1. Introduction
The thermodynamic model for natural gas hydrate formation conditions can be traced back to the van der Waals and Platteeuw (vdW-P) model, which was proposed based on classical adsorption theory by van der Waals and Platteeuw [
1]. Subsequently, many efforts have been conducted for the improvement of the vdW-P model. Notably, Parrish and Prausnitz [
2] made a significant contribution by introducing an empirical method to compute the Langmuir constant. This improvement gave rise to the classical P-P model, greatly simplifying the computation of the Langmuir constant and successfully extending the application of the vdW-P model to multi-component systems. However, in alcohol-containing systems, polar substances like water/alcohol can undergo complex self-association and cross-association interactions, affecting component fugacities, activities, and other thermophysical properties, resulting in dislocation of the gas hydrate formation conditions. Merely improving the thermodynamic model for hydrate formation may not yield the desired predictive accuracy. The traditional models attempt to address the impact of polar molecules by combining hydrate thermodynamic models, EoSs, and activity models [
3,
4]. However, this method complicates the modeling process and is inconvenient for application, making it challenging to accurately depict various molecular interactions and their impact on gas hydrate formation [
5].
In the process of solving thermodynamic models for natural gas hydrate and multiphase equilibrium calculations, various phase equilibrium parameters such as fugacity and activity, as well as thermophysical parameters, including compressibility factor and enthalpy, are involved. EoS can calculate these parameters over a wide range of temperature and pressure, serving as the foundation of most thermodynamic models. In order to describe liquid phase non-ideality, many attempts have been made to improve EoS. Masoudi and Tohidi et al. [
6] employed the modified Patel-Teja EoS to predict hydrate formation conditions in multi-component systems containing alcohols. The results showed higher predictive accuracy within the pressure range of 50 MPa. Delavar and Haghtalab et al. [
7] utilized the Chen-Guo model, Soave-Redlich-Kwong (SRK) EoS, and Huron-Vidal (HV) mixing rules to describe the hydrate, gas, and aqueous phases in systems containing acidic gases, alcohols, and electrolytes. The predicted results were in good agreement with experimental data. Promisingly, there is potential to develop more reliable and widely applicable natural gas hydrate prediction models by enhancing equations of state or introducing correction factors.
Due to the incapacity of traditional EoS to represent hydrogen bonding interactions among polar substances like water and alcohols, significant deviation can arise in activity calculation. Kontogeorgis et al. [
8] introduced an associating term within the SRK equation of state to describe hydrogen bonding interactions, resulting in the development of the CPA EoS. For associating fluids, the most critical aspect is the study of associating models. For the associating models of polar molecules like water and alcohols, the most widely applied are the 2B and 4C associating models proposed by Huang and Radosz in 1990 [
9]. These models consider hydrogen and oxygen atoms individually as electron, proton donors or acceptors. In recent years, Kruger and Qvistgaard et al. [
10,
11] have proposed various associating models for ethylene glycol (MEG) and glycerol (TEG). These models consider the dipole sites formed by hydrogen and oxygen atoms on one side of the molecular structure, which exhibit a higher probability of association and provide a better description of the associating interactions among polar molecules. With the development of new associating models and bonding methods, further evaluation of the performance of these associating models is necessary. The application of CPA in predicting hydrate formation conditions in alcohol-containing systems is subject to further research. Additionally, when CPA EoS introduces new associating models, it is imperative to adjust the binary interaction coefficients.
In conclusion, we have proposed a novel unified thermodynamic framework based on the P-P model and the CPA EoS for calculating the hydrate formation condition in multicomponent systems containing gas/water/alcohol. Within this framework, we have replaced the traditional activity model with the association term in the CPA EoS, effectively characterizing non-ideality in solution. Furthermore, various association models used for alcohol substances such as MEOH, MEG, DEG, and TEG are evaluated. Due to the consideration of multiple association models in our modeling process, it was necessary to adjust the existing binary interaction coefficients. Therefore, we collected a substantial amount of vapor-liquid phase equilibrium experimental data and conducted regression on these parameters. Finally, experimental data are used to validate the model.
