Numerical simulation of sulfur particle agglomeration at bends of high sulfur natural gas gathering pipelines based on Euler–PBM coupling

Abstract

Sulfur deposition can result in an increase in the wall thickness of high-sulfur natural gas gathering pipelines, leading to issues like unstable pipeline flow. It is crucial to reveal the aggregation of sulfur particles at key locations of high-sulfur natural gas gathering pipelines to predict the location and amount of sulfur deposition in the pipelines. In this paper, the Euler–PBM (Population balance model) coupling is used to establish a numerical simulation model of gas–solid two-phase pipe flow accompanied by sulfur particle agglomeration in the pipe bends, focusing on the influence of sulfur particle volume fraction, pipe inclination angle and inlet flow velocity on sulfur particles agglomeration behavior. The results show that the sulfur particles have a significant agglomeration effect at the bend of the collecting pipeline, and the agglomeration growth occurs to different degrees throughout the bend, and the main area of sulfur particles agglomeration is near the top wall of the pipeline, followed by other areas near the wall of the pipeline. When the inlet volume fraction of sulfur particles was increased from 0.05 to 0.25%, and the inclination angle of the pipe was increased from 30° to 90°, the distribution range of sulfur particle size after agglomeration became wider, and the maximum size of sulfur particles was 187.56 μm, and the effect of sulfur particle agglomeration was enhanced; the inlet flow rate was increased from 3.0 to 9.0 m/s, and the reduction range of sulfur particle size after agglomeration was 5.68–9.87 μm. The maximum particle size of sulfur particles also decreased, and the effect of sulfur particle agglomeration was weakened.

Introduction

As global warming intensifies, reducing carbon emissions has become a crucial priority. Now the most effective way to achieve this goal is by replacing non-clean energy sources, especially coal, with natural gas. High-sulfur natural gas is widely distributed in the world, and the development and utilization of high-sulfur natural gas is one of the major ways to reduce carbon emissions. Different from the conventional natural gas development, the formation of elemental sulfur is inevitable with the formation of H₂S under the condition of high temperature and high pressure¹. When high-sulfur gas flows into a surface gathering pipeline, the pressure and temperature inside the gathering pipeline are relatively low, and the pressure and temperature along the pipeline will change continuously. Monomorphic sulfur will transform directly from the gaseous state to the solid state to form sulfur nuclei, which will grow and agglomerate with the natural gas flow in the pipeline, and then transport with the natural gas in the pipeline². These nuclei will continuously grow and agglomerate along with the flow of natural gas in the pipeline, resulting in sulfur deposition in the gathering pipeline. Sulfur deposition in the pipeline can lead to blockages in the pipeline and corrosion of the pipeline and equipment^3,4. Corrosion product ${\text{FeS}}$ is capable of spontaneous combustion, which can result in the leakage of high sulfur-containing natural gas and explosions, and other major safety accidents, resulting in harm to public life, health, and safety, as well as significant economic losses. When the sulfur particles formed in the flow field are not all deposited on the inner wall of the pipeline, they will be transported along with the gas phase in the pipeline, which is a typical gas–solid two-phase flow. Therefore, accurately describing the gas–solid two-phase flow in high-sulfur natural gas is critical to precisely predict the location and amount of sulfur deposition.

For the problem of gas–solid two-phase flow, the most commonly used methods are experimental measurements and numerical simulation. In experimental aspect, Tang et al.⁵ applied the method to the study of the acceleration performance of tracer particles in different structures of inlet flue. Chen et al.⁶ applied the PIV technique to the study of the gas–liquid two-phase flow characteristics in the bulging tower. Ruan et al.⁷ applied the PIV technology to the study of the gas–solid two-phase flow characteristics in the asymptotic tube to obtain the law of influence of the gas flow rate, the position of the particle inlet, and the particle size on the particle flow rate, particle inlet position and particle size on gas flow rate. And Luo et al.⁸ successfully applied the PIV technology to the solid particle motion characteristics under gas–liquid-solid three-phase coexistence. Due to the highly toxic nature of H₂S gas, there are still a lot of problems to be solved in applying these test methods to the study of sulfur particle motion laws in gathering pipelines. The gas–solid two-phase flow process is affected by many factors. It brings a lot of difficulties to experimental measurements, such as the uneven distribution of solid-phase particle concentration in the cross-section, the great change in the difference between the velocity of gas and solid phases, and the difficulty of capturing the invariance of the flow pattern and other problems⁹. Therefore, numerical simulation is the main research method used at home and abroad.

