A Study on Elemental Sulfur Equilibrium Content in Mixtures of Methane, Carbon Dioxide, and Hydrogen Sulfide under Conditions of Natural Gas Pipeline Transmission

Abstract

The effect of gathering pipeline pressure, temperature, and key components on the solubility of monomeric sulfur in high-sulfur-content natural gas is directly related to the prediction and prevention of sulfur deposition in surface gathering pipelines. Based on our previous study on a prediction model of sulfur solubility in gas with a new formula for the binary interaction coefficient between sulfur and H₂S, a new gas–solid thermodynamic phase equilibrium solubility prediction model for monomeric sulfur in high-sulfur-content natural gas was improved based on the gas–solid phase equilibrium principle considering both physical and chemical solution mechanisms. Two new expressions for binary interaction coefficients between sulfur and CO₂ and CH₄, considering both temperature and solvent density, are proposed in this new solubility prediction model. In this paper, the main factors, such as the gathering pipeline pressure, gathering pipeline temperature, H₂S, and CO₂, affecting the solubility law of elemental sulfur in high-sulfur-content natural gas are investigated. The results show that the total solubility of elemental sulfur in high-sulfur-bearing natural gas tends to decrease with an increase in the gathering temperature, in which the increase in temperature promotes physical solution, and the physical solution mechanism prevails. Conversely, chemical solution is promoted, and the chemical solution mechanism prevails. With an increase in the gathering pressure, the total solubility of elemental sulfur in high-sulfur-content gas tends to increase, where the physical solubility decreases slightly at first and then increases continuously, with a pressure inflection point of about 2.0 MPa, and the pressure increase has a significant promoting effect on the chemical solubility of elemental sulfur. The increase in the H₂S concentration promotes the solution of elemental sulfur in the gas phase in general and significantly promotes the chemical solution of elemental sulfur. The effect on elemental sulfur solubility can be neglected when the molar concentration of CO₂ in the gas phase does not exceed 10%.

1. Introduction

At present, many high-sulfur-content natural gas fields are being developed and utilized around the world. Different from conventional natural gas, some elemental sulfur is often dissolved in high-sulfur-content natural gas exploited from high-temperature and high-pressure strata. In the process of high-sulfur-content natural gas gathering, with changes in the pressure, temperature, and gas components in the pipeline, the elemental sulfur in the gas phase easily undergoes phase transformation and precipitates in the form of a solid phase. Under certain conditions, solid-phase sulfur will be deposited in the pipeline. Once sulfur deposition occurs in the gathering pipeline, it will not only cause problems such as blockages and internal corrosion in the pipeline system [1,2], but will also cause safety problems in the equipment attached to the gathering pipeline [3,4]. It affects safe transmission by the pipeline and brings huge economic losses to enterprises.

The pressure, temperature, and gas quality components are the dominant factors affecting the solubility of elemental sulfur in high-sulfur-content natural gas [5,6]. In high-temperature and high-pressure formation environments, elemental sulfur can stabilize in high-sulfur-content natural gas by physical or chemical solution. When high-sulfur-content gas reaches the surface gathering system, the gas-phase elemental sulfur concentration will change with changes in pressure and temperature. Once it is above the saturation solubility of elemental sulfur in the natural gas, excess monomeric sulfur in the gas phase will precipitate out.

The essence of the gas–solid phase equilibrium of elemental sulfur in high-sulfur-content natural gas in gathering pipelines is the elemental sulfur solution equilibrium in the gas phase. Sulfur deposition in the pipeline is only possible when the gas-phase elemental sulfur concentration exceeds its solubility in high-sulfur-content natural gas. In addition, the elemental sulfur solution equilibrium directly affects the precipitation of sulfur crystal nuclei and calculations related to the agglomerative growth of sulfur particles. Therefore, the prediction of the phase equilibrium solubility of elemental sulfur in high-sulfur-content natural gas is a prerequisite and fundamental condition for the study of elemental sulfur deposition. The current research on elemental sulfur solubility prediction in high-sulfur-content natural gas mainly includes experimental studies and theoretical prediction model studies [7].

Kennedy and Wieland (1960), Roof (1971), Smith et al. (1970), Swift et al. (1976), Brunner et al. (1980, 1988), Migdisov et al. (1998), Sun and Chen (2003), Serin et al. (2010), and Cloarec et al. (2012) carried out numerous studies on the solubility of solid monomeric sulfur in natural gas and high-sulfur-content natural gas [8,9,10,11,12,13,14,15,16,17]. They enriched the experimental data on the solubility of elemental sulfur. However, the amount of experimental data obtained under the current experimental conditions is still very limited. Experimental data on the gathering pressure and temperature are especially relatively lacking [18,19]. The highly toxic nature of H₂S, the special physical properties of sulfur-containing natural gas, and the sensitivity of sulfur’s solubility to external factors are all issues to be considered. Therefore, it is still quite difficult to predict elemental sulfur’s solubility and study the mechanism of elemental sulfur’s solubility solely through experimental studies.

The freezing point of elemental sulfur is not stable in high-sulfur-content natural gas, but mainly changes with the pressure and H₂S concentration of the gas mixture [20]. Yang (2006) determined the change rule of the freezing point of elemental sulfur under different concentrations of H₂S in natural gas and pressures by experimental tests. The lowest freezing point reached 363.15 K in the range of common natural gas transmission pressures (P ≤ 15 MPa) [21]. The temperature in the transmission pipelines of high-sulfur-content natural gas is usually the ambient temperature, which rarely exceeds the minimum freezing point of elemental sulfur (363.15 K). In other words, elemental sulfur will only exist as a gas or a solid phase in gas transmission pipelines. According to the theory related to thermodynamic phase equilibrium, the solution process of elemental sulfur in highly sulfurous acidic natural gas can be regarded as a gas–solid phase equilibrium solution problem for elemental sulfur. Based on this, a thermodynamic prediction model can be developed to determine elemental sulfur’s solubility.

Gu et al. (1993) related the solution process of elemental sulfur in H₂S and a H₂S-containing acid gas mixture to the principle of solid extraction in supercritical fluids and developed a gas–solid-phase equilibrium thermodynamic model based on the PR equation of state [22,23]. The model of Karan et al. (1998) treated all elemental sulfur as S₈ to establish a sulfur solubility prediction model [24]. The model can be used to predict the solubility of elemental sulfur in a mixture of H₂S, CO₂, CH₄, and N₂ species [25]. Guo and Du (2007) developed a gas–liquid–solid three-phase-equilibrium thermodynamic model based on a three-phase flash algorithm to predict sulfur’s solubility under high-temperature and -pressure conditions in the formation [26]. Serin et al. (2007, 2008, 2009), from the University of Pau, France, developed a new model for the prediction of elemental sulfur’s solubility based on the relationship between the gas–liquid–solid-phase thermodynamic equilibrium, correlating the fugacity of solid-phase sulfur and liquid-phase sulfur [19,27,28,29]. Karan et al. (1998), Heidemann et al. (2001), and Sun et al. (2003) considered the binary interaction coefficient of elemental sulfur with H₂S, CO₂, and CH₄ as a constant [15,24,25]. Among them, the models developed by Sun et al. (2003) and Karan et al. (1998) considered only the physical solution mechanism [15,24], while the model by Heidemann et al. (2001) considered only the chemical solution mechanism [25]. Cézac et al. (2007) considered both chemical and physical solution mechanisms and developed a binary interaction coefficient regression model [19]. In summary, the thermodynamic model predicts the solubility of elemental sulfur from the solution mechanism. It has a broad application prospect in theory. However, there are still some problems in the current application of elemental sulfur solubility prediction under the gathering conditions. The first is the calculation of solid-phase sulfur’s fugacity at the gathering and transportation temperatures. The second is the lack of a unified model for calculating the binary interaction coefficients between elemental sulfur and H₂S, CO₂, and CH₄, especially a thermodynamic model applicable to the gathering and transportation conditions and considering both the physical and chemical solution mechanisms of elemental sulfur.

