# Preliminary Test of Gas Wave Ejector in Coalbed Methane Well Site and Impact Analysis of Fluid Medium Characteristics on Its Pressure Port Design

Authors: Yiming Zhao; Dapeng Hu; Yang Yu; Haoran Li; Qing Feng
Journal: International Journal of Energy Research
 (2023)

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## Abstract

Gas wave ejector (GWE) is an efficient energy conversion device which can recover and utilize excess pressure energy of high pressure fluid. It has significant application value in the sustainable development of energy resources, including the exploitation and transportation of natural gas. The first practical application test of GWE in the natural gas industry is completed in this study. The mechanisms and regulations of the effect of actual gas properties on the pressure port design of GWE were also investigated. The primary conclusions are as follows: the composition of working medium significantly affects the propagation velocity of pressure waves. The wave propagation velocity increases as methane purity (φCH4) rises. The equipment efficiency was decreased by up to 23% of the optimal value as a result of the variation in port position brought on by the increase in φCH4 from 0.7 to 1.0. The variation of working pressure ratio has a greater impact on the device performance than the pressure value when the pressure port design kept unchanged. The intake temperature fluctuation in the range of -30°C to 50°C has no obvious influence on the ideal port design. In the coal-bed methane well site industrial test, GWE accomplishes a recovery of low-pressure well by roughly 8.83×103Nm3/d, demonstrating its excellent ejection capacity. In addition, the variable speed test conducted under different working conditions shows that the equipment performance can be effectively enhanced by adjusting the rotation speed as the equipment structure remains unchanged with the changing working pressure/pressure ratio. In the field test range of this study, the increase of equipment ejection rate obtained by speed adjustment can rise up to 12% of the corresponding value at the designed rotation speed.


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## Body

## 1. Introduction

The efficient recovery and utilization of the excess energy of high-pressure fluid are one of the crucial means for energy-saving and emission reduction, which has a broad application prospect in the sustainable development of energy resources, such as the fast-growing natural gas industry [1]. As a cleaner energy source, natural gas is very important for ensuring the sustainable and development of economy and society [2]. With the improvement of drilling and other mining technologies [3], the unconventional gas resource, such as the shale gas, has been effectively mined [4]. However, the pressure inconsistency between gas wells is still one of the significant issues in the extraction, gathering, and transportation of natural gas. In order to meet the unified demand pressure for gathering and transportation, gas from the high-pressure wells such as newly opened wells and tight gas wells usually requires throttling and decompression treatment, which not only generates throttling effect resulting in blockages of valves and pipelines but also causes significant pressure energy waste [5]. However, the gathering and transportation requirements are frequently not met by low-pressure gas wells, such as late wells or coalbed methane wells. A quantity of external power is consumed when utilizing the compressor for pressurization mining. Meanwhile, the discharging and burning of the gas from low-pressure wells will contaminate the environment and waste resources [6]. Therefore, transferring the excess pressure energy form high-pressure gas wells to the low-pressure gas wells for pressurization mining can not only reduce the energy dissipation of high-pressure gas but also increase the output of low-pressure wells, which is an efficient technology to solve the problem of commingling production of gases with different pressures. It is beneficial to promote energy conservation, emission reduction, and sustainable development of the energy resources [7].The expander-booster compressor and static ejector are the common pressure energy transfer technologies applied in the industry recently. The expander-booster compressor mainly uses the impeller and other mechanical components to complete the pressure energy transmission, which commonly operates at a high rotational speed. As a result, its capacity for carrying impurities is poor, and the equipment is expensive for manufacturing and maintenance [8]. In addition, the pressure energy of high-pressure gas must be converted twice before being transferred to the low-pressure gas, which has a limited impact on the overall working efficiency of the equipment. The static ejector primarily uses the suction and turbulent diffusion of the high-speed jet (which is created by the high-pressure gas from nozzle) to enroll the low-pressure gas into the equipment, and the energy transfer process is accomplished through the direct contact and mixing of the two gases with different pressures. Static ejectors have the benefits of simple structure, low cost, and low sensitivity to impurities [9, 10]. However, the isentropic efficiency of static ejector is low due to the viscous dissipation in the turbulent mixing [11], and its pressurization capacity is limited when the gas pressure ratio between gases is small [12].The gas wave ejector (GWE) is a kind of wave rotor technology. Its energy transfer mode is the same as the static ejector for direct contact transmission, and its structure corresponds to the category of rotating equipment like the expander-booster compressor. As shown in Figure1, the core components of GWE include a wave rotor composed of identical channels and fixed end plates with pressure ports. The wave rotor rotates between the fixed end plates when the equipment is in operation, and different pressure gases enter to the channels through pressure ports to create pressure waves in the channels. These pressure waves serve as the medium for achieving energy transfer between the gases with different pressures [13]. Wave rotor technology including GWE has the advantages of high efficiency, simple structure, low manufacturing and maintenance cost, low rotating speed, and strong liquid carrying capacity. It has been successfully applied in many fields such as gas turbine topping cycle [14, 15], gas wave refrigeration [16], and wave rotor combustor [17, 18].Figure 1
Schematic diagram of core composition of gas wave ejector.The research and development of GWE commenced relatively late in comparison to the pressure wave supercharger whose first prototype test was finished in the 1940s [19]. In 1958, Spalding of Power Jets first proposed the concept of GWE and completed part of the theoretical work. The first trials of this technique were performed in the mid-1960s by Kentfield [20, 21] et al. of Imperial College London. Their experiments preliminary confirmed that the isentropic efficiency of GWE was closed to the turbine machinery and demonstrated the influence of some operating parameters on the equipment performance. However, Kentfield only conducted restricted experimental research on the equipment performance without in-depth theoretical and numerical analyses because of the constrained experimental conditions and numerical calculation capabilities. In the 1990s, Wilson [22] of NASA established an experimental platform for three-port wave rotor equipment, in order to evaluate their calculation software for the wave rotor design. Through these experiments, Wilson et al. proved the precision of their wave rotor design software and acquired a quantitative calculation technique for evaluating the various energy losses in the wave rotor equipment. In the 2000s, Kharazi et al. [23, 24] at the University of Michigan started to investigate the application of GWE to replace the condenser and primary compressor in the R718 refrigeration cycle, and it was discovered that this might substantially simplify the system construction. Since 2010, Hu et al. [25] from Dalian University of Technology have been studying GWE through the combination of numerical simulation and experiment. They obtained the characteristics of gas flow and gas waves in the GWE rotor channels and acquired the influence laws of rotational speed and other parameters on equipment performance. They also proposed several methods, such as the collaborative production mode, for improving the GWE performance.Although the performance advantages of wave rotor technology attract researchers to conduct amounts of research and development work for GWE, the above-mentioned theoretical and experimental researches (except for the GWE in R718 cycle) are all completed under the condition ofair<1MPa (approximately ideal gas) as the fluid medium. The equipment design and practical application performance in real gas medium have not been analyzed and investigated on the background of the natural gas industry which is suitable for GWE applications. According to the basic working principle of GWE, reasonable pressure port design based on the ideal wave system is the first premise to guarantee the effectiveness of equipment. And the parameters of pressure wave propagation, especially propagation velocity, primarily influence port design. Because the gas medium characteristics, such as composition and working pressure ratio, affect the propagation velocity of pressure waves, the medium characteristics of natural gas are intimately related to the design of pressure ports. In order to address the issues mentioned above, this study carried out research on the design of GWE pressure port in real gas medium and performed the first industrial application of wave rotor technology in the natural gas industry (extraction of coalbed methane gas) to preliminarily test the practical application performance of GWE. The relevant results are useful to the improvement of GWE design theory and have some reference and guiding significance for the promotion and application of GWE in industrial production with actual gas as medium.


## 2. Research Methodology

### 2.1. Testing System and Operational Process

The real scene of the GWE testing system of the coalbed methane well site is shown in Figure2(a), and the construction of system is clearly demonstrated in Figure 2(b). In addition to the GWE testing equipment, the system mainly consisted of the high- and low-pressure pipelines connected to the high- and low-pressure Christmas trees, respectively, and the medium-pressure exhausting pipeline was connected to the well-site gathering and transmission lines. A gas-liquid separator was placed before the high-pressure inlet of the GWE equipment, which was used for purifying the high-pressure intake gas. There was no separation device set up for the low-pressure gas, since the low-pressure well had been discontinued, and the data of low-pressure gas composition and flow rate could not be obtained accurately. The pressure regulating valves and antivibration pressure gauges were equipped on each pipeline in the system. The high-pressure pipeline and the gathering pipeline both had the integrated flowmeters installed to continuously monitor the pressure, temperature, and flow rate of the high- and medium-pressure gas. GWE equipment was driven by a motor, whose speed was regulated and monitored by a frequency converter. The parameters of the monitoring and controlling devices in the system are shown in Table 1.Figure 2
GWE testing system at the coalbed methane well site: (a) real scene of testing system and (b) schematic diagram of system.Table 1
Parameters of components used in the gas wave ejector experimental system.
InstrumentationMeasuring/controlling rangeMeasurement accuracyIntegrated flow meter-50~200°C±0.5%0~6/10 MPa±0.5%0~2500 Nm3/h±0.5%Frequency convertor7.5 kW/0~50 Hz0.01 HzDriving motor5.5 kW/0~50 Hz/Pressure indicator0~6/10 MPa±1.6%The experimental procedure is as follows: firstly, nitrogen was used to replace the gas in the system to avoid mixing air into the natural-gas collection and transmission pipelines of the system. Secondly, the equipment rotational speed was accelerated to the design value by the frequency converter before natural gas entering into the testing platform. Thirdly, the valve of medium-pressure exhausting pipeline was opened to connect the testing platform with the gathering and exhausting pipeline of well site. Fourthly, the high-pressure Christmas tree was switched to connect with the high-pressure pipeline of the testing system, and the high-pressure control valve was slowly opened, allowing the high-pressure gas to pass through the GWE equipment and enter to the gathering pipeline; Then, the low-pressure gas Christmas tree was switched to connect with the low-pressure pipeline, and the low-pressure regulating valve was fully opened. Finally, the high-pressure valve was adjusted until the pressure of high-pressure gas reached to the predetermined value of the working condition, establishing a stable ejection and pressurization process.The GWE testing equipment was composed of rotational parts like spindle and wave rotor as shown in Figure3(b), as well as the fixed portions, such as the exhausting end plate and angle modulation plate as shown in Figure 3(c). In order to conduct an auxiliary comparison analysis, the performance experiments of the equipment under air medium were carried out in this study on the air testing platform as shown in Figure 3(a). The driving, testing instrumentations, and fundamental operation procedure of the air testing system were mostly identical with the coalbed methane mining test (without gas displacement).Figure 3
GWE testing system in air medium and part of the core equipment components.


