# Research on Slug Flow Elimination Method Based on Kinetic Energy Conversion

Authors: Ruiquan Liao; Donghui Zhou; Zhihui Wang; Weixia Yang; Xingkai Zhang
Journal: Geofluids
 (2023)

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## Abstract

Slug flow is one of the most common flow patterns in the petrochemical industry. It will affect the normal operation of oil well surface pipelines and connected equipment, especially gas well multiphase flowmeters. The extant slug flow traps are complex in structure and limited in application sites. To reduce the influence of slug flow on gas-liquid two-phase measurement, a slug flow elimination device is designed based on the kinetic energy conversion method. The gas-liquid two-phase flow law inside the device and its energy loss are investigated using a combination of indoor experiments and numerical simulations. This study evaluates the device’s working performance, including the flow pattern, pressure fluctuation, velocity distribution, and energy loss. The results show that the flow rate and pressure fluctuation of the gas-liquid two-phase flow are weakened after the device. And the flow pattern changes from intermittent slug flow to gas-liquid continuous flow. The pressure drop calculation method for the device is developed based on the share of different structures in the total pressure drop, with a prediction error of 20%. The slug elimination device is designed to provide a flow pattern basis for metering equipment, improve metering accuracy, and further promote the development of multiphase metering technology.


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## Body

## 1. Introduction

The fluids produced from oil wells may be a gas-liquid two-phase flow in many cases. Slug flow is a common gas-liquid two-phase flow pattern. Slug flow will affect the stable operation of the oil well surface pipeline and the equipment connected to the pipeline. In particular, slug flow will cause large measurement errors in the real-time measurement of oil well production [1]. A large number of studies have been conducted to investigate the kinematic characteristics of slug flow [2–5]. Based on the existing theories, the optimal method to reduce or eliminate the slug flow is sought [6]. Meng et al. [7] and Zheng et al. [8] installed a cyclone upstream of the gas-liquid measurement device to eliminate the slug so that the flow through the measurement device is annular. They concluded that the cyclone can force the downstream flow pattern into an annular flow and obtained a stable differential pressure, thus obtaining high measurement accuracy. Passive section slug control can also be achieved by using many devices or techniques, commonly installing flow pattern adjusters [9, 10], multiple riser combinations [11], gas lift method [12], bubble crushers [13], mixing devices [14], and slug flow traps [15], etc. In general, according to the site production needs, different types of slug flow elimination devices can be chosen [16]. If the slug flow can be effectively eliminated, it will not only provide a good flow basis for gas-liquid measurement but also improve the stability of equipment operation [17, 18].To eliminate the measurement error caused by a sudden change in the flow pattern, the experimental study of the slug flow elimination method under different working conditions is the most convincing. Experimental studies are limited to some extent due to the disadvantages such as the high cost of experimental equipment and low reusability. Researchers began to seek more accurate and predictive tools for the study of gas-liquid two-phase flow laws [19]. Among them, Computational Fluid Dynamics (CFD) methods are coming into the researchers’ view, and they will be a valuable addition to the toolbox [20].The purpose of this study is to eliminate or weaken the slug flow and provide a stable flow pattern for gas-liquid two-phase measurements while increasing the stability of the instrument operation. Based on the principle of kinetic energy conversion and turbulent diffusion [21], a new slug flow elimination method is put forward to change the distribution pattern of gas and liquid phases [22]. This method can effectively reduce the slug flow frequency and achieve the purpose of weakening or eliminating the slug flow. The performance of the slug flow elimination device is tested by experimental and numerical simulation, and the pressure drop calculation model of the device is also established.


## 2. Model Building and Methodology

### 2.1. Slug Flow Model

When the gas-liquid two-phase flow flows through the horizontal- or upward-inclined pipeline, the liquid moves forward under the action of a pressure gradient to overcome gravity and friction. When the pressure gradient or gas-carrying role is large enough, the liquid has a large kinetic energy and can flow together with the gas. On the contrary, the impact of gravity and friction will move the liquid behind the gas, causing the liquid to accumulate in the pipe, and then generating hydraulic slug flow. Therefore, if the energy of the liquid is increased by certain means, the slug flow can be eliminated.As shown in Figure1, the total length of a hydraulic slug flow unit is LS, which mainly consists of a liquid film zone and a liquid slug zone. Assume that the length of the liquid slug zone is LL and the length of the liquid film zone is LT, where the liquid film zone consists of Taylor bubbles and liquid film. The hydraulic section slug flow is accompanied by the accumulation of the liquid phase at the head of the slug and the loss of the liquid phase at its tail during the movement.Figure 1
Geometric model of the slug flow in the inner section of the slightly inclined pipe.Taking the slug flow unit as the object of study, the continuity equation is established based on the conservation of mass of the slug flow per unit of time.(1)Liquid:vslρle=vlltsHlsρl+vftfHlfρl,(2)Gas:vsgρge=vlgts1−Hlsρg+vTtf1−Hlfρg,wherevsl and vsg represent the superficial velocity of liquid and gas, respectively, m/s. vll and vf represent the liquid phase velocity in the liquid slug and liquid film zones, respectively, m/s. Hls and Hlf represent the liquid holdup in the liquid slug and liquid film zones, respectively, dimensionless. e is the frequency of the slug flow, Hz. vlg and vT are the bubble velocity and Taylor bubble velocity in the liquid slug zone, respectively, m/s. ρl and ρg represent the density of the liquid and gas, respectively, kg/m3. ts and tf represent the time for the liquid slug and Taylor bubble to pass a point, respectively, s.
(3)ts=LLvll,(4)tf=Lfvf.Substituting Equations (3) and (4) into Equations (1) and (2), respectively, we get
(5)vsle=LLHlsvllvla+LfHlf,(6)vsge=LLvlgvla1−Hls+Lfvf1−HlfvT.The Taylor bubble motion process causes the interconversion of the liquid slug and liquid film zones, and in the steady state, the mass exchange is(7)HlsvT−vll=HlfvT−vlf.A certain amount of liquid holdup is the basic condition for the formation of slug flow. The liquid holdup in the liquid slug zone is much larger than that in the liquid film zone in the slug flow unit. The liquid holdup of the slug is calculated according to the relationship proposed by Gregory [23], namely,
(8)Hls=11+vm/8.661.39.The velocity of the mixture (vm) in the slug flow unit can be expressed as
(9)vm=vsl+vsg.


### 2.2. Structure and Working Principle

The elimination device designed and processed in this study mainly consists of an impacting tee, an ejector, a liquid accumulation riser, a liquid storage pipe, and a gas pipe, and the structure is shown in Figure2(a). Among them, the main function of the impacting tee is to separate the liquid slug area and liquid film area of the slug flow unit, so that the gas-liquid two phases are reorganized at the Venturi throat. When the slug flow passes through the inlet impacting tee, the kinetic energy of the liquid along the axial direction disappears and flows into the liquid storage pipe by its gravity. The impacting tee has little effect on the gas superficial velocity, and the gas in the Taylor bubble enters the horizontal gas pipe. The role of the ejector is to convert part of the gas energy into the liquid. The gas velocity increases and pressure decrease as it flows through the nozzle. Low pressure draws the liquid in the accumulation riser into the mix chamber (Figure 2(b)), achieving gas-liquid kinetic energy exchange so that the liquid velocity increases. By recombining the gas-liquid phases through the ejector, the energy distribution of each phase in the pipe and the liquid content distribution in the cross-section are changed. When the liquid content of the treated section does not meet the conditions for the formation of slug flow [24], the slug disappears.Figure 2
Schematic design of the main structure of the slug flow elimination device (the blue arrow is the flow direction).
(a)
Schematic diagram of the main structure of the device(b)
Structural diagram of the ejector(c)
Structure diagram of the drainage pipe inlet