For numerical simulation, based on the basic characteristics of the gas–solid two-phase flow of sulfur-containing particles, it is generally necessary to model the gas phase and the sulfur particle phase separately. The gas phase can usually be regarded as a continuous phase, and the corresponding mathematical model is established based on Euler coordinates. The key to the modelling of gas–solid two-phase flow with sulfur particles lies in the modelling of the particulate phase and the coupling and solution of the particulate phase model with the continuous phase. Chai et al.¹⁰ used a population balance model to simulate the particle aggregation behavior and investigated the effects of the initial average particle size of flocculated particles and the volume fraction of flocculated particles on the aggregation behavior of particles in a grid flocculation cell using a CFD–PBM coupling method. Balakin et al.^11,12 applied the PBM to study the nucleation, growth, aggregation and fragmentation of hydrate particles, and based on this the PBM and CFD coupling method was applied. Based on coupling PBM and CFD to study the particle size distribution law of hydrate after agglomeration under turbulent conditions. Liu¹³ simulated the upward flat elbow section, horizontal upward elbow section and anti-Z elbow section of the pipeline, and found that the main location of sulfur deposition was at the flat section of the upward flat elbow section, the outside of the elbow of the horizontal upward elbow section, and the outside of the lower elbow of the anti-Z elbow section, as well as at the flat section of the pipeline after the upper elbow. Veluswamy et al.¹⁴ simulated the sulfur deposition in the vicinity of ball valves and their connected pipelines using a continuous-discrete joint model, and the results showed that the larger the valve opening, the closer the downstream of the valve, the closer the sulfur particles were to the valve when they started to be deposited. Zhu et al.¹⁵ used a single-phase natural gas pipeline model to calculate the temperature and pressure distributions along the route and then used this model as a basis for the calculation of a long-track sulfur solubility, deposition, and the area of sulfur particle deposition. particle deposition region. Chen et al.¹⁶ used the Reynolds stress model to describe the gas phase, and concluded that the deposition rate of sulfur particles at the ball valve was positively correlated with the gas flow rate when the ball valve opening and particle size were certain; whereas, the deposition rate of sulfur particles increased with the increase of particle size when the valve opening and the gas flow rate were constant. Based on the Reynolds stress model, Li et al.¹⁷ analyzed the sulfur deposition law and influencing factors of cage-type throttle valves. The results show that sulfur deposition mainly occurs in the lower edge of the outer side of the throttle orifice facing the incoming flow, the edge of the valve sleeve, and the inner side of the valve spool, among which the deposition in the lower edge of the outer side of the throttle orifice is the most serious. However, they did not consider the coupling between gas-phase and solid-phase flow parameters, which has a large error with the sulfur deposition in the actual pipeline, and it is difficult to objectively reflect the sulfur deposition in the high-sulfur gathering pipeline.

In conclusion, the gas–solid two-phase flow in high-sulfur natural gas gathering pipelines is a complex unsteady gas–solid two-phase flow accompanied by nucleation, growth and ablation, gas–solid phase transition and crystallization of monomorphic sulfur particles, and the existing methods have not yet been less investigated for the agglomeration and coalescence of sulfur particles in the process of gas–solid two-phase flow. Especially the effect of sulfur particle agglomeration in the bends, which is the key deposition site of the gathering pipeline. To this end, a computational fluid dynamics model for the gas phase of high-sulfur natural gas and the solid phase of sulfur particles in the Euler coordinate system is established based on the Navie–Stokes equations for gas–solid two-phase flow, which is called the Euler model here; and then the PBM is used to describe the agglomeration of sulfur particles in the pipeline under the effect of turbulent shear. The Euler model and the PBM are coupled to establish a numerical simulation model of the gas–solid two-phase pipe flow accompanied by sulfur particle agglomeration in this chapter. In this chapter to analyze the different factors such as bend inclination angle, inlet velocity and inlet concentration of sulfur particles on the agglomeration behavior of sulfur particles, in the pipeline. This will provide some theoretical guidance for the accurate prediction and prevention of sulfur deposition in the bends of gathering pipelines.

Numerical method

Model assumption

The following assumptions were made for this study.

• (1) Assume that the particle phase is treated as a proposed fluid phase.

• (2) It is assumed that the sulfur particles are mainly subject to gas–solid resistance, particle inertial force and gravity. Buoyancy, pressure gradient force, and spurious mass force on the sulfur particles are not included.

• (3) It is assumed that the inlet sulfur particle sizes are all set to 1.0 μm.

Mathematical model

Control equations based on Euler coordinates

High-sulfur natural gas gas-phase in the pipeline flow process follows the three laws of mass conservation, momentum conservation and energy conservation, and the corresponding control equations are the continuity equation, momentum conservation equation and energy conservation equation, respectively, as shown in Eqs. (1)–(6)¹⁸:

Gas-phase continuity equation:

$$\frac{{\partial \varepsilon_{{\text{g}}} \rho_{{\text{g}}} }}{\partial t} + \nabla \cdot \left( {\varepsilon_{{\text{g}}} \rho_{{\text{g}}} {\mathbf{u}}_{{\text{g}}} } \right) = \dot{m}_{{\text{g}}}$$

(1)

Solid-phase continuity equation:

$$\frac{{\partial \varepsilon_{{\text{s}}} \rho_{{\text{s}}} }}{\partial t} + \nabla \cdot \left( {\varepsilon_{{\text{s}}} \rho_{{\text{s}}} {\mathbf{u}}_{{\text{s}}} } \right) = \dot{m}_{{\text{s}}}$$

(2)

Gas-phase momentum conservation equation:

$$\frac{{\partial \left( {\varepsilon_{{\text{g}}} \rho_{{\text{g}}} {\mathbf{u}}_{{\text{g}}} } \right)}}{\partial t} + \varepsilon_{{\text{g}}} \rho_{{\text{g}}} {\mathbf{u}}_{{\text{g}}} \nabla \cdot \left( {{\mathbf{u}}_{{\text{g}}} } \right) + \left( {{\mathbf{u}}_{{\text{g}}} \cdot \nabla } \right)\varepsilon_{{\text{g}}} \rho_{{\text{g}}} {\mathbf{u}}_{{\text{g}}} = \nabla \cdot {{\varvec{\uptau}}}_{{\text{g}}} + \varepsilon_{{\text{g}}} \rho_{{\text{g}}} {\mathbf{g}} + f_{i} + {\dot{\mathbf{M}}}_{{\text{g}}}$$

(3)

Solid-phase momentum conservation equation:

$$\frac{{\partial \left( {\varepsilon_{{\text{s}}} \rho_{{\text{s}}} {\mathbf{u}}_{{\text{s}}} } \right)}}{\partial t} + \varepsilon_{{\text{s}}} \rho_{{\text{s}}} {\mathbf{u}}_{{\text{s}}} \nabla \cdot \left( {{\mathbf{u}}_{{\text{s}}} } \right) + \left( {{\mathbf{u}}_{{\text{s}}} \cdot \nabla } \right)\varepsilon_{{\text{s}}} \rho_{{\text{s}}} {\mathbf{u}}_{{\text{s}}} = \nabla \cdot {{\varvec{\uptau}}}_{{\text{s}}} + \varepsilon_{{\text{s}}} \rho_{{\text{s}}} {\mathbf{g}} - f_{i} + {\dot{\mathbf{M}}}_{{\text{s}}}$$

