# Simulation Study on the Influence of Gas Mole Fraction and Aqueous Activity under Phase Equilibrium

^{1}

^{2}

^{*}

## Abstract

**:**

_{4}, CO

_{2}, N

_{2}and O

_{2}, the gas mole fraction in aqueous phase as one of phase equilibrium conditions was proposed, and a simplified correlation of the gas mole fraction was established. The gas mole fraction threshold maintaining three-phase equilibrium was obtained by phase equilibrium data regression. The UNIFAC model, the predictive Soave-Redlich-Kwong equation and the Chen-Guo model were used to calculate aqueous phase activity, the fugacity of gas and hydrate phase, respectively. It showed that the predicted phase equilibrium pressures are in good agreement with published phase equilibrium experiment data, and the percentage of Absolute Average Deviation Pressures are given. The water activity, gas mole fraction in aqueous phase and the fugacity coefficient in vapor phase are discussed.

## 1. Introduction

_{2}capture and storage (CCS) technology [7]. In the last couple of years, vast quantities of natural gas hydrate in the permafrost and deep seabed was found, which is twice as much as the amount of the other fossil fuels combined under a conservative estimate [8]; it makes natural gas hydrate as a kind of potential energy possible. However, gas hydrates can block oil and gas pipelines with high pressure and low temperature inside subsea oil and gas flow line [9]. Furthermore, the methane trapped in gas hydrates is a potent greenhouse gas [10]. In order to solve these problems, scholars conducted a lot of studies and found that adding thermodynamic inhibitors can effectively change the conditions of hydrate formation into higher pressure and lower temperature. On the contrary, the formation conditions of hydrate can be changed to lower pressure and higher temperature by adding thermodynamic promoters. Regardless of whether inhibiting or promoting hydrate formation, it is necessary to predict phase equilibrium conditions for the above-mentioned applications, and it is important to use reliable and accurate predictive models for predicting hydrate phase equilibria [11].

_{4}/CO

_{2}/H

_{2}S/N

_{2}), in which parameters were determined from the phase equilibrium data. Moreover, among the models of aqueous phase activity, the UNIQUAC model [23] and the modified UNIFAC model [24] were constantly used to calculate aqueous phase. Delavar and Haghtalab [25,26] used the Chen-Guo and UNIQUAC models, referring the Soave-Redlich-Kwong-Huron-Vidal equation of state (SRK-HV EoS) conjunction with the Henry’s law, to calculate the gas hydrate formation conditions. Dehaghani and Karami [27] employed the predictive Soave-Redlich-Kwong equation of state (PSRK-EoS) along with the modified Huron-Vidal (MHV1) missing rule and UNIQUAC model to calculate fugacity and activity coefficient of water in equilibrated fluid phases. Klauda and Sander [19,28] applied the modified UNIFAC model and PSRK-EoS coupled with the classical mixing rules, and the results obtained were in good agreement with the experimental data.

_{4}, CO

_{2}, N

_{2}or O

_{2}) and distilled water system; the mixed gas system and the additive system will be further studied in future work. Finally, the results calculated are compared with the experimental data in literatures, and the calculated fugacity coefficient of vapor phase and water activity are given.

## 2. Thermodynamic Framework

#### 2.1. Thermodynamic Model of Vapor Phase

_{m}represents the mole volume, which is obtained by solving the cubic equation derived from Equation (3), and the value is the same as the largest real root of the equation [35]; a and b are parameters of PSRK.

_{i}and b

_{i}of pure component i can be calculated from the critical properties T

_{c,i}and P

_{c,i}.

_{r}= T/T

_{c}; the pure fluid parameter c

_{1}is taken from the study of Holderbaum and Gmehling [35]. The PSRK mixing rule is written as:

_{i}stands for the activity coefficient of component i calculated by UNIFAC model; the recommended value of A

_{1}= -0.64663 in PSRK model. The activity coefficient γ

_{i}is a correction factor that accounts for deviations of real systems from that of an ideal solution, which can be estimated from chemical models (such as UNIFAC). Thus, the fugacity coefficient is given by:

_{i}is the fugacity coefficient of component i; z = Pv

_{m}/RT.