## 2.1. Slug Flow Model

When the gas-liquid two-phase flow flows through the horizontal- or upward-inclined pipeline, the liquid moves forward under the action of a pressure gradient to overcome gravity and friction. When the pressure gradient or gas-carrying role is large enough, the liquid has a large kinetic energy and can flow together with the gas. On the contrary, the impact of gravity and friction will move the liquid behind the gas, causing the liquid to accumulate in the pipe, and then generating hydraulic slug flow. Therefore, if the energy of the liquid is increased by certain means, the slug flow can be eliminated.As shown in Figure1, the total length of a hydraulic slug flow unit is LS, which mainly consists of a liquid film zone and a liquid slug zone. Assume that the length of the liquid slug zone is LL and the length of the liquid film zone is LT, where the liquid film zone consists of Taylor bubbles and liquid film. The hydraulic section slug flow is accompanied by the accumulation of the liquid phase at the head of the slug and the loss of the liquid phase at its tail during the movement.Figure 1
Geometric model of the slug flow in the inner section of the slightly inclined pipe.Taking the slug flow unit as the object of study, the continuity equation is established based on the conservation of mass of the slug flow per unit of time.(1)Liquid:vslρle=vlltsHlsρl+vftfHlfρl,(2)Gas:vsgρge=vlgts1−Hlsρg+vTtf1−Hlfρg,wherevsl and vsg represent the superficial velocity of liquid and gas, respectively, m/s. vll and vf represent the liquid phase velocity in the liquid slug and liquid film zones, respectively, m/s. Hls and Hlf represent the liquid holdup in the liquid slug and liquid film zones, respectively, dimensionless. e is the frequency of the slug flow, Hz. vlg and vT are the bubble velocity and Taylor bubble velocity in the liquid slug zone, respectively, m/s. ρl and ρg represent the density of the liquid and gas, respectively, kg/m3. ts and tf represent the time for the liquid slug and Taylor bubble to pass a point, respectively, s.
(3)ts=LLvll,(4)tf=Lfvf.Substituting Equations (3) and (4) into Equations (1) and (2), respectively, we get
(5)vsle=LLHlsvllvla+LfHlf,(6)vsge=LLvlgvla1−Hls+Lfvf1−HlfvT.The Taylor bubble motion process causes the interconversion of the liquid slug and liquid film zones, and in the steady state, the mass exchange is(7)HlsvT−vll=HlfvT−vlf.A certain amount of liquid holdup is the basic condition for the formation of slug flow. The liquid holdup in the liquid slug zone is much larger than that in the liquid film zone in the slug flow unit. The liquid holdup of the slug is calculated according to the relationship proposed by Gregory [23], namely,
(8)Hls=11+vm/8.661.39.The velocity of the mixture (vm) in the slug flow unit can be expressed as
(9)vm=vsl+vsg.


## 2.2. Structure and Working Principle

The elimination device designed and processed in this study mainly consists of an impacting tee, an ejector, a liquid accumulation riser, a liquid storage pipe, and a gas pipe, and the structure is shown in Figure2(a). Among them, the main function of the impacting tee is to separate the liquid slug area and liquid film area of the slug flow unit, so that the gas-liquid two phases are reorganized at the Venturi throat. When the slug flow passes through the inlet impacting tee, the kinetic energy of the liquid along the axial direction disappears and flows into the liquid storage pipe by its gravity. The impacting tee has little effect on the gas superficial velocity, and the gas in the Taylor bubble enters the horizontal gas pipe. The role of the ejector is to convert part of the gas energy into the liquid. The gas velocity increases and pressure decrease as it flows through the nozzle. Low pressure draws the liquid in the accumulation riser into the mix chamber (Figure 2(b)), achieving gas-liquid kinetic energy exchange so that the liquid velocity increases. By recombining the gas-liquid phases through the ejector, the energy distribution of each phase in the pipe and the liquid content distribution in the cross-section are changed. When the liquid content of the treated section does not meet the conditions for the formation of slug flow [24], the slug disappears.Figure 2
Schematic design of the main structure of the slug flow elimination device (the blue arrow is the flow direction).
(a)
Schematic diagram of the main structure of the device(b)
Structural diagram of the ejector(c)
Structure diagram of the drainage pipe inlet


## 3. Experimental System

### 3.1. Experimental Procedure

Water and air are used as media for the experiment. The experimental fluid is supplied through a water pump and an air compressor. The high-pressure gas in the storage tank is dried and processed into the slug flow elimination device. The separated gas is directly discharged into the air, and the water is recycled. The experimental flow is as shown in Figure3. In this experiment, an “L” riser is set at the inlet to simulate the slug flow, so that the flow pattern entering the separation device is a slug flow. The experimental setup is shown in Figure 4(d).Figure 3
Experimental flow chart.Figure 4
Physical model and grid structure (the blue dotted box is the “L” riser).
(a)
Overall grid division of the device(b)
Inlet grid structure(c)
Ejector grid structure(d)
Physical drawing of device


### 3.2. Experimental Equipment

The liquid flow rate is given by adjusting the pump inverter to change the liquid flow rate into the mixer. The gas flowmeter used in this experiment is a thermal gas mass flowmeter with a measurement range of 5 m3/h to 400 m3/h and an accuracy of ±1.5%. The liquid flow rate is measured by a Coriolis force mass flowmeter with a range of 0 to 1.5 kg/s and an accuracy of ±0.1%. Pressure sensors are installed at the inlet and outlet of the elimination device to record the pressure change before and after entering the device, to judge the effect of slug flow elimination. The pressure sensors at the inlet and outlet have a range of 0~0.6 MPa and an accuracy of ±0.1%. The output of the flowmeter and differential pressure sensor is a 4-20 mA current signal, and all parameters are collected and stored using an NI PCI-6220 64-bit multifunctional data acquisition card with a 200 Hz data acquisition frequency.


### 3.3. Numerical Simulation

Slug flow is an intermittent flow between gas-liquid phases and requires specific operating conditions to be formed. Therefore, this simulation utilizes the user-defined function (UDF) to customize the gas-liquid flow at the inlet to ensure that the two phases can form a slug flow before entering the device. According to the experimental findings, the loading and unloading processes of the compressor lead to a sinusoidal variation of the gas flow [25]. So the inlet mass flow curve is defined as a sinusoidal fluctuation law during the numerical simulation. The numerical simulation results are verified by experiments with the same structure.


### 3.4. Physical Model and Grid

When building the physical model, the inlet and outlet pipe lengths were increased to observe the slug flow elimination effect. The inner diameters of both outlet and inlet pipes are 60 mm, the inner diameters of both descending and inducing pipes are 25 mm, and the throat diameter ratio of the ejector is 0.4. To improve the mesh quality and enhance the calculation accuracy, the structure is first divided into blocks, and the structured meshing method is used for the regular structure. The ejector and the impacting tee use the tetrahedral meshing method, and the mesh structures are shown as follows in Figures4(a)–4(c).


### 3.5. Multiphase Flow Model

The VOF model is chosen for this simulation to simulate the gas-liquid distribution law with the following control equation.(10)∂αq∂t+v⟶⋅∇αq=Sαqρq.By default, the source term at the right of Eq. (10) is zero, except when a constant or a user-defined mass source is specified for each phase. The volumetric ratio equation is not solved for the main phase. The calculation of the volume ratio for the main phase is based on the following constraint.
(11)∑q=1nαq=1.Solving for a single momentum equation over the entire zone, the resulting velocity field is shared by all phases. As shown below, the momentum equation depends on the volume ratio of all phases ofρ and μ.
(12)∂∂tρv⟶+∇⋅ρv⟶v⟶=−∇p+∇⋅μ∇v⟶+∇v⟶T+ρg⟶+F⟶.In this simulation, the gas is used as the primary phase and the liquid as the secondary phase, and the density in each grid cell is calculated using the following equation.(13)ρ=ρgαq+ρl1−αq,wherev⟶ is the velocity of the fluid, αq is the q-th phase volume fraction, ρq is the q-th density, p is the static pressure, μ is the fluid’s dynamic viscosity, and F⟶ is the surface tension between the two phases.