We proposed three-parameter temperature-dependent equations for the binary interaction coefficients between sulfur and H₂S, CO₂, and CH₄ in natural gas [6]. The accuracy of the prediction of sulfur solubility using the experimental data for H₂S, CO₂, and CH₄ of the thermodynamic model with the improved binary interaction coefficients was greatly improved. Later, in order to further consider the effect of chemical dissolution on sulfur’s solubility in high-sulfur-content natural gas and to apply the thermodynamic model to the condition of the gas transmission temperature, a new vapor pressure expression of elemental sulfur and a new formula for the binary interaction coefficient between sulfur and H₂S considering both the temperature and solvent density are proposed [30]. This new model can adapt to lower temperatures, and the threshold value can be as low as 273.15 K. However, this latest model considers sulfur’s solubility only in pure H₂S. Sulfur’s physical and chemical solution mechanisms in the transmission conditions of natural gas mixtures and their controlling factors are not studied. To further elucidate the intrinsic solution mechanism of elemental sulfur in high-sulfur-content natural gas under gathering conditions and its variation law, this paper will further investigate the influence law of four main factors, namely the gathering temperature (273.15 K ≤ T ≤ 333.15 K), gathering pressure (0.5 MPa ≤ P ≤ 15 MPa), and H₂S and CO₂ concentrations in the gas phase, on elemental sulfur’s solubility and solution mechanism. The average concentrations of sulfur and carbon dioxide in natural gas are unusually below 0.1% and 10% in transmission pipelines [7,9]. To further reveal the intrinsic solution mechanism of elemental sulfur in high-sulfur-content natural gas under gathering conditions and its variation law, this paper will further investigate the influence law of four main factors, namely the gathering temperature (273.15 K ≤ T ≤ 333.15 K), gathering pressure (0.5 MPa ≤ P ≤ 15 MPa), and the H₂S (≤20%vol) and CO₂ (≤10%vol) concentrations in the gas phase, on elemental sulfur’s solubility and solution mechanism.

2. Modeling and Analysis

2.1. Model Assumption

Elemental sulfur’s solubility is directly related to the determination of elemental sulfur’s deposition and the prediction of its deposition volume in high-sulfur-content natural gas-gathering pipelines. The solution of elemental sulfur in sour gas under gathering conditions can be regarded as an elemental sulfur gas–solid phase equilibrium problem. In this paper, the classical gas–solid-phase equilibrium theoretical model is used to describe it. Before establishing the corresponding mathematical model, it is defined that the volume content of H₂S in the gas components does not exceed 20%, the volume content of CO₂ does not exceed 10%, and the remaining components are CH₄. Considering the actual situation of the gathering pipeline, the following assumptions are made:

(1) The S₈ contents of surface gathering system sediments are 94% and 93.5%, as shown in experiments by Wang (2012) and Li (2013), respectively [31,32]. Elemental sulfur is composed entirely of S8 molecules in this article.

(2) According to the experimental results regarding sulfur’s freezing point under different concentrations of H₂S by Yang (2006) [21], elemental sulfur exists in two phases, solid phase and gas phase, and no other phases exist.

(3) After the phase analysis of gas mixture with H₂S (≤20%vol), CO₂ (≤20%vol), and CH₄, all high-sulfur-content natural gas exists in the form of the gas phase.

According to the studies by Heidemann et al. (2001) and Cézac et al. (2007), the allotropes of elemental sulfur include 8 substances, such as S₁ and S₂ to S₈, and the corresponding polysulfide hydrogen includes 8 compounds, such as H₂S₂ and H₂S₃ to H₂S₉ [19,25]. Therefore, theoretically, the sulfur-containing substances in high-sulfur-content natural gas should include allotropes of elemental sulfur, H₂S, and polysulfide hydrogen.

Therefore, regarding the chemical solubility of elemental sulfur, the gas phase includes the following two main chemical reaction equations, as shown in Equations (1) and (2). It is generally believed that, when the pressure and temperature increase, the reaction moves toward the production of S₈ and polysulfide hydrogen H₂S_{x + 1}, and the chemical solubility of elemental sulfur increases. When the pressure and temperature decrease, the reaction moves to the left and the decomposition of polysulfide hydrogen causes more S₈ to appear in the gas phase. When the concentration of S₈ in the gas phase exceeds the critical solubility at this point, elemental sulfur will undergo supersaturated precipitation to produce solid-phase sulfur, which will further lead to the occurrence of sulfur deposition in the gathering pipeline.

As this paper mainly focuses on the gathering pipeline conditions, according to the theoretical findings of Heidemann et al. (2001), Li (2012), and the experimental test results of Li et al. (2012) and Wang (2013), it is clear that only S₈ exists in the gathering pipeline, and there is a very small chance of other elemental sulfur isomers existing [25,31,32,33]. Therefore, the chemical transformation between isomers of sulfur molecules in Equation (1) is not considered in the set transport conditions, and the chemical solution mechanism of elemental sulfur is shown in Equation (2), where x = 8.

$${\mathrm{S}}_x\underset{P, T\downarrow }{\overset{P, T\uparrow }{\rightleftarrows }}{\mathrm{S}}_8 x=1, \dots , 7$$

(1)

$${\mathrm{H}}_2\mathrm{S}+\frac{x}{8}{\mathrm{S}}_8\underset{P, T\downarrow }{\overset{P, T\uparrow }{\rightleftarrows }}{\mathrm{H}}_2{\mathrm{S}}_{x+1} x=1, \dots , 8$$

(2)

2.2. Model Study on the Solution Mechanism of Elemental Sulfur in High-Sulfur-Content Natural Gas under Gathering Conditions

The solution of elemental sulfur in high-sulfur-content natural gas is considered to be mainly in the form of a physical solution, chemical solution, or a combination of both dissolution methods, but the solution mechanism of elemental sulfur under gathering pipeline conditions is still unclear. In order to make the established solubility model more general and to further explore the solution mechanism of elemental sulfur in natural gas under gathering conditions, based on the experimental data of elemental sulfur’s solubility, a phase equilibrium solubility prediction model was established. Both the chemical and physical solution mechanisms of elemental sulfur are considered in this model. At this point, the solubility of elemental sulfur is equal to the sum of its physical and chemical solubility, as shown in Equation (3).

$$y_{{\mathrm{S}}_8}=y_{{\mathrm{S}}_8}^{\mathrm{phy}}+y_{{\mathrm{S}}_8}^{\mathrm{chem}}$$

(3)

where $y_{{\mathrm{S}}_8}$ is the total solubility of elemental sulfur, mol/mol; $y_{{\mathrm{S}}_8}^{\mathrm{phy}}$ is the physical solubility, mol/mol; and $y_{{\mathrm{S}}_8}^{\mathrm{chem}}$ is the chemical solubility, also equal to the molar fraction of H₂S₉ in the gas phase, mol/mol.

2.2.1. Physical Solution Mechanism Model of Elemental Sulfur

Elemental sulfur exists only in two forms, gas phase and solid phase, considering the conditions of the gathering pressure and temperature. According to the thermodynamic phase equilibrium theory, the physical solution equilibrium of elemental sulfur needs to satisfy equal fugacity of the elemental sulfur in the gas phase and the solid phase [34], as shown in Equation (4).

$$f_{{\mathrm{S}}_8}^{\mathrm{S}}\left(T,P\right)=f_{{\mathrm{S}}_8}^{\mathrm{V}}\left(T,P,y_{{\mathrm{S}}_8}^{\mathrm{phy}}\right)$$

(4)

where $f_{{\mathrm{S}}_8}^{\mathrm{S}}$ is the solid-phase sulfur’s fugacity, Pa, and $f_{{\mathrm{S}}_8}^{\mathrm{V}}$ is the gas-phase sulfur’s fugacity, Pa.

