Predicting sulfur solubility in hydrogen sulfide, carbon dioxide, and methane with an improved thermodynamic model

During development of high sulfur-content natural gas fields, gaseous sulfur is likely to precipitate and deposit in the reservoir and transmission pipelines owing to changes in the temperature, pressure, and gas components. It is important to accurately predict the elemental sulfur solubility in hydrogen sulfide, carbon dioxide, and methane because these are the three main components of high-sulfur-content natural gas. The binary interaction coefficients between sulfur and hydrogen sulfide, carbon dioxide, and methane are the key parameters for predicting the sulfur solubility with a thermodynamic model. In this work, we show that the binary interaction coefficients are not constant, but temperature dependent. Three-parameter temperature-dependent equations for the binary interaction coefficients between sulfur and solvents are proposed. The corresponding regression equations for calculating the binary interaction coefficients between sulfur and hydrogen sulfide, carbon dioxide, and methane are obtained using experimental sulfur solubility data. The average relative errors of the sulfur solubility predicted using the experimental data in hydrogen sulfide, carbon dioxide, and methane using the thermodynamic model with the improved binary interaction coefficients are 6.30%, 1.69%, and 4.34%, and the average absolute relative errors are 7.90%, 13.12%, and 14.98%, respectively. Comparing the improved binary interaction coefficients with four other sets of reported values shows that the solubility values predicted by the thermodynamic model with improved binary interaction coefficients fit the experimental data better.


Introduction
High-sulfur-content natural gas contains hydrogen sulde, mercaptans, sulfoethers, and other sulfurous substances, with hydrogen sulde gas comprising the majority of all sulfurous substances. There are many high-sulfur-content gas elds. For example, the hydrogen sulde volume contents in natural gas mixtures are 15% to 18% in the Puguang Gas Field (China). 1 High-sulfur-content natural gas elds supply not only clean energy, but also raw materials for sulfur series products. 2 However, deposited elemental sulfur may cause pore formation, wellbore blockage, and even transmission pipeline blockage and corrosion with changes in pressure, temperature, and gas components. [3][4][5] This can lead to the normal production of the gas eld being severely inhibited. [6][7][8][9] In recent years, the problem of sulfur deposition has received extensive attention. Notably, elemental sulfur solubility in high-sulfur-content natural gas is the key factor determining whether sulfur deposition occurs. [10][11][12] Many researchers have measured the sulfur solubility in hydrogen sulde (H 2 S), carbon dioxide (CO 2 ), and methane (CH 4 ), because high-H 2 S-content natural gas mainly comprises these three gas components. Kennedy and Wieland 3 were the rst to measure elemental sulfur solubility in pure H 2 S, CO 2 , CH 4 and their mixtures in 1960. Their work showed that elemental sulfur solubility increases with increasing pressure and temperature. Roof found that the solubility in H 2 S rst increases and then decreases with increasing temperature. 13 Swi, Brunner, and Gu et al. extended the pressure and temperature range for sulfur solubility in H 2 S. 14-16 Serin et al. developed an experimental apparatus for measuring elemental sulfur solubility in CO 2 . 17 Using the same experimental apparatus, Cloarec et al. 18 obtained experimental data of sulfur solubility in CH 4 .
The above experimental results of solubility in H 2 S, CO 2 , and CH 4 provide the basis for establishing a solubility predicting model, and strongly support the development of high-sulfurcontent natural gas elds. [19][20][21][22][23][24] The binary interaction coefficients between sulfur and H 2 S, CO 2 , and CH 4 in natural gas are important parameters when using a thermodynamic predicting model based on the equation of state (EoS). As shown in Table 1, Heidemann 25 reported that the binary interaction coefficients between sulfur and H 2 S, CO 2 , or CH 4 aer regression of the experimental data were 0.0812, 0.135, and 0.155, respectively. 25 In Sun's model, 26 the binary interaction coefficients between sulfur and H 2 S, CO 2 , and CH 4 are 0.0758, 0.190, and 0.115, respectively. However, Gu 16 and Cézac et al. 27 suggested that the binary interaction coefficients are related to the gas mixture temperature. Based on the experimental data of Roof, Brunner, and their own work, Gu 16 reported binary interaction coefficients between sulfur and H 2 S, CO 2 , and CH 4 of 316.3-374.8 K, 363.2 and 383.2 K, and 383.2 K, respectively. Cézac et al. 27 proposed three equations for calculating the binary interaction coefficients between sulfur and H 2 S, CO 2 , and CH 4 based on analysis of experimental data. The binary interaction coefficients from the above studies are considerably different, and it is unclear which set represents better experimental data.
This work aims to evaluate differences in the binary interaction coefficients between sulfur and H 2 S, CO 2 , and CH 4 . We investigated the relationships between the binary interaction coefficients and temperature to extend the range and improve the accuracy of predicting the sulfur solubility in H 2 S, CO 2 , and CH 4 using a thermodynamic model based on the Peng-Robinson (PR) EoS. Based on experimental data analysis, new threeparameter temperature-dependent equations for calculating the binary interaction coefficients are proposed. The equation parameters are obtained by regression analysis of the experimental data. Furthermore, we compared the solubility results calculated using the thermodynamic model with the binary interaction coefficients reported by Heidemann, Sun, Cézac, Gu, and in this work.