### 2.2. Numerical Simulation Method

#### 2.2.1. Numerical Model

In this study, the commercial CFD software FLUENT was used to compute the flow and wave characteristics in the rotor channels, which are difficult to monitor experimentally. In order to improve the calculation accuracy, the three-dimensional numerical model shown in Figure4 was used in this study to replace the two-dimensional and quasitwo-dimensional models commonly employed in the wave rotor technologies [26]. The calculation domains of the model included high-, medium- and low-pressure ports, wave rotor channels, and the gaps between channels and ports. The outer boundary surfaces of each port were the calculation boundaries of the model. The high- and low-pressure port boundaries were set as the pressure inlet, and the medium-pressure port boundary was the pressure outlet. The overlapping end faces between ports, gaps, and channels were set as interfaces so as to realize the gas flow among each domain. Other wall surfaces of the channels and ports were set as adiabatic smooth wall surfaces. In addition, periodic boundary conditions were also set in the model to realize the periodic reciprocating motion of the rotor channels. It eliminated the need for the integrated rotor in model, which improved the computational efficiency.Figure 4
Diagram of GWE numerical model and mesh generation.In this study, hexahedral grids were used for meshing the model. Mesh encryption was conducted close to the interfaces between pressure ports, channels, and gaps due to the large gradient of parameters in the process of connection and disconnection between them. And the grids were refined on each wall of channels and ports to simulate the unsteady flow more accurately.The gas flow processes in the computational model satisfy the following governing equations:Mass conservation equation:(1)∂ρ∂t+∇⋅ρu⟶=0,where t represents time and u⟶ is the velocity vector.Momentum conservation equation:(2)∂ρu⟶∂t+∇⋅ρu⟶u⟶=−∇p+∇⋅τ+F⟶,where F⟶ represents external volume forces, and τ represents the viscous force tensor.Energy conservation equation:(3)∂∂tρE+∇⋅ρu⟶E=∇⋅k∇T+∇⋅u⟶⋅τ+u⟶⋅F⟶,where E represents the total energy of gas, and k represents the effective thermal conductivity.The National Institute of Standards and Technology (NIST) real gas model was employed in this study, in order to precisely calculate the flow and wave laws in the rotor channels under actual gas medium. This model can dynamically read the relevant property parameters in REFPROP (international authoritative software for calculating the physical properties of working medium) to ensure the accuracy of computation. The SAS-SST turbulence model was used to calculate the unsteady turbulence in the model, which has excellent computational ability for strong swirling flow and strong contact flow (typical gas flow phenomena in the wave rotor channels) [27], and its applicability in the field of wave rotor technology has also been experimentally tested by PIV technology [28]. In the calculation process, equations (1)-(3) and the equations of turbulent kinetic energy and specific dissipation rate in the SAS-SST model were solved simultaneously (in which the transfer property parameters such as gas density were extracted from REFPROP in real time), so as to obtain the distributions of the gas state parameters in the unsteady flow process. Other relevant settings for the calculation model are shown in Table 2. In conclusion, a proper combination of calculation methods including the turbulence model, real gas model, and discretization method was established and verified according to gas flow characteristics in this study (Section 2.2.2 below for details). And the specific equations in SAS-SST and the detailed description of the NIST model as well as the AUSM method can be found in theoretical manual of FLUENT.Table 2
Fluent solver settings in this research.
PropertyParameterDimension of model3DSolver typeDensity-basedTimeTransientDensityNISTEnergy modelActivatedTurbulence modelSST-SASConvective flux typeAUSMSpatial discretizationFlow parametersSecond order upwindTurbulent kinetic energySecond order upwindSpecific dissipation rateSecond order upwindTransient formulationSecond order implicit


#### 2.2.2. Grid Independence and Model Verification

In order to improve the calculation efficiency as much as possible while maintaining the computation correctness, the comparative calculation of models with different grid numbers was conducted under the same working condition in this study, and the results are shown in Figure5. The model can more precisely represent the velocity and pressure fluctuations caused by the vortex and reflection of compression waves at the inlet end of channel when the grid number of a single channel reaches to 3.6×104. The gas velocity and pressure after shock wave can be calculated more accurately with this grid scale. The distribution of parameters in channel remains essentially unchanged as the grid number of single channel increases to 8.0×104, which proves that the grid encryption has no impact on the calculation results. Therefore, 3.6×104 grids of single channel were set as the grid division standard in this research, and the total grid number of the model under this standard was close to 8.0×105.Figure 5
Grid independence verification: (a) velocity distribution comparison and (b) pressure distribution comparison.
(a)(b)The expansion ratioα and compression ratio β are defined to represent the working conditions of GWE according to its working mechanism and application background. The specific expressions of the two are as follows:
(4)α=phtplt,β=pmsplt,where pht and plt represent the total pressures of high-pressure and low-pressure gas, respectively, and pms is the back pressure of the equipment medium-pressure outlet. The ejection rate ξ and efficiency η are usually used as the performance evaluation parameters of this device, and their expression are
(5)ξ=mlpmhp,η=mlpTltpmt/pltγ−1/γ−1mhpTht1−pmt/phtγ−1/γ,where mlp and mhp represent the mass flow rate of low-pressure and high-pressure gas, respectively; Tht and Tlt represent the temperature of high-pressure and low-pressure gas, respectively; pht, pmt, and plt represent the total pressure of high-pressure, medium-pressure, and low-pressure gas, respectively.In this study, the air experiment platform was used to verify the numerical model, and the results are shown in Figure6. Because the adiabatic and smooth walls were employed, and the negative effects caused by the gas leakage between the channels and the equipment gas cavities were ignored in the numerical simulation, the ejection rate of the equipment obtained by numerical calculation is slightly higher than the experimental value under each working condition. However, the difference between the two values is within 4.9%. In addition, the variation law of ejection rate with working conditions obtained by simulation is basically consistent with the experimental results, which demonstrates the accuracy of the numerical model used in this research.Figure 6
Verification of the numerical model.


## 2.1. Testing System and Operational Process

The real scene of the GWE testing system of the coalbed methane well site is shown in Figure2(a), and the construction of system is clearly demonstrated in Figure 2(b). In addition to the GWE testing equipment, the system mainly consisted of the high- and low-pressure pipelines connected to the high- and low-pressure Christmas trees, respectively, and the medium-pressure exhausting pipeline was connected to the well-site gathering and transmission lines. A gas-liquid separator was placed before the high-pressure inlet of the GWE equipment, which was used for purifying the high-pressure intake gas. There was no separation device set up for the low-pressure gas, since the low-pressure well had been discontinued, and the data of low-pressure gas composition and flow rate could not be obtained accurately. The pressure regulating valves and antivibration pressure gauges were equipped on each pipeline in the system. The high-pressure pipeline and the gathering pipeline both had the integrated flowmeters installed to continuously monitor the pressure, temperature, and flow rate of the high- and medium-pressure gas. GWE equipment was driven by a motor, whose speed was regulated and monitored by a frequency converter. The parameters of the monitoring and controlling devices in the system are shown in Table 1.Figure 2
GWE testing system at the coalbed methane well site: (a) real scene of testing system and (b) schematic diagram of system.Table 1
Parameters of components used in the gas wave ejector experimental system.
InstrumentationMeasuring/controlling rangeMeasurement accuracyIntegrated flow meter-50~200°C±0.5%0~6/10 MPa±0.5%0~2500 Nm3/h±0.5%Frequency convertor7.5 kW/0~50 Hz0.01 HzDriving motor5.5 kW/0~50 Hz/Pressure indicator0~6/10 MPa±1.6%The experimental procedure is as follows: firstly, nitrogen was used to replace the gas in the system to avoid mixing air into the natural-gas collection and transmission pipelines of the system. Secondly, the equipment rotational speed was accelerated to the design value by the frequency converter before natural gas entering into the testing platform. Thirdly, the valve of medium-pressure exhausting pipeline was opened to connect the testing platform with the gathering and exhausting pipeline of well site. Fourthly, the high-pressure Christmas tree was switched to connect with the high-pressure pipeline of the testing system, and the high-pressure control valve was slowly opened, allowing the high-pressure gas to pass through the GWE equipment and enter to the gathering pipeline; Then, the low-pressure gas Christmas tree was switched to connect with the low-pressure pipeline, and the low-pressure regulating valve was fully opened. Finally, the high-pressure valve was adjusted until the pressure of high-pressure gas reached to the predetermined value of the working condition, establishing a stable ejection and pressurization process.The GWE testing equipment was composed of rotational parts like spindle and wave rotor as shown in Figure3(b), as well as the fixed portions, such as the exhausting end plate and angle modulation plate as shown in Figure 3(c). In order to conduct an auxiliary comparison analysis, the performance experiments of the equipment under air medium were carried out in this study on the air testing platform as shown in Figure 3(a). The driving, testing instrumentations, and fundamental operation procedure of the air testing system were mostly identical with the coalbed methane mining test (without gas displacement).Figure 3
GWE testing system in air medium and part of the core equipment components.


## 2.2. Numerical Simulation Method

### 2.2.1. Numerical Model

In this study, the commercial CFD software FLUENT was used to compute the flow and wave characteristics in the rotor channels, which are difficult to monitor experimentally. In order to improve the calculation accuracy, the three-dimensional numerical model shown in Figure4 was used in this study to replace the two-dimensional and quasitwo-dimensional models commonly employed in the wave rotor technologies [26]. The calculation domains of the model included high-, medium- and low-pressure ports, wave rotor channels, and the gaps between channels and ports. The outer boundary surfaces of each port were the calculation boundaries of the model. The high- and low-pressure port boundaries were set as the pressure inlet, and the medium-pressure port boundary was the pressure outlet. The overlapping end faces between ports, gaps, and channels were set as interfaces so as to realize the gas flow among each domain. Other wall surfaces of the channels and ports were set as adiabatic smooth wall surfaces. In addition, periodic boundary conditions were also set in the model to realize the periodic reciprocating motion of the rotor channels. It eliminated the need for the integrated rotor in model, which improved the computational efficiency.Figure 4
Diagram of GWE numerical model and mesh generation.In this study, hexahedral grids were used for meshing the model. Mesh encryption was conducted close to the interfaces between pressure ports, channels, and gaps due to the large gradient of parameters in the process of connection and disconnection between them. And the grids were refined on each wall of channels and ports to simulate the unsteady flow more accurately.The gas flow processes in the computational model satisfy the following governing equations:Mass conservation equation:(1)∂ρ∂t+∇⋅ρu⟶=0,where t represents time and u⟶ is the velocity vector.Momentum conservation equation:(2)∂ρu⟶∂t+∇⋅ρu⟶u⟶=−∇p+∇⋅τ+F⟶,where F⟶ represents external volume forces, and τ represents the viscous force tensor.Energy conservation equation:(3)∂∂tρE+∇⋅ρu⟶E=∇⋅k∇T+∇⋅u⟶⋅τ+u⟶⋅F⟶,where E represents the total energy of gas, and k represents the effective thermal conductivity.The National Institute of Standards and Technology (NIST) real gas model was employed in this study, in order to precisely calculate the flow and wave laws in the rotor channels under actual gas medium. This model can dynamically read the relevant property parameters in REFPROP (international authoritative software for calculating the physical properties of working medium) to ensure the accuracy of computation. The SAS-SST turbulence model was used to calculate the unsteady turbulence in the model, which has excellent computational ability for strong swirling flow and strong contact flow (typical gas flow phenomena in the wave rotor channels) [27], and its applicability in the field of wave rotor technology has also been experimentally tested by PIV technology [28]. In the calculation process, equations (1)-(3) and the equations of turbulent kinetic energy and specific dissipation rate in the SAS-SST model were solved simultaneously (in which the transfer property parameters such as gas density were extracted from REFPROP in real time), so as to obtain the distributions of the gas state parameters in the unsteady flow process. Other relevant settings for the calculation model are shown in Table 2. In conclusion, a proper combination of calculation methods including the turbulence model, real gas model, and discretization method was established and verified according to gas flow characteristics in this study (Section 2.2.2 below for details). And the specific equations in SAS-SST and the detailed description of the NIST model as well as the AUSM method can be found in theoretical manual of FLUENT.Table 2
Fluent solver settings in this research.
PropertyParameterDimension of model3DSolver typeDensity-basedTimeTransientDensityNISTEnergy modelActivatedTurbulence modelSST-SASConvective flux typeAUSMSpatial discretizationFlow parametersSecond order upwindTurbulent kinetic energySecond order upwindSpecific dissipation rateSecond order upwindTransient formulationSecond order implicit