(4)

Gas-phase energy conservation equation:

$$\frac{{\partial \varepsilon_{{\text{g}}} \rho_{{\text{g}}} e_{{\text{g}}} }}{\partial t} + \nabla \cdot \left( {\varepsilon_{{\text{g}}} \rho_{{\text{g}}} e_{{\text{g}}} {\mathbf{u}}_{{\text{g}}} } \right){ = }\nabla \cdot \left( {\varepsilon_{{\text{g}}} \rho_{{\text{g}}} k_{{\text{g}}} \nabla T_{{\text{g}}} } \right) + {{\varvec{\uppsi}}}_{{\text{g}}} + q_{{\text{g}}}$$

(5)

Solid-phase energy conservation equation:

$$\frac{{\partial \varepsilon_{{\text{s}}} \rho_{{\text{s}}} e_{{\text{s}}} }}{\partial t} + \nabla \cdot \left( {\varepsilon_{{\text{s}}} \rho_{{\text{s}}} e_{{\text{s}}} {\mathbf{u}}_{{\text{s}}} } \right){ = }\nabla \cdot \left( {\varepsilon_{{\text{s}}} k_{{\text{s}}} \nabla T_{{\text{s}}} } \right) + {{\varvec{\uppsi}}}_{{\text{s}}} - q_{{\text{s}}}$$

(6)

where $\varepsilon_{{\text{g}}}$ and $\varepsilon_{{\text{s}}}$ are the gas and solid phase volume fractions, respectively, and $\varepsilon_{{\text{g}}} + \varepsilon_{{\text{s}}} { = }1.0$; $\rho_{{\text{g}}}$ and $\rho_{{\text{s}}}$ are the gas and solid phase densities, respectively, kg/m³; ${\mathbf{u}}_{{\text{g}}}$ and ${\mathbf{u}}_{{\text{s}}}$ are the gas and solid phase velocity vectors, respectively, m/s; $\dot{m}_{{\text{g}}}$ and $\dot{m}_{{\text{s}}}$ are the gas and solid phase mass source terms, respectively, ${\text{kg/(m}}^{{3}} \;{\text{s)}}$, and $\dot{m}_{{\text{g}}} + \dot{m}_{{\text{s}}} = 0$; ${{\varvec{\uptau}}}_{{\text{g}}}$ and ${{\varvec{\uptau}}}_{{\text{s}}}$ are the tangential stress tensor for the gas and solid phases, respectively, N/m²; ${\mathbf{g}}$ are the mass force vectors; $f_{i}$ are the interaction forces between the sulfur particles and the gas phase within the system, and the resistance (trailing force), false mass force, pressure gradient force, and Basset force, etc.; ${\dot{\mathbf{M}}}_{{\text{g}}}$ and ${\dot{\mathbf{M}}}_{{\text{s}}}$ are the momentum transfer vectors of the gas and solid phases, respectively, N/m³; $e_{{\text{g}}}$ and $e_{{\text{s}}}$ are the energies per unit volume of the gas and solid phases, respectively, including the internal energy and the kinetic energy; ${{\varvec{\uppsi}}}{ = }2\mu {\mathbf{S}}:{\mathbf{S}}$, ${\mathbf{S}}$ denotes the strain rate tensor; $q_{{\text{g}}}$ and $q_{{\text{s}}}$ is the source term of the energy transfer of the gas phase and the solid phase.

When solving the system of equations, the second term at the left end of the equation for the conservation of momentum in the gas and solid phases is expanded to include a second-order correlation term related to velocity, which prevents the closure of the system with the mass conservation equation. To make the system of equations closed, a new turbulence equation needs to be introduced. Considering the simulation accuracy and computer processing capability, the RNG $k - \varepsilon$ turbulence model is selected for simulation in this paper¹⁹.

Gas–solid two-phase traction model

In this paper, the simulation is solved by Ansys Fluent 2024R1 (The URL of Fluent is as follows: https://learningcommunity.ansys.com/index.php/commercial-customers/), for the population balance model (PBM) there are commonly the following four trailing coefficient models, as shown in Table 1. Among them, the Grace model and Tomiyama model are commonly used for gas–liquid two-phase flow, which is used when there are more bubble shapes. The Morsi–Alexander model applies to a wide range of Reynolds flow problems, but its computational stability is poor. The Schiller–Naumann model is the default selected default in the PBM model, which can be applied to the flow of larger Reynolds numbers. At the same time, it can also guarantee considerable accuracy. Therefore, the Schiller-Naumann model is chosen to calculate the flow resistance between sulfur particles and the high sulfur gas phase.

Table 1 Commonly used traction coefficient models.