#### 2.2. Thermodynamic Model of Hydrate Phase

_{i}denotes the mole fraction of the basic hydrate formed by gas component i, and z

_{i}= 1 for pure gas; θ

_{j}represents the fraction of the linked cavities occupied by the gas component j; α is the ratio of linked cavities and basic cavities [39], which equals 1/3 for sI hydrates and 2 for sII hydrates, respectively.

_{j}denotes the fugacity of component j in vapor phase calculated by PSRK method; c

_{j}stands for the rigorous Langmuir constant, which is calculated from the Lennard-Jones potential function.

_{i}

^{0}represents the fugacity of component i in vapor phase in equilibrium with the unfilled pure basic hydrate i (∑θ

_{j}= 0) [21]. According to the Chen-Guo model, it can be calculated as:

_{ij}is the binary interaction coefficient which stands for the interplays between gas molecule i in the basic hydrate and gas molecule j in the linked cavities. A

_{i}′, B

_{i}′ and C

_{i}′ are the Antoine constants, as reported by Chen and Guo [17]. β is the function of water volume difference between that in the unfilled basic hydrate phase and the water phase, and the large cavity number per water molecule [20], β = 0.4242 K/bar for sI hydrates, β = 1.0224 K/bar for sII hydrates. a

_{w}is the activity of water in aqueous phase, which is calculated by the UNIFAC method. For sI and sII hydrates, λ

_{2}= 3/23 and λ

_{2}= 1/17, respectively.

#### 2.3. Thermodynamic Model of Aqueous Phase

_{i}is the Henry’s constant of component i, given by the Krichevsky-Kasarnovsky correlation [36,40]; ${\overline{{V}_{i}}}^{\infty}$ is the infinite partial molar volume of the component i in water, given by Heidemann and Prausnitz [29].

_{i}and M

_{p}are the molecular weight of component i and the promoter (inhibitor); wt% stands for the weight percentage of the promoter (inhibitor) in aqueous phase.

_{i}represents the mole fraction of water, gas component and the promoter (inhibitor) in aqueous solutions of a unit mass.

_{i}and θ

_{i}are the volume and surface area fraction of component i, respectively; r

_{i}and q

_{i}are the volume and surface area parameters of component i, respectively. They are calculated by the van der Waals volumes R

_{k}and surface areas Q

_{k}of the individual group k using equations as follows:

_{k}parameters and surface areas parameters Q

_{k}of group k are listed in Table 1.

_{k}stands for the residual activity coefficient of functional group k; ln${\Gamma}_{k}^{\left(i\right)}$ is the residual activity coefficient of group k in the reference solution containing only component i. Both lnΓ

_{k}and ln${\Gamma}_{k}^{\left(i\right)}$ are calculated as:

_{m}and X

_{m}are the surface area fraction and the mole fraction of group m in the mixture, respectively. The group interaction parameter Ψ

_{nm}proposed by Sander et al. [30] is described as:

_{nm}and u

_{mm}are the adjustable group interaction parameters (energy parameters). For each group–group interaction, the two parameters have the relation of u

_{nm}= u

_{mn}. The gas–gas group interaction-energy parameters u

_{nm}and temperature range are given in Table 2.

## 3. Calculation Procedure

## 4. Results and Discussion

_{i}and the infinitely diluted partial molar volume ${\overline{{V}_{i}}}^{\infty}$ are employed to calculate the gas mole fraction in aqueous phase. Heidemann and Prausnitz [29] provided a correlation for solving ${\overline{{V}_{i}}}^{\infty}$ as follows:

_{11}represents the cohesive energy for water, which was evaluated at each temperature from thermodynamic properties tabulated; h

^{0}is the molar enthalpy at the given temperature but at zero pressure, and ${v}_{w}^{s}$ is the molar volume of the saturated liquid [29].