### 3.6. Turbulence Model

The RNGk−ε model is capable of simulating moderately complex flows such as jet impingement, separated flows, secondary flows, and cyclonic flows. The model is derived from the transient N-S equation using the mathematical method of the “renormalization group.” The resolvability is converted from the standard k−ε model, and new functions or terms appear in the equations, whose turbulent kinetic energy and dissipation rate equations are
(14)ρDkDt=∂∂xiαkμeff∂k∂xi+Gk+Gb−ρε−YM,(15)ρDεDt=∂∂xiαεμeff∂ε∂xi+C1εεkGk+C3εGb−C2ερε2k−R,whereGk is the turbulent kinetic energy due to the velocity gradient; Gb is the turbulent kinetic energy due to buoyancy; YM is the fluctuation due to the diffusion of the transition in compressible turbulence; C1ε, C2ε, and C3ε are constants; and αK and αε are the inverse of the effective turbulent Prandtl number for the turbulent kinetic energy k and the dissipation rate ε, respectively.


### 3.7. Boundary Conditions

The inlet and outlet boundary types are set as the mass flow inlet and pressure outlet, respectively. The inlet pressure is 0.1 MPa, and the inlet mass flow rate is assigned by UDF. The turbulence definition method selects the turbulence intensity and hydraulic diameter; the no-slip wall condition is used; i.e., the wall velocity is 0. The gas-liquid surface tension is set to 0.0072 N/m. The parameters such as phase content, pressure, and phase velocity in the flow direction are recorded during the simulation.


## 3.1. Experimental Procedure

Water and air are used as media for the experiment. The experimental fluid is supplied through a water pump and an air compressor. The high-pressure gas in the storage tank is dried and processed into the slug flow elimination device. The separated gas is directly discharged into the air, and the water is recycled. The experimental flow is as shown in Figure3. In this experiment, an “L” riser is set at the inlet to simulate the slug flow, so that the flow pattern entering the separation device is a slug flow. The experimental setup is shown in Figure 4(d).Figure 3
Experimental flow chart.Figure 4
Physical model and grid structure (the blue dotted box is the “L” riser).
(a)
Overall grid division of the device(b)
Inlet grid structure(c)
Ejector grid structure(d)
Physical drawing of device


## 3.2. Experimental Equipment

The liquid flow rate is given by adjusting the pump inverter to change the liquid flow rate into the mixer. The gas flowmeter used in this experiment is a thermal gas mass flowmeter with a measurement range of 5 m3/h to 400 m3/h and an accuracy of ±1.5%. The liquid flow rate is measured by a Coriolis force mass flowmeter with a range of 0 to 1.5 kg/s and an accuracy of ±0.1%. Pressure sensors are installed at the inlet and outlet of the elimination device to record the pressure change before and after entering the device, to judge the effect of slug flow elimination. The pressure sensors at the inlet and outlet have a range of 0~0.6 MPa and an accuracy of ±0.1%. The output of the flowmeter and differential pressure sensor is a 4-20 mA current signal, and all parameters are collected and stored using an NI PCI-6220 64-bit multifunctional data acquisition card with a 200 Hz data acquisition frequency.


## 3.3. Numerical Simulation

Slug flow is an intermittent flow between gas-liquid phases and requires specific operating conditions to be formed. Therefore, this simulation utilizes the user-defined function (UDF) to customize the gas-liquid flow at the inlet to ensure that the two phases can form a slug flow before entering the device. According to the experimental findings, the loading and unloading processes of the compressor lead to a sinusoidal variation of the gas flow [25]. So the inlet mass flow curve is defined as a sinusoidal fluctuation law during the numerical simulation. The numerical simulation results are verified by experiments with the same structure.


## 3.4. Physical Model and Grid

When building the physical model, the inlet and outlet pipe lengths were increased to observe the slug flow elimination effect. The inner diameters of both outlet and inlet pipes are 60 mm, the inner diameters of both descending and inducing pipes are 25 mm, and the throat diameter ratio of the ejector is 0.4. To improve the mesh quality and enhance the calculation accuracy, the structure is first divided into blocks, and the structured meshing method is used for the regular structure. The ejector and the impacting tee use the tetrahedral meshing method, and the mesh structures are shown as follows in Figures4(a)–4(c).


## 3.5. Multiphase Flow Model

The VOF model is chosen for this simulation to simulate the gas-liquid distribution law with the following control equation.(10)∂αq∂t+v⟶⋅∇αq=Sαqρq.By default, the source term at the right of Eq. (10) is zero, except when a constant or a user-defined mass source is specified for each phase. The volumetric ratio equation is not solved for the main phase. The calculation of the volume ratio for the main phase is based on the following constraint.
(11)∑q=1nαq=1.Solving for a single momentum equation over the entire zone, the resulting velocity field is shared by all phases. As shown below, the momentum equation depends on the volume ratio of all phases ofρ and μ.
(12)∂∂tρv⟶+∇⋅ρv⟶v⟶=−∇p+∇⋅μ∇v⟶+∇v⟶T+ρg⟶+F⟶.In this simulation, the gas is used as the primary phase and the liquid as the secondary phase, and the density in each grid cell is calculated using the following equation.(13)ρ=ρgαq+ρl1−αq,wherev⟶ is the velocity of the fluid, αq is the q-th phase volume fraction, ρq is the q-th density, p is the static pressure, μ is the fluid’s dynamic viscosity, and F⟶ is the surface tension between the two phases.


## 3.6. Turbulence Model

The RNGk−ε model is capable of simulating moderately complex flows such as jet impingement, separated flows, secondary flows, and cyclonic flows. The model is derived from the transient N-S equation using the mathematical method of the “renormalization group.” The resolvability is converted from the standard k−ε model, and new functions or terms appear in the equations, whose turbulent kinetic energy and dissipation rate equations are
(14)ρDkDt=∂∂xiαkμeff∂k∂xi+Gk+Gb−ρε−YM,(15)ρDεDt=∂∂xiαεμeff∂ε∂xi+C1εεkGk+C3εGb−C2ερε2k−R,whereGk is the turbulent kinetic energy due to the velocity gradient; Gb is the turbulent kinetic energy due to buoyancy; YM is the fluctuation due to the diffusion of the transition in compressible turbulence; C1ε, C2ε, and C3ε are constants; and αK and αε are the inverse of the effective turbulent Prandtl number for the turbulent kinetic energy k and the dissipation rate ε, respectively.


## 3.7. Boundary Conditions

The inlet and outlet boundary types are set as the mass flow inlet and pressure outlet, respectively. The inlet pressure is 0.1 MPa, and the inlet mass flow rate is assigned by UDF. The turbulence definition method selects the turbulence intensity and hydraulic diameter; the no-slip wall condition is used; i.e., the wall velocity is 0. The gas-liquid surface tension is set to 0.0072 N/m. The parameters such as phase content, pressure, and phase velocity in the flow direction are recorded during the simulation.