PBM-based modelling of sulfur particle agglomeration

After precipitation of the sulfur nuclei, sulfur particles with a particle size of about 1 μm are first formed in a short time by Brownian condensation. Subsequently, these small sulfur particles are transported along with the natural gas in the pipeline. Due to turbulence and diffusion of the sulfur particles, the particles are further agglomerated and deposited on the inner wall of the pipeline. To describe the agglomeration process of small sulfur particles and the variation of particle size scales, it is necessary to introduce the Population Balance Equation (PBE). Population Balance Model (Population Balance Model) is a kind of numerical density of continuous equations, and its constitutive equation is shown in Eq. (7).

$$\frac{{\partial n\left( {\xi ,x,t} \right)}}{\partial t} + \left\langle {u_{i} } \right\rangle \frac{{\partial n\left( {\xi ,x,t} \right)}}{{\partial x_{i} }} - \frac{\partial }{{\partial x_{i} }}\left[ {\left( {\Gamma_{t} + \Gamma } \right)\frac{{\partial n\left( {\xi ,x,t} \right)}}{{\partial x_{i} }}} \right] = - \frac{\partial }{{\partial \xi_{j} }}n\left( {\xi ,t} \right)\xi_{j} + h\left( {\xi ,t} \right)$$

(7)

where h is a source term representing the effect of the microscopic behavior of sulfur particles (e.g. agglomeration, fragmentation or growth) on the solid-phase sulfur particle system; $\xi$ is a specified vector of sulfur particle state properties; n is the number density function of the sulfur particles; $u_{i}$ is the Reynolds’ mean velocity in the its direction; $x_{i}$ is the spatial coordinate in the its direction; $\Gamma$ is the molecular diffusivity; and $\Gamma_{t}$ is the turbulent diffusivity.

The flow in the space of $\xi$ is defined as:

$$\xi_{j} = \frac{{d\xi_{j} }}{dt},\quad j \in 1,2, \ldots ,N$$

(8)

Assuming that the microscopic behavior of sulfur particles is solely influenced by mechanical action, the particle size distribution function using volume as the internal coordinate can be simplified to Eq. (9). $\beta \left( {v{,}\;\overline{v}} \right)$ In this case, it represents turbulent condensation nuclei, m³/s.

Kinetic modelling of sulfur particle growth

Müller²⁴ derived the transport equation for polydisperse particles based on Smoluchowski’s²⁵ transport equation for monodisperse particles as shown in Eq. (9). This equation can also be used as the controlling equation for the variation of the growth size of sulfur particles with time.

$$\frac{{\partial n\left( {v,t} \right)}}{\partial t} = \frac{1}{2}\int_{0}^{v} {\beta \left( {v - \overline{v},\overline{v}} \right)} \;n\left( {v - \overline{v},t} \right)\;n\left( {\overline{v},t} \right){\text{d}}\overline{v} - n\left( {v,t} \right)\int_{0}^{\infty } {\beta \left( {v,\overline{v}} \right)} \;n\left( {\overline{v},t} \right){\text{d}}\overline{v}$$

(9)

where, $n\left( {v,t} \right)$ is the distribution of particle concentrations, denotes the number of sulfur particles of volume v at time t; $\beta \left( {v,\overline{v}} \right)$ is the Brownian condensation kernel, which represents the collision frequency function of two sulfur particles of the same or different sizes; The first integral on the right side of Eq. (9) represents the process of generating a particle of volume $v - \overline{v}$ by collision of a particle of volume $v$ with a particle of volume $\overline{v}$. Because the counting is repeated twice, the coefficient 1/2 is multiplied in front of it; whereas the second integral represents the process of loss of volume caused by collision of a particle of volume $v$ with other particles. Calculation of condensation nuclei is mainly a reference to the literature of Park²⁶ and Otto²⁷.

In this paper, the Luo model²⁸, which comes with Fluent software, is chosen to calculate the agglomeration behavior among sulfur particles, and the effect of Brownian condensation on the agglomeration of sulfur particles is no longer considered in the model because the vast majority of sulfur particles targeted in the pipeline conveying process are larger than 1 μm in particle size. In Luo’s model, related parameters had been adjusted to parameters of properties of sulfur particles. Finally, to calculate the results to get the size distribution of the particles after agglomeration, the PBM of sulfur particles was solved by the discrete method^29,30,31.

Euler gas–solid multiphase flow model coupled with PBM

Different from the traditional Euler gas–solid two-phase flow model, the model in this paper not only considers the mutual coupling between solid-phase particles and the gas phase but also considers the solid-phase particles agglomeration and condensation under the action of the flow field. Additionally, it considers how changes in particle size distribution affect the flow field. This approach better reflects the real-world conditions of sulfur deposition in collection and transportation systems. The block diagram illustrating the Euler multiphase flow model coupled with the PBM is presented below (Fig. 1).

Figure 1

Block diagram illustrating the Euler multiphase flow model coupled with the PBM³².

Applicability and reliability analysis of simulation models

At present, there is a lack of experimental research results on the gas–solid two-phase flow of sulfur particles in the pipeline. Therefore, the applicability and reliability of the simulation model can only be verified by comparing the field sulfur deposition trend and the simulation results of this model with the calculated values of the literature, so as to indirectly verify the applicability and reliability of the model.

Analysis of the applicability of the simulation model

The model employed in this study simulates the agglomeration growth of sulfur particles post-passage through a fixed orifice plate throttle valve. The pressure drop is 2.0 MPa, with a pipe diameter of 74.3 mm, an orifice plate diameter of 37.15 mm, and an orifice plate thickness of 7.43 mm. The diameters of the inlet sulfur particles are all 1.0 μm, and the volume fraction of the inlet sulfur particles is 0.1%.

As shown in Fig. 2, the upper cloud is the result of the model simulation, and it can be seen that after the throttle valve, the sulfur particles in the outlet of the orifice plate around have the significant agglomeration effect. Comparison with the physical picture of sulfur deposition, it can be seemed that the clustering position obtained from the simulation is very close to the actual clustering position of sulfur particles. Indirectly, this paper shows that the simulation model to simulate the agglomeration effect of sulfur particles in the pipeline is applicable.

Figure 2

Comparison of simulation results of sulfur particle agglomeration with the actual situation for fixed orifice plate throttle valve^33,34.