3. Regression of Binary Interaction Coefficients for Alcohol-Water-Gas
The SRK EoS employs binary interaction coefficients (
kij) from the single-fluid mixing rules to describe intermolecular forces. The CPA EoS, upon introducing association terms, necessitates the adjustment of the pre-existing binary interaction coefficients. These coefficients directly affect the precision of phase equilibrium calculation [
5,
13]. This paper involves binary interaction coefficients between methane-solvent and solvent-solvent molecules. The Britt-Luecke algorithm is employed to regress vapor-liquid equilibrium experimental data [
14]. The objective function used is shown in Equation (16).
where
is the calculated value;
is the experimental value;
is the variance.
To evaluate the predictive accuracy of the model, the Average Relative Deviation (ARD) between the model predictions and experimental values is defined as shown in Equation (17).
Many researchers believe that in gas molecule mixtures, the binary interaction coefficients depend to some extent on temperature [
15]. When considering temperature, the form of the regression equation for binary interaction coefficients mainly depends on the actual relationship between the coefficients and temperature for different substances. We considered the distribution of experimental data and conducted a comparative analysis of various equation forms, such as linear equations and quadratic equations, among others. The linear equation was ultimately chosen because it best approximated the actual distribution of the experimental data, as shown in Equation (18).
where
A and
B are the Regression coefficients.
The regression yielded the binary interaction coefficients between methane and alcohol/water molecules, as shown in
Table 5 and
Table 6. It should be noted that, for the purpose of comparing the predictive performance of the new and old association models, the superscript “a” in the table represents the use of the 4C association model for that component, while the superscript “b” indicates the use of the 4F model.
4. Results and Discussion
In the previous section, the models for the association of alcohol and water molecules, the mixing rules, and the binary interaction coefficients for alcohol, water, and gas molecules were established. Subsequently, the CPA EoS will be combined with the P-P hydrate formation condition model, and a comparative analysis will be conducted using experimental data to assess the predictive performance of the model for hydrate formation conditions in alcohol-containing systems.
4.1. Prediction of Component Fractions in Methane-Alcohol-Water Systems
In the methane-alcohol-water system, there are both cross-association interactions between alcohol and water molecules and self-association interactions of alcohol and water molecules, which significantly affect the accuracy of EOS. Therefore, component fractions will be predicted separately in both the methane-alcohol/water binary system and the methane-alcohol-water ternary system to validate the predictive accuracy of CPA EoS and the applicability of the association model and mixing rules.
4.1.1. Prediction of Component Fractions in Methane-Alcohol/Water Binary Systems
To validate the predictive capabilities of the proposed model for component fractions in the methane-alcohol/water binary system, binary phase equilibrium data for the methane-MeOH/MEG/DEG/TEG/H
2O systems were collected. Within the temperature range of 273.15 to 444.26 K and pressure range of 0.1 to 106.6 MPa, methane mole fraction in the liquid phase was predicted using the CPA EoS in combination with single-fluid and CR-1 mixing rules. Prediction errors are shown in
Table 7, and some computational results are presented in
Figure 1. It can be seen that the average relative deviation of the mole fraction of methane in the liquid phase predicted by the proposed model in the methane-alcohol/water system ranges from 3.16% to 6.42%. The reason for the significant error is the low solubility of methane in the liquid phase, especially at low pressures, where the deviation is more pronounced. For example, in the methane-MEG system, at a temperature of 398 K and a pressure of 0.33 MPa, the experimental value of methane’s molar fraction in the liquid phase is 0.0005, while the predicted value is 0.0007, resulting in only a difference of 0.0002. However, this small difference leads to a relative deviation of 44%. Therefore, except for a few data points at low pressure, CPA EoS exhibits high accuracy in predicting the liquid phase composition for the methane-alcohol/water system. In the methane-alcohol system, due to the presence of self-association of alcohol molecules, the suitability of the association model is more critical. For the methane-MEG/TEG systems, the 4F association model demonstrates superior predictive performance.
4.1.2. Prediction of Component Fractions in Methane-Alcohol-Water Ternary Systems
To validate the predictive capabilities of the proposed model for component fractions in the methane-alcohol-water ternary system, ternary phase equilibrium data for the methane-MeOH/MEG/DEG/TEG-H
2O systems were collected. Methane molar fractions in the liquid phase were predicted in the temperature range of 278.15 to 342.69 K and pressure range of 0.1 to 46.92 MPa. Prediction errors are shown in
Table 8, and some computational results are presented in
Figure 2. In the methane-alcohol-water ternary system, the average relative deviation of the model’s predictions for methane molar fractions in the liquid phase ranged from 4.14% to 5.25%. This indicates that the model exhibits a high level of accuracy in predicting gas-liquid phase equilibrium in the methane-alcohol-water system. Furthermore, for the gas-liquid phase equilibrium predictions in the methane-MEG/TEG-water ternary systems, the performance of the 4F association model remains superior to the 4C association model.