## 4. Results and Discussion

### 4.1. Flow Pattern

Flow pattern variation is the main cause of pressure in the pipe [26]. As Figure 5(a) shows, the inlet is a slug flow, and bubbles are found in the liquid slug in the slug flow unit. The closer to the end of the liquid slug, the larger the bubble is. This is caused by the pressure between the liquid film and the liquid slug becoming smaller. When the slug flow enters the device, a large amount of liquid is initially separated into the descending pipe under the small gas-liquid ratio condition, and a small amount of liquid is carried to the horizontal pipe by the gas. The preliminary separation of the gas and liquid is obvious, in which a small amount of liquid in the gas pipe flows into the liquid collection pipe through the descending pipe. The gas imparts kinetic energy to the liquid in the mix chamber of the ejector, which improves the velocity of the liquid and reduces the accumulation of liquid in the riser. After passing through the elimination device, the liquid slug and Taylor bubble are separated, and the flow pattern formed after the second mixing is as Figure 5(b) shows. From Figure 5(b), it can be seen that the flow pattern at the outlet has changed from slug flow to transition flow and gradually transformed to annular flow. Due to gravity, the liquid phase at the outlet is deposited at the bottom of the pipe, forming a small wave, and a fine liquid flow is formed around the pipe wall, which temporarily fails to form a complete liquid film.Figure 5
Fluid motion at the inlet and outlet of the experimental setup.
(a)
Inlet flow pattern (Ql=0.42m3/h, Qg=41.6m3/h)(b)
Outlet flow pattern (Ql=0.42m3/h, Qg=41.6m3/h)The liquid holdup is one of the main variables describing the slug flow. Therefore, the magnitude of the liquid holdup not only affects the pressure fluctuation but will also determine the slug flow elimination effect. According to the Gregory et al. [27] study, the liquid holdup of the pipe cross-section capable of slug flow is 48%. The experimental results show that when the gas-liquid ratio is greater than 380 m3/m3, it is difficult to form slug flow in the pipe. However, a large gas-liquid ratio still leads to a liquid-phase impact on the impacting tee and a significant wave pattern in the outlet pressure.When the kinetic energy of the gas is small, it cannot pump the liquid in the accumulation riser in time, causing it to accumulate (Figure6(a)). When the gas cannot continuously transfer its kinetic energy to the liquid, the liquid entering the priming chamber appears to oscillate at the inlet of the drainage pipe (Figure 6(b)). Figure 7 shows the distribution of the liquid phase at t=2.2s under different gas-liquid ratio conditions. As the gas-liquid ratio gradually increases, the initial separation effect of the impacting tee on the two-phase gas-liquid decreases, and a vortex is generated at the elbow. More liquid enters the gas pipe. The liquid phase entering the gas pipe makes the flow rate at the nozzle fluctuate, and the energy conversion between the gas and liquid phases is then affected. When the gas-liquid ratio is greater than 120 m3/m3, a complete slug flow is not formed at the inlet due to the reduced liquid content in the pipe. It can be seen from the outlet pressure curve that the gradual increase of the gas-liquid ratio increases the outlet pressure amplitude. Due to the small liquid content rate, a full pipe flow cannot be formed in the accumulation riser, and the intermittent supply of liquid from the riser to the inlet pipe results. Therefore, the pressure fluctuations generated at a gas-liquid ratio greater than 120 m3/m3 cannot be used to define the applicability of the device.Figure 6
Liquid state of the accumulation riser with a different gas-liquid ratio.Figure 7
Gas-liquid distribution under different gas-liquid ratios (t=2.2s).


### 4.2. Pressure Fluctuation

From the experimental phenomena, it is clear that the slug flow pressure signal has the following characteristics.(1)
The pressure fluctuation time in the liquid film area is long, the amplitude is small, and there are obvious fluctuations in the pressure rise phase with a step-up pattern(2)
The pressure amplitude in the liquid slug area is large, has a short duration, and shows a vertical wave, and the pressure reaches the highest point when a precipitous drop occurs [25].Combining Figures5 and 8(a), it can be seen that the pressure at the outlet depends on the various flow patterns. Under non-slug flow conditions, both mass flow and pressure tend to stabilize, which can further improve the accuracy of the gas-liquid two-phase measurement device and its stability. From Figure 8(b), which shows various patterns of the inlet pressure, it can be seen that the various patterns of simulated and measured pressures are almost the same at the same time nodes. The two have a high degree of agreement. Due to the loading and unloading of the air compressor during the experiment, which leads to the unstable gas-liquid flow, it will form as shown in Figure 8(b) (green dashed line). That is, the measured pressure value occasionally shows large fluctuations at the position of the trough, and the pressure measurement is slightly higher than that of the simulated value. Comparing the variation laws of pressure, flow pattern, and other parameters under experimental and numerical simulation conditions, respectively, it is verified that the numerical simulation method can accurately describe the gas-liquid two-phase flow pattern and its motion law.Figure 8
Variation of flow and pressure at the inlet and outlet.
(a)
Mass flow rate at inlet and outlet (GLR=100m3/m3)(b)
Experimental and simulated pressure curvesThe relationship between the void fraction and pressure at the cross-section 400 mm from the inlet before the liquid slug enters the impacting tee (GLR=60m3/m3) was simulated in this study, as shown in Figure 9. The graph shows that the pipe pressure reaches its maximum value when the void fraction reaches its minimum value, i.e., when the cross-sectional liquid content is at its maximum. When the cross-sectional liquid content increases, the pressure starts to build up in the Taylor bubble at the moment when the liquid slug blocks the gas passage. When the pressure at the end of the slug is greater than the pushing force required for the movement of the slug, the slug is discharged. At this point, the Taylor bubble acts as a “piston” and pushes the liquid slug to move. Combined with Figure 9, it can be seen that the larger the gas-liquid ratio, the larger the void ratio, and the smaller the average cross-sectional liquid content, the smaller the pressure in the pipe and the smaller the amplitude of pressure fluctuation.Figure 9
Variation curve of cross-sectional void ratio and pressure.Combined with the flow pattern variation at the inlet and outlet, it shows that the magnitude of the pressure fluctuation depends on the velocity of the slug flow, and the length of the slug flow determines the duration of the pressure peak. From the results of Figure10, it can be seen that the higher the liquid phase flow rate, the higher the pressure and the greater the amplitude of its fluctuation under the same gas flow rate. The smaller the gas-liquid ratio, the longer the duration of the pressure wave. The pressure fluctuations at the inlet and outlet have obvious differences. Among them, the high-frequency pressure signal at the inlet shows obvious slug flow characteristics, but the inlet slug flow pressure signal occasionally shows abnormal fluctuations at the trough during the experiment due to the influence of compressor loading (Figure 10(a) marked by a red solid line). During the experiment, it was found that when the liquid slug reached the top of the riser, the pressure reached a peak at that time. When the slug is discharged momentarily, the pressure suddenly decreases. As the gas-liquid ratio gradually decreases, the liquid slug stagnates at the riser and a small amount of liquid then flows back, causing the pressure signal at the inlet to fluctuate at the crest, as shown in Figures 10(b) and 10(c) marked by the red solid line. Combined with Figure 8(a), which shows the mass flow rate variation law, it can be seen that the stability of the outlet pressure depends on whether its mass flow rate is stable.Figure 10
Pressure fluctuation curve of the inlet and outlet with a different gas-liquid ratio.
(a)
GLR=100m3/m3(b)
GLR=66.56m3/m3(c)
GLR=49.88m3/m3


### 4.3. Velocity Distribution

From the internal streamline diagram of the ejector (Figure11), it can be seen that the velocity is smaller near the pipe wall and is larger in the center where the liquid content is small. The velocity before entering the ejector is about 1.15 m/s and reaches 13.41 m/s inside the mix chamber (GLR=40m3/m3). The sudden increase in velocity caused a sharp drop in pressure in the mix chamber due to the reduced-flow cross-section of the nozzle. To maintain the local pressure balance, the pressure in the induced chamber also decreases simultaneously. The liquid in the accumulation riser is pumped into the mix chamber by negative pressure and then enters the expansion pipe after being accelerated by high-speed gas, forming a non-slug flow downstream. As the gas-liquid ratio increases, the flow rate in the mix chamber gradually increases, and the suction force continues to increase.Figure 11
Streamline diagram of the ejector.Due to the existence of friction between the liquid and the pipe wall, the flow rate gradually decreases from the center to the wall. Among them is the location of the thin layer near the inner wall, which is a thin layer of resistance; this thin layer is called the boundary layer. In the flow through the throat, the fluid increases the velocity and decreases the pressure, maintaining the original surface layer. Downstream of the throat, the flow area increases, the kinetic energy of the fluid is converted into pressure energy, and the boundary layer is subject to the force of the opposite direction of the mainstream, resulting in fluid backflow. Beyond the boundary layer of the fluid to maintain the original direction of advance, the formation of the vortex phenomenon as shown in Figure8 results in the separation of the boundary layer.


