3.1. Binary system CH4 + CO2
The CH4 + CO2 vapor-liquid equilibrium curve was first simulated at 250 K and the results were compared to experimental data provided by Wei et al. (1995). Additionally, the values predicted by Do et al. (2010) with Monte Carlo simulations were also used for comparison. The results are presented in Fig. 2.
It is important to say Do et al. chose an explicit hydrogen force field to represent methane, the TraPPE-EH, and used the EPM force field for CO2. They also used a spherical cutoff of 12 A with tail correction even for the electrostatic contribution. This approach towards the electrostatic interactions reduces the computational time, allowing the use of a more rigorous force field, like the TraPPE-EH. However, this simple truncation can introduce relevant prediction errors associated with incorrect representation of the electrostatic interactions. It is convenient to remember that the electrostatic interactions decay slower with the distance than Lennard-Jones interactions, what makes a simple truncation a riskier approach for electrostatic interactions.
It is possible to see that the DSF predictions were superior to those reported by Do et al. for the dew point section of the equilibrium curve. One could expect the rigorous force field makes up for the less rigorous electrostatic calculation method, but in this region of the curve, it is not what is observed. The DSF method, more rigorous than a simple truncation, performed better even with a united-atom force field. Both approaches overestimated slightly the methane content for that region.
For the bubble point curve, the predictions from Do et al. were superior, being very close to the experimental data. DSF simulations, although with inferior performance, could correctly describe the equilibrium. Considering the whole curve, the GERG-2008 equation of state (Kunz and Wagner 2012) provided the best description of the equilibrium. It is important to remember that this equation was parameterized considering 20 main components present in natural gas streams, therefore its predictivity is restricted when introducing other components, limiting the search for new solutions in the specialized industry.
The CH4 + CO2 binary equilibrium was also simulated at 270 K and the results were compared to the experimental data reported by Wei et al. (1995) and to Monte Carlo simulations provided by Ramdin et al. (2016). They are shown in Fig. 3. In this case, the authors used the same force fields as the present work, but the electrostatic calculations were done by the Ewald method.
For this temperature, the comparison is more direct, given that the same force fields were used. The simulations of the present work and those by Ramdin et al. differ mostly by the electrostatic interactions calculation method and by the cutoff radius, as Ramdin et al. used the Ewald method and 14 A, respectively. For both the bubble point and dew point curves, DSF simulations provided better predictions. This result may sound unexpected, give the more rigorous character of the Ewald method. Some factors may explain this, like the cutoff radius. Preliminary simulations for this work using the same methodology showed better results for a cutoff radius of 12 A rather than 14 A, which may point at a systematic overestimation of the methane content with the increase of the cutoff radius.
On the other hand, Monte Carlo methods are inherently influenced by some degree of stochasticity. It can be seen that DSF predictions were not vastly superior, thus when the uncertainties are considered, it is reasonable to say the predictions are approximately equivalent. Unfortunately, Ramdin et al. don’t provide such uncertainties. The uncertainties of the present work, as well as the values provided by the simulations, are provided in Supplementary Information.
Finally, it is possible that small differences are found when using different simulators, as shown by Gowers et al. (2017). Ramdin et al. used the RASPA simulator (Dubbeldam et al. 2017). Anyway, the DSF simulations were able to predict competently the vapor-liquid equilibria of the CH4 + CO2 binary system, being quantitatively comparable to experimental data, equation of state and Monte Carlo simulations with Ewald method.
Another property that is important for the design of new processes is the density. The quality of the predictions for the density was also investigated and is represented in Figs. 4 and 5. Arai et al. (1971) reported the experimental density behavior for this system at 253 and 273 K. All the data shown in Figs. 4 and 5 seen to be in qualitative and quantitative agreement with the values presented by Arai et al. Unfortunately, they didn’t provide the exact values found in experiments, thus a more in-depth, accurate comparison is difficult. Like for the description of the equilibria compositions, the DSF simulations could describe the property correctly, with quality comparable to more rigorous methods, and the data, as well as the uncertainties, can be found in Supplementary Information.