(1) Solid-phase sulfur fugacity calculation

According to the classical gas–solid phase equilibrium theory, the solid-phase sulfur fugacity can be calculated using Equation (5).

$$f_{{\mathrm{S}}_8}^{\mathrm{S}}\left(T,P\right)={\varphi }_{{\mathrm{S}}_8}^{\mathrm{sat}}{P}_{{\mathrm{S}}_8}^{\mathrm{sat}}\exp \frac{V_{{\mathrm{S}}_8}^{\mathrm{S}}\left(P-P_{{\mathrm{S}}_8}^{\mathrm{sat}}\right)}{\mathrm{R}T}$$

(5)

where ${\varphi }_{{\mathrm{S}}_8}^{\mathrm{sat}}$ is the fugacity coefficient of saturated sulfur vapor, and the saturated sulfur vapor is extremely low at the collector temperature, when the fugacity coefficient can be regarded as 1; $P_{{\mathrm{S}}_8}^{\mathrm{sat}}$ is the saturated vapor pressure of S₈, Pa; $V_{{\mathrm{S}}_8}^{\mathrm{S}}$ is the molar volume of S₈, m³/mol; P is the pressure, Pa; T is the temperature, K; and R is the gas constant, J/(kg·K).

(2) Gas-phase sulfur fugacity calculation

The gas-phase fugacity of elemental sulfur can be calculated using Equation (6).

$$f_{{\mathrm{S}}_8}^{\mathrm{V}}\left(T,P,y_{{\mathrm{S}}_8}^{\mathrm{phy}}\right)=y_{{\mathrm{S}}_8}^{\mathrm{phy}}\cdot {\varphi }_{{\mathrm{S}}_8}^{\mathrm{V}}\cdot P$$

(6)

where ${\varphi }_{{\mathrm{S}}_8}^{\mathrm{V}}$ is the fugacity coefficient of gas-phase S₈.

Equation (7) for calculating the physical solubility of elemental sulfur can be obtained by connecting Equations (4) and (6).

$$y_{{\mathrm{S}}_8}^{\mathrm{phy}}=\frac{{\varphi }_{{\mathrm{S}}_8}^{\mathrm{sat}}{P}_{{\mathrm{S}}_8}^{\mathrm{sat}}}{{\varphi }_{{\mathrm{S}}_8}^{\mathrm{V}}P}\exp \frac{V_{{\mathrm{S}}_8}^{\mathrm{S}}\left(P-P_{{\mathrm{S}}_8}^{\mathrm{sat}}\right)}{\mathrm{R}T}$$

(7)

2.2.2. Chemical Solution Mechanism Model of Elemental Sulfur

According to the chemical reaction equilibrium relationship, when the reaction reaches equilibrium in the system, the following relationship is satisfied [34]:

$${{\displaystyle \prod _i^n\left(\frac{f_i^{\mathrm{V}}}{P^{\Theta }}\right)}}^{{\upsilon }_{ik}}={{\displaystyle \prod _i^n\left(y_i^{\mathrm{V}}{\varphi }_i^{\mathrm{V}}\right)}}^{{\upsilon }_{ik}}=\exp \left[-\frac{{\displaystyle \sum _{i=1}^n{\upsilon }_{ik}{\mu }_i^{\Theta }}}{\mathrm{R}T}\right]$$

(8)

For the actual situation in this paper, as mentioned above, only S₈ in the system is considered to be involved in the chemical reaction; thus, Equation (2) can be simplified to Equation (9).

$${\mathrm{H}}_2{\mathrm{S}+\mathrm{S}}_8\underset{P, T\downarrow }{\overset{P, T\uparrow }{\rightleftarrows }}{\mathrm{H}}_2{\mathrm{S}}_9$$

(9)

When the chemical reaction equilibrium of Equation (9) is reached, the combined Equation (8) system needs to satisfy the following relationship:

$$\frac{y_{{\mathrm{H}}_2{\mathrm{S}}_9}^{\mathrm{V}}{\varphi }_{{\mathrm{H}}_2{\mathrm{S}}_9}^{\mathrm{V}}}{\left(y_{{\mathrm{H}}_2\mathrm{S}}^{\mathrm{V}}{\varphi }_{{\mathrm{H}}_2\mathrm{S}}^{\mathrm{V}}\right)\cdot \left(y_{{\mathrm{S}}_8}^{\mathrm{phy}}{\varphi }_{{\mathrm{S}}_8}^{\mathrm{V}}\right)}=\exp \left[-\frac{\left({\mu }_{{\mathrm{H}}_2{\mathrm{S}}_9}^{\Theta }-{\mu }_{{\mathrm{S}}_8}^{\mathsf{\Theta }}-{\mu }_{{\mathrm{H}}_2\mathrm{S}}^{\mathsf{\Theta }}\right)}{\mathrm{R}T}\right]$$

(10)

where $y_{{\mathrm{H}}_2{\mathrm{S}}_9}^{\mathrm{V}}$ is the molar fraction of H₂S₉ in the gas phase and its value is equal to the chemical solubility of S₈, mol/mol; ${\varphi }_{{\mathrm{H}}_2{\mathrm{S}}_9}^{\mathrm{V}}$ is the fugacity coefficient of H₂S₉ in the gas phase; $y_{{\mathrm{H}}_2\mathrm{S}}^{\mathrm{V}}$ is the molar fraction of H₂S in the gas phase, mol/mol; ${\varphi }_{{\mathrm{H}}_2\mathrm{S}}^{\mathrm{V}}$ is the fugacity coefficient of H₂S in the gas phase; and ${\mu }_{{\mathrm{H}}_2{\mathrm{S}}_9}^{\mathsf{\Theta }}$, ${\mu }_{{\mathrm{S}}_8}^{\Theta }$, and ${\mu }_{{\mathrm{H}}_2\mathrm{S}}^{\mathsf{\Theta }}$ are the standard chemical potentials of H₂S₉, S₈, and H₂S, respectively.

After further sorting out Equation (10), the chemical solubility of elemental sulfur can be obtained, as follows:

$$y_{{\mathrm{S}}_8}^{\mathrm{chem}}=y_{{\mathrm{H}}_2{\mathrm{S}}_9}^{\mathrm{V}}=\frac{\left(y_{{\mathrm{H}}_2\mathrm{S}}^{\mathrm{V}}{\varphi }_{{\mathrm{H}}_2\mathrm{S}}^{\mathrm{V}}\right)\cdot \left(y_{{\mathrm{S}}_8}^{\mathrm{phy}}{\varphi }_{{\mathrm{S}}_8}^{\mathrm{V}}\right)}{{\varphi }_{{\mathrm{H}}_2{\mathrm{S}}_9}^{\mathrm{V}}}\exp \left[-\frac{\left({\mu }_{{\mathrm{H}}_2{\mathrm{S}}_9}^{\Theta }-{\mu }_{{\mathrm{S}}_8}^{\mathsf{\Theta }}-{\mu }_{{\mathrm{H}}_2\mathrm{S}}^{\mathsf{\Theta }}\right)}{\mathrm{R}T}\right]$$

(11)

2.3. Calculation Process of Important Parameters

2.3.1. Important Parameters in Physical Dissolution

As shown in Equation (7), the basic parameters involved in the calculation of the physical dissolution equilibrium of elemental sulfur include the molar volume of solid-phase sulfur, the saturation vapor pressure of elemental sulfur, and the gas-phase fugacity coefficient of elemental sulfur S₈.

(1)

Calculation of the molar volume of solid-phase sulfur $V_{{\mathrm{S}}_8}^{\mathrm{S}}$

The molar volume of solid-phase sulfur can be calculated according to Equation (12):

$$V_{{\mathrm{S}}_8}^{\mathrm{S}}=\frac{M}{{\rho }_{{\mathrm{S}}_8}}$$

(12)

where M is the molar mass of S₈, g/mol, and ${\rho }_{{\mathrm{S}}_8}$ is the density of sulfur, kg/m³. According to the study Shuai and Meisen (1995), the density of elemental sulfur is taken as 2070 kg/m³ when the temperature is less than 433.15 K [35].