### 2.2.2. Grid Independence and Model Verification

In order to improve the calculation efficiency as much as possible while maintaining the computation correctness, the comparative calculation of models with different grid numbers was conducted under the same working condition in this study, and the results are shown in Figure5. The model can more precisely represent the velocity and pressure fluctuations caused by the vortex and reflection of compression waves at the inlet end of channel when the grid number of a single channel reaches to 3.6×104. The gas velocity and pressure after shock wave can be calculated more accurately with this grid scale. The distribution of parameters in channel remains essentially unchanged as the grid number of single channel increases to 8.0×104, which proves that the grid encryption has no impact on the calculation results. Therefore, 3.6×104 grids of single channel were set as the grid division standard in this research, and the total grid number of the model under this standard was close to 8.0×105.Figure 5
Grid independence verification: (a) velocity distribution comparison and (b) pressure distribution comparison.
(a)(b)The expansion ratioα and compression ratio β are defined to represent the working conditions of GWE according to its working mechanism and application background. The specific expressions of the two are as follows:
(4)α=phtplt,β=pmsplt,where pht and plt represent the total pressures of high-pressure and low-pressure gas, respectively, and pms is the back pressure of the equipment medium-pressure outlet. The ejection rate ξ and efficiency η are usually used as the performance evaluation parameters of this device, and their expression are
(5)ξ=mlpmhp,η=mlpTltpmt/pltγ−1/γ−1mhpTht1−pmt/phtγ−1/γ,where mlp and mhp represent the mass flow rate of low-pressure and high-pressure gas, respectively; Tht and Tlt represent the temperature of high-pressure and low-pressure gas, respectively; pht, pmt, and plt represent the total pressure of high-pressure, medium-pressure, and low-pressure gas, respectively.In this study, the air experiment platform was used to verify the numerical model, and the results are shown in Figure6. Because the adiabatic and smooth walls were employed, and the negative effects caused by the gas leakage between the channels and the equipment gas cavities were ignored in the numerical simulation, the ejection rate of the equipment obtained by numerical calculation is slightly higher than the experimental value under each working condition. However, the difference between the two values is within 4.9%. In addition, the variation law of ejection rate with working conditions obtained by simulation is basically consistent with the experimental results, which demonstrates the accuracy of the numerical model used in this research.Figure 6
Verification of the numerical model.


## 2.2.1. Numerical Model

In this study, the commercial CFD software FLUENT was used to compute the flow and wave characteristics in the rotor channels, which are difficult to monitor experimentally. In order to improve the calculation accuracy, the three-dimensional numerical model shown in Figure4 was used in this study to replace the two-dimensional and quasitwo-dimensional models commonly employed in the wave rotor technologies [26]. The calculation domains of the model included high-, medium- and low-pressure ports, wave rotor channels, and the gaps between channels and ports. The outer boundary surfaces of each port were the calculation boundaries of the model. The high- and low-pressure port boundaries were set as the pressure inlet, and the medium-pressure port boundary was the pressure outlet. The overlapping end faces between ports, gaps, and channels were set as interfaces so as to realize the gas flow among each domain. Other wall surfaces of the channels and ports were set as adiabatic smooth wall surfaces. In addition, periodic boundary conditions were also set in the model to realize the periodic reciprocating motion of the rotor channels. It eliminated the need for the integrated rotor in model, which improved the computational efficiency.Figure 4
Diagram of GWE numerical model and mesh generation.In this study, hexahedral grids were used for meshing the model. Mesh encryption was conducted close to the interfaces between pressure ports, channels, and gaps due to the large gradient of parameters in the process of connection and disconnection between them. And the grids were refined on each wall of channels and ports to simulate the unsteady flow more accurately.The gas flow processes in the computational model satisfy the following governing equations:Mass conservation equation:(1)∂ρ∂t+∇⋅ρu⟶=0,where t represents time and u⟶ is the velocity vector.Momentum conservation equation:(2)∂ρu⟶∂t+∇⋅ρu⟶u⟶=−∇p+∇⋅τ+F⟶,where F⟶ represents external volume forces, and τ represents the viscous force tensor.Energy conservation equation:(3)∂∂tρE+∇⋅ρu⟶E=∇⋅k∇T+∇⋅u⟶⋅τ+u⟶⋅F⟶,where E represents the total energy of gas, and k represents the effective thermal conductivity.The National Institute of Standards and Technology (NIST) real gas model was employed in this study, in order to precisely calculate the flow and wave laws in the rotor channels under actual gas medium. This model can dynamically read the relevant property parameters in REFPROP (international authoritative software for calculating the physical properties of working medium) to ensure the accuracy of computation. The SAS-SST turbulence model was used to calculate the unsteady turbulence in the model, which has excellent computational ability for strong swirling flow and strong contact flow (typical gas flow phenomena in the wave rotor channels) [27], and its applicability in the field of wave rotor technology has also been experimentally tested by PIV technology [28]. In the calculation process, equations (1)-(3) and the equations of turbulent kinetic energy and specific dissipation rate in the SAS-SST model were solved simultaneously (in which the transfer property parameters such as gas density were extracted from REFPROP in real time), so as to obtain the distributions of the gas state parameters in the unsteady flow process. Other relevant settings for the calculation model are shown in Table 2. In conclusion, a proper combination of calculation methods including the turbulence model, real gas model, and discretization method was established and verified according to gas flow characteristics in this study (Section 2.2.2 below for details). And the specific equations in SAS-SST and the detailed description of the NIST model as well as the AUSM method can be found in theoretical manual of FLUENT.Table 2
Fluent solver settings in this research.
PropertyParameterDimension of model3DSolver typeDensity-basedTimeTransientDensityNISTEnergy modelActivatedTurbulence modelSST-SASConvective flux typeAUSMSpatial discretizationFlow parametersSecond order upwindTurbulent kinetic energySecond order upwindSpecific dissipation rateSecond order upwindTransient formulationSecond order implicit


## 2.2.2. Grid Independence and Model Verification

In order to improve the calculation efficiency as much as possible while maintaining the computation correctness, the comparative calculation of models with different grid numbers was conducted under the same working condition in this study, and the results are shown in Figure5. The model can more precisely represent the velocity and pressure fluctuations caused by the vortex and reflection of compression waves at the inlet end of channel when the grid number of a single channel reaches to 3.6×104. The gas velocity and pressure after shock wave can be calculated more accurately with this grid scale. The distribution of parameters in channel remains essentially unchanged as the grid number of single channel increases to 8.0×104, which proves that the grid encryption has no impact on the calculation results. Therefore, 3.6×104 grids of single channel were set as the grid division standard in this research, and the total grid number of the model under this standard was close to 8.0×105.Figure 5
Grid independence verification: (a) velocity distribution comparison and (b) pressure distribution comparison.
(a)(b)The expansion ratioα and compression ratio β are defined to represent the working conditions of GWE according to its working mechanism and application background. The specific expressions of the two are as follows:
(4)α=phtplt,β=pmsplt,where pht and plt represent the total pressures of high-pressure and low-pressure gas, respectively, and pms is the back pressure of the equipment medium-pressure outlet. The ejection rate ξ and efficiency η are usually used as the performance evaluation parameters of this device, and their expression are
(5)ξ=mlpmhp,η=mlpTltpmt/pltγ−1/γ−1mhpTht1−pmt/phtγ−1/γ,where mlp and mhp represent the mass flow rate of low-pressure and high-pressure gas, respectively; Tht and Tlt represent the temperature of high-pressure and low-pressure gas, respectively; pht, pmt, and plt represent the total pressure of high-pressure, medium-pressure, and low-pressure gas, respectively.In this study, the air experiment platform was used to verify the numerical model, and the results are shown in Figure6. Because the adiabatic and smooth walls were employed, and the negative effects caused by the gas leakage between the channels and the equipment gas cavities were ignored in the numerical simulation, the ejection rate of the equipment obtained by numerical calculation is slightly higher than the experimental value under each working condition. However, the difference between the two values is within 4.9%. In addition, the variation law of ejection rate with working conditions obtained by simulation is basically consistent with the experimental results, which demonstrates the accuracy of the numerical model used in this research.Figure 6
Verification of the numerical model.


## 3. Pressure Port Design under Real Gas Medium

### 3.1. Pressure Port Design Process

The ideal wave system in GWE under the fluid medium of natural gas, such as coalbed methane (pressure less than 10MPa and temperature higher than -30°C), is identical with that of atmospheric air [29]. The ideal wave diagram (which is commonly adopted in the technical analysis and design of wave rotor technologies [15, 19, 25]) of GWE is shown in Figure 7. Under the real gas medium, the port design process of GWE based on wave diagram is as follows: in order to make the shock wave S generated with the channel connecting to the high-pressure port (used for pressurizing the low-pressure gas) reflect to form the reflected expansion wave E1 to reduce the gas pressure in channels, the medium-pressure port should be fully opened precisely when the shock wave arrives at the outlet end of the channel. In order to make the E1 reflect at the inlet end of the channel to further reduce the gas pressure, the high-pressure port should be closed before the head of E1 reaches the inlet end of the channel. The expansion wave system E2 is generated when the high-pressure port is closed, and it propagates to the outlet end of the channel. The pressure in channel drops sharply to form a deep expansion region after E1 and E2 pass through. When the pressure in this region drops below the pressure of low-pressure gas, the low-pressure port can be opened to allow the low-pressure gas to flow into the rotor channels forming the incident compression wave C1. The gas exhausting velocity is gradually decreased by E2, and the exhausting process of medium-pressure gas stops when E2 reaches to the outlet end of channel. At that point, the medium-pressure port should be closed to prevent the backflow of the medium-pressure gas. The compression wave C2 is generated to propagate to the inlet end of the channel when the medium-pressure port is closed, which increases gas pressure in channel to exceed the value of low-pressure gas. Therefore, the low-pressure port should be closed before C2 arrives at the inlet end of the channel to prevent gas backflow.Figure 7
Ideal wave system diagram of GWE.