### 4.4. Pressure Drop Model

Through experiments and simulations, it can be seen that the pressure loss generated by this device mainly comes from the ejector, the inlet of the drainage pipe, the impacting tee, and the heavy pressure drop. The total pressure drop of the slug flow elimination device is(16)ΔP=ΔPo+ΔP1+ΔPv+ΔPh,whereΔPo is the pressure drop generated by the impacting tee, kPa. ΔP1 is the pressure drop generated at the inlet of the drainage pipe, kPa. ΔPv is the pressure drop generated by the ejector, kPa. ΔPh is the repositioning pressure difference from the height difference, kPa.Under the same conditions of the gas flow rate, the total pressure drop increases gradually as the liquid flow rate increases. As Figure12(a) shows, the liquid-phase flow rate is related to the total pressure drop. According to the study, it is known that the frictional pressure drops upstream and downstream of the Venturi throat are equal under the condition of symmetry between the constriction and expansion sections [28]. In this study, due to the complex structure of the ejector, the differential pressure generated in the convergent section is significantly larger than that in the divergent section (as Figure 12(b)). From Figure 12(b), it can be seen that an increase in the gas-liquid ratio leads to a significant increase in the total pressure drop at constant pressure conditions. The differential pressure of the gas-liquid mixture passing through the Venturi nozzle is related to the magnitude of the flow rate and increases with increase of the gas-liquid ratio [29, 30].Figure 12
(a) Pressure drop variation at different liquid phase flow conditions; (b) differential pressure variation curves at different cross-sections (ΔP1 sudden contraction structure, ΔP2 Venturi-tapering section, ΔP3 Venturi-expansion section).
(a)(b)


### 4.5. Model Building

#### 4.5.1. Venturi Differential Pressure

Based on the experimental and numerical simulation results, the pressure drop calculation model of this device is established. The pressure drop generated by the Venturi nozzle mainly includes an accelerated pressure drop and friction pressure drop [31]. Among them, the acceleration pressure drop accounts for the largest proportion and increases with the increase of the liquid content. The next is the frictional pressure drop, which is mainly generated at the nozzle. The frictional pressure drop is mainly generated at the nozzle. The frictional pressure drop increases due to the reduction of the flow channel. The experimental and simulation results show that the frictional pressure drop accounts for a small percentage of the total pressure drop. Based on the installation location of the ejector and the focus of the study, the frictional pressure drop and the repositioning pressure drop are neglected in this calculation.The accelerated pressure drop in the axial position of the Venturi [32] can be calculated by the following equation.
(17)ΔPv=Gmvg,thAφ=ΔP2+ΔP3,where the gas velocity in the Venturi throat can be expressed as the following equation.(18)vg,th=0.135XLMFrg1.27ρlρg0.806,whereGm represents the mass flow rate of the mixture, kg/s; vg,th is the Venturi throat gas velocity, m/s; A represents the cross-sectional area, m2; φ represents the cross-sectional gas content, dimensionless; and XLM and Frg represent the Lockhart-Martinelli number and gas Froude number, respectively, dimensionless.


#### 4.5.2. Differential Pressure of Drainage Pipe Inlet

The pressure drop at the inlet of the drainage pipe is mainly caused by the sudden shrinkage of the pipe diameter, as shown in the structure in Figure2(c). Therefore, in this study, the pressure drop between section 1 and section 2 is investigated using the pressure drop calculation method of the abruptly shrunken pipe. Due to the small experimental pressure, it is assumed that there is no significant change in the density of the gas-liquid phase and no mass exchange.The continuity equations for the gas and liquid phases on cross-section 1 and cross-section 2 are obtained from the conservation of mass.(19)Ggs=ρgAgvgs=ρgAsφs,(20)Gls=ρlAlvls=ρgvlsAs1−φs,whereGg and Gl represent the mass flow rate of the gas and liquid, respectively, kg/s; Ag and Al represent the cross-section of the gas and liquid, respectively, m2; φs represents the gas content of a section, dimensionless; vgs and vls represent the flow rate of the gas and liquid of a section, respectively, m/s; and s represents any section of section 1 and section 2.Since the section is very short and friction is neglected, the sum of the external forces on the two gas-liquid phases along the axial direction isA2P1−P2; then, the momentum equation is
(21)P1−P2=1A2Gg2vg2+Gl2vl2−Gg1vg1−Gl1vl1.The subscript 1 represents the inlet section of the protruding joint and 2 represents the outlet section of the protruding joint.From the analysis of simulation results, it is found that the streamline of the gas-liquid mixture is bent downstream of the sudden contraction pipe (Figure2(c), section t). Therefore, the resistance loss of the sudden contraction pipe is mainly generated from section t to section 2 [33]. Based on the specificity of the structure, it is assumed that the gas content of the drainage pipe is equal to that of section 2; i.e., φt=φ2=φ. From the momentum equation, the pressure drop of the sudden contraction pipe can be calculated as
(22)ΔP1=0.235GmρlA121−α21−φ+α2φρlρg,whereα is the dryness fraction, dimensionless. Gm is the gas-liquid two-phase mass flow rate, kg/s.


#### 4.5.3. Impacting Tee Differential Pressure

The pressure loss from the impacting tee is calculated according to the following Eq. (23) calculation [34].
(23)ΔPo=λQm22A2ρl1+xρgρl−1,Where the local resistance factorλ is calculated using the following equation.
(24)λ=cλo.The correction factorc is obtained using the empirical equation:
(25)c=1+0.75x1−x1+ρl/ρg1−ρg/ρl1+xρl/ρg−1.


#### 4.5.4. Heavy Pressure Drop

According to the operating principle of the device, the repositioning pressure drop is mainly generated in the rising pipe section of the drainage pipe and the impacting tee. The calculation equation is(26)ΔPh=2ρmgΔh.Since the diameter of the drainage pipe is much smaller than the accumulation riser when the gas-liquid mixture enters the drainage pipe, the flow rate of the liquid phase suddenly increases, and when the superficial velocity of the gas-liquid two phases are equal, the slip ratio is 1. Therefore, referring to Eq. (12), the following equation is used to calculate the mixing density.
(27)ρm=ρgφ+ρl1−φ.


### 4.6. Accuracy Verification

As per Figure13(a), it can be seen that under atmospheric conditions, the Venturi nozzle pressure drop has the largest proportion in the total pressure drop, while the heavy pressure drop has the smallest proportion. In small-volume conditions, the pressure drop caused by sudden changes in the diameter of the drainage pipe is the largest. The pressure drop of the Venturi nozzle increases with the liquid flow rate, and its average proportion is 52.82% of the total pressure drop. In the large liquid volume and small air volume conditions, the pressure drop at the inlet of the drainage pipe produced a maximum percentage of 95.83%.Figure 13
Comparative analysis of the error of pressure drop calculation under different gas-liquid ratio conditions.
(a)
Proportion of local pressure drop(b)
Comparison of pressure drop calculation errorAccording to the calculation results, it can be seen that the pressure drop calculation method obtained from this experiment can truly reflect the motion of the fluid. The error of the pressure drop calculation increases gradually with the decrease in the gas flow rate, as shown in Figure13(b). The relative error is the smallest when the liquid flow rate is 1.67 m3/h. As the gas flow rate decreases gradually, the relative error increases gradually. The gas-liquid rate is small, and the liquid phase is mostly deposited at the bottom of the accumulation riser due to gravity. As shown in Figure 7(c), the liquid level is higher than the inlet of the drainage pipe. The height difference of the liquid level provides energy for the liquid movement, and the gravitational potential energy is ignored in the calculation. This is the main reason for the large calculation results and the large error in the total pressure drop. When the gas flow rate increases, the Venturi produces a larger proportion of the pressure drop. The calculation accuracy of the total pressure drop depends on the Venturi nozzle. Under a large gas flow rate, the flow rate at the Venturi nozzle changes little, the measured results match the calculated results to a high coincidence, and the calculation error is small.