Therefore, the molar volume of solid-phase sulfur is as follows:

$$V_{{\mathrm{S}}_8}^{\mathrm{S}}=\frac{32.064\times 8 \mathrm{g}/\mathrm{mol}}{2070 {\mathrm{kg}/\mathrm{m}}^3}=1.2392\times {10}^{-4} {\mathrm{m}}^3/\mathrm{mol}$$

(13)

(2)

Calculation of elemental sulfur saturation vapor pressure $P_{{\mathrm{S}}_8}^{\mathrm{sat}}$

Elemental sulfur’s saturation vapor pressure $P_{{\mathrm{S}}_8}^{\mathrm{sat}}$ is an important parameter in the calculation of elemental sulfur’s solution equilibrium. This section uses two modified elemental sulfur saturation vapor pressure correlation equations obtained from the literature [30].

When T ≤ 368.65 K, elemental sulfur is orthogonal sulfur, and its sulfur saturation vapor pressure can be calculated using Equation (14).

$$\ln P_{{\mathrm{S}}_8}^{\mathrm{sat}}=33.33-\frac{12834.49}{T+8.87}$$

(14)

When T > 368.65 K, elemental sulfur is monoclinic sulfur, and its sulfur saturation vapor pressure can be calculated using Equation (15).

$$\ln P_{{\mathrm{S}}_8}^{\mathrm{sat}}=17.65-\frac{3628.87}{T-171.05}$$

(15)

(3)

Calculation of the elemental sulfur gas phase fugacity coefficient ${\varphi }_{{\mathrm{S}}_8}^{\mathrm{V}}$

The most commonly used method for calculating the elemental sulfur gas phase fugacity coefficient is the EOS method. The van der Waals type equation can be expanded into a cubic polynomial of the volume, as shown in Equation (16). This type of equation of state can not only be solved analytically, but also be solved numerically easily, so this cubic equation of state is widely used in engineering practice. In this paper, the PR equation of state is chosen for the calculation of the gas-phase fugacity coefficient ${\varphi }_{{\mathrm{S}}_8}^{\mathrm{V}}$ [36].

$$P=\frac{\mathrm{R}T}{V-b}-\frac{a}{\left(V+\epsilon b\right)\left(V+\sigma b\right)}$$

(16)

where R is the gas constant, taken as 8.314 J/(mol·K); T is the temperature, K; V is the molar volume of the substance, m³/mol; a and b are the parameters of the equation of state; and $\epsilon$ and $\sigma$ are constants related to the type of equation of state, depending on the type of equation of state. As the equation of state chosen in this paper is the PR equation of state, there are $\epsilon =1-\sqrt{2}$, $\sigma =1+\sqrt{2}$.

For pure components, a and b can be expressed using Equations (17) and (20).

$$a_i=\mathsf{\psi }\frac{\alpha \left(T_{\mathrm{r}i}\right){\mathrm{R}}^2{T}_{\mathrm{c}i}^2}{P_{\mathrm{c}i}}$$

(17)

$$\alpha \left(T_{\mathrm{r}i}\right)={\left[1+\left(0.37464+1.54226{\omega }_i-0.26992{\omega }_i^2\right)\left(1-T_{\mathrm{r}i}^{0.5}\right)\right]}^2$$

(18)

$$T_{\mathrm{r}i}=\frac{T}{T_{\mathrm{c}i}}$$

(19)

$$b_i=\Omega \frac{R{T}_{\mathrm{c}i}}{P_{\mathrm{c}i}}$$

(20)

where $P_{\mathrm{c}i}$ is the critical pressure of component i, Pa; $T_{\mathrm{c}i}$ is the critical temperature of component i, K; $T_{\mathrm{r}i}$ is the comparison temperature of component i; ${\omega }_i$ is the eccentricity factor of component i; and $\mathsf{\psi }$ and $\Omega$ are constants related to the type of equation of state. For the PR equation of state, $\mathsf{\psi }=0.45742$ and $\Omega =0.07780$. The critical parameters and eccentric factors of the main components of high-sulfur-content natural gas involved in the equations are shown in

Table 1. Critical parameters for major natural gas components.

Components	${\mathit{P}}_{\mathbf{c}\mathit{i}}$/MPa	${\mathit{T}}_{\mathbf{c}\mathit{i}}$/K	${\mathit{\omega }}_{\mathit{i}}$
S8	5.0	1025.0	0.4439
H2S	8.963	373.5	0.094
CO2	7.383	304.2	0.224
CH4	4.599	190.6	0.012

.

For natural gas mixtures, parameters a and b need to be calculated according to certain mixing rules, where mixture parameter b is calculated using a linear mixing rule, as shown in Equation (21).

$$b={\displaystyle \sum _i{y}_i{b}_i}$$

(21)

Mixture a uses a quadratic mixing rule, as shown in Equation (22). In this paper, the binary interaction coefficient calculation model improved by Li et al. (2019) is used [30].

$$a={\displaystyle \sum _i{\displaystyle \sum _j{y}_i{y}_j{a}_{ij}}}$$

(22)

$$a_{ij}=\left(1-k_{ij}\right)\sqrt{a_{ii}{a}_{jj}}$$

(23)

where $y_i$ and $y_j$ are the molar fractions of component i and component j, respectively, and $k_{ij}$ is the binary interaction coefficient between component i and component j.

The gas-phase fugacity coefficient of elemental sulfur S₈ is a key parameter for calculating the gas-phase escape of elemental sulfur. It directly affects the accuracy of physical solution equilibrium calculations. The dissolved-phase equilibrium calculated using the cubic type equation of state can be calculated using Equation (24) [34].

$${\varphi }_{{\mathrm{S}}_8}^{\mathrm{V}}=\exp \left[\frac{b_{{\mathrm{S}}_8}}{b}\left(Z-1\right)-\ln \left(Z-\beta \right)-{\overline{q}}_{{\mathrm{S}}_8}I\right]$$

(24)

In the equation, Z is the compression factor of the gas mixture, and when the gas mixture is vapor or vapor-like, the compression factor can be calculated using Equation (25).

$$Z=1+\beta -q\beta \frac{Z-\beta }{\left(Z+\epsilon \beta \right)\left(Z+\sigma \beta \right)}$$

(25)

When the gas mixture is liquid or liquid-like, the compression factor can be calculated using Equation (26).

$$Z=\beta +\left(Z+\epsilon \beta \right)\left(Z+\sigma \beta \right)\frac{1+\beta -Z}{q\beta }$$

(26)

In the equation, $\beta$, $q$, ${\overline{q}}_{{\mathrm{S}}_8}$, and $I$ are calculated using Equations (27) to (30), respectively.

$$\beta =\frac{bP}{\mathrm{R}T}$$

(27)

$$q=\frac{a}{b\mathrm{R}T}$$

(28)

$${\overline{q}}_{{\mathrm{S}}_8}=q\left[\frac{2{\displaystyle \sum _j{y}_j{a}_{j{\mathrm{S}}_8}}}{a}-\frac{b_{{\mathrm{S}}_8}}{b}\right]$$

(29)

$$I=\frac{1}{\sigma -\epsilon }\ln \left[\frac{Z+\sigma \beta }{Z+\epsilon \beta }\right]$$

(30)

2.3.2. Important Parameters in Chemical Solution

As shown in Equation (12), the basic parameters involved in the elemental sulfur chemical equilibrium calculation include the molar fraction of H₂S $y_{{\mathrm{H}}_2\mathrm{S}}^{\mathrm{V}}$, the physical solubility of elemental sulfur S₈ $y_{{\mathrm{S}}_8}^{\mathrm{phy}}$, the gas-phase fugacity coefficient of S₈ ${\varphi }_{{\mathrm{S}}_8}^{\mathrm{V}}$, the gas-phase fugacity coefficient of H₂S ${\varphi }_{{\mathrm{H}}_2\mathrm{S}}^{\mathrm{V}}$, the gas-phase fugacity coefficient of H₂S₉ ${\varphi }_{{\mathrm{H}}_2{\mathrm{S}}_9}^{\mathrm{V}}$, and the standard chemical potentials ${\mu }_{{\mathrm{H}}_2{\mathrm{S}}_9}^{\mathsf{\Theta }}$, ${\mu }_{{\mathrm{S}}_8}^{\Theta }$, and ${\mu }_{{\mathrm{H}}_2\mathrm{S}}^{\mathsf{\Theta }}$ of H₂S₉, S₈, and H₂S, respectively. The molar fraction of H₂S $y_{{\mathrm{H}}_2\mathrm{S}}^{\mathrm{V}}$ is given according to the actual situation, the physical solubility of elemental sulfur $y_{{\mathrm{S}}_8}^{\mathrm{phy}}$ is calculated from the physical solution equilibrium, and the gas-phase fugacity coefficient of S₈ ${\varphi }_{{\mathrm{S}}_8}^{\mathrm{V}}$ is calculated from the physical solution equilibrium calculation.