Reliability analysis of the simulation model

The agglomeration pattern of sulfur particles in a straight pipe section is used as an example to verify the reliability of the model in this paper. The pipe has a diameter of 74.3 mm and a length of 11.145 m. The inlet gas flow rate is 8.0 m/s, and the inlet sulfur particle sizes are set at 1.0 μm. In addition, the single variables are the volumetric concentrations of the sulfur particles in the inlet gas mixture, which are 0.025%, 0.05%, 0.075%, and 0.10%, respectively. Since the maximum size of sulfur particles after agglomeration represents the range of sulfur particle size distribution after agglomeration as well as the agglomeration strength of sulfur particles, the maximum size of sulfur particles after agglomeration is taken as the key index for the reliability analysis of the model.

Figure 3 shows the comparison between the calculated values of the simulation model and the literature values³³, from the figure it can be seen that the calculated values of the simulation model in this paper are generally smaller than the literature values, which is because the literature³³ does not given the gas components, and the model used in this paper is caused by the differences in the gas components. However, through observation, it is not difficult to find that the calculated values of this paper’s model for the maximum sulfur particle size after agglomeration and the variation trend of literature values are completely consistent. This indirectly indicates that the simulation model in this paper is reliable.

Figure 3

Comparison curve of the simulated values of this paper with the literature values.

Physical model

Geometric model

This paper aims to simulate the gas–solid two-phase flow in the bend pipe with large turbulence intensity and easy to occur sulfur deposition. According to China's national “Code for design of gas transmission pipeline engineering” requirements³⁵, the radius of curvature of the bend is 6 d and the inner diameter of the pipe d is 200 mm. To ensure the stability of the flow field, in and out of the straight section of the section of the pipe is taken as 1.0 m. Take the pipe inclination angle of 60° as an example, the bend pipe geometric model is shown in Fig. 4.

Figure 4

Geometric model of a 60° bend.

Meshing

To better capture the agglomeration behavior of cross-section sulfur particles, the boundary layer mesh is divided by a stepwise encryption method from the wall to the center of the pipe cross-section. The thickness of the initial layer is 1.0 mm, consisting of 12 grid layers, and the asymptotic rate is set at 1.2. In addition, the simulation model in this paper controls the number of grids by controlling the maximum face grid size. The grid numbers are 650,000, 1,030,000, 1,600,000, 2,290,000 and 2,810,000, respectively. Through the analysis of the simulation calculations of the flow field at different grid scales, the results show that when the maximum face grid size is 6.5 mm and the number of grids is 1.6 million, the velocity and pressure fields tend to be stable with the further increase of the number of grids. Therefore, through the verification of the irrelevance of the number of grids, it is determined that the number of grids is 1.6 million, and the maximum face grid size is 6.5 mm. Figures 5 and 6 show the mesh division schematic diagram of the pipe section and the local enlargement of the grid, respectively.

Figure 5

Schematic diagram of mesh division.

Figure 6

Local enlargement of meshing.

Determination of boundary conditions and simulation variables

In this simulation, the inlet is set to be velocity-inlet, where the inlet is set to be the gas phase of natural gas and sulfur particles. The outlet is the free outflow boundary (outflow), and the pipe wall is the wall boundary (wall). According to the gas fractions of ${\text{Puguang}}$ gas field, a famous high sulfur gas field in China³⁴, the natural gas fractions were set as 16.5% H₂S by volume, 9% CO₂ by volume, and 74.5% CH₄ by volume. According to the results of Liu et al.³⁶, which showed that under throttling conditions, nanoscale sulfur nuclei can grow into micron-sized sulfur particles in a very short period of time (no more than 1.0 s) and the results of previous research on turbulent condensation³⁷, it was determined that the inlet sulfur particle sizes were all set to 1.0 μm.

The difference in pressure and temperature of the on-site gathering pipeline will directly affect the solubility of elemental sulfur in high sulfur-containing natural gas, and the amount of elemental sulfur precipitation after elemental sulfur supersaturation is directly related to the solubility of elemental sulfur. Through the certain conversion, it can be obtained the concentration of sulfur particles caused by changing the pressure and temperature in the gas phase in the pipe. That is, the pressure and temperature of the pipe affect the agglomeration of sulfur particles by acting on the volumetric concentration of sulfur particles. The fluctuation of the flow rate will directly affect the turbulence intensity of the fluid in the pipe, and the turbulence intensity affects the particle agglomeration and particle size distribution by acting on the particle agglomeration kernel function. The study shows that different pipeline undulations will cause a certain degree of reversal of the flow direction in the pipe, thus affecting the agglomeration and deposition of sulfur particles. In summary, the simulation focused on selected factors influencing the volume concentration of sulfur particles in the gas phase, including gas flow rate, pipe inclination, and three other factors. According to the literature data, the specific values of each parameter are shown in Table 2. A total of 75 sets of conditions were simulated for three different pipe inclinations, five different inlet flow velocities and five different inlet sulfur particle volume concentrations.

Table 2 Analogue control parameter value.

Effect of different parameters on the agglomeration of sulfur particles

To understand the agglomeration law of sulfur particles in different bending sections of the gathering pipeline in more detail, the five selected sections are the entrance section of the bending section, the middle section of the bending section, the exit section of the bending section, the straight section of the pipe section 0.5 m from the exit of the bending section, and the straight section of the pipe section 1.0 m from the exit of the bending section. As shown in Fig. 7, the five sections are labelled T1-B1, T2-B2, T3-B3, T4-B4 and T5-B5, where the letter T stands for the top of the pipe and B stands for the bottom of the pipe.

Figure 7

Position of section selection.