Governing equations
The model is based on thermodynamic gas-solid phase equilibrium theory, which assumes perfect mixing of the sulfur and gas components. As mentioned by Gu, Sun, and Heidemann, solid sulfur should be treated as single molecule S 8 . 16,25,26 The phase equilibrium condition of the gas-solid system requires the fugacity of gaseous sulfur to be the same as that of the solid phase, as expressed by eqn (1): 28

Solid phase fugacity
The solid sulfur fugacity is related to the saturation vapor pressure of solid sulfur, as expressed by eqn (2): 28 The sulfur saturation vapor pressure is always small in the gas-solid phase equilibrium system, so f sat S8 ¼ 1.0 in this model. 28 Shuai and Meisen 29 reported regression equations of the sulfur saturation vapor pressure at different temperatures.

Elemental sulfur fugacity in the gas phase
The elemental sulfur fugacity in the gas phase can be expressed with eqn (6): 28 2.3.1. Peng-Robinson (PR) EoS. According to eqn (6), f V S8 is the key parameter for calculating the gaseous sulfur fugacity, and can be described by the PR EoS. The basic form of the PR EoS is shown in eqn (7): 30 where a and b are the parameters of PR EoS. When gas mixtures are single component, a and b can be expressed as follows:      Table 2 shows the critical parameters and acentric factors of S 8 , H 2 S, CO 2 , and CH 4 . 27,28 Aer the solid and gas phase sulfur reach the equilibrium state, the gas phase is composed of a solvent gas component and gaseous sulfur. The a and b parameters need be calculated using the mixing rule according to the values of the single components. In this model, b is calculated by the classic mixing rule, as follows: The a parameter of the gas mixture components will be discussed in detail in Section 3.
The fugacity coefficient of gaseous sulfur f V S8 can be expressed as: Z is calculated using Relevant parameters in eqn (15) are expressed by eqn (16)- In this work, this equation was solved using the Newton-Raphson method. The elemental sulfur solubilities in H 2 S, CO 2 , and CH 4 can be calculated easily using eqn (1)- (19).

Proposed method
The classical mixing rule for a is: 28 where k ij are the binary interaction coefficients between S 8 and the gas solvents (H 2 S, CO 2 , and CH 4 ). Eqn (20) shows that k ij are important parameters. The reported binary interaction coefficients between S 8 and H 2 S, CO 2 , and CH 4 obtained by regression of the sulfur solubility experimental data are shown in Table 1. The binary interaction coefficients reported by Sun and Heidemann are constant. 25,26 However, as mentioned by Gu 16 and Cézac, 27 the binary interaction coefficients are temperature dependent. Therefore, we proposed a new mixing rule for a to calculate the gaseous S 8 fugacity: Therefore, The k ij values with temperature were regressed based on the experimental data of Kennedy, Roof, Brunner, Serin, and Gu, 3,13,15-17 and the calculated diagram is shown in Fig. 1.
According to Eslamimanesh's study of the consistency of the experimental results of sulfur solubility in H 2 S, only $45% of the values are reliable. 20 To obtain an equation for the binary interaction coefficient between sulfur and H 2 S, we chose the 14 sets of experimental data at temperatures of 316.26, 338.71, and 363.15 K reported by Roof and Gu. 13,16 Fig. 2 shows the binary interaction coefficient between S 8 and H 2 S calculated at temperatures of 316.26-363.15 K. To better t the k ij value, we obtained the average k ij values at the same temperatures shown in Fig. 3, and a new equation and tting curve of k S 8 -H 2 S with temperature were obtained. The adjusted R 2 value (R adj 2 ) was 0.896, which showed that the tting precision was high. The new equation for the binary interaction coefficient between S 8 and H 2 S is: The new equation for the binary interaction coefficient between S 8 and CO 2 is: The new equation for the binary interaction coefficient between S 8 and CH 4 is:

Predicting the sulfur solubility using the proposed binary interaction coefficients
The sulfur solubility in H 2 S predicted by the thermodynamic model using the proposed binary interaction coefficient between sulfur and H 2 S obtained by eqn (23) (21)) and (23).
There are 14 sets of experimental data in Table 5. The RE, ARE, and AARE values were calculated using eqn (26)-(28): Based on Table 5, the total ARE and AARE values for predicting the sulfur solubility were 6.30% and 7.90%, respectively. Fig. 8 and 9 show the ARE and AARE values of sulfur solubility in H 2 S calculated from the experimental data at three different temperatures (316.26, 338.71, and 363.15 K). Fig. 8 shows that the ARE values were positive at temperatures of 316.26, 338.71, and 363.15 K. The lowest ARE value was 5.69% at 316.26 K, with a highest ARE value of 7.64% at 363.15 K. Fig. 9 shows that the greatest AARE value was 12.61% at 363.15 K. The other AARE values were all below 7%. Considering the critical temperature of H 2 S (373.5 K), the large deviation at 363.15 K could be due to the kinetic characteristics of the solvent molecule (H 2 S) being inuenced by external factors near the critical temperature and pressure. 16 The predicted sulfur solubility in CO 2 using the proposed binary interaction coefficient, along with the experimental data of Kennedy, Gu, and Serin, in the temperature range 333.15-394.26 K at pressures of 15.1-41.4 MPa are shown in Table 6. 3,16,17 Parameter a with the proposed mixing rule (eqn (21)) was calculated using eqn (24). There were 32 sets of experimental data included in Table 6. Based on these calculated results, the total ARE and AARE values of the predicted sulfur solubility in CO 2 were 1.69% and 13.12%, respectively. Fig. 10 and 11 show the ARE and AARE values of sulfur solubility in CO 2 calculated using the experimental data for six temperature ranges from 333.15-394.26 K. The lowest ARE value was 0.41% at 394.26 K, and the highest ARE value was 8.22% at 383.15 K (Fig. 10). Fig. 11 shows that the highest AARE value was 18.23% at 383.15 K.
The predicted sulfur solubility in CH 4 using the proposed binary interaction coefficient, along with the experimental data of Kennedy and Gu in the temperature range 338.71-394.26 K and pressures range 6.89-41.4 MPa, are shown in Table 7. 3,16 Parameter a with the proposed mixing rule (eqn (21)) was calculated with eqn (25). There are 17 sets of experimental data in Table 7. Based on the calculated results in Table 7, the total ARE and AARE values calculated with eqn (27) and (28) were 4.34% and 14.98%, respectively. Fig. 12 and 13 show the ARE and AARE values of sulfur solubility in CH 4 calculated using the    experimental data for four temperatures in the range 338.71-394.26 K. Fig. 12 shows that the lowest ARE value was 2.42% at 383.15, while the highest ARE value was 8.32% at 366.48 K. The highest AARE value was 17.53% at 383.15 K (Fig. 13). Combined with the results of the calculated sulfur solubilities in CO 2 and CH 4 , the main reason for the large deviation could be that the solubility values in CO 2 and CH 4 are too small.  16,27 Here, we compared the sulfur solubilities predicted using these different k S 8 -H 2 S values with our proposed k S 8 -H 2 S model (eqn (23)).       Fig. 15 shows a comparison of the sulfur solubility in CO 2 calculated using the thermodynamic model with ve different k S 8 -CO 2 values at temperatures ranging from 333.15 to 394.26 K. When k S 8 -CO 2 was 0.135, the difference between the calculated sulfur solubility and the experimental data was the largest at 383.15 K. Based on Fig. 15, the accuracy of the predicted sulfur solubility seemed to be acceptable using all of the other four k S 8 -CO 2 values except 0.135.