(1)

The gas-phase fugacity coefficient of H₂S ${\varphi }_{{\mathrm{H}}_2\mathrm{S}}^{\mathrm{V}}$

Similar to the gas-phase fugacity coefficient of S₈ ${\varphi }_{{\mathrm{S}}_8}^{\mathrm{V}}$, the gas-phase fugacity coefficient of H₂S ${\varphi }_{{\mathrm{H}}_2\mathrm{S}}^{\mathrm{V}}$ can be expressed as Equation (31).

$${\varphi }_{{\mathrm{H}}_2\mathrm{S}}^{\mathrm{V}}=\exp \left[\frac{b_{{\mathrm{H}}_2\mathrm{S}}}{b}\left(Z-1\right)-\ln \left(Z-\beta \right)-{\overline{q}}_{{\mathrm{H}}_2\mathrm{S}}I\right]$$

(31)

(2)

The gas-phase fugacity coefficient of H₂S₉ ${\varphi }_{{\mathrm{H}}_2{\mathrm{S}}_9}^{\mathrm{V}}$

The gas-phase fugacity coefficient of H₂S₉ ${\varphi }_{{\mathrm{H}}_2{\mathrm{S}}_9}^{\mathrm{V}}$ can be expressed as Equation (32).

$${\varphi }_{{\mathrm{H}}_2{\mathrm{S}}_9}^{\mathrm{V}}=\exp \left[\frac{b_{{\mathrm{H}}_2{\mathrm{S}}_9}}{b}\left(Z-1\right)-\ln \left(Z-\beta \right)-{\overline{q}}_{{\mathrm{H}}_2{\mathrm{S}}_9}I\right]$$

(32)

When considering the chemical solution of elemental sulfur, the calculation of parameters $a_{{\mathrm{H}}_2{\mathrm{S}}_9}$ and $b_{{\mathrm{H}}_2{\mathrm{S}}_9}$ requires physical parameters, such as the critical parameters of H₂S₉ and the eccentric factor according to Equation (32). As H₂S₉ is not stable, it is difficult to use experiments to measure its relevant physical parameters. A method for the indirect calculation of the physical parameters of associated molecules was proposed by Heidemann et al. (2001, 1976) [25,37]. They considered the reactants in the chemical reaction equation as the source substances of the products and the corresponding stoichiometric coefficients in the equation as the weighting coefficients, and calculated the physical parameters of the reactants by weighting them in order to determine the relevant state parameters of the products, as shown in Equations (33)–(35).

$$a_i={\displaystyle \sum _{j=1}^{N_{\mathrm{S}}}{\nu }_{ji}{a}_j}$$

(33)

$$b_i={\displaystyle \sum _{j=1}^{N_S}{\nu }_{ji}{b}_j}$$

(34)

$$a_{ij}={\displaystyle \sum _{k=1}^{N_{\mathrm{S}}}{\displaystyle \sum _{l=1}^{N_{\mathrm{S}}}{\nu }_{ki}{\nu }_{lj}}}{a}_{kl}$$

(35)

For H₂S₉ involved in chemical reaction (9) in this paper, the source substances of H₂S₉ are H₂S and S₈. The stoichiometric coefficients are both 1, so 1 mol of H₂S and 1 mol of S₈ are required to produce 1 mol of H₂S₉. Therefore the equation of the state parameters of H₂S₉ according to the method proposed by Heidemann et al. (2001, 1976) can be calculated from the relevant parameters of H₂S and S₈ [25,37].

$$a_{{\mathrm{H}}_2{\mathrm{S}}_9}=a_{{\mathrm{H}}_2\mathrm{S}}+a_{{\mathrm{S}}_8}$$

(36)

$$b_{{\mathrm{H}}_2{\mathrm{S}}_9}=b_{{\mathrm{H}}_2\mathrm{S}}+b_{{\mathrm{S}}_8}$$

(37)

Similarly, for H₂S₉, the interaction coefficient term with other components can be expressed using Equation (38).

$$a_{{\mathrm{H}}_2{\mathrm{S}}_9, j}=a_{{\mathrm{H}}_2\mathrm{S}, j}+a_{{\mathrm{S}}_8, j}=\left(1-k_{{\mathrm{H}}_2\mathrm{S}, j}\right)\sqrt{a_{{\mathrm{H}}_2\mathrm{S}}{a}_j}+\left(1-k_{{\mathrm{S}}_8, j}\right)\sqrt{a_{{\mathrm{S}}_8}{a}_j}$$

(38)

In summary, the gas-phase fugacity coefficient of H₂S₉ ${\varphi }_{{\mathrm{H}}_2{\mathrm{S}}_9}^{\mathrm{V}}$ and other interaction coefficient terms of H₂S₉ with other high-content natural gas components can be calculated.

(3)

The standard chemical potentials ${\mu }_{{\mathrm{H}}_2{\mathrm{S}}_9}^{\mathsf{\Theta }}$, ${\mu }_{{\mathrm{S}}_8}^{\Theta }$, and ${\mu }_{{\mathrm{H}}_2\mathrm{S}}^{\mathsf{\Theta }}$ for H₂S₉, S₈, and H₂S

To facilitate the solution of the model and the preparation of the calculation program, Heidemann et al. (2001) fitted the standard chemical potentials of H₂S₉, S₈, and H₂S as a function of temperature when the temperature is 300–1300 K, as expressed by Equation (39) [25].

$$\frac{{\mu }_i^{\Theta }}{\mathrm{R}T}=\frac{c_1}{T}+c_2\ln T+c_3+c_{4}T+c_5{T}^2+c_6{T}^3$$

(39)

The constant parameters corresponding to each component in Equation (39) are shown in

Table 2. Standard chemical potential constants.

Constants	c1 × 10−3	c2	c3	c4 × 103	c5 × 106	c6 × 109
H2S	−3.36061	−1.40815	−11.7803	−7.81173	4.85597	−1.68973
S8	6.66384	−16.1969	60.6753	−7.37600	2.28741	−0.368491
H2S9	−5.68656	−71.6492	377.350	99.3619	−49.2157	11.2677

.

3. Results and Discussion

3.1. Model Verification

The thermodynamic solubility prediction models established by Karan, Gu, and Sun et al. [15,22,24], respectively, ignore the chemical dissolution of elemental sulfur in gas mixtures, which is inconsistent with the actual situation. Therefore, Heidemann’s model considering chemical dissolution was selected in this paper to compare the predicted results of the solubility of elemental sulfur in H₂S with the improved model in this paper. Figure 1 shows a comparison of sulfur’s solubility in the H₂S prediction results at temperatures of 316.26 K and 338.71 K using Heidemann’s model and our model with experimental data [9,25]. Upon comparing the results of model calculation with experimental data, the overall relative deviations and corresponding absolute deviations are better than those of Heidemann’s model. The results show that our model can better fit the experimental data of elemental sulfur in hydrogen sulfide, especially at temperatures such as those in gas transmission pipelines.