Sulfur particles volume fraction

The effect of different inlet sulfur particle volume concentrations on the agglomeration characteristics of sulfur particles was compared to the cross-section of the elbow with an inlet flow rate of 6 m/s and an inclination angle of 60°. As shown in Fig. 8, the typical sites and general trends of sulfur particle agglomeration in the same cross-section are similar. Some agglomeration occurs in the T1-B1 cross-section. However, the overall effect of sulfur particles agglomeration is not obvious, and the sulfur particle size is mainly close to the initial sulfur particle size. T2-B2 cross-section of the sulfur particles have occurred obvious agglomeration phenomenon, which is mainly distributed in the top of the pipeline near the wall area. From left to right, three cross sections, T3-B3, T4-B4 and T5-B5, showed significant agglomeration of sulfur particles. The results show that the particle size of the agglomerated sulfur particles becomes larger and larger with the increase of the volume concentration of the inlet sulfur particles, and the curved pipe section enhances the agglomeration of the sulfur particles. In addition, the larger-sized sulfur particles after agglomeration mainly appeared on the inner wall of the ring pipe, with the top of the pipe being more significant.

Figure 8

Sulfur particles size distribution after agglomeration in the elbow cross-section under different inlet sulfur particles volume concentration.

Cloud maps of sulfur particle size and sulfur particle velocity distribution in the longitudinal section of the bend with a pipe inclination angle of 60°, inlet sulfur particle volume concentration and inlet velocity of 0.15% and 6.0 m/s, respectively, were intercepted as shown in Fig. 9, respectively. It can be seen that the typical location of sulfur particle agglomeration occurs in the wall region near the top of the pipe, followed by the near-wall region at the bottom of the pipe. Because of the influence of the bending section of the elbow, the velocity at the top near-wall region of the pipe is significantly lower than the inlet initial flow velocity. The centrifugal force caused by the relative velocity difference causes more sulfur particles to gather in the low-velocity region, which increases the time of sulfur particle agglomeration and provides the necessary conditions for sulfur particle agglomeration, which is why sulfur particles are mainly agglomerated near the top of the pipeline near the wall region. Secondly, due to the frictional resistance near the wall, the flow velocity at the bottom of the pipe near the wall is also relatively small, which also provides certain conditions for the sulfur particles to agglomerate in the other main agglomeration location.

Figure 9

Sulfur particle size distribution cloud and sulfur particle velocity distribution cloud after agglomeration.

Figure 10 illustrates the distribution of sulfur particle sizes following the agglomeration process within the bend. As the volume fraction of sulfur particles at the inlet increases from 0.05 to 0.25%, the size of sulfur particles after agglomeration also increases. However, the growth in size of sulfur particles due to agglomeration, the total volume fraction of sulfur particles remains constant. Although the agglomeration grows larger sulfur particles, since the total volume fraction of sulfur particles remains the same, this results in a relatively small percentage of sulfur particles of different sizes for larger volume fractions of sulfur particles.

Figure 10

Particle size distribution of sulfur particles after agglomeration at different inlet sulfur particle volume concentrations.

As shown in Fig. 11, when the inlet flow rate is 6.0 m/s, the maximum sulfur particle size in the pipeline increases significantly with the increase of the inlet sulfur particle volume concentration at three different pipeline inclination angles. As shown in Fig. 12, when the inclination angle of the pipeline is 60°, the maximum sulfur particle size in the pipeline increases with the increase of the volume concentration of the inlet sulfur particles at three different flow rates, with a range of 31.5–189.6 μm. Since the particle size of the inlet sulfur particles is determined to be 1.0 μm, the larger the concentration of sulfur particles in the pipeline, the greater the number of sulfur particles of the corresponding 1.0 μm initial particle size. The higher the number of sulfur particles, the higher the number of particles between the particles. The larger the number of sulfur particles, the greater the chance of agglomeration and coalescence between the particles, and thus the larger the size of the sulfur particles and the wider their distribution.

Figure 11

Trend of maximum particle size of sulfur particles after agglomeration at an inlet flow rate of 6 m/s.

Figure 12

Trend of maximum particle size of sulfur particles after agglomeration at 60° pipe inclination.

As shown in Fig. 13a, the larger volume concentration of sulfur particles is mainly distributed at the bottom of the pipe in the curved section, which indicates that the high-risk area for sulfur particle deposition is at the bottom of the pipe in the curved section of the gathering pipeline. Firstly, this phenomenon arises due to the inertia of the sulfur particles, which alters the flow direction within the curved pipe. Unlike gas-phase mediums, sulfur particles cannot smoothly adjust their flow direction smoothly. Consequently, these particles collide with the bottom wall of the pipe, resulting in a greater volume fraction of sulfur particles being present at the bottom of the curved pipe. Secondly this is because the solid-phase sulfur particles have a density of 2070 kg/m³ while the gas-phase density is often less than 100 kg/m³, and this huge density difference makes the buoyancy force provided by the gas-phase to the sulfur particles much smaller than the gravity force generated by the sulfur particles themselves. As a result, the large-size sulfur particles agglomerated at the top of the pipeline near the wall tend to move towards the bottom of the pipeline in the direction of gravity, and are eventually deposited at the bottom of the pipeline. Due to the existence of certain frictional resistance near the wall, the fluid flow velocity is lower near the pipe wall, and the sulfur particles are more likely to agglomerate and grow to form larger-sized sulfur particles in the relatively low-speed region.

Figure 13

Sulfur Particle Volume Concentration Distribution Cloud on Pipe Walls.

On the other hand, since the top of the pipe near the wall is the most significant region for sulfur particles to agglomerate, the top of the pipe in the curved section shown in Fig. 13b is a low-risk region for sulfur particle deposition due to the neglect of the adsorption effect on the wall of the pipe in the model, coupled with the effect of turbulence and the gravitational force of the sulfur particles.

Overall, the predominant factor influencing the higher volume fraction of sulfur particles at the bottom of the bend is the inertia of these particles. Gravity's influence is relatively minor.