### 3.2. Influence of Real Gas Characteristics on Wave Propagation

According to the port design method mentioned above, the offset distancewhm between the open position of the high-pressure port xho and the optimal opening position of the medium-pressure port xmoo (x that represents the port position gradually increases along the direction of channel rotation) as shown in Figure 7 is as follows:
(6)whm=xmoo−xho=πdn6×104.lvs,where d, l, and n represent the middle diameter (mm), length (mm), and rotational speed (r/min) of the rotor channels, respectively. And vs represents the average propagation velocity of shock wave (m/s). According to Equation (6), whm (mm) is mainly determined by vs when the structure and rotational speed of rotor channels are unchanged. Similarly, the other positions of pressure ports in the device are also mainly determined by the pressure wave propagation velocity in channels. Therefore, analyzing how the real gas medium affects the pressure port design is equivalent to studying its influence on the velocity of pressure wave propagation. To emphatically investigate the influence law of medium parameters on the shock wave velocity (shock wave that is generated earliest in the wave system with maximum intensity has the greatest impact on the equipment performance), the numerical models with various gas parameters were computed and compared, including gas pressure, pressure ratio, and composition.


#### 3.2.1. Real Gas Composition

The thermodynamic characteristics of gas medium, such as specific heat ratioγ, are closely related to the composition of gas medium. Therefore, the differences in fluid properties lead to variations of sound speed c and shock wave propagation velocity vs. The pressure port design for air is not suitable for NIST typical natural gas (composition: 88.8% methane, 6.5% nitrogen, 4.7% ethane, and other gases). And it will result in the wall reflection of shock wave shown in Figure 8(b), which destroys the ideal wave system of the air medium shown in Figure 8(a) and forms inoperative equipment.Figure 8
Comparison of pressure contours in different fluid medium: (a) air and (b) typical natural gas.According to the pertinent studies, the fluid medium used in the field test of this research (coalbed methane) is primarily composed of methane with a mass fraction generally ranging from 74.4% to 92.5%. The rest of components are mainly nitrogen with a fraction of 6.9%~24.6% and some other elements like carbon dioxide and heavy hydrocarbons whose total proportion is usually less than 1% [30]. As shown in Figure 9(a), the propagation distance in methane medium is much greater than that of air at the same stage with αin=1.6 (the ratio of pressures between channel and pressure ports before connecting). It indicates that the shock wave propagation velocity is relatively larger in the medium of methane. In addition, the shock wave propagation velocity is unidentical for methane with varying purities, demonstrating that the methane purity difference has impact on the velocity of pressure wave propagation. According to Figure 9(a), the shock wave propagates faster with purer methane (the medium used in the numerical calculation is the mixture of methane and nitrogen). However, the gap between the natural gases with different purities is much smaller than that between natural gas and air. As shown in Figure 9(b), when αin=1.6, the mean shock velocity vs approximately increases linearly with the increment of methane purity φCH4. vs is higher with the larger αin when φCH4 remains unchanged, and the growth rate of vs with φCH4 is also relatively larger.Figure 9
Comparison of shock wave propagation with different methane purities: (a) distance of shock wave propagation within the same time and (b) variation of mean shock propagation velocity with methane purity.
(a)(b)Under various compositions of the gas medium, the optimal opening and closing positions of the pressure ports are different because of the variations in the pressure wave propagation velocity. The best opening position of medium-pressure portxmoo of φCH4=0.7 is later than that of φCH4=1.0 as the shock wave propagates more slowly with lower methane purity. As a result, the difference between xmoo of the two different φCH4 (Δxmo) reaches 1.19 mm with α=1.6 and β =1.15 (shown in Table 3), which is about 24.3% of the channel width. Δxmo of different φCH4 is positive and decreases with increasing φCH4, which indicates that using pure methane as fluid medium in the pressure port design of GWE industrial equipment will cause the medium-pressure port to open earlier. Additionally, a lower φCH4 results in a larger negative divergence. The Δxmo between natural gas and air reaches to 42.4%-66.7% of the channel width. Therefore, utilizing air in the design of the structural parameters of GWE for industrial production of natural gas is inaccurate, though air can be used to study the function mechanism and performance law of GWE. In addition to the opening position of medium-pressure port xmo, the opening position of low-pressure port xlo is another critical port-design parameter that significantly impacts the device performance 36. The low-pressure port should be opened as the reflected expansion wave arrives at the inlet end of channel, as a result of which, the difference of the optimal low-pressure port opening position calculated under different medium compositions, defined as Δxlo, is affected both by the velocities of shock wave and expansion wave. Under the compared working conditions, Δxlo between φCH4=0.7 and 1.0 reaches 3.9 mm (around 79.6% of the channel width), which is larger than Δxmo. The calculation results show that the largest relative reduction ratio of ejection rate is about 23% when the port-design deviations reach to the above-mentioned values of φCH4=0.7~1.0. Therefore, it is necessary to obtain the actual medium composition for the accurate pressure port design of the industrial GWE equipment.Table 3
Relative deviations of medium-pressure port opening position atα=1.60, β=1.15, and plt=2.6MPa.
Mass fraction of CH4Mass fraction of N2Mass fraction of airΔxmo (mm)0.70.30.01.190.80.20.00.860.90.10.00.481.00.00.00.000.00.01.03.27


#### 3.2.2. Pressure of the Equipment Inlet Gas

The pressure of inlet gases is another major factor affecting the design of GWE pressure ports. When air is employed as the gas medium, the gas thermodynamic parameters, such as the specific heat ratioγ, vary with increasing gas pressure from 0.101 MPa to 2.60 MPa as shown in Table 4. However, there is a very minor variation in the average propagation velocity of shock wave vs. Therefore, the offset between the optimal opening positions of high- and medium-pressure ports whm remains basically unchanged. As shown in Figure 10, the equipment ejection rates are nearly identical with different pressure values, when the pressure port structure and pressure ratio remain unchanged. It supports the analysis result that the pressure port design is almost not affected by the pressure value in air medium within a certain range. However, both the relative variations of γ and vs are higher with the same pressure change in natural gas medium. As a result, the whm difference of plt=0.101~2.60MPa is significantly larger than that of air medium. The natural gas exploitation usually does not operate under atmospheric or negative pressure conditions in consideration of the pressure characteristics of gas wells and subsequent transportation demands. Therefore, plt=1.50MPa and plt=4.00MPa were chosen for comparative analysis in this study. As shown in Table 4, the results indicate that the maximum relative variation of whm is about 24.5% of the channel width in the typical industrial pressure range of plt=1.50~4.00MPa. When the pressure ratio and port design are left unaltered, the influence of the pressure variation within this range on the equipment performance is similar to that in the component variation as previously mentioned. The change of efficiency caused by the pressure value variation within above range reaches up to about 24% of the optimal efficiency under the calculated pressure ratio.Table 4
Comparison of calculation results under different pressure conditions with the identical pressure ratios ofα=1.60 and β=1.15.
Working conditions (MPa)Gas mediumγ (Hp gas)vs (m/s)whm (mm)plt=0.101NIST air1.403340.317.7plt=2.60NIST air1.477346.117.5plt=0.101NIST natural gas1.309382.416.0plt=1.50NIST natural gas1.391406.215.1plt=2.60NIST natural gas1.468424.914.3plt=4.00NIST natural gas1.592433.813.9Figure 10
Variation of the ejection rateξ with plt under the air medium of α=1.55 and β=1.10.Compared with the conditions with constant compression ratio and changing pressure values, the variation of pressure ratio within a certain pressure range not only changes the gas thermodynamic parameters, such as the specific heat ratioγ, but also impacts on the gas pressure in the stable pressure region (where the channels are closed at both ends as shown in Figure 7) [27, 29]. It alters the incident pressure ratio αin (αin=pht/ps, where ps is the average pressure in the stable pressure region) when the channel is connected to the high-pressure port, which has an impact on the propagation velocities of shock wave and other pressure waves. As a result, the influence of pressure ratio variation has a greater impact on the pressure port design. As shown in Figure 11(a), when pht and pms are fixed, ps rises with the increase of plt, causing αin of GWE to rise monotonically with the increase of plt. But when plt and pms remain unchanged, as depicted in Figure 11(b), ps and αin both rise with the increase of pht. As shown in Figure 11(c), the change of αin causes the variation of the shock wave propagation velocity vs. However, since αin is not the only factor determining vs, the change rate of vs with αin differs between the two variable working conditions. When pht is changed alone, vs rises monotonically with the increase of αin, and its variation rate is relatively higher.Figure 11
Variation laws of pressure in stable pressure regionps and incident pressure ratio αin under different incident gas pressures: (a) variation law of ps and αin with plt at pht=4.15MPa and pms=3.00MPa, (b) variation law of ps and αin with pht at plt=2.60MPa and pms=3.00MPa, and (c) variation of shock wave average propagation velocity vs with αin at pms=3.00MPa.
(a)(b)(c)The variation ofvs mentioned above inevitably results in different optimal opening positions of the medium-pressure port xmoo under various working conditions. As shown in Table 5, whenphtis reduced from 4.15 MPa (the reference working condition No. 7) to 3.12 MPa (No. 1), thexmoobecomes significant later because of the decrease ofvs, and the xmoo difference between the working conditions No. 1 and No. 7 reaches to 4.56 mm, about 93.1% of the channel width (Δxmo>0 when xmoo is later than that of the reference working condition, and Δxmo<0 when xmoo is earlier than the reference condition). As shown in Figure 12(a), when the optimal pressure port structure under the reference working condition No. 7 is adopted in the working condition No. 1, the rotor channel starts to connect with the medium-pressure port before the shock wave arriving at its outlet end. A reversed compression wave (RCW) moving toward the inlet end of channel is generated since the gas pressure at the channel outlet end is lower than the pressure of medium-pressure port at the beginning of connection. When RCW moves to the inlet end of channel, the gas in channels flows back to the low-pressure port and obstructs the inflow of low-pressure gas, which negatively affects the ejection performance of equipment. Under the working condition No. 12, an increase of incident pressure ratio results in an improvement of vs in comparison to the reference working condition. As shown in Figure 12(b), under the working condition No. 12, the channel is not connected with the medium-pressure port designed for the reference working condition when the head of shock wave reaches the outlet end of channel. As a result, the shock wave (which is actually part of the compression waves) reflects at the wall, and the reflected compression wave (RFCW) propagates to the inlet end of channel. As shown in Table 5, the xmoo difference between working condition No. 12 and No. 7 is -2.02 mm, which is about 41.2% of the channel width. However, the reflection of shock wave shown in Figure 12(b) is not very strong, since the existence of gap makes the end of channel not strictly closed and the channel is connected to the medium-pressure port immediately after the shock wave arriving. Therefore, the RFCW strength is not very strong, and the influence of RFCW on the equipment performance is relatively minimal when it reaches the inlet end of channel. It is because that it has a limited obstructive effect on the entrance of low-pressure gas without resulting in the gas backflow.Table 5
Comparison of calculation results under different pressure ratios.
Numberpht (MPa)plt (MPa)pms (MPa)αβΔxmo (mm)ps>pms13.122.603.001.201.154.56No23.122.803.001.111.073.89No33.512.403.001.461.252.40No43.512.603.001.351.151.50No53.512.803.001.251.07/Yes64.152.403.001.731.250.22No7 (ref.)4.152.603.001.601.150.00No84.152.803.001.481.07/Yes94.552.403.001.901.25-1.14No104.552.603.001.751.15/Yes114.552.803.001.631.07/Yes125.202.403.002.171.25-2.02No135.202.603.002.001.15/Yes145.202.803.001.861.07/YesFigure 12
Pressure and velocity distribution in the computational model with the optimal pressure port design for the reference working condition: (a)pht=3.12MPa, plt=2.60MPa, and pms=3.00MPa. (b) pht=5.20MPa, plt=2.40MPa, and pms=3.00MPa.
(a)(b)Both the pressures of the high- and low-pressure gases in working condition No. 14 are higher than those in the reference working condition as shown in Table5, which leads to a significant pressure increment in the channels of the stable pressure region. As shown in Figure 13(b), the medium-pressure port in working condition No. 14 has a higher gas pressure than the pressure in channel before shock wave passing through. As a result, there is no RCW generated even the channel completely connects to the medium-pressure port earlier than the shock wave arriving at the channel outlet end. And the equipment performance in this case does not deviate significantly from the optimum value. Therefore, the condition No. 14 with ps>pms has no strict optimal port opening position xmoo, and the working condition Nos. 5, 8, 10, 11, and 13 shown in Table 5 are similar to the condition No. 14 with ps>pms.Figure 13
Pressure distribution in the rotor channels and medium-pressure port in different working conditions: (a)pht=4.15MPa, plt=2.60MPa, and pms=3.00MPa. (b) pht=5.20MPa, plt=2.80MPa, and pms=3.00MPa.Based on the findings of above-mentioned analysis, when the design of the equipment pressure port remains unchanged, the decrease of the expansion ratioα caused by the pressure drop of the high-pressure gas has the most significant impact on the equipment performance. The port position deviation shown in Table 5 can result in a reduction of about 70% in equipment efficiency, when the relative expansion ratio ψexψex=α−1 is decreased by 66.7% (from working condition No. 7 to No. 1). Therefore, the pressures of high- and low-pressure gases need to be tracked and monitored in actual industrial production. When the gas well pressure lowers to a certain level with the increase of mining time, the pressure ports need to be replaced to maintain the high efficiency of equipment operation. In contrast, when ψex is increased by 95% (from working condition No. 12 to No. 7), the port position deviation only causes a drop of about 21% in the equipment efficiency. In addition, the internal wave system is not seriously damaged when the pressure in the stable pressure region is higher than the pressure in the medium-pressure port with the change of gas intake pressure, and the equipment efficiency does not decrease significantly compared with the optimal value under the corresponding pressure ratio.