Figure 2 shows a comparison of sulfur’s solubility in the CO₂ prediction results at temperatures of 316.26 K and 338.71 K using Karan’s model, Sun’s model, Cézac’s model, Heidemann’s model, and our model with experimental data [8,15,16,19,24,25]. After the results of model calculation were compared with experimental data, the overall relative deviations and corresponding absolute deviations were better than those of other models. The overall relative deviations and corresponding absolute deviations of sulfur’s solubility in CH₄ using our model are better than those of other models.

3.2. Calculated Case Description

Based on the prediction model of elemental sulfur’s phase equilibrium solubility under the gathering conditions established in the previous section, the procedure flow of the elemental sulfur solubility calculation in high-sulfur-content natural gas was designed, and the corresponding calculation program was prepared by using C#2010. Using the conditions of the collecting temperature (273.15 K ≤ T ≤ 333.15 K) and collecting pressure (0.5 MPa ≤ P ≤ 15 MPa) as a background, this study calculates and analyzes the variation relationship between elemental sulfur’s total solubility, physical solubility, and chemical solubility and the gathering temperature, gathering pressure, and concentrations of H₂S and CO₂ in the gas phase in high-sulfur-content natural gas containing the three main components of H₂S (volume fraction ≤ 20%), CO₂ (volume fraction ≤ 10%), and CH₄ (volume fraction ≥ 70%). Moreover, the solution mechanism of elemental sulfur in high-sulfur-content natural gas and its variation law are studied deeply.

In order to further study the solution mechanism and solubility variation law of elemental sulfur in high-sulfur-content natural gas mixtures under the gathering conditions, six high-sulfur-content natural gas mixtures with different components were firstly set up, as shown in

Table 3. High-sulfur natural gas mixture components.

No.	Gas 1	Gas 2	Gas 3	Gas 4	Gas 5	Gas 6
Molar Content of Components/%
H2S	0.5	2	5	10	15	20
CO2	5	5	5	5	5	5
CH4	94.5	93	90	85	80	75

. The components included three gas components, H₂S, CO₂, and CH₄, with the molar percentages of H₂S ranging from 0.5% to 20%, the molar percentage of CO₂ remaining at a constant value of 5%, and the molar percentages of CH₄ ranging from 94.5% to 75%. Combined with the actual conditions of the gathering conditions, in this study, the gathering temperature range studied was determined to be 273.15–333.15 K, and the gathering pressure range studied was 0.5–15.0 MPa. According to the conclusions of the phase analysis in Section 2.1, elemental sulfur exists in only two phases, gas and solid, in the pressure and temperature range covered in the study case, while the high-sulfur-content natural gas always remains in the gas phase.

3.3. Analysis of the Effect of Temperature on the Solution Mechanism of Elemental Sulfur in High-Sulfur-Content Natural Gas

3.3.1. The Effect of Temperature on the Total Solubility of Elemental Sulfur

Figure 3 shows the graph of the total solubility of elemental sulfur with temperature in gas mixture 3. The curves with different colors in the graph represent the total solubility of elemental sulfur under different pressure conditions. It can be seen that the total solubility of elemental sulfur had a similar trend to temperature, decreasing first and then increasing in general. Under lower-pressure conditions (P ≤ 4.0 MPa), the total elemental sulfur solubility first changed little and then increased continuously as the temperature increased from 273.15 K to 333.15 K. However, when P ≥ 6.0 MPa, the total elemental sulfur solubility first decreased significantly and then increased slightly. This is because, when the temperature was lower, the lower the saturated vapor pressure of the gas phase of elemental sulfur at the corresponding temperature, the fewer sulfur molecules enter the gas phase, resulting in lower physical solubility. However, the lower the temperature, the more stable H₂S₉ is, so the sulfur molecules are more likely to combine with H₂S molecules to form H₂S₉. Thus, the chemical solubility increases. This eventually led to a corresponding increase in the total solubility of elemental sulfur.

In addition, the saturation vapor pressure of elemental sulfur increased with increasing temperature, which caused an increase in physical solubility. However, at the same time, the increase in temperature caused a decrease in solvent density and also caused the decomposition of H₂S₉, causing a decrease in chemical solubility and ultimately causing a decrease in total solubility. These two processes have opposite effects on the total solubility as the temperature increases, and when the two produce an equilibrium point, the lower point of total solubility shown in Figure 3 occurs. Therefore, the lower the total solubility at a lower temperature, the more conducive to the formation of polymer H₂S₉, and the chemical solubility occupies a larger proportion. On the contrary, the greater the total solubility at a higher temperature, the more the physical solubility occupies a larger proportion because of the significant increase in elemental sulfur molecules in the gas phase.

3.3.2. The Effect of Temperature on the Physical Solubility of Elemental Sulfur

Figure 4 shows a graph of the physical solubility of elemental sulfur with temperature in gas 3. When P ≤ 9.0 MPa, the physical solubility of elemental sulfur tended to increase significantly with temperature; when P ≥ 12.0 MPa, the physical solubility of elemental sulfur decreased slowly with the increase in temperature and then increased slowly. In general, the physical solubility did not change much at this time.

As the temperature increased, the saturation vapor pressure of elemental sulfur increased and the diffusion of sulfur molecules was enhanced, making it easier for the solid-phase sulfur molecules to diffuse into the gas phase. This was the main reason for the increase in the physical solubility of elemental sulfur. When P ≤ 9.0 MPa, the intermolecular distance was a little larger compared with that above 9.0 MPa, which also facilitated the diffusion of solid-phase sulfur molecules.

When P > 9.0 MPa, the increase in temperature increased the density of the gas mixture, the molecular spacing decreased, and the molecules in the dense phase of the gas phase hindered the diffusion of sulfur molecules. Thus, there was a certain degree of reduction in physical solubility. The increase in temperature drove the diffusion of solid-phase sulfur molecules, but the effect of temperature was a secondary factor at this point. As the temperature continued to increase, the saturated vapor pressure of elemental sulfur continued to increase; at this time, the temperature-induced molecular diffusion continued to increase. The result was that the diffusion of solid-phase sulfur molecules began to gradually dominate, and the physical solubility of elemental sulfur increased with increasing temperature. At P > 9.0 MPa, an extreme point of physical solubility appeared, which was the result of the balance between the effect of higher pressure inducing a high-density gas phase to hinder the diffusion of sulfur molecules and the effect of higher temperature to promote the diffusion of sulfur molecules.

3.3.3. The Effect of Temperature on the Chemical Solubility of Elemental Sulfur

Figure 5 shows a graph of the chemical solubility of elemental sulfur with temperature in gas 3. It can be seen that the trends of chemical solubility in the six gas mixtures with temperature were exactly the same. The chemical solubility of elemental sulfur in high-sulfur-content natural gas tended to decrease gradually with increasing temperature. The higher the pressure, the greater the decrease in chemical solubility with increasing temperature.

According to the mechanism of chemical solution of S₈ and H₂S, the essence of chemical dissolution of S₈ was due to the chemical reaction of S₈ with H₂S to produce polymeric polysulfide hydrogen H₂S₉. Due to the poor thermal stability of H₂S₉, a lower temperature is conducive to the stable existence of the polymer. Conversely, at a higher temperature, H₂S₉ is not stable under heat, which promotes the decomposition of H₂S₉ into H₂S and S₈. As shown in the figure, the chemical solubility decreased with the increase in temperature.

On the other hand, it can be seen from the graph that the solubility curve with higher pressure at the same temperature was always at the upper part of the curve with lower pressure. This was because an increase in pressure created a tendency for the distance between molecules to decrease, and the opportunity for contact and collision between S₈ and H₂S increased, thus facilitating the polymerization of S₈ and H₂S to form polysulfide hydrogen H₂S₉. Therefore, we can infer that the lower the transport temperature and higher the pressure, the more favorable the chemical solution of elemental sulfur in high-sulfur-content natural gas.

3.3.4. Analysis of the Effect of Temperature on the Solution Mechanism of Elemental Sulfur

The previous paper analyzed the law of the effect of gathering temperature on physical solubility and chemical solubility. This section further analyzes the law of the influence of gathering temperature on the physical and chemical solution mechanisms of elemental sulfur in high-sulfur-content natural gas mixtures; that is, it investigates which dissolution mode is the main solution mechanism of elemental sulfur in high-sulfur-content natural gas under the influence of different temperatures.