Pipe flow rate

In the same way, according to the five positions identified in Fig. 7, the cloud diagrams of sulfur particle size distribution after the agglomeration of sulfur particles in different cross sections of the elbow pipe were selected respectively. The study compared the impact of various inlet flow velocities on the coalescence characteristics of sulfur particles within the same pipe section. As shown in Fig. 14, the typical sites and general trends of sulfur particles agglomerated in the same cross-section are also similar. The difference is that as the inlet flow rate increases, the cross-sectional area of the pipeline occupied by the agglomerated sulfur particles of larger sizes is gradually smaller. This is because the higher the airflow velocity, the easier the sulfur particles are carried. This leaves relatively little time for agglomeration to occur in the pipe. This indicates that the higher the airflow velocity, the less favorable it is for the agglomeration and growth of sulfur particles in the pipe. That is, the increase in airflow flow rate tends to inhibit the sulfur deposition in the gathering pipeline, which is consistent with the law of the influence of airflow flow rate on sulfur deposition in the gathering pipeline in the field³⁴.

Figure 14

Sulfur particle size distribution in the elbow cross-section after agglomeration at different inlet flow rates.

As shown in Fig. 15, when the inclination angle of the pipeline is 60° and the initial volume concentration of the inlet sulfur particles is 0.15%, the larger the particle size of the aggregated sulfur particles, the smaller the proportion of the total sulfur particles in the horizontal comparison. Vertical comparison, with the increase of the inlet flow rate, the distribution range of sulfur particles after agglomeration tends to become narrower. The larger the flow rate, the less time the sulfur particles have to agglomerate in the pipe, and the corresponding range of sulfur particle size distribution and maximum sulfur particle size are smaller. However, the reduction is small because the inlet sulfur particle size of 1.0 μm and the initial volume concentration of the inlet sulfur particles are determined first so that the initial number of sulfur particles of the corresponding 1.0 μm size is the same. The number of sulfur particles dominates. The small difference in the distribution probabilities between the particle size intervals suggests that the increase in the inlet flow rate from 3.0 to 9.0 m/s has little effect on the distribution of sulfur particles after the aggregation of sulfur particles.

Figure 15

Particle size distribution of sulfur particles after agglomeration at different inlet flow rates.

In Fig. 16, when the inlet sulfur particle concentration is 0.15%, the maximum size of sulfur particles in the pipeline decreases notably with increasing inlet flow rate across three distinct pipeline inclination angles. The reduction is most pronounced at a 90° pipeline inclination angle, with a maximum decrease of 12.46 μm, and least significant at a 60° inclination angle, with a minimum decrease of 9.87 μm. In Fig. 17 illustrates that when the volume concentration of inlet sulfur particles remains below 0.20%, the maximum size of sulfur particles in the pipeline tends to decrease with rising inlet flow rate. The range of reduction is 5.68–9.87 μm. In addition, the figure shows that when the volume concentration of inlet sulfur particles is 0.25%, the maximum sulfur particle size in the pipeline tends to decrease first and then increase. When the inlet flow rate increased from 3.0 to 6.0 m/s, the maximum particle size of sulfur particles decreased from 189.61 to 160.67 μm; and when the inlet flow rate increased from 6.0 to 9.0 m/s, the maximum particle size of sulfur particles increased from 160.67 to 168.58 μm. This was because when the inlet flow rate was lower than 6.0 m/s, the increase in flow rate caused the sulfur particle. This is since when the inlet flow rate is lower than 6.0 m/s, the increase of flow rate makes the sulfur particles agglomeration time in the pipeline decrease, which leads to the decrease of the maximum size of sulfur particles with the increase of flow rate. When the flow velocity is higher than 6.0 m/s, the turbulence intensity in the pipeline increases significantly with the increase of flow velocity, which, together with the larger number of sulfur particles, leads to the enhancement of the turbulent agglomeration effect. Thus, the maximum sulfur particle size in the pipeline tends to increase.

Figure 16

Trend of maximum particle size of sulfur particles after agglomeration at 0.15% volume fraction of inlet sulfur particles.

Figure 17

Trend of maximum size of sulfur particles after agglomeration at pipe inclination angle of 60°

Pipe inclination

As shown in Fig. 18, the cloud of sulfur particle size distribution after the agglomeration of sulfur particles on the section of the elbow was intercepted at the same five positions as determined in Fig. 7, and the particle size distributions of sulfur particles with different inlet sulfur particle volume concentrations in the five cross-sections were shown from the left to the right, respectively. From the top to the bottom, the particle size distributions of the five cross-sections at the same pipe inclination angle are shown. Different colors in the graph represent different particle size distributions, where the closer the color is to blue, the closer the particle size is to the initial setup of sulfur particles at the inlet, and the closer the color is to red, the closer the sulfur particle size is to the maximum sulfur particle size after agglomeration under the working condition. The initial particle size of sulfur particles in the inlet boundary was set to 1.0 μm, and from the cloud view of the distribution of sulfur particles at each cross-section, the sulfur particles in the elbow under different flow velocity conditions also showed significant sulfur particle agglomeration phenomenon.

Figure 18

Sulfur particle size distribution in the elbow cross-section after agglomeration at different pipeline inclination angles.

As shown in Fig. 18, When the inlet flow rate is 6.0 m/s and the initial volume concentration of inlet sulfur particles is 0.15%, the larger size sulfur particles after agglomeration mainly appeared on the inner wall of the ring pipe, with the top of the pipe being the most significant. As shown in Fig. 19, the larger the inclination angle of the pipe, the smaller the proportion of initial-size sulfur particles in the total number of sulfur particles, and the larger the maximum size of sulfur particles formed after agglomeration; the larger the inclination angle of the pipe, the larger the proportion of larger-size sulfur particles in the pipe bends. In summary, the increase of pipeline inclination angle from 30° to 90° is conducive to promoting the agglomeration of sulfur particles in the pipe bend. Due to the bend section radius of curvature being certain, with the increase in pipe inclination, the longer the length of the bent pipe section; pipe inclination is greater, but also makes the inlet and outlet fluid flow direction of the greater folding. The combined effect of these two factors makes the sulfur particles in the pipeline with the increase in the inclination of the pipeline, the effect of agglomeration in the pipeline is stronger.