#### 3.2.3. Temperature of the Equipment Inlet Gas

In addition to the pressure of gas, temperature variation also has an impact on the thermodynamic and transfer properties of gas, such as the adiabatic index and sound speed, which affects the shock wave velocity and the optimal design for the pressure ports. However, as shown in Table6, the variation of Δxmo is only 1.27~1.37 mm (less than 28% of the channel width), even when the temperatures of high-pressure gas Thp and low-pressure gas Tlp both change from 10°C to -30°C or 50°C at the same time. Δxmo is relatively smaller as Thp or Tlp changes alone. The maximum Δxmo is only 0.55 mm, about 11.2% of the channel width, when Tlp is changed alone within the study range. Because the actual gas temperature in actual industrial production is usually less than the research scope shown in Table 6 and gas temperature fluctuation is small during the production process, the port design of the industrial GWE device can adapt to the temperature fluctuations in the actual industrial application without seriously affecting the equipment efficiency.Table 6
Comparison of calculation results in the condition ofα=1.60, β=1.15, and plt=2.6MPa.
Thp (°C)Tlp (°C)γ (Hp gas)γ (Lp gas)Δxmo (mm)10101.4691.3980.0010-301.4691.4770.30-30101.6311.3980.8810501.4691.348-0.5550101.3891.398-1.06-30-301.6311.4771.2750501.3891.348-1.37


## 3.1. Pressure Port Design Process

The ideal wave system in GWE under the fluid medium of natural gas, such as coalbed methane (pressure less than 10MPa and temperature higher than -30°C), is identical with that of atmospheric air [29]. The ideal wave diagram (which is commonly adopted in the technical analysis and design of wave rotor technologies [15, 19, 25]) of GWE is shown in Figure 7. Under the real gas medium, the port design process of GWE based on wave diagram is as follows: in order to make the shock wave S generated with the channel connecting to the high-pressure port (used for pressurizing the low-pressure gas) reflect to form the reflected expansion wave E1 to reduce the gas pressure in channels, the medium-pressure port should be fully opened precisely when the shock wave arrives at the outlet end of the channel. In order to make the E1 reflect at the inlet end of the channel to further reduce the gas pressure, the high-pressure port should be closed before the head of E1 reaches the inlet end of the channel. The expansion wave system E2 is generated when the high-pressure port is closed, and it propagates to the outlet end of the channel. The pressure in channel drops sharply to form a deep expansion region after E1 and E2 pass through. When the pressure in this region drops below the pressure of low-pressure gas, the low-pressure port can be opened to allow the low-pressure gas to flow into the rotor channels forming the incident compression wave C1. The gas exhausting velocity is gradually decreased by E2, and the exhausting process of medium-pressure gas stops when E2 reaches to the outlet end of channel. At that point, the medium-pressure port should be closed to prevent the backflow of the medium-pressure gas. The compression wave C2 is generated to propagate to the inlet end of the channel when the medium-pressure port is closed, which increases gas pressure in channel to exceed the value of low-pressure gas. Therefore, the low-pressure port should be closed before C2 arrives at the inlet end of the channel to prevent gas backflow.Figure 7
Ideal wave system diagram of GWE.


## 3.2. Influence of Real Gas Characteristics on Wave Propagation

According to the port design method mentioned above, the offset distancewhm between the open position of the high-pressure port xho and the optimal opening position of the medium-pressure port xmoo (x that represents the port position gradually increases along the direction of channel rotation) as shown in Figure 7 is as follows:
(6)whm=xmoo−xho=πdn6×104.lvs,where d, l, and n represent the middle diameter (mm), length (mm), and rotational speed (r/min) of the rotor channels, respectively. And vs represents the average propagation velocity of shock wave (m/s). According to Equation (6), whm (mm) is mainly determined by vs when the structure and rotational speed of rotor channels are unchanged. Similarly, the other positions of pressure ports in the device are also mainly determined by the pressure wave propagation velocity in channels. Therefore, analyzing how the real gas medium affects the pressure port design is equivalent to studying its influence on the velocity of pressure wave propagation. To emphatically investigate the influence law of medium parameters on the shock wave velocity (shock wave that is generated earliest in the wave system with maximum intensity has the greatest impact on the equipment performance), the numerical models with various gas parameters were computed and compared, including gas pressure, pressure ratio, and composition.


### 3.2.1. Real Gas Composition

The thermodynamic characteristics of gas medium, such as specific heat ratioγ, are closely related to the composition of gas medium. Therefore, the differences in fluid properties lead to variations of sound speed c and shock wave propagation velocity vs. The pressure port design for air is not suitable for NIST typical natural gas (composition: 88.8% methane, 6.5% nitrogen, 4.7% ethane, and other gases). And it will result in the wall reflection of shock wave shown in Figure 8(b), which destroys the ideal wave system of the air medium shown in Figure 8(a) and forms inoperative equipment.Figure 8
Comparison of pressure contours in different fluid medium: (a) air and (b) typical natural gas.According to the pertinent studies, the fluid medium used in the field test of this research (coalbed methane) is primarily composed of methane with a mass fraction generally ranging from 74.4% to 92.5%. The rest of components are mainly nitrogen with a fraction of 6.9%~24.6% and some other elements like carbon dioxide and heavy hydrocarbons whose total proportion is usually less than 1% [30]. As shown in Figure 9(a), the propagation distance in methane medium is much greater than that of air at the same stage with αin=1.6 (the ratio of pressures between channel and pressure ports before connecting). It indicates that the shock wave propagation velocity is relatively larger in the medium of methane. In addition, the shock wave propagation velocity is unidentical for methane with varying purities, demonstrating that the methane purity difference has impact on the velocity of pressure wave propagation. According to Figure 9(a), the shock wave propagates faster with purer methane (the medium used in the numerical calculation is the mixture of methane and nitrogen). However, the gap between the natural gases with different purities is much smaller than that between natural gas and air. As shown in Figure 9(b), when αin=1.6, the mean shock velocity vs approximately increases linearly with the increment of methane purity φCH4. vs is higher with the larger αin when φCH4 remains unchanged, and the growth rate of vs with φCH4 is also relatively larger.Figure 9
Comparison of shock wave propagation with different methane purities: (a) distance of shock wave propagation within the same time and (b) variation of mean shock propagation velocity with methane purity.
(a)(b)Under various compositions of the gas medium, the optimal opening and closing positions of the pressure ports are different because of the variations in the pressure wave propagation velocity. The best opening position of medium-pressure portxmoo of φCH4=0.7 is later than that of φCH4=1.0 as the shock wave propagates more slowly with lower methane purity. As a result, the difference between xmoo of the two different φCH4 (Δxmo) reaches 1.19 mm with α=1.6 and β =1.15 (shown in Table 3), which is about 24.3% of the channel width. Δxmo of different φCH4 is positive and decreases with increasing φCH4, which indicates that using pure methane as fluid medium in the pressure port design of GWE industrial equipment will cause the medium-pressure port to open earlier. Additionally, a lower φCH4 results in a larger negative divergence. The Δxmo between natural gas and air reaches to 42.4%-66.7% of the channel width. Therefore, utilizing air in the design of the structural parameters of GWE for industrial production of natural gas is inaccurate, though air can be used to study the function mechanism and performance law of GWE. In addition to the opening position of medium-pressure port xmo, the opening position of low-pressure port xlo is another critical port-design parameter that significantly impacts the device performance 36. The low-pressure port should be opened as the reflected expansion wave arrives at the inlet end of channel, as a result of which, the difference of the optimal low-pressure port opening position calculated under different medium compositions, defined as Δxlo, is affected both by the velocities of shock wave and expansion wave. Under the compared working conditions, Δxlo between φCH4=0.7 and 1.0 reaches 3.9 mm (around 79.6% of the channel width), which is larger than Δxmo. The calculation results show that the largest relative reduction ratio of ejection rate is about 23% when the port-design deviations reach to the above-mentioned values of φCH4=0.7~1.0. Therefore, it is necessary to obtain the actual medium composition for the accurate pressure port design of the industrial GWE equipment.Table 3
Relative deviations of medium-pressure port opening position atα=1.60, β=1.15, and plt=2.6MPa.
Mass fraction of CH4Mass fraction of N2Mass fraction of airΔxmo (mm)0.70.30.01.190.80.20.00.860.90.10.00.481.00.00.00.000.00.01.03.27