The solubility changes of elemental sulfur in gas 1 and gas 4 were selected as a representative case to analyze the law of temperature’s influence on the mechanism of physical and chemical solution. As shown in Figure 6, the proportion of the total solubility accounted for by physical solubility increased from less than 20% to nearly 100% as the temperature increased from 273.15 K to 333.15 K for elemental sulfur’s solubility in gas 1, for example. The proportion of chemical solubility to total solubility followed an opposite trend. This indicates that temperature had a significant effect on the change in elemental sulfur’s solution mechanism. There was also a similar trend of variation in gas 4.

Figure 7 shows the change curves of physical solubility and chemical solubility as a percentage of total solubility in the same system under the same conditions. The solid line in the graph represents the variation curve of physical solubility as a percentage of total solubility with temperature, and the dashed line of the corresponding color represents the variation curve of chemical solubility as a percentage of total solubility with temperature under the same conditions. It can be seen that the proportion of chemical solubility increased as the temperature became closer to 273.15 K and approached 100%. As the temperature increased to 333.15 K, the percentage of chemical solubility due to the decomposition of H₂S₉ gradually decreased and approached 0%. In contrast, the changing process of the proportion of physical solubility was the opposite; as the temperature gradually increased from 273.15 K, the proportion of physical solubility gradually increased from near 0% to close to 100%. In addition, it can be observed from the graph that the curves of the proportion of the two types of solubility always intersected at a certain temperature, and the higher the pressure, the higher the temperature corresponding to the intersection point. This was because the higher the pressure, the higher the proportion of chemical solubility, and the higher the temperature, the higher the physical solubility, so the higher the pressure, the higher the temperature when the two solubilities reach equality. In conclusion, chemical solution is the main solution mechanism of elemental sulfur in high-sulfur-content natural gas at lower gathering temperatures. Conversely, physical solution is the main solution mechanism of elemental sulfur.

3.4. Analysis of the Effect of Pressure on the Solution Mechanism of Elemental Sulfur in High-Sulfur-Content Natural Gas

3.4.1. Analysis of the Effect of Pressure on the Total Solubility of Elemental Sulfur

Figure 8 shows the variation law of the total solubility of elemental sulfur in gas 3 in the gathering pressure range. As the pressure increased from 0.5 MPa to 15 MPa, the total solubility of elemental sulfur showed a trend of slowly decreasing and then continuously increasing. The pressure corresponding to the turning point of the total solubility of the elemental sulfur is about 2.0 MPa.

It can be seen from the figure that, when P ≥ 2.0 MPa, the lower the temperature, the greater the increase in the total solubility with increasing pressure. The second half of the total solubility curve is in the upper part of the graph at lower temperatures. This is because the lower temperature and higher pressure significantly promoted the chemical solution of elemental sulfur. In addition, we can see that the curve with a lower temperature is not mostly at the upper part, and the increase in sulfur solubility was greater with the increase in pressure under the condition of lower temperatures. In other words, at higher temperatures, more solid sulfur molecules would be converted into vapor sulfur molecules; eventually, the solubility of sulfur would have to increase.

When the pressure was 0.5–2.0 MPa, the increase in gathering pressure promoted the chemical solution of elemental sulfur; however, on the other hand, when the temperature was constant, the increase in pressure caused the density of gas-phase molecules to increase, the molecular spacing to decrease, and the intermolecular repulsion to cause the gas-phase sulfur molecules to become oversaturated, thus reducing the physical solubility of elemental sulfur. The increase in chemical solubility was not enough to offset the decrease in physical solubility caused by the increase in the gas phase molecular density.

3.4.2. Analysis of the Effect of Pressure on the Physical Solubility of Elemental Sulfur

Figure 9 shows the variation in the physical solubility of elemental sulfur in gas 3 with pressure. It can be seen that, as the pressure gradually increased from 0.5 MPa to 15 MPa, the physical solubility of elemental sulfur first decreased and then increased, and the pressure corresponding to the turn in physical solubility was about 2.0 MPa.

When the pressure was 0.5–2.0 MPa, the density of gas-phase molecules increased with the increase in pressure, while the distance between gas-phase molecules decreased. This led to an increase in intermolecular repulsion, causing the gas-phase sulfur molecules to become supersaturated, and, finally, the physical solubility of elemental sulfur in high-sulfur-content natural gas decreased. At low temperatures, the physical solubility was always lower. At low temperatures, it was difficult to convert solid sulfur molecules into the gas phase. There was only a small amount of elemental sulfur in the gas phase, and sulfur molecules tended to combine with hydrogen sulfide to produce H₂S₉, a relatively stable compound at low temperatures.

After the pressure exceeded 2.0 MPa, the physical solubility of elemental sulfur increased significantly as the pressure increased to 15.0 MPa. This was because the increase in chemical solubility of elemental sulfur increased with the increase in pressure. For ease of analysis, the mechanism of the chemical dissolution of elemental sulfur can be viewed as the polymerization of 1 mol of S₈ and 1 mol of H₂S to form 1 mol of H₂S₉, indicating that the increase in chemical solubility drove a decrease in the number of molecules in the gas phase. This effect continued to increase with increasing pressure in the subsequent pressure range, resulting in a decrease in the total number of molecules in the gas phase. The intermolecular distance increased again in the direction favoring the diffusion of solid-phase sulfur molecules into the gas phase. This was the main reason why the physical solubility of elemental sulfur increased with further increases in pressure after P ≥ 2.0 MPa. It is worth noting that the physical solubility of elemental sulfur increased significantly with the increase in pressure at lower temperatures. The distance between solvent molecules would be smaller under conditions of greater pressure. H₂S₉ has to break down to produce more elemental sulfur and hydrogen sulfide. As the temperature was stable and it was difficult to change the elemental sulfur molecules change from the gas phase to the solid phase, they instead existed in the gas phase as a stable form of physical dissolution.

3.4.3. Analysis of the Effect of Pressure on the Chemical Solubility of Elemental Sulfur

Figure 10 shows the variation curves of the chemical solubility of elemental sulfur in gas 3 with pressure. It can be seen that, with the gradual increase in pressure from 0.5 MPa to 15 MPa, the chemical solubility of elemental sulfur shows a gradual increase with increasing of temperature. This is because the higher the pressure, the higher the density and the smaller the spacing between S₈ molecules and H₂S molecules. The chances for contact and collision bonding between the two increase, thus promoting the generation of polymeric polysulfide H₂S₉.

When the temperature was lower, the trend of increasing chemical solubility of elemental sulfur was more significant with increasing pressure. This was because the structure of the polymer H₂S₉ produced by the chemical solution of elemental sulfur was more stable at lower temperatures. Namely, the lower the temperature, the more favorable the chemical solution of elemental sulfur. In addition, it can be seen from the figure that the increase with pressure of chemical solution showed a monotonic increasing trend, and the increase was greater at higher pressures. Therefore, the increase in the gathering pressure had a significant promotion effect on the chemical solution of elemental sulfur.

3.4.4. Analysis of the Effect of Pressure on the Dissolution Mechanism of Elemental Sulfur

Based on the results of the previous analysis, considering that the trends of physical and chemical solubility changes of elemental sulfur in six types of high-sulfur-content natural gas are very close, this section selects only the solubility mechanism of elemental sulfur in gas 4 as an example to analyze the law of the influence of gathering pressure on the solubility mechanism of elemental sulfur. Figure 11a,b show the curves of physical solubility as a percentage of the total solubility and chemical solubility of elemental sulfur in gas 4 with the pressures, respectively. Regarding the trend of the solubility percentages, the percentage of physical solubility gradually decreased with increasing pressure, while the percentage of chemical solubility gradually increased.

In addition, the lower the temperature, the smaller the change in the solubility percentage caused by the pressure change. As shown in Figure 11 and Figure 12, when the temperature was 273.15 K, neither the physical nor chemical solubility percentages changed much, and the chemical solution mechanism was dominant at this time; when the temperature was 333.15 K, the physical solution mechanism was dominant.

