Application of Peng-Robinson Equation of State for Calculating Solid-Vapor and Solid-Liquid Equilibrium of CH4-CO2 System

In this study, the performances of Peng-Robinson Equation of State combined with the classical mixing rules were evaluated to calculate solid-vapor equilibrium (SVE) and solid-liquid equilibrium (SLE) of CH4-CO2 system. The evaluation was performed by comparing the calculated values with the literature data. In the calculation of SVE, the new binary interaction parameter values (kij) of CH4-CO2 mixtures were proposed in this work based on the experimental data. The proposed kij obtained in this work might increase the accuracy of Peng-Robinson Equation of State by reducing average absolute deviation in the temperatures between calculated values and literature data from (2.18% to 0.26%), (0.88% to 0.70%) and (0.61% to 0.44%) at CO2 composition of 1%, 1.91% and 2.93%, respectively. Significant improvement was found at CO2 composition of 1%. In the calculation of SLE, new parameters were not proposed since the calculation using the existing (literature) kij gives good results with an average absolute deviation of 0.5%.


Introduction
The increasing number of population and technology advancement encourages the world's rising energy consumption.One of the widely used energy is natural gas.The advantages of the using natural gas compared to other energy sources are energy generated from natural gas is more efficient, cleaner, and environmentally friendly.In addition, natural gas is also colorless, odorless, non-toxic, and non-corrosive (van Egteren, 1993).The composition of raw natural gas is depending on the reservoir source location.It is generally containing of 30% to 90% methane, with other light hydrocarbons (i.e., ethane, propane and other heavier hydrocarbons).On the other hand, some contaminants, such as water, carbon dioxide (CO 2 ), hydrogen sulfide, helium and nitrogen, are also contained in the natural gas (Scholes et al., 2012).CO 2 is one of greenhouse gas that causes global warming.CO 2 is also an acidic gas and reacts with water to form acidic compounds, namely strong carbonic acid (H 2 CO 3 ) which can cause corrosion in piping systems.
Recently, there are so many methods to distribute natural gas.One of the promising methods used in the distribution of natural gas is liquefied natural gas (LNG).By using this method, the volume of natural gas can be reduced significantly after the liquefaction process that can make the natural gas transportation more efficient (Stringari et al., 2014).In the production of LNG, CO 2 must be removed from natural gas.One of CO 2 removal technology is controlled freeze out method.In this method, CO 2 is removed by utilizing behaviour of phase transitions among vapor, liquid and solid phases.To rational design and optimization condition of this type of separation method, solid-vapor equilibrium (SVE) and solid-liquid equilibrium (SLE) of CH 4 -CO 2 system are necessary.In this work, The performance of Peng-Robinson Equation of State (Peng and Robinson, 1976) combined with the classical mixing rules in calculating phase equilibrium for SVE and SLE of CH 4 -CO 2 system was evaluated.

Calculation Method
In this work, the standard form of Peng-Robinson Equation of State (PR-EoS) was used for the prediction of solid-vapor and solid-liquid equilibrium for binary mixtures of CH 4 -CO 2 : Eq. ( 1) in cubic form can be written as the following form: where The values of a m and b m of the mixtures were calculated using the classical mixing rules to apply PR-EoS to the mixtures based on the following description: ( ) ( ) where z i are the mole fraction of component i, ω i are the accentric factor of component i and k ij are the binary interaction parameter characterizing molecular interactions between molecules i and j.

Solid-Vapor Equilibrium
The fugacity coefficient of a component in the mixture can be calculated using PR-EoS by the following equation: Due to the solid phase is almost containing of CO 2 only, the assumption of pure solid CO 2 formation was used.So, the equilibrium relationship between solid and vapor phase in the CO 2 freezing phenomena can be described below: ( ) The CO 2 vapor pressure was calculated using the exponential equation as described below: 8 2 3108.29.44 10 exp Although Eq. ( 11) derived from equating partial fugacity describes the necessary condition for solid CO 2 formation, but it is not sufficient to show the stability of the produced solids.So, the following condition should also be checked in this case: TP T T ≤ (13) where T TP is the triple point temperature (for CO 2 , T TP = 216.55K).There are several cases where solids formation is thermodynamically predicted, but the temperature is too high for a stable solid (ZareNezhad and Eggeman, 2006).