_{2}and O

_{2}systems is mainly related to the temperature increment.

_{4}, CO

_{2}, N

_{2}and O

_{2}in aqueous phase under phase equilibrium condition is shown in Figure 3, Figure 4 and Figure 5. In this work, the gas mole fraction was considered as one of the factors affecting the phase equilibrium. It represents the ratio of the number of gas molecules maintaining the three-phase equilibrium to the number of all molecules in aqueous phase.

^{−3}. Therefore, there may exist a threshold value for the gas mole fraction in aqueous phase. In other words the hydrate will form when the gas mole fraction in aqueous phase reaches a certain threshold value. Furthermore, for methane hydrate, the results in this work are in agreement with the views of Walsh et al. [47] and Guo and Rodger [48]. Walsh et al. suggested that the threshold value of gas mole fraction triggering hydrate formation calculated by the molecular dynamics (MD) simulation was 1.5 × 10

^{−3}. The threshold value is a reasonable explanation for reducing the temperature or increasing the pressure, which could effectively promote the formation of hydrate. This is because lowering the temperature or increasing the pressure will enhance the gas dissolution, which in turn causes the gas mole fraction in aqueous phase exceeding the threshold value, then hydrate forms.

_{2}system reached its maximum value at about 279.5 K. However, the maximum water activity in the CO

_{2}system is only about 0.5658, which is probably because of the effect of the carbonic acid. Nevertheless, the activity of water in aqueous phase decreases almost linearly with temperature increase in nitrogen and oxygen systems, as shown in Figure 5.

_{4}and CO

_{2}. It can be seen the predicted results for all the gas systems are in excellent agreement with the experimental data. It should be noted that the type of carbon dioxide hydrate structure was set to sI, and, because the carbon dioxide gas molecule is too big to be encaged in the linked cavities, the filling rate of the gas molecules in the linked cavities, θ

_{j}, was set to 0, as described by Chen and Guo [16].

_{2}and O

_{2}are displayed in Figure 7. The predicted phase equilibrium pressures are in good agreement with the experiment. It is especially noteworthy that, when calculating oxygen and nitrogen hydrate, the hydrate structure was set to sII, which was based on the ideas proposed by Chen and Guo [16]. This is because the gas molecules of N

_{2}and O

_{2}are small and have a high filling rate in the connected cavities.

^{−3}at 0 °C. A possible reason is that part of carbon dioxide molecules in aqueous phase react with water to form carbonic acid. When the temperature is above 0 °C and below 0 °C, the pressure increment and the temperature decrement become a dominant factor that results in more stability for the carbonic acid and less solubility of carbon dioxide, respectively. However, this analysis should be proved by further study.

## 5. Conclusions

_{4}, CO

_{2}, N

_{2}or O

_{2}in pure water systems. The gas mole fraction in aqueous phase is one of the factors that affect the phase equilibrium of gas hydrate proposed in this work. The gas mole fraction threshold value maintaining the three-phase equilibrium was obtained by reversed phase equilibrium data. Meanwhile, in order to obtain the water activity in aqueous phase, the correlation of the gas mole fraction threshold value in aqueous phase was fitted though UNIFAC model. The calculated water activity can effectively improve the accuracy of the prediction results, and the predicted results of this work are in good agreement with the experimental data reported in the references.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Calculation procedure for the prediction of phase equilibrium pressures at given temperatures.

**Figure 8.**The experimental data and the mole fraction of carbon dioxide in aqueous phase for CO

_{2}+ water systems under the phase equilibrium. The experimental data were reported by Ma et al. [42].