### 3.2.2. Pressure of the Equipment Inlet Gas

The pressure of inlet gases is another major factor affecting the design of GWE pressure ports. When air is employed as the gas medium, the gas thermodynamic parameters, such as the specific heat ratioγ, vary with increasing gas pressure from 0.101 MPa to 2.60 MPa as shown in Table 4. However, there is a very minor variation in the average propagation velocity of shock wave vs. Therefore, the offset between the optimal opening positions of high- and medium-pressure ports whm remains basically unchanged. As shown in Figure 10, the equipment ejection rates are nearly identical with different pressure values, when the pressure port structure and pressure ratio remain unchanged. It supports the analysis result that the pressure port design is almost not affected by the pressure value in air medium within a certain range. However, both the relative variations of γ and vs are higher with the same pressure change in natural gas medium. As a result, the whm difference of plt=0.101~2.60MPa is significantly larger than that of air medium. The natural gas exploitation usually does not operate under atmospheric or negative pressure conditions in consideration of the pressure characteristics of gas wells and subsequent transportation demands. Therefore, plt=1.50MPa and plt=4.00MPa were chosen for comparative analysis in this study. As shown in Table 4, the results indicate that the maximum relative variation of whm is about 24.5% of the channel width in the typical industrial pressure range of plt=1.50~4.00MPa. When the pressure ratio and port design are left unaltered, the influence of the pressure variation within this range on the equipment performance is similar to that in the component variation as previously mentioned. The change of efficiency caused by the pressure value variation within above range reaches up to about 24% of the optimal efficiency under the calculated pressure ratio.Table 4
Comparison of calculation results under different pressure conditions with the identical pressure ratios ofα=1.60 and β=1.15.
Working conditions (MPa)Gas mediumγ (Hp gas)vs (m/s)whm (mm)plt=0.101NIST air1.403340.317.7plt=2.60NIST air1.477346.117.5plt=0.101NIST natural gas1.309382.416.0plt=1.50NIST natural gas1.391406.215.1plt=2.60NIST natural gas1.468424.914.3plt=4.00NIST natural gas1.592433.813.9Figure 10
Variation of the ejection rateξ with plt under the air medium of α=1.55 and β=1.10.Compared with the conditions with constant compression ratio and changing pressure values, the variation of pressure ratio within a certain pressure range not only changes the gas thermodynamic parameters, such as the specific heat ratioγ, but also impacts on the gas pressure in the stable pressure region (where the channels are closed at both ends as shown in Figure 7) [27, 29]. It alters the incident pressure ratio αin (αin=pht/ps, where ps is the average pressure in the stable pressure region) when the channel is connected to the high-pressure port, which has an impact on the propagation velocities of shock wave and other pressure waves. As a result, the influence of pressure ratio variation has a greater impact on the pressure port design. As shown in Figure 11(a), when pht and pms are fixed, ps rises with the increase of plt, causing αin of GWE to rise monotonically with the increase of plt. But when plt and pms remain unchanged, as depicted in Figure 11(b), ps and αin both rise with the increase of pht. As shown in Figure 11(c), the change of αin causes the variation of the shock wave propagation velocity vs. However, since αin is not the only factor determining vs, the change rate of vs with αin differs between the two variable working conditions. When pht is changed alone, vs rises monotonically with the increase of αin, and its variation rate is relatively higher.Figure 11
Variation laws of pressure in stable pressure regionps and incident pressure ratio αin under different incident gas pressures: (a) variation law of ps and αin with plt at pht=4.15MPa and pms=3.00MPa, (b) variation law of ps and αin with pht at plt=2.60MPa and pms=3.00MPa, and (c) variation of shock wave average propagation velocity vs with αin at pms=3.00MPa.
(a)(b)(c)The variation ofvs mentioned above inevitably results in different optimal opening positions of the medium-pressure port xmoo under various working conditions. As shown in Table 5, whenphtis reduced from 4.15 MPa (the reference working condition No. 7) to 3.12 MPa (No. 1), thexmoobecomes significant later because of the decrease ofvs, and the xmoo difference between the working conditions No. 1 and No. 7 reaches to 4.56 mm, about 93.1% of the channel width (Δxmo>0 when xmoo is later than that of the reference working condition, and Δxmo<0 when xmoo is earlier than the reference condition). As shown in Figure 12(a), when the optimal pressure port structure under the reference working condition No. 7 is adopted in the working condition No. 1, the rotor channel starts to connect with the medium-pressure port before the shock wave arriving at its outlet end. A reversed compression wave (RCW) moving toward the inlet end of channel is generated since the gas pressure at the channel outlet end is lower than the pressure of medium-pressure port at the beginning of connection. When RCW moves to the inlet end of channel, the gas in channels flows back to the low-pressure port and obstructs the inflow of low-pressure gas, which negatively affects the ejection performance of equipment. Under the working condition No. 12, an increase of incident pressure ratio results in an improvement of vs in comparison to the reference working condition. As shown in Figure 12(b), under the working condition No. 12, the channel is not connected with the medium-pressure port designed for the reference working condition when the head of shock wave reaches the outlet end of channel. As a result, the shock wave (which is actually part of the compression waves) reflects at the wall, and the reflected compression wave (RFCW) propagates to the inlet end of channel. As shown in Table 5, the xmoo difference between working condition No. 12 and No. 7 is -2.02 mm, which is about 41.2% of the channel width. However, the reflection of shock wave shown in Figure 12(b) is not very strong, since the existence of gap makes the end of channel not strictly closed and the channel is connected to the medium-pressure port immediately after the shock wave arriving. Therefore, the RFCW strength is not very strong, and the influence of RFCW on the equipment performance is relatively minimal when it reaches the inlet end of channel. It is because that it has a limited obstructive effect on the entrance of low-pressure gas without resulting in the gas backflow.Table 5
Comparison of calculation results under different pressure ratios.
Numberpht (MPa)plt (MPa)pms (MPa)αβΔxmo (mm)ps>pms13.122.603.001.201.154.56No23.122.803.001.111.073.89No33.512.403.001.461.252.40No43.512.603.001.351.151.50No53.512.803.001.251.07/Yes64.152.403.001.731.250.22No7 (ref.)4.152.603.001.601.150.00No84.152.803.001.481.07/Yes94.552.403.001.901.25-1.14No104.552.603.001.751.15/Yes114.552.803.001.631.07/Yes125.202.403.002.171.25-2.02No135.202.603.002.001.15/Yes145.202.803.001.861.07/YesFigure 12
Pressure and velocity distribution in the computational model with the optimal pressure port design for the reference working condition: (a)pht=3.12MPa, plt=2.60MPa, and pms=3.00MPa. (b) pht=5.20MPa, plt=2.40MPa, and pms=3.00MPa.
(a)(b)Both the pressures of the high- and low-pressure gases in working condition No. 14 are higher than those in the reference working condition as shown in Table5, which leads to a significant pressure increment in the channels of the stable pressure region. As shown in Figure 13(b), the medium-pressure port in working condition No. 14 has a higher gas pressure than the pressure in channel before shock wave passing through. As a result, there is no RCW generated even the channel completely connects to the medium-pressure port earlier than the shock wave arriving at the channel outlet end. And the equipment performance in this case does not deviate significantly from the optimum value. Therefore, the condition No. 14 with ps>pms has no strict optimal port opening position xmoo, and the working condition Nos. 5, 8, 10, 11, and 13 shown in Table 5 are similar to the condition No. 14 with ps>pms.Figure 13
Pressure distribution in the rotor channels and medium-pressure port in different working conditions: (a)pht=4.15MPa, plt=2.60MPa, and pms=3.00MPa. (b) pht=5.20MPa, plt=2.80MPa, and pms=3.00MPa.Based on the findings of above-mentioned analysis, when the design of the equipment pressure port remains unchanged, the decrease of the expansion ratioα caused by the pressure drop of the high-pressure gas has the most significant impact on the equipment performance. The port position deviation shown in Table 5 can result in a reduction of about 70% in equipment efficiency, when the relative expansion ratio ψexψex=α−1 is decreased by 66.7% (from working condition No. 7 to No. 1). Therefore, the pressures of high- and low-pressure gases need to be tracked and monitored in actual industrial production. When the gas well pressure lowers to a certain level with the increase of mining time, the pressure ports need to be replaced to maintain the high efficiency of equipment operation. In contrast, when ψex is increased by 95% (from working condition No. 12 to No. 7), the port position deviation only causes a drop of about 21% in the equipment efficiency. In addition, the internal wave system is not seriously damaged when the pressure in the stable pressure region is higher than the pressure in the medium-pressure port with the change of gas intake pressure, and the equipment efficiency does not decrease significantly compared with the optimal value under the corresponding pressure ratio.


### 3.2.3. Temperature of the Equipment Inlet Gas

In addition to the pressure of gas, temperature variation also has an impact on the thermodynamic and transfer properties of gas, such as the adiabatic index and sound speed, which affects the shock wave velocity and the optimal design for the pressure ports. However, as shown in Table6, the variation of Δxmo is only 1.27~1.37 mm (less than 28% of the channel width), even when the temperatures of high-pressure gas Thp and low-pressure gas Tlp both change from 10°C to -30°C or 50°C at the same time. Δxmo is relatively smaller as Thp or Tlp changes alone. The maximum Δxmo is only 0.55 mm, about 11.2% of the channel width, when Tlp is changed alone within the study range. Because the actual gas temperature in actual industrial production is usually less than the research scope shown in Table 6 and gas temperature fluctuation is small during the production process, the port design of the industrial GWE device can adapt to the temperature fluctuations in the actual industrial application without seriously affecting the equipment efficiency.Table 6
Comparison of calculation results in the condition ofα=1.60, β=1.15, and plt=2.6MPa.
Thp (°C)Tlp (°C)γ (Hp gas)γ (Lp gas)Δxmo (mm)10101.4691.3980.0010-301.4691.4770.30-30101.6311.3980.8810501.4691.348-0.5550101.3891.398-1.06-30-301.6311.4771.2750501.3891.348-1.37