Solid-Liquid Equilibrium
The fugacity coefficient of a component in the mixture can be calculated using PR-EoS by the following equation: Assuming solid phase is pure CO 2 , the solubility of solid CO 2 (2) in the CH 4 (1) can be described by the following equilibrium relationship: ( ) To find the solubility of CO 2 , Eq. ( 15) can be simplified into the following equation (Shen et al., 2012):

Results and Discussion
In this work, the calculation of frost points have been made using PR-EoS for the binary CH 4 -CO 2 system with the CO 2 compositions of 1%; 1.91%; and 2.93%.The results were compared with the experimental data determined by Le and Trebble (2007).The best fitted binary interaction parameters based on the classical mixing rule of PR-EoS obtained in this work for the calculation of SVE of CH 4 -CO 2 system are presented in Table 1 together with the literature k ij data as comparison.The frost points calculated using the binary interaction parameters obtained in this work and using those obtained by ZareNezhad and Eggeman (2006) were compared with the experimental data (Le and Trebble, 2007) as shown in Table 2.As shown in this table, the significant improvement of accuracy was found for CO 2 concentration of 1% by reducing AAD from (2.18% to 0.26%).However for other CO 2 concentrations studied, the improvements were not significant because the calculation using existing k ij also gave a little bit same magnitude of errors with the experimental data.In addition, the proposed present k ij values in this work gave high accuracy for the correlation at lower CO 2 concentration.The illustrations of the improvements were presented in Figures 1 to 3.  (Le and Trebble, 2007) for SVE of CH 4 (1) + CO 2 (2) system for x 2 = 1% Figure 2. Comparison of calculated frost points using present and literature k ij with experimental data (Le and Trebble, 2007) for SVE of CH 4 (1) + CO 2 (2) system for x 2 = 1.91% Figure 3.Comparison of calculated frost points using present and literature k ij with experimental data (Le and Trebble, 2007) for SVE of CH 4 (1) + CO 2 (2) system for x 2 = 2.93% PR-EoS was also applied to calculate SLE in term of solubility of CO 2 (x 2 ) for CH 4 (1) + CO 2 (2) system with k ij proposed by ZareNezhad and Eggeman (2006).The results were compared with the experimental data obtained by Shen et al. (2012) and giving a good accuracy with average absolute deviation of 0.5% between calculated and experimental solubility data.The comparison was presented in Figure 4.

Conclusion
The proposed k ij obtained in this work might improve the accuracy of Peng-Robinson Equation of State combined with classical mixing rules in calculation of SVE of CH 4 (1) + CO 2 (2) system by reducing average absolute deviation (AAD) from (2.18% to 0.26%) for the CO 2 composition of 1%, (0.88% to 0.70%) for the CO 2 composition of 1.91%, and (0.61% to 0.44%) for the CO 2 composition of 2.93%.The significant improvement was found for CO 2 concentration of 1%.The calculation of SLE based in the literature k ij gave absolute average deviation of 0.5% between calculated and literature data for solubility of CO 2 in CH 4 -CO 2 system.

Figure 4 .
Figure 4. Comparison between calculated solubility of CO 2 (2) in CH 4 (1) using the Peng-Robinson Equation of State and experimental data obtained by Shen et al. (2012)

Table 1 .
The best fitted binary interaction parameters obtained in this work for SVE of CH 4 -CO 2 system

Table 2 .
(Le and Trebble, 2007)tion between calculated and experimental(Le and Trebble, 2007)frost point data using present and literature k ij for SVE of CH 4 -CO 2 system Comparison of calculated frost points using present and literature k ij with experimental data