Figure 19

Particle size distribution of sulfur particles after agglomeration at different pipe inclinations.

Determine the inlet sulfur particle volume concentration of 0.15%, as shown in Fig. 20, the maximum sulfur particle size in the pipeline under the three different inlet flow rate conditions with the increase of pipeline inclination angle showed an increasing trend, of which the maximum size of sulfur particles increased by 8.4 μm, 8.6 μm and 7.28 μm at the inlet flow rates of 3.0 m/s, 6.0 m/s and 9.0 m/s. Determine the inlet flow rate of 6.0 m/s, as shown in Fig. 21, with the increase of pipeline inclination angle under different conditions of inlet sulfur particle volume concentration also showed an increasing trend. 6.0 m/s, as shown in Fig. 21, with the increase of pipeline inclination angle, the maximum sulfur particle size in the pipeline under the condition of different inlet sulfur particle volume concentrations also showed an increasing trend. When the volume concentration of inlet sulfur particles is 0.05%, 0.10%, 0.15%, 0.20% and 0.25%, the increase in the maximum particle size of sulfur particles is 2.16 μm, 2.60 μm, 8.60 μm, 8.38 μm and 27.56 μm, respectively, and it can be seen that, when the volume concentration of the inlet sulfur particles is not more than 0.10%, the pipeline inclination angle does not have much influence on the maximum size of the sulfur particles after the agglomeration. The influence is not significant.

Figure 20

Trend of maximum particle size of sulfur particles after agglomeration at 0.15% volume fraction of inlet sulfur particles.

Figure 21

Trend of maximum particle size of sulfur particles after agglomeration at an inlet flow rate of 6.0 m/s.

Discussion and results

This study aims to explore the agglomeration-growth-fragmentation dynamics of sulfur particles in high-sulfur natural gas gathering pipelines post-formation. Using the PBM model, we established the agglomeration-growth law of sulfur particles influenced by gas-phase turbulence. The study analysis reveals that sulfur particle deposition results from a combination of two primary factors. As detailed in “Sulfur particles volume fraction”, particle deposition in bends is influenced by both particle agglomeration and inertia. Particle inertia causes a higher concentration of sulfur particles at the bend’s bottom, while the bend itself creates a stagnation region where particle agglomeration predominates near the top. This interplay, along with sulfur particle adsorption by the pipe wall, can lead to simultaneous deposition at both the top and bottom of the pipe. Determining the dominant factor remains a complex scientific challenge, as particle location and deposition are intricately linked to fluid flow rate, gas phase composition, pipe wall roughness, wall adsorption effects, and pipe flow structure. Ongoing research aims to uncover the sulfur particle deposition mechanisms promptly, providing a robust theoretical foundation for effective sulfur deposition management.

Conclusions

A numerical simulation model of gas–solid two-phase pipe flow accompanied by sulfur particle agglomeration was established by the coupling method of the Euler model and PBM. The simulation analyses the influence of different sulfur particle inlet volume concentrations (range from 0.05 to 0.25%), gas flow inlet velocity (range from 3.0 to 9.0 m/s) and different pipeline inclination angles (range from 30° to 90°) on the agglomeration of sulfur particles in the elbow pipe. In addition, the initial particle size of sulfur particles is set to 1 micron, and the influence of turbulence on particle transport is mainly considered. The conclusions of this paper are as follows.

• (1) The sulfur particles in the pipe under the action of turbulent shear have a significant agglomeration effect, in which the location of the key agglomeration is closely related to the flow field in the pipe. Due to the role of the wall friction, the focus of the sulfur particles is mainly distributed in the near pipe wall area. In addition, due to the bend section of the pipe, the fluid flow direction is reversed, resulting in the bend section of the inner near-wall region producing a more obvious section of the relatively low-velocity zone, and the sulfur particles in the region of the strongest effect of agglomeration.

• (2) The results of the volume concentration of sulfur particles in the pipe bend show that, due to the great difference between the density of sulfur particles and that of the gas phase, sulfur particles tend to be deposited on the wall of the pipe along the direction of gravity, and therefore the high-risk area for sulfur particle deposition is on the wall of the outer part of the pipe bend.

• (3) At a certain inlet velocity and pipe inclination, with the increase of the inlet volume concentration of sulfur particles, the wider the particle size distribution of sulfur particles after agglomeration, the larger the maximum particle size of sulfur particles, and the stronger the effect of sulfur particles agglomeration in the pipe bend; at a certain inlet flow rate and inlet volume concentration of sulfur particles, with the increase of pipe inclination, the distribution of sulfur particles and the largest sulfur particles after agglomeration are consistent with the trend of the single factor. With the increase of inlet velocity and inlet volume concentration of sulfur particles, the trends of the particle size distribution and the largest sulfur particle size after agglomeration were the same as those of the single-factor increase in inlet volume concentration of sulfur particles, but the effect of agglomeration was weaker than that of the inlet volume concentration of sulfur particles. At a certain pipe inclination angle and when the inlet volume concentration of sulfur particles does not exceed 0.20%, with the increase of inlet flow rate, the range of sulfur particle size distribution becomes narrower, and the maximum size of sulfur particles is 187.56 μm. However, when the inlet volume concentration of sulfur particles is 0.25%, with the increase of the inlet flow rate, the maximum size of sulfur particles after agglomeration first decreases and then increases.