4.2. Prediction of Gas-Hydrate Formation Conditions in Gas-Alcohol-Water Systems
Based on the analysis in
Section 4.1, it is evident that the combination of CPA EoS with association models and mixing rules provides an effective description of intermolecular interactions in alcohol-containing systems, demonstrating high predictive accuracy for CH
4 component fractions. In this section, we will combine the P-P hydrate formation condition model to evaluate the applicability of the proposed model for predicting hydrate formation conditions in various systems. Furthermore, to comprehensively evaluate the advantages of this model, traditional Peng-Robinson (PR) EoS and SRK EoS will be introduced for comparative analysis. For polar systems, in order to provide more compelling comparative results, the HV mixing rules were introduced into the PR and SRK EoS to replace the van der Waals mixing rules.
4.2.1. Prediction of Gas-Hydrate Formation Conditions in Gas-Water Binary Systems
In the gas-water binary system, a comparative analysis is conducted to assess the predictive accuracy of the PR-HV, SRK-HV, and CPA EOS combined with the P-P model for hydrate formation conditions. We primarily focus on CH
4/CO
2 hydrates and use the hydrate formation condition data measured by Nakano [
28], Dyadin [
29], and Takenouchi [
30] as reference values. Within the pressure range of 13 to 186 MPa, we employed three models to predict the CH
4/CO
2 hydrate formation temperatures. The error analysis is presented in
Table 9 and comparison between predicted values and experimental data is depicted in
Figure 3. As shown in
Table 9, the CPA EOS, which considers molecular association, exhibits a lower deviation between predicted and experimental values compared to the PR-HV/SRK-HV EOS combined with the P-P model. Due to methane being an inert gas and not participating in molecular association, the primary interaction in the methane-water system is the self-association of water molecules. In
Figure 3a, it can be observed that the combination of the PR-HV EOS with the P-P model overestimates the hydrate formation pressure, while the combination of the CPA/SRK-HV EOS with the P-P model both performs well. In
Figure 3b, the combination of the PR-HV/SRK-HV state equations with the P-P model exhibits a larger predictive deviation. This indicates that in the CO
2-H
2O system, the cross-association between carbon dioxide and water amplifies the computational deviation of these two models. In contrast, the method proposed in this work demonstrates high predictive accuracy in both CH
4-H
2O and CO
2-H
2O systems.
4.2.2. Prediction of Gas-Hydrate Formation Conditions in Gas-Alcohol-Water Ternary Systems
To comprehensively compare the predictive abilities of the three models for hydrate formation conditions in the “methane-alcohol-water” ternary system, we collected multiple sets of hydrate formation experimental data. The data included four types of alcohol solutions: MeOH, MEG, DEG, and TEG, with alcohol concentrations ranging from 6 to 65 wt%. The comparison between the calculated values and the experimental values is shown in
Figure 4, and the calculation errors are presented in
Table 10.
From
Table 10, it can be observed that, even though the HV mixing rules account for the interactions of polar molecules, the average relative deviations of PR-HV and SRK-HV still range from 4.32% to 9.80% and 3.9% to 9.34%, respectively. However, the average relative deviations of the CPA EoS predictions are 1.93% to 4.78%, indicating a significant improvement in accuracy. From
Figure 4, it can be observed that the three models generally exhibit consistent predictive trends. The main deviations occur under conditions of high alcohol concentration or elevated pressure. Taking the prediction results of methane-methanol-water system hydrate formation conditions in
Figure 4a as an example, at a methanol mass concentration of 65 wt%, all three models have overestimated the hydrate formation pressure. For example, at a temperature of 240.3 K, the experimental hydrate formation pressure is 10.31 MPa, while the predicted values for the three models, CPA EoS, PR-HV, SRK-HV combined with the P-P model, are 15.16 MPa, 16.41 MPa, and 16.06 MPa. However, in
Figure 3b, the prediction errors remain within a normal range, with no sudden significant increase in errors. Furthermore, we examined the sources of the experimental data and found that these data were obtained from different references. Therefore, we believe that the significant prediction errors in
Figure 3a may be attributed to experimental errors inherent in the data source. From an overall perspective, the CPA EoS, due to its consideration of molecular association, exhibits better predictive performance.