Comparison of binary interaction
As shown in Table 1, the k S 8 -CO 2 values obtained by Sun and Heidemann were 0.190 and 0.135. 25,26 Gu et al. 16 related the k S 8 -CO 2 parameter to the temperature, and reported k S 8 -CO 2 values of 0.2107 and 0.1993 at 363.2 and 383.2 K. 16 Cézac 27 suggested that k S 8 -CO 2 was temperature dependent and proposed an equation for calculating k S 8 -CO 2 , as shown in Table 1. A comparison of the total ARE and AARE values of the calculated sulfur solubility in CO 2 with different k S 8 -CO 2 values is shown in Table 3. From Table  3, the deviations of the solubility values calculated using the proposed k S 8 -CO 2 from the experimental data were smaller than those using the other four k S 8 -CO 2 values.
A comparison of the ARE and AARE values for different k S 8 -CO 2 at temperatures of 333.15 to 394.26 K is shown in Fig. 16. Fig. 16(a) shows that ARE is the closest to 0% with the proposed k S 8 -CO 2 , ARE < 0 when k S 8 -CO 2 is 0.190, and ARE is both negative and positive for the k S 8 -CO 2 reported by Cézac. 27 Using the k S 8 -CO 2 and k S 8 -CO 2 values reported by Cézac,27 the ARE values were À1.67% and À1.57% at 363.15 K, and 4.34% and 0.55% at 366.48 K, respectively. As shown Fig. 16(b), the AARE values were smaller using the proposed k S 8 -CO 2 than using the other two k S 8 -CO 2 values. At 338.71 K, the AARE values were 7.72% and 6.82% using the proposed k S 8 -CO 2 and k S 8 -CO 2 ¼ 0.190, respectively. Using the proposed k S 8 -CO 2 , the k S 8 -CO 2 reported by Sun, and the k S 8 -CO 2 reported by Cézac, the AARE values were 14.63%, 14.18%, and 14.64% at 363.15 K, and 11.14%, 8.18%, and 8.40% at 366.48 K, respectively. Fig. 16 shows that the accuracy of the predicted solubility using k S 8 -CO 2 reported by Cézac was higher than that using the k S 8 -CO 2 proposed in this paper at 363.15 and 366.48 K. However, the total deviation was still the lowest using the proposed binary interaction coefficient between sulfur and CO 2 ( Table 6 and Fig. 16).
4.2.3. Comparison of sulfur solubility in CH 4 with different k S 8 -CH 4 . As shown in Table 1, the k S 8 -CH 4 values reported by Heidemann and Sun are 0.155 and 0.115, respectively. 25, 26 Gu et al. related k S 8 -CH 4 to temperature, but only reported the k S 8 -CH 4 value at 383.2 K. 16 Cézac et al. 27 proposed a temperature-dependent equation for calculating k S 8 -CH 4 ( Table 1). The total ARE and AARE values of the calculated sulfur solubility in CH 4 using the different values of k S 8 -CH 4 are shown in Table 4. The lowest ARE and AARE values of 4.34% and 14.98% were obtained using the proposed k S 8 -CH 4 . When k S 8 -CH 4 ¼ 0.155, the deviations between the calculated solubility values and the experimental data were the largest. From Table 4, the deviations between the solubility values calculated using the proposed k S 8 -CH 4 and the experimental data were smaller than those using the other four k S 8 -CH 4 . Fig. 17 shows a comparison of the predicted sulfur solubility in CH 4 using the thermodynamic model with ve different k S 8 -CH 4 at temperatures ranging from 338.71 to 394.26 K. Fig. 17(c) shows the calculated sulfur solubility in CH 4 with ve different k S 8 -CH 4 at 383.15 K, because k S 8 -CH 4 ¼ 0.1345 reported by Gu was only suitable at 383.2 K. Therefore, Fig. 17(a), (b), and (d) only show the predicted solubility with four different k S 8 -CH 4 .
As shown in Fig. 17, the predicted solubility values were negative, except for that calculated using the proposed k S 8 -CH 4 . When k S 8 -CH 4 ¼ 0.115 at 394.26 K, the calculated solubility in CH 4 was close to the experimental data ( Fig. 17(d)). Using k S 8 -CH 4 ¼ 0.115 and the proposed k S 8 -CH 4 , the ARE values were 5.25% and 3.46%, while the AARE values were 12.11% and 12.63%, respectively. This indicated that the predicted solubility at 394.26 K using the k S 8 -CH 4 value reported by Sun was also acceptable. 26

Conclusions
We have proposed new three-parameter equations to calculate the binary interaction coefficients between sulfur and H 2 S, CO 2 , and CH 4 . The parameters in these three equations were obtained by regression of the experimental sulfur solubility data in H 2 S, CO 2 , and CH 4 . The relationship between the binary interaction coefficients and temperature was quadratic. The binary interaction coefficients k S 8 -H 2 S , k S 8 -CO 2 , and k S 8 -CH 4 were expressed by eqn (23)- (25), and the R adj 2 values of the new tting equations were 0.896, 0.966, and 0.933, respectively, which indicated that the new three-parameter equations were reliable. By comparing the experimental solubility data with the solubility results predicted using the binary interaction coefficients proposed by this work and those reported by Sun, Heidemann, Gu, and Cézac, the total ARE and AARE values were signicantly smaller when using the proposed binary interaction coefficients than those when using the other four. However, the accuracy of the predicted sulfur solubility in CO 2 using k S 8 -CO 2 reported by Cézac was slightly higher than that using k S 8 -CO 2 proposed in this article at temperatures of 363.15 and 366.48 K. Furthermore, the calculated solubility in CH 4 using k S 8 -CH 4 reported by Sun was acceptable.
In general, the predicted results were satisfactory considering the ARE and AARE values obtained using the proposed binary interaction coefficients. The accuracies of the predicted sulfur solubilities in H 2 S, CO 2 , and CH 4 using the thermodynamic model with the proposed binary interaction coefficients based on the PR EoS were greatly improved, and the thermodynamic model was suitable for a wide range of temperatures and pressures.