Main Group | Sub Group | Number | R_{k} | Q_{k} |
---|---|---|---|---|

H_{2}O | H_{2}O | 4 | 1.506 | 1.732 |

CO_{2} | CO_{2} | 56 | 2.592 | 2.522 |

CH_{4} | CH_{4} | 57 | 2.244 | 2.312 |

O_{2} | O_{2} | 58 | 1.764 | 1.910 |

N_{2} | N_{2} | 60 | 1.868 | 1.970 |

Gas | Temperature Range (K) | Gas (56, 57, 58, 60) ^{a} |
---|---|---|

CO_{2} | 280–475 | 84.2 |

CH_{4} | 275–375 | −80 |

O_{2} | 250–330 | −260 |

N_{2} | 210–330 | −250 |

^{a}56, 57, 58 and 60 are the group numbers of CO

_{2}, CH

_{4}, O

_{2}and N

_{2}in the UNIFAC group parameter list, respectively.

**Table 3.**Constants for the calculation of gas–water interaction-energy parameters in the temperature range 273–348K.

Gas | u_{0} | u_{1}(×10^{−5}) |
---|---|---|

CO_{2} | 980.1 | −1.6895 |

CH_{4} | 1059.8 | −2.3172 |

O_{2} | 1259.9 | −3.0295 |

N_{2} | 1260.4 | −2.7416 |

Guests | a | b (×10^{5}) | c (×10^{7}) | N_{p} | R-Square |
---|---|---|---|---|---|

CH_{4} | 0.01713 | −2.58066 | −24.6488 | 500 | 0.99924 |

CO_{2} | 0.00454 | 0.403588 | −49.4532 | 200 | 0.89458 |

N_{2} | 0.0226 | −5.50785 | 1.49362 | 409 | 1 |

O_{2} | 0.02468 | −5.90016 | 1.33948 | 200 | 0.99999 |

**Table 5.**The phase equilibrium pressure and temperature range of experimental data and the Average Absolute Deviation in Pressure (AADP) for predicted results.

Guests | Temperature (K) | P-Range (bar) | N_{p} | Reference | AADP (%) |
---|---|---|---|---|---|

CH_{4} | 273.27–289.9 | 26.33–159.52 | 66 | [41] | 1.0634 |

276.81–281.3 | 37.79–62.02 | 5 | This work | 1.6317 | |

CO_{2} | 270.15–283.15 | 10.19–45.05 | 41 | [42] | 1.5911 |

N_{2} | 273.15–291.05 | 160.09–958.53 | 34 | [43] | 1.0224 |

274.55–283.05 | 190.93–453.55 | 3 | [44] | 1.4142 | |

279.30–284.00 | 303.00–500.00 | 3 | [45] | 1.5697 | |

O_{2} | 273.15 | 121.50 | 1 | [43] | 1.6786 |

273.78–284.55 | 138.21–441.30 | 4 | [44] | 1.8001 | |

286.27–291.18 | 527.13–953.65 | 6 | [46] | 0.9462 |

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**MDPI and ACS Style**

Zhao, W.; Wu, H.; Wen, J.; Guo, X.; Zhang, Y.; Wang, R.
Simulation Study on the Influence of Gas Mole Fraction and Aqueous Activity under Phase Equilibrium. *Processes* **2019**, *7*, 58.
https://doi.org/10.3390/pr7020058

**AMA Style**

Zhao W, Wu H, Wen J, Guo X, Zhang Y, Wang R.
Simulation Study on the Influence of Gas Mole Fraction and Aqueous Activity under Phase Equilibrium. *Processes*. 2019; 7(2):58.
https://doi.org/10.3390/pr7020058

**Chicago/Turabian Style**

Zhao, Weilong, Hao Wu, Jing Wen, Xin Guo, Yongsheng Zhang, and Ruirui Wang.
2019. "Simulation Study on the Influence of Gas Mole Fraction and Aqueous Activity under Phase Equilibrium" *Processes* 7, no. 2: 58.
https://doi.org/10.3390/pr7020058