## 3.2.1. Real Gas Composition

The thermodynamic characteristics of gas medium, such as specific heat ratioγ, are closely related to the composition of gas medium. Therefore, the differences in fluid properties lead to variations of sound speed c and shock wave propagation velocity vs. The pressure port design for air is not suitable for NIST typical natural gas (composition: 88.8% methane, 6.5% nitrogen, 4.7% ethane, and other gases). And it will result in the wall reflection of shock wave shown in Figure 8(b), which destroys the ideal wave system of the air medium shown in Figure 8(a) and forms inoperative equipment.Figure 8
Comparison of pressure contours in different fluid medium: (a) air and (b) typical natural gas.According to the pertinent studies, the fluid medium used in the field test of this research (coalbed methane) is primarily composed of methane with a mass fraction generally ranging from 74.4% to 92.5%. The rest of components are mainly nitrogen with a fraction of 6.9%~24.6% and some other elements like carbon dioxide and heavy hydrocarbons whose total proportion is usually less than 1% [30]. As shown in Figure 9(a), the propagation distance in methane medium is much greater than that of air at the same stage with αin=1.6 (the ratio of pressures between channel and pressure ports before connecting). It indicates that the shock wave propagation velocity is relatively larger in the medium of methane. In addition, the shock wave propagation velocity is unidentical for methane with varying purities, demonstrating that the methane purity difference has impact on the velocity of pressure wave propagation. According to Figure 9(a), the shock wave propagates faster with purer methane (the medium used in the numerical calculation is the mixture of methane and nitrogen). However, the gap between the natural gases with different purities is much smaller than that between natural gas and air. As shown in Figure 9(b), when αin=1.6, the mean shock velocity vs approximately increases linearly with the increment of methane purity φCH4. vs is higher with the larger αin when φCH4 remains unchanged, and the growth rate of vs with φCH4 is also relatively larger.Figure 9
Comparison of shock wave propagation with different methane purities: (a) distance of shock wave propagation within the same time and (b) variation of mean shock propagation velocity with methane purity.
(a)(b)Under various compositions of the gas medium, the optimal opening and closing positions of the pressure ports are different because of the variations in the pressure wave propagation velocity. The best opening position of medium-pressure portxmoo of φCH4=0.7 is later than that of φCH4=1.0 as the shock wave propagates more slowly with lower methane purity. As a result, the difference between xmoo of the two different φCH4 (Δxmo) reaches 1.19 mm with α=1.6 and β =1.15 (shown in Table 3), which is about 24.3% of the channel width. Δxmo of different φCH4 is positive and decreases with increasing φCH4, which indicates that using pure methane as fluid medium in the pressure port design of GWE industrial equipment will cause the medium-pressure port to open earlier. Additionally, a lower φCH4 results in a larger negative divergence. The Δxmo between natural gas and air reaches to 42.4%-66.7% of the channel width. Therefore, utilizing air in the design of the structural parameters of GWE for industrial production of natural gas is inaccurate, though air can be used to study the function mechanism and performance law of GWE. In addition to the opening position of medium-pressure port xmo, the opening position of low-pressure port xlo is another critical port-design parameter that significantly impacts the device performance 36. The low-pressure port should be opened as the reflected expansion wave arrives at the inlet end of channel, as a result of which, the difference of the optimal low-pressure port opening position calculated under different medium compositions, defined as Δxlo, is affected both by the velocities of shock wave and expansion wave. Under the compared working conditions, Δxlo between φCH4=0.7 and 1.0 reaches 3.9 mm (around 79.6% of the channel width), which is larger than Δxmo. The calculation results show that the largest relative reduction ratio of ejection rate is about 23% when the port-design deviations reach to the above-mentioned values of φCH4=0.7~1.0. Therefore, it is necessary to obtain the actual medium composition for the accurate pressure port design of the industrial GWE equipment.Table 3
Relative deviations of medium-pressure port opening position atα=1.60, β=1.15, and plt=2.6MPa.
Mass fraction of CH4Mass fraction of N2Mass fraction of airΔxmo (mm)0.70.30.01.190.80.20.00.860.90.10.00.481.00.00.00.000.00.01.03.27


## 3.2.2. Pressure of the Equipment Inlet Gas

The pressure of inlet gases is another major factor affecting the design of GWE pressure ports. When air is employed as the gas medium, the gas thermodynamic parameters, such as the specific heat ratioγ, vary with increasing gas pressure from 0.101 MPa to 2.60 MPa as shown in Table 4. However, there is a very minor variation in the average propagation velocity of shock wave vs. Therefore, the offset between the optimal opening positions of high- and medium-pressure ports whm remains basically unchanged. As shown in Figure 10, the equipment ejection rates are nearly identical with different pressure values, when the pressure port structure and pressure ratio remain unchanged. It supports the analysis result that the pressure port design is almost not affected by the pressure value in air medium within a certain range. However, both the relative variations of γ and vs are higher with the same pressure change in natural gas medium. As a result, the whm difference of plt=0.101~2.60MPa is significantly larger than that of air medium. The natural gas exploitation usually does not operate under atmospheric or negative pressure conditions in consideration of the pressure characteristics of gas wells and subsequent transportation demands. Therefore, plt=1.50MPa and plt=4.00MPa were chosen for comparative analysis in this study. As shown in Table 4, the results indicate that the maximum relative variation of whm is about 24.5% of the channel width in the typical industrial pressure range of plt=1.50~4.00MPa. When the pressure ratio and port design are left unaltered, the influence of the pressure variation within this range on the equipment performance is similar to that in the component variation as previously mentioned. The change of efficiency caused by the pressure value variation within above range reaches up to about 24% of the optimal efficiency under the calculated pressure ratio.Table 4
Comparison of calculation results under different pressure conditions with the identical pressure ratios ofα=1.60 and β=1.15.
Working conditions (MPa)Gas mediumγ (Hp gas)vs (m/s)whm (mm)plt=0.101NIST air1.403340.317.7plt=2.60NIST air1.477346.117.5plt=0.101NIST natural gas1.309382.416.0plt=1.50NIST natural gas1.391406.215.1plt=2.60NIST natural gas1.468424.914.3plt=4.00NIST natural gas1.592433.813.9Figure 10
Variation of the ejection rateξ with plt under the air medium of α=1.55 and β=1.10.Compared with the conditions with constant compression ratio and changing pressure values, the variation of pressure ratio within a certain pressure range not only changes the gas thermodynamic parameters, such as the specific heat ratioγ, but also impacts on the gas pressure in the stable pressure region (where the channels are closed at both ends as shown in Figure 7) [27, 29]. It alters the incident pressure ratio αin (αin=pht/ps, where ps is the average pressure in the stable pressure region) when the channel is connected to the high-pressure port, which has an impact on the propagation velocities of shock wave and other pressure waves. As a result, the influence of pressure ratio variation has a greater impact on the pressure port design. As shown in Figure 11(a), when pht and pms are fixed, ps rises with the increase of plt, causing αin of GWE to rise monotonically with the increase of plt. But when plt and pms remain unchanged, as depicted in Figure 11(b), ps and αin both rise with the increase of pht. As shown in Figure 11(c), the change of αin causes the variation of the shock wave propagation velocity vs. However, since αin is not the only factor determining vs, the change rate of vs with αin differs between the two variable working conditions. When pht is changed alone, vs rises monotonically with the increase of αin, and its variation rate is relatively higher.Figure 11
Variation laws of pressure in stable pressure regionps and incident pressure ratio αin under different incident gas pressures: (a) variation law of ps and αin with plt at pht=4.15MPa and pms=3.00MPa, (b) variation law of ps and αin with pht at plt=2.60MPa and pms=3.00MPa, and (c) variation of shock wave average propagation velocity vs with αin at pms=3.00MPa.
(a)(b)(c)The variation ofvs mentioned above inevitably results in different optimal opening positions of the medium-pressure port xmoo under various working conditions. As shown in Table 5, whenphtis reduced from 4.15 MPa (the reference working condition No. 7) to 3.12 MPa (No. 1), thexmoobecomes significant later because of the decrease ofvs, and the xmoo difference between the working conditions No. 1 and No. 7 reaches to 4.56 mm, about 93.1% of the channel width (Δxmo>0 when xmoo is later than that of the reference working condition, and Δxmo<0 when xmoo is earlier than the reference condition). As shown in Figure 12(a), when the optimal pressure port structure under the reference working condition No. 7 is adopted in the working condition No. 1, the rotor channel starts to connect with the medium-pressure port before the shock wave arriving at its outlet end. A reversed compression wave (RCW) moving toward the inlet end of channel is generated since the gas pressure at the channel outlet end is lower than the pressure of medium-pressure port at the beginning of connection. When RCW moves to the inlet end of channel, the gas in channels flows back to the low-pressure port and obstructs the inflow of low-pressure gas, which negatively affects the ejection performance of equipment. Under the working condition No. 12, an increase of incident pressure ratio results in an improvement of vs in comparison to the reference working condition. As shown in Figure 12(b), under the working condition No. 12, the channel is not connected with the medium-pressure port designed for the reference working condition when the head of shock wave reaches the outlet end of channel. As a result, the shock wave (which is actually part of the compression waves) reflects at the wall, and the reflected compression wave (RFCW) propagates to the inlet end of channel. As shown in Table 5, the xmoo difference between working condition No. 12 and No. 7 is -2.02 mm, which is about 41.2% of the channel width. However, the reflection of shock wave shown in Figure 12(b) is not very strong, since the existence of gap makes the end of channel not strictly closed and the channel is connected to the medium-pressure port immediately after the shock wave arriving. Therefore, the RFCW strength is not very strong, and the influence of RFCW on the equipment performance is relatively minimal when it reaches the inlet end of channel. It is because that it has a limited obstructive effect on the entrance of low-pressure gas without resulting in the gas backflow.Table 5
Comparison of calculation results under different pressure ratios.
Numberpht (MPa)plt (MPa)pms (MPa)αβΔxmo (mm)ps>pms13.122.603.001.201.154.56No23.122.803.001.111.073.89No33.512.403.001.461.252.40No43.512.603.001.351.151.50No53.512.803.001.251.07/Yes64.152.403.001.731.250.22No7 (ref.)4.152.603.001.601.150.00No84.152.803.001.481.07/Yes94.552.403.001.901.25-1.14No104.552.603.001.751.15/Yes114.552.803.001.631.07/Yes125.202.403.002.171.25-2.02No135.202.603.002.001.15/Yes145.202.803.001.861.07/YesFigure 12
Pressure and velocity distribution in the computational model with the optimal pressure port design for the reference working condition: (a)pht=3.12MPa, plt=2.60MPa, and pms=3.00MPa. (b) pht=5.20MPa, plt=2.40MPa, and pms=3.00MPa.
(a)(b)Both the pressures of the high- and low-pressure gases in working condition No. 14 are higher than those in the reference working condition as shown in Table5, which leads to a significant pressure increment in the channels of the stable pressure region. As shown in Figure 13(b), the medium-pressure port in working condition No. 14 has a higher gas pressure than the pressure in channel before shock wave passing through. As a result, there is no RCW generated even the channel completely connects to the medium-pressure port earlier than the shock wave arriving at the channel outlet end. And the equipment performance in this case does not deviate significantly from the optimum value. Therefore, the condition No. 14 with ps>pms has no strict optimal port opening position xmoo, and the working condition Nos. 5, 8, 10, 11, and 13 shown in Table 5 are similar to the condition No. 14 with ps>pms.Figure 13
Pressure distribution in the rotor channels and medium-pressure port in different working conditions: (a)pht=4.15MPa, plt=2.60MPa, and pms=3.00MPa. (b) pht=5.20MPa, plt=2.80MPa, and pms=3.00MPa.Based on the findings of above-mentioned analysis, when the design of the equipment pressure port remains unchanged, the decrease of the expansion ratioα caused by the pressure drop of the high-pressure gas has the most significant impact on the equipment performance. The port position deviation shown in Table 5 can result in a reduction of about 70% in equipment efficiency, when the relative expansion ratio ψexψex=α−1 is decreased by 66.7% (from working condition No. 7 to No. 1). Therefore, the pressures of high- and low-pressure gases need to be tracked and monitored in actual industrial production. When the gas well pressure lowers to a certain level with the increase of mining time, the pressure ports need to be replaced to maintain the high efficiency of equipment operation. In contrast, when ψex is increased by 95% (from working condition No. 12 to No. 7), the port position deviation only causes a drop of about 21% in the equipment efficiency. In addition, the internal wave system is not seriously damaged when the pressure in the stable pressure region is higher than the pressure in the medium-pressure port with the change of gas intake pressure, and the equipment efficiency does not decrease significantly compared with the optimal value under the corresponding pressure ratio.