3.2. Binary system CO2 + H2O
In the case of the CO2 + H2O binary, the vapor-liquid equilibrium was simulated at 348.15K and compared to experimental data (Zawisza and Boguslawa 1981; Wiebe 1941; Coan and King Jr. 1971; Wiebe and Gaddy 1939), as well as Monte Carlo simulations by Vorholz et al. (2000) and the predictive Soave-Redlich-Kwong and Schwartzentruber-Renon equations of state (AspenTech 2017). The results are presented in Figs. 6 and 7. Vorholz simulated this system using the EPM and SPC potentials for carbon dioxide and water, respectively.
Like what was seen for the first system at 250 K, the DSF simulations, despite not giving the most accurate predictions among all analyzed, were capable of both correctly representing the behavior of the system and providing predictions quantitatively comparable to Ewald simulations and equation of state. There is, though, a slight underestimation of the CO2 mole fraction that increases with increasing pressure, what can also be observed for the simulation data by Vorholz et al (2000).
For the study of these data, it is important to point certain controversy regarding the experimental values. Aasen et al. (2017) suggested that those bubble point molar fractions reported by Wiebe and Gaddy (1939) are, actually, slightly under the real values. They point to this trend through the evaluation and comparison of other literature reports. On the other hand, Diamond and Akinfiev (2003), doing an analysis of data quality from several references, attributed the highest level of confidence to the values published by Wiebe and Gaddy. Another relevant point is that the study by Aasen et al. comprised only the temperatures of 298.1 and 323 K and the alleged deviations seem to decrease with increasing temperature. Hence, it is expected that the deviations, already small even for lower temperatures, are even smaller for 348.15 K, if they really exist.
For the CO2-rich region, the experimental data present a notable behavior of stabilization close to 0.99 CO2 mole fraction with increasing pressure. For even higher pressures, even a decrease in this fraction can be observed, as shown by Aasen et al. Besides providing predictions with deviations comparable to other approaches, the DSF simulations were capable of correctly representing the mole fraction stabilization behavior, what doesn’t happen when using the equation of state, for instance.
The density of each phase was also simulated and is shown as a function of pressure in Fig. 8. The predictions by other approaches are provided as well. What draws attention is the severe disagreement between the PSRK equation of state and the simulations for the liquid phase density. Though, the incapability of reliably predicting the liquid phase density is a well-known flaw of the SRK equation of state and its modifications (Ashour et al. 2011). Because of that, the Peng-Robinson (PR) equation of state was also included in the comparison. The exact values of the simulations and the uncertainties can be found in the Supplementary Information.
3.3. Binary system CH4 + H2O
The prediction of CH4 + H2O equilibrium turned out to be especially tricky, because the non-ideality of the system results in extremely low mutual solubilities. This fact resulted in the need of increasing the number of the molecules in the simulation boxes to avoid finite size effects, what, ultimately, led to simulations multiple times more computationally demanding. The results are presented in Figs. 9 and 10. All values and uncertainties are presented in Supplementary Information.
For the water-rich region of the curve, the behavior of the system was well-represented and predictions close to experimental data were obtained. For the methane-rich region, although some bigger deviations can be seen, the overall pressure-composition behavior was correctly represented, which can be considered satisfactory, especially considering the particular complexity of the system. Here is convenient to remember that the electronegativity difference between oxygen and hydrogen promotes intense polarization effects and hydrogen bonds, what leads to the formation of association structures. Moreover, these structures can have a variable number of water molecules. Hence, the correct representation of the water molecule and their association structures in the bulk is not trivial, especially with rigid potentials (Shvab 2014). The utilization of non-rigid potentials, though, increase simulation times. The challenge increases even further when far from ideal mixtures are simulated. Even a heavily parameterized equation of state like GERG-2008 presents some difficulty in keeping the prediction accuracy seen before, for less challenging systems, like for CH4 + CO2.
The underestimation of the water content in methane, though, is consistent with what was verified by Errington et al. (1998). The same was observed for water-ethane solubility. Bolton et al. (2009) evaluated the water solubility in decane and found the same trend for the classic Lorentz-Berthelot mixing rule and many of its modifications, especially for temperatures under 450 K. Moreover, their simulations showed the same behavior is found for longer hydrocarbons, up to 300 carbons.
The comparison between the densities provided by equations of state and Monte Carlo simulations for the CH4 + H2O system is given in Fig. 11, showing again an almost perfect agreement between simulations and equation of state. The exact values and uncertainties are presented in Supplementary Information.