### 4.7. Model Correction

To further improve the calculation accuracy of the pressure drop under different gas-liquid ratios, the total pressure drop calculation method of the slug flow elimination device is optimized based on the pressure drop ratio of different structures. The modified pressure drop calculation method is obtained as follows.(28)ΔP=0.804+0.957ΔPo+ΔP1+ΔPh+0.421ΔPv1.5.As Figure14 shows, the pressure drop gradually decreases as the gas-liquid ratio increases. The accuracy of the modified pressure drop calculation method is significantly improved, and the error with the measured pressure drop is kept within 20%, among which 69.31% of the data points are within 10%.Figure 14
Relative error distribution of pressure drop calculation after correction.


## 4.1. Flow Pattern

Flow pattern variation is the main cause of pressure in the pipe [26]. As Figure 5(a) shows, the inlet is a slug flow, and bubbles are found in the liquid slug in the slug flow unit. The closer to the end of the liquid slug, the larger the bubble is. This is caused by the pressure between the liquid film and the liquid slug becoming smaller. When the slug flow enters the device, a large amount of liquid is initially separated into the descending pipe under the small gas-liquid ratio condition, and a small amount of liquid is carried to the horizontal pipe by the gas. The preliminary separation of the gas and liquid is obvious, in which a small amount of liquid in the gas pipe flows into the liquid collection pipe through the descending pipe. The gas imparts kinetic energy to the liquid in the mix chamber of the ejector, which improves the velocity of the liquid and reduces the accumulation of liquid in the riser. After passing through the elimination device, the liquid slug and Taylor bubble are separated, and the flow pattern formed after the second mixing is as Figure 5(b) shows. From Figure 5(b), it can be seen that the flow pattern at the outlet has changed from slug flow to transition flow and gradually transformed to annular flow. Due to gravity, the liquid phase at the outlet is deposited at the bottom of the pipe, forming a small wave, and a fine liquid flow is formed around the pipe wall, which temporarily fails to form a complete liquid film.Figure 5
Fluid motion at the inlet and outlet of the experimental setup.
(a)
Inlet flow pattern (Ql=0.42m3/h, Qg=41.6m3/h)(b)
Outlet flow pattern (Ql=0.42m3/h, Qg=41.6m3/h)The liquid holdup is one of the main variables describing the slug flow. Therefore, the magnitude of the liquid holdup not only affects the pressure fluctuation but will also determine the slug flow elimination effect. According to the Gregory et al. [27] study, the liquid holdup of the pipe cross-section capable of slug flow is 48%. The experimental results show that when the gas-liquid ratio is greater than 380 m3/m3, it is difficult to form slug flow in the pipe. However, a large gas-liquid ratio still leads to a liquid-phase impact on the impacting tee and a significant wave pattern in the outlet pressure.When the kinetic energy of the gas is small, it cannot pump the liquid in the accumulation riser in time, causing it to accumulate (Figure6(a)). When the gas cannot continuously transfer its kinetic energy to the liquid, the liquid entering the priming chamber appears to oscillate at the inlet of the drainage pipe (Figure 6(b)). Figure 7 shows the distribution of the liquid phase at t=2.2s under different gas-liquid ratio conditions. As the gas-liquid ratio gradually increases, the initial separation effect of the impacting tee on the two-phase gas-liquid decreases, and a vortex is generated at the elbow. More liquid enters the gas pipe. The liquid phase entering the gas pipe makes the flow rate at the nozzle fluctuate, and the energy conversion between the gas and liquid phases is then affected. When the gas-liquid ratio is greater than 120 m3/m3, a complete slug flow is not formed at the inlet due to the reduced liquid content in the pipe. It can be seen from the outlet pressure curve that the gradual increase of the gas-liquid ratio increases the outlet pressure amplitude. Due to the small liquid content rate, a full pipe flow cannot be formed in the accumulation riser, and the intermittent supply of liquid from the riser to the inlet pipe results. Therefore, the pressure fluctuations generated at a gas-liquid ratio greater than 120 m3/m3 cannot be used to define the applicability of the device.Figure 6
Liquid state of the accumulation riser with a different gas-liquid ratio.Figure 7
Gas-liquid distribution under different gas-liquid ratios (t=2.2s).


## 4.2. Pressure Fluctuation

From the experimental phenomena, it is clear that the slug flow pressure signal has the following characteristics.(1)
The pressure fluctuation time in the liquid film area is long, the amplitude is small, and there are obvious fluctuations in the pressure rise phase with a step-up pattern(2)
The pressure amplitude in the liquid slug area is large, has a short duration, and shows a vertical wave, and the pressure reaches the highest point when a precipitous drop occurs [25].Combining Figures5 and 8(a), it can be seen that the pressure at the outlet depends on the various flow patterns. Under non-slug flow conditions, both mass flow and pressure tend to stabilize, which can further improve the accuracy of the gas-liquid two-phase measurement device and its stability. From Figure 8(b), which shows various patterns of the inlet pressure, it can be seen that the various patterns of simulated and measured pressures are almost the same at the same time nodes. The two have a high degree of agreement. Due to the loading and unloading of the air compressor during the experiment, which leads to the unstable gas-liquid flow, it will form as shown in Figure 8(b) (green dashed line). That is, the measured pressure value occasionally shows large fluctuations at the position of the trough, and the pressure measurement is slightly higher than that of the simulated value. Comparing the variation laws of pressure, flow pattern, and other parameters under experimental and numerical simulation conditions, respectively, it is verified that the numerical simulation method can accurately describe the gas-liquid two-phase flow pattern and its motion law.Figure 8
Variation of flow and pressure at the inlet and outlet.
(a)
Mass flow rate at inlet and outlet (GLR=100m3/m3)(b)
Experimental and simulated pressure curvesThe relationship between the void fraction and pressure at the cross-section 400 mm from the inlet before the liquid slug enters the impacting tee (GLR=60m3/m3) was simulated in this study, as shown in Figure 9. The graph shows that the pipe pressure reaches its maximum value when the void fraction reaches its minimum value, i.e., when the cross-sectional liquid content is at its maximum. When the cross-sectional liquid content increases, the pressure starts to build up in the Taylor bubble at the moment when the liquid slug blocks the gas passage. When the pressure at the end of the slug is greater than the pushing force required for the movement of the slug, the slug is discharged. At this point, the Taylor bubble acts as a “piston” and pushes the liquid slug to move. Combined with Figure 9, it can be seen that the larger the gas-liquid ratio, the larger the void ratio, and the smaller the average cross-sectional liquid content, the smaller the pressure in the pipe and the smaller the amplitude of pressure fluctuation.Figure 9
Variation curve of cross-sectional void ratio and pressure.Combined with the flow pattern variation at the inlet and outlet, it shows that the magnitude of the pressure fluctuation depends on the velocity of the slug flow, and the length of the slug flow determines the duration of the pressure peak. From the results of Figure10, it can be seen that the higher the liquid phase flow rate, the higher the pressure and the greater the amplitude of its fluctuation under the same gas flow rate. The smaller the gas-liquid ratio, the longer the duration of the pressure wave. The pressure fluctuations at the inlet and outlet have obvious differences. Among them, the high-frequency pressure signal at the inlet shows obvious slug flow characteristics, but the inlet slug flow pressure signal occasionally shows abnormal fluctuations at the trough during the experiment due to the influence of compressor loading (Figure 10(a) marked by a red solid line). During the experiment, it was found that when the liquid slug reached the top of the riser, the pressure reached a peak at that time. When the slug is discharged momentarily, the pressure suddenly decreases. As the gas-liquid ratio gradually decreases, the liquid slug stagnates at the riser and a small amount of liquid then flows back, causing the pressure signal at the inlet to fluctuate at the crest, as shown in Figures 10(b) and 10(c) marked by the red solid line. Combined with Figure 8(a), which shows the mass flow rate variation law, it can be seen that the stability of the outlet pressure depends on whether its mass flow rate is stable.Figure 10
Pressure fluctuation curve of the inlet and outlet with a different gas-liquid ratio.
(a)
GLR=100m3/m3(b)
GLR=66.56m3/m3(c)
GLR=49.88m3/m3


