization, M.M.G.; and supervision, A.H.M. All authors have read and agreed to the published version of the manuscript.

The authors declare no competing financial interest.

Supplementary Material

ao1c03214_si_001.pdf (1.6MB, pdf)

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