The solid and dashed lines in Figure 12 indicate the variation patterns of the proportion of physical and chemical solubility with pressure, respectively. It can be seen that, when the temperature was less than 293.15 K, the chemical solubility mechanism of elemental sulfur always dominated with the change in pressure; when the temperature was greater than 303.15 K, the chemical solubility and physical solubility were equal at a certain pressure.

3.5. Analysis of the Effect of H₂S Concentration on the Dissolution Mechanism of Elemental Sulfur in High-Sulfur-Bearing Natural Gas

(1)

Temperature of 293.15 K

Figure 13a, b show the relationship curves between the physical solubility of elemental sulfur as a percentage of total solubility, and chemical solubility as a percentage of total solubility and the molar fraction of H₂S, respectively, at a temperature of 293.15 K. The variation pattern of the curves shown in the figure once again demonstrates the promoting effect of increasing H₂S concentration on the chemical solution mechanism of elemental sulfur and the inhibiting effect on the physical solution mechanism of elemental sulfur when the temperature is constant. When the molar fraction of H₂S gradually increased from 0.5% to 20%, the proportion of physical solubility gradually decreased to 0%; on the contrary, the proportion of chemical solubility gradually increased to 100%.

The short, dashed line in Figure 14 represents the percentage of chemical solubility of elemental sulfur, and the solid line with the same color as the dashed line represents the percentage of physical solubility of elemental sulfur. Overall, as the molar fraction of H₂S in the gas increased, the proportion of chemical solubility increased. Only when the molar content of H₂S was small and the pressure was low did the proportion of physical solubility dominate. In addition, above this H₂S molar content, the main solution mechanisms of elemental sulfur were dominated by chemical solution.

(2)

Pressure of 6.0 MPa

Figure 15a,b show the variation curves of the physical and chemical solubility as a percentage of the total solubility versus the molar fraction of H₂S under 6.0 MPa pressure conditions, respectively. The variation pattern of the curves shown in the figure indicated that the increase in the H₂S concentration had a promoting effect on the chemical dissolution mechanism of elemental sulfur and an inhibiting effect on the physical dissolution mechanism of elemental sulfur. As the molar fraction of H₂S gradually increased from 0.5% to 20%, the percentage of physical solubility gradually decreased, but the reduction was not always large. Especially when the temperature approached 273.15 K and 333.15 K, the changes in the physical and chemical solubility percentages were less than 10% and 20%, respectively.

In Figure 16, the short, dashed line represents the percentage of chemical solubility of elemental sulfur and the solid line with the same color as the dashed line represents the percentage of physical solubility of elemental sulfur. Overall, the larger the molar fraction of H₂S in the gas, the higher the percentage of chemical solubility. However, the increase in the percentage of chemical solubility was also limited by the temperature. As shown in Figure 16, when the temperature was greater than 323.15 K, the percentage of chemical solubility was still less than 50%, even though the molar fraction of H₂S gradually increased from 0% to 20%. This was because the tendency of the increase in temperature to promote the physical solution mechanism still dominated.

3.6. Analysis of the Effect of CO₂ Concentration on the Solution Mechanism of Elemental Sulfur in High-Sulfur-Content Natural Gas

The previous section analyzed the law of the effect of the variation in the H₂S concentration of the gas components on the solubility and solution mechanism of elemental sulfur. The results show that, as the concentration of H₂S in the gas mixture increased, the total solubility of elemental sulfur also increased, to some extent. According to the experimental data, the solubility of elemental sulfur in single-component CO₂ was significantly greater than that of elemental sulfur in single-component CH₄ under the same conditions. Therefore, it is necessary to analyze the law of the effect of the CO₂ concentration in gas mixtures on the solubility of elemental sulfur.

As shown in

Table 4. High-sulfur-content natural gas mixture components.

NO.	Gas 7	Gas 8	Gas 9	Gas 10	Gas 11	Gas 12
Molar Content of Components/%
H2S	5	5	5	5	5	5
CO2	0.5	2	4	6	8	10
CH4	94.5	93	91	89	87	85

, six gas mixtures with different molar concentrations of their components were set. The molar concentration of H₂S in the gas was 5% and remained fixed. The molar concentration of CO₂ ranged from 0.5 to 10%, and that of CH₄ ranged from 85 to 94.5%. This was because the solubility of elemental sulfur in the single-component CO₂ was significantly greater than the solubility of elemental sulfur in the single-component CH₄ under the same conditions. When the molar concentrations of CO₂ and CH₄ in the gas mixture changed by the same magnitude, it can still be assumed that the main reason for the change in elemental sulfur solubility was the change in the molar concentration of CO₂ in the gas.

Figure 17a,b show the variation curves between the physical and chemical solubility of elemental sulfur as percentages of the total solubility and the molar fraction of CO₂, respectively, at 293.15 K. It can be seen that the percentages of the physical and chemical solubility of elemental sulfur in the total solubility differed under different pressure conditions. However, the proportion of the physical and chemical solubilities of elemental sulfur in the total solubility remained essentially the same as the molar fraction of CO₂ increased from 0.5% to 10%. It shows that the effect of the CO₂ concentration on the solution mechanism of elemental sulfur was not significant when the CO₂ content in the gas was varied in the low concentration range (molar fraction not exceeding 10%).

Figure 18 shows the variation curves of physical and chemical solubility as percentages of the total solubility versus the molar fraction of CO₂ under 6.0 MPa pressure conditions, respectively. The overall trend of the curve change is consistent with Figure 17. The ratios of the physical and chemical solubilities of elemental sulfur to the total solubility remained essentially the same as the molar fraction of CO₂ increased from 0.5% to 10% at a pressure of 6.0 MPa. It can be inferred that the effect of the CO₂ concentration changes on the solubility and solution mechanism of elemental sulfur in high-sulfur-content natural gas was negligible at low concentrations (molar fraction not exceeding 10%) of CO₂ concentration in the gas.

4. Conclusions

In order to further study the law of the variation of elemental sulfur’s solubility and solution mechanism under gathering conditions, based on the established comprehensive prediction model of elemental sulfur solubility, this paper analyzed the law of the influence of four main factors, including temperature, pressure, and the H₂S and CO₂ concentrations, on elemental sulfur’s solution equilibrium.

The results show that:

(1) The solution of elemental sulfur in high-sulfur-content gas is accompanied by both physical and chemical solution. At lower temperatures, the predominant mechanism is chemical solution; at higher temperatures, the predominant mechanism is physical solution. At certain temperatures, the two reach equilibrium. In addition, the lower the temperature and the higher the pressure, the greater the solubility of elemental sulfur in natural gas.

(2) With the increase in pressure, the chemical solubility of elemental sulfur monotonically increased, and the increase became larger and larger.

As the pressure increased from 0.5 MPa to 15 MPa, the physical solubility of elemental sulfur first decreased slightly with the increase in pressure and then increased gradually. The pressure inflection point was about 2.0 MPa.

(3) As the molar concentration of H₂S in the gas increased from 0.5% to 20%, the chemical solubility of elemental sulfur increased significantly and contributed to the increase in total solubility. However, the physical solubility of elemental sulfur showed a tendency to decrease.

(4) When the molar concentration of CO₂ in the gas mixture ranged from 0.5 to 10%, the change in CO₂ concentration had a negligible effect on elemental sulfur’s solubility and solution mechanism.

(5) After analysis, we found that the behavior of sulfur deposition was influenced by elemental sulfur’s solubility in the gas phase, nucleation and particle growth, and gas and sulfur particle interaction in the process of high-sulfur natural gas transmission. Elemental sulfur solubility and its controlling factors are discussed in detail in this work. However, accurate prediction of the probability of sulfur deposition in the pipeline of high-sulfur natural gas transmission is a meaningful topic. Next, we will study the mechanism of sulfur nucleation and particle growth, gas and sulfur particle interactions, and other related issues further.