## 4.3. Velocity Distribution

From the internal streamline diagram of the ejector (Figure11), it can be seen that the velocity is smaller near the pipe wall and is larger in the center where the liquid content is small. The velocity before entering the ejector is about 1.15 m/s and reaches 13.41 m/s inside the mix chamber (GLR=40m3/m3). The sudden increase in velocity caused a sharp drop in pressure in the mix chamber due to the reduced-flow cross-section of the nozzle. To maintain the local pressure balance, the pressure in the induced chamber also decreases simultaneously. The liquid in the accumulation riser is pumped into the mix chamber by negative pressure and then enters the expansion pipe after being accelerated by high-speed gas, forming a non-slug flow downstream. As the gas-liquid ratio increases, the flow rate in the mix chamber gradually increases, and the suction force continues to increase.Figure 11
Streamline diagram of the ejector.Due to the existence of friction between the liquid and the pipe wall, the flow rate gradually decreases from the center to the wall. Among them is the location of the thin layer near the inner wall, which is a thin layer of resistance; this thin layer is called the boundary layer. In the flow through the throat, the fluid increases the velocity and decreases the pressure, maintaining the original surface layer. Downstream of the throat, the flow area increases, the kinetic energy of the fluid is converted into pressure energy, and the boundary layer is subject to the force of the opposite direction of the mainstream, resulting in fluid backflow. Beyond the boundary layer of the fluid to maintain the original direction of advance, the formation of the vortex phenomenon as shown in Figure8 results in the separation of the boundary layer.


## 4.4. Pressure Drop Model

Through experiments and simulations, it can be seen that the pressure loss generated by this device mainly comes from the ejector, the inlet of the drainage pipe, the impacting tee, and the heavy pressure drop. The total pressure drop of the slug flow elimination device is(16)ΔP=ΔPo+ΔP1+ΔPv+ΔPh,whereΔPo is the pressure drop generated by the impacting tee, kPa. ΔP1 is the pressure drop generated at the inlet of the drainage pipe, kPa. ΔPv is the pressure drop generated by the ejector, kPa. ΔPh is the repositioning pressure difference from the height difference, kPa.Under the same conditions of the gas flow rate, the total pressure drop increases gradually as the liquid flow rate increases. As Figure12(a) shows, the liquid-phase flow rate is related to the total pressure drop. According to the study, it is known that the frictional pressure drops upstream and downstream of the Venturi throat are equal under the condition of symmetry between the constriction and expansion sections [28]. In this study, due to the complex structure of the ejector, the differential pressure generated in the convergent section is significantly larger than that in the divergent section (as Figure 12(b)). From Figure 12(b), it can be seen that an increase in the gas-liquid ratio leads to a significant increase in the total pressure drop at constant pressure conditions. The differential pressure of the gas-liquid mixture passing through the Venturi nozzle is related to the magnitude of the flow rate and increases with increase of the gas-liquid ratio [29, 30].Figure 12
(a) Pressure drop variation at different liquid phase flow conditions; (b) differential pressure variation curves at different cross-sections (ΔP1 sudden contraction structure, ΔP2 Venturi-tapering section, ΔP3 Venturi-expansion section).
(a)(b)


## 4.5. Model Building

### 4.5.1. Venturi Differential Pressure

Based on the experimental and numerical simulation results, the pressure drop calculation model of this device is established. The pressure drop generated by the Venturi nozzle mainly includes an accelerated pressure drop and friction pressure drop [31]. Among them, the acceleration pressure drop accounts for the largest proportion and increases with the increase of the liquid content. The next is the frictional pressure drop, which is mainly generated at the nozzle. The frictional pressure drop is mainly generated at the nozzle. The frictional pressure drop increases due to the reduction of the flow channel. The experimental and simulation results show that the frictional pressure drop accounts for a small percentage of the total pressure drop. Based on the installation location of the ejector and the focus of the study, the frictional pressure drop and the repositioning pressure drop are neglected in this calculation.The accelerated pressure drop in the axial position of the Venturi [32] can be calculated by the following equation.
(17)ΔPv=Gmvg,thAφ=ΔP2+ΔP3,where the gas velocity in the Venturi throat can be expressed as the following equation.(18)vg,th=0.135XLMFrg1.27ρlρg0.806,whereGm represents the mass flow rate of the mixture, kg/s; vg,th is the Venturi throat gas velocity, m/s; A represents the cross-sectional area, m2; φ represents the cross-sectional gas content, dimensionless; and XLM and Frg represent the Lockhart-Martinelli number and gas Froude number, respectively, dimensionless.


### 4.5.2. Differential Pressure of Drainage Pipe Inlet

The pressure drop at the inlet of the drainage pipe is mainly caused by the sudden shrinkage of the pipe diameter, as shown in the structure in Figure2(c). Therefore, in this study, the pressure drop between section 1 and section 2 is investigated using the pressure drop calculation method of the abruptly shrunken pipe. Due to the small experimental pressure, it is assumed that there is no significant change in the density of the gas-liquid phase and no mass exchange.The continuity equations for the gas and liquid phases on cross-section 1 and cross-section 2 are obtained from the conservation of mass.(19)Ggs=ρgAgvgs=ρgAsφs,(20)Gls=ρlAlvls=ρgvlsAs1−φs,whereGg and Gl represent the mass flow rate of the gas and liquid, respectively, kg/s; Ag and Al represent the cross-section of the gas and liquid, respectively, m2; φs represents the gas content of a section, dimensionless; vgs and vls represent the flow rate of the gas and liquid of a section, respectively, m/s; and s represents any section of section 1 and section 2.Since the section is very short and friction is neglected, the sum of the external forces on the two gas-liquid phases along the axial direction isA2P1−P2; then, the momentum equation is
(21)P1−P2=1A2Gg2vg2+Gl2vl2−Gg1vg1−Gl1vl1.The subscript 1 represents the inlet section of the protruding joint and 2 represents the outlet section of the protruding joint.From the analysis of simulation results, it is found that the streamline of the gas-liquid mixture is bent downstream of the sudden contraction pipe (Figure2(c), section t). Therefore, the resistance loss of the sudden contraction pipe is mainly generated from section t to section 2 [33]. Based on the specificity of the structure, it is assumed that the gas content of the drainage pipe is equal to that of section 2; i.e., φt=φ2=φ. From the momentum equation, the pressure drop of the sudden contraction pipe can be calculated as
(22)ΔP1=0.235GmρlA121−α21−φ+α2φρlρg,whereα is the dryness fraction, dimensionless. Gm is the gas-liquid two-phase mass flow rate, kg/s.


### 4.5.3. Impacting Tee Differential Pressure

The pressure loss from the impacting tee is calculated according to the following Eq. (23) calculation [34].
(23)ΔPo=λQm22A2ρl1+xρgρl−1,Where the local resistance factorλ is calculated using the following equation.
(24)λ=cλo.The correction factorc is obtained using the empirical equation:
(25)c=1+0.75x1−x1+ρl/ρg1−ρg/ρl1+xρl/ρg−1.


### 4.5.4. Heavy Pressure Drop

According to the operating principle of the device, the repositioning pressure drop is mainly generated in the rising pipe section of the drainage pipe and the impacting tee. The calculation equation is(26)ΔPh=2ρmgΔh.Since the diameter of the drainage pipe is much smaller than the accumulation riser when the gas-liquid mixture enters the drainage pipe, the flow rate of the liquid phase suddenly increases, and when the superficial velocity of the gas-liquid two phases are equal, the slip ratio is 1. Therefore, referring to Eq. (12), the following equation is used to calculate the mixing density.
(27)ρm=ρgφ+ρl1−φ.


## 4.5.1. Venturi Differential Pressure

Based on the experimental and numerical simulation results, the pressure drop calculation model of this device is established. The pressure drop generated by the Venturi nozzle mainly includes an accelerated pressure drop and friction pressure drop [31]. Among them, the acceleration pressure drop accounts for the largest proportion and increases with the increase of the liquid content. The next is the frictional pressure drop, which is mainly generated at the nozzle. The frictional pressure drop is mainly generated at the nozzle. The frictional pressure drop increases due to the reduction of the flow channel. The experimental and simulation results show that the frictional pressure drop accounts for a small percentage of the total pressure drop. Based on the installation location of the ejector and the focus of the study, the frictional pressure drop and the repositioning pressure drop are neglected in this calculation.The accelerated pressure drop in the axial position of the Venturi [32] can be calculated by the following equation.
(17)ΔPv=Gmvg,thAφ=ΔP2+ΔP3,where the gas velocity in the Venturi throat can be expressed as the following equation.(18)vg,th=0.135XLMFrg1.27ρlρg0.806,whereGm represents the mass flow rate of the mixture, kg/s; vg,th is the Venturi throat gas velocity, m/s; A represents the cross-sectional area, m2; φ represents the cross-sectional gas content, dimensionless; and XLM and Frg represent the Lockhart-Martinelli number and gas Froude number, respectively, dimensionless.


## 4.5.2. Differential Pressure of Drainage Pipe Inlet

The pressure drop at the inlet of the drainage pipe is mainly caused by the sudden shrinkage of the pipe diameter, as shown in the structure in Figure2(c). Therefore, in this study, the pressure drop between section 1 and section 2 is investigated using the pressure drop calculation method of the abruptly shrunken pipe. Due to the small experimental pressure, it is assumed that there is no significant change in the density of the gas-liquid phase and no mass exchange.The continuity equations for the gas and liquid phases on cross-section 1 and cross-section 2 are obtained from the conservation of mass.(19)Ggs=ρgAgvgs=ρgAsφs,(20)Gls=ρlAlvls=ρgvlsAs1−φs,whereGg and Gl represent the mass flow rate of the gas and liquid, respectively, kg/s; Ag and Al represent the cross-section of the gas and liquid, respectively, m2; φs represents the gas content of a section, dimensionless; vgs and vls represent the flow rate of the gas and liquid of a section, respectively, m/s; and s represents any section of section 1 and section 2.Since the section is very short and friction is neglected, the sum of the external forces on the two gas-liquid phases along the axial direction isA2P1−P2; then, the momentum equation is
(21)P1−P2=1A2Gg2vg2+Gl2vl2−Gg1vg1−Gl1vl1.The subscript 1 represents the inlet section of the protruding joint and 2 represents the outlet section of the protruding joint.From the analysis of simulation results, it is found that the streamline of the gas-liquid mixture is bent downstream of the sudden contraction pipe (Figure2(c), section t). Therefore, the resistance loss of the sudden contraction pipe is mainly generated from section t to section 2 [33]. Based on the specificity of the structure, it is assumed that the gas content of the drainage pipe is equal to that of section 2; i.e., φt=φ2=φ. From the momentum equation, the pressure drop of the sudden contraction pipe can be calculated as
(22)ΔP1=0.235GmρlA121−α21−φ+α2φρlρg,whereα is the dryness fraction, dimensionless. Gm is the gas-liquid two-phase mass flow rate, kg/s.


## 4.5.3. Impacting Tee Differential Pressure

The pressure loss from the impacting tee is calculated according to the following Eq. (23) calculation [34].
(23)ΔPo=λQm22A2ρl1+xρgρl−1,Where the local resistance factorλ is calculated using the following equation.
(24)λ=cλo.The correction factorc is obtained using the empirical equation:
(25)c=1+0.75x1−x1+ρl/ρg1−ρg/ρl1+xρl/ρg−1.


## 4.5.4. Heavy Pressure Drop

According to the operating principle of the device, the repositioning pressure drop is mainly generated in the rising pipe section of the drainage pipe and the impacting tee. The calculation equation is(26)ΔPh=2ρmgΔh.Since the diameter of the drainage pipe is much smaller than the accumulation riser when the gas-liquid mixture enters the drainage pipe, the flow rate of the liquid phase suddenly increases, and when the superficial velocity of the gas-liquid two phases are equal, the slip ratio is 1. Therefore, referring to Eq. (12), the following equation is used to calculate the mixing density.
(27)ρm=ρgφ+ρl1−φ.


## 4.6. Accuracy Verification

As per Figure13(a), it can be seen that under atmospheric conditions, the Venturi nozzle pressure drop has the largest proportion in the total pressure drop, while the heavy pressure drop has the smallest proportion. In small-volume conditions, the pressure drop caused by sudden changes in the diameter of the drainage pipe is the largest. The pressure drop of the Venturi nozzle increases with the liquid flow rate, and its average proportion is 52.82% of the total pressure drop. In the large liquid volume and small air volume conditions, the pressure drop at the inlet of the drainage pipe produced a maximum percentage of 95.83%.Figure 13
Comparative analysis of the error of pressure drop calculation under different gas-liquid ratio conditions.
(a)
Proportion of local pressure drop(b)
Comparison of pressure drop calculation errorAccording to the calculation results, it can be seen that the pressure drop calculation method obtained from this experiment can truly reflect the motion of the fluid. The error of the pressure drop calculation increases gradually with the decrease in the gas flow rate, as shown in Figure13(b). The relative error is the smallest when the liquid flow rate is 1.67 m3/h. As the gas flow rate decreases gradually, the relative error increases gradually. The gas-liquid rate is small, and the liquid phase is mostly deposited at the bottom of the accumulation riser due to gravity. As shown in Figure 7(c), the liquid level is higher than the inlet of the drainage pipe. The height difference of the liquid level provides energy for the liquid movement, and the gravitational potential energy is ignored in the calculation. This is the main reason for the large calculation results and the large error in the total pressure drop. When the gas flow rate increases, the Venturi produces a larger proportion of the pressure drop. The calculation accuracy of the total pressure drop depends on the Venturi nozzle. Under a large gas flow rate, the flow rate at the Venturi nozzle changes little, the measured results match the calculated results to a high coincidence, and the calculation error is small.


## 4.7. Model Correction

To further improve the calculation accuracy of the pressure drop under different gas-liquid ratios, the total pressure drop calculation method of the slug flow elimination device is optimized based on the pressure drop ratio of different structures. The modified pressure drop calculation method is obtained as follows.(28)ΔP=0.804+0.957ΔPo+ΔP1+ΔPh+0.421ΔPv1.5.As Figure14 shows, the pressure drop gradually decreases as the gas-liquid ratio increases. The accuracy of the modified pressure drop calculation method is significantly improved, and the error with the measured pressure drop is kept within 20%, among which 69.31% of the data points are within 10%.Figure 14
Relative error distribution of pressure drop calculation after correction.


## 5. Conclusion

According to the motion law of slug flow, the elimination device designed based on the kinetic energy conversion method can better solve the problem of large pressure fluctuations caused by slug flow. In this study, the flow field of the slug flow through the elimination device is analyzed. The energy losses generated during the operation of the device are evaluated. A pressure drop calculation method for this device is developed based on the share of different structures in the total pressure drop. The relative error of the pressure drop model is controlled within 20%, which meets the process production requirements. The optimal and most economical operating range of the device is to maintain the gas-liquid ratio within 120 m3/m3 (under operating conditions). The device can provide stable flow patterns and pressure conditions for gas-liquid two-phase flow measurements for the oil well.



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*Source: 7731591-2023-01-30.xml*