## 3.2.3. Temperature of the Equipment Inlet Gas

In addition to the pressure of gas, temperature variation also has an impact on the thermodynamic and transfer properties of gas, such as the adiabatic index and sound speed, which affects the shock wave velocity and the optimal design for the pressure ports. However, as shown in Table6, the variation of Δxmo is only 1.27~1.37 mm (less than 28% of the channel width), even when the temperatures of high-pressure gas Thp and low-pressure gas Tlp both change from 10°C to -30°C or 50°C at the same time. Δxmo is relatively smaller as Thp or Tlp changes alone. The maximum Δxmo is only 0.55 mm, about 11.2% of the channel width, when Tlp is changed alone within the study range. Because the actual gas temperature in actual industrial production is usually less than the research scope shown in Table 6 and gas temperature fluctuation is small during the production process, the port design of the industrial GWE device can adapt to the temperature fluctuations in the actual industrial application without seriously affecting the equipment efficiency.Table 6
Comparison of calculation results in the condition ofα=1.60, β=1.15, and plt=2.6MPa.
Thp (°C)Tlp (°C)γ (Hp gas)γ (Lp gas)Δxmo (mm)10101.4691.3980.0010-301.4691.4770.30-30101.6311.3980.8810501.4691.348-0.5550101.3891.398-1.06-30-301.6311.4771.2750501.3891.348-1.37


## 4. Experimental Results and Analysis

The first industrial application test of GWE equipment was carried out in coalbed methane well site (as shown in Figure14), in order to obtain the practical application effect of GWE in natural gas industrial exploitation and to verify the accuracy of pressure port design as well as the influence law of port design on equipment performance. The reference conditions for the pressure port design of testing equipment are shown in Table 7. The equipment was not equipped with total pressure measuring meters inside to ensure production safety. Therefore, the gas flow rate and ejection rate, which are the most important data in the actual production of the gas field, were chosen as the equipment performance evaluation index.Figure 14
Coalbed methane well site for GWE test.Table 7
Reference parameters for pressure port design of GWE testing equipment.
Design parametersValueDesign parametersValuepht3.3 MPapms2.1 MPaThp10°CTmp10°Cplt1.90 MPaf49 HzTlp10°CφCH488%The variations of gas flow rates and ejection rate with the pressure of high-pressure gas under the designed rotational frequency are shown in Figure15. GWE can successfully suck and pressurize low-pressure gas in the condition of pms=2.1MPa and pht=2.4MPa, demonstrating the application advantage of GWE at a small expansion ratio. It also proves the great capacity of GWE to adapt to the changing working conditions without modifying the device structure. The dimensions of pressure ports and rotor channel remained unchanged throughout the testing process, as a result of which, the high-pressure gas flow rate mhp continuously rises with the increase of pht as shown in Figure 15. The energy input rises with the increase of mhp and pht, which enhances the ejection capacity of GWE. Therefore, both the low-pressure gas flow rate mlp and ejection rate ξ are also increased. mhp and ξ can reach about 2.52×104Nm3/d and 35%, respectively, with pht=3.26MPa (close to the reference value for pressure port design). The equipment can realize the production recovery of low-pressure well for about 8.83×103Nm3/d under this condition, and plt is also reduced to 1.90~ 1.95 MPa (plt fluctuates in the process of low-pressure well production, and the average value is about 1.93 MPa).Figure 15
The variation of gas well production and ejection rate with high pressure value at the design rotational frequency of 49 Hz.The difference inmhp between experiment and numerical simulation results is only about 6% of the experimental value at pht=3.26MPa, which proves the accuracy of the design for the gas flow rate of the testing equipment. However, the ejection rate of the equipment obtained in the field test only reaches about 65% of the numerical simulation, which is significantly less than the average value of 91% in the air medium as shown in Figure 10. This is primarily because numerous fluid impurities contained in the field test are not considered in the numerical simulation. The amount of impurities (including water and coal ash) entrained in the gas from the field test wells is constantly changing as shown in Figure 16. As a result, the impurity level significantly varied from that for the system design, and the cleanliness of the equipment gas intake is difficult to be guaranteed during the field testing procedure. Although the solid and liquid impurities are difficult to block the rotor channels which are open at both ends [16, 20], the movement of impurities in the narrow channel requires a certain amount of gas energy. It is more difficult for the high-pressure gas to suck the low-pressure gas containing impurities (which is “heavier” than the clean gas and requires more driving force for flowing process), so that the ejection rate value is significantly lower than the numerical simulation results. In addition, the accumulation of solid and liquid impurities in the gas cavity and other equipment components increases the resistance to gas flow inside the equipment, which also increases energy loss and affects the testing performance of equipment. There were lots of impurities in the gas produced from the testing gas wells, and both gas pressures and impurity content kept fluctuating throughout the entire experiment process. However, the GWE testing equipment maintained a long-term effective and stable operation, which proves its superior impurity carrying capacity and anti-fluctuation ability.Figure 16
Impurity discharge in the test system.As shown in Figure17(a), the high-pressure gas flow rate mhp initially declines and then rises with the increase of the rotational frequency f when the pressure of high-pressure gas pht is closed to that for the port design as shown in Table 7. The low-pressure gas flow rate mlp increases first and then decreases with the increase of f, peaking at f=49Hz, which is the designed value. The equipment ejection rate ξ is maximum at f=49Hz as shown in Figure 17(b), which proves the accuracy of the pressure port design for the actual gas medium in this study (the average methane purity of the gas from the high-pressure well is around 85%). When the equipment rotation frequency deviates from the design value, the pressure waves, such as shock wave, may not reach the channel ends precisely at the moments of opening or closure of the pressure ports. It generates the redundant pressure waves, including the RCW and RFCW mentioned above, which destroys the ideal wave system, obstructs the inflow of low-pressure gas, and lowers the equipment performance. The experimental results demonstrate that ξ decreases by 36.1% compared with the optimal value when the actual frequency of equipment deviates from the design reference value by 22.4%.Figure 17
Variation laws ofmhp, mlp, and ξ with the rotational frequency f: (a) mhp and mlp at pht=3.26~3.30MPa, (b) ξ at pht=3.26~3.30MPa, (c) mhp and mlp at pht=3.00~3.03MPa, (d) ξ at pht=3.00~3.03MPa, (e) mhp and mlp at pht=2.69~2.72MPa, and (f) ξ at pht=2.69~2.72MPa.
(a)(b)(c)(d)(e)(f)Whenpht drops to around 3.00 MPa as shown in Figure 17(c), the variation laws of mhp and mlp with f are essentially the same as those at pht=3.26~3.30MPa, and ξ reaches its maximum when f=45Hz. According to the numerical analysis, the propagation velocity of shock wave slows down compared with the working condition for design when the expansion ratio α decreases with declining pht. And the channel entirely connects to the medium-pressure port before the shock wave reaches the outlet end of channel when the equipment runs at the design frequency. In contrast, when f is lower than the design value to a certain extent, the medium-pressure port can be completely opened just at the moment of shock wave arriving, which avoids or mitigates the formation of RCW. As a result, the equipment can achieve a higher ξ at the rotational frequency slightly lower than the design value.Whenpht further decreases to around 2.7 MPa, as shown in Figure 17(e), mhp and mlp monotonically change with the increase of f. As shown in Figure 17(f), the actual frequency for getting the highest ξ under the condition of pht=2.7MPa is also lower than the designed value for the same reason as pht=3.0MPa.Under the design frequencyf=49Hz, the ejection rate ξ at pht=3.00MPa and pht=2.70MPa is about 30.9% and 22.2%, respectively. In comparison, ξ reaches 32.2% with pht=3.00MPa and f=45Hz, and it is 24.8% with pht=2.70MPa and f=38Hz. The experimental results show that the equipment performance can be effectively improved by adjusting the rotational frequency when the equipment structure remains unchanged with the varying gas pressure/pressure ratio. The increment of ξ (relative to the design frequency) obtained by the frequency adjustment can rise up to 12% of that at the design frequency within the field testing range in this study.


## 5. Conclusion

Gas wave ejector is an effective pressure energy recovery technology that enables fluid energy conversion and has numerous potential applications in the sustainable development of energy resources. The first practical application test of GWE at the coalbed methane well site was accomplished in this study, which also examined in detail the influence mechanism and law of the real fluid medium characteristics on the wave system and pressure port design. The key conclusions are as following:(1)
The real gas composition has a certain influence on the propagation velocity of the pressure waves. The shock wave propagation velocity increases with increasing methane purity (φCH4) of intake gases. The difference of medium-pressure port location (Δxmo) and the difference of low-pressure port location (Δxlo) between φCH4=0.7 and φCH4=1.0 can reach to 24.3% and 79.6% of the channel width, respectively. The efficiency reduction caused by these port position deviations can reach about 23% of the optimal value(2)
The maximumΔxmo is about 24.5% of the channel width when the pressure ratio remains constant with the pressure of low-pressure gas plt varying from 1.50 to 4.00 MPa, and the change in equipment efficiency caused by this deviation is no more than 24% of the optimal value. However, when plt remains constant with changing pressure ratios, the decrease of expansion ratio results in the formation of reversed compression wave (RCW) in channel, which has a relatively negative impact on equipment performance. The port position deviation can lead to a 70% drop in device efficiency when ψex falls by 66.7%; however, the efficiency decline only accounts for roughly 21% of the optimal value as ψex rising by 95%(3)
The maximumΔxmo induced by the change in gas intake temperature within the region of -30°C to 50°C is only approximately 11.2% of the channel width. As a result, in the actual application of GWE, the equipment performance is not obviously affected by the intake temperature variation within a specific range(4)
In the GWE practical application test at the coalbed methane well site, the GWE equipment successfully resumes the production of discontinued low-pressure well by about8.83×103Nm3/d, which demonstrates the considerable suction and pressurization ability of this technology. The deviation of high-pressure gas flow rate between the experiment and numerical simulation results is only about 6% of the experimental value, which proves the precision of the design for the equipment handling capacity. In addition, the stable operation of the equipment under the condition of containing water and coal ash in the intake gas also proves the strong ability to carry impurities of this equipment(5)
In the variable rotation frequency test, the ejection rate of the equipment is the highest at the design frequency employed for the pressure port design, which proves the accuracy of the port design for real gas in this study. When the working pressure ratio changes, the appropriate rotation frequency adjustment can make the opening and closing of the medium-pressure port match better with the pressure wave propagation. It reduces the influence of negative pressure waves like RCW and enhances the application performance of the equipment. Within the test range, the relative increase of the equipment ejection rate obtained by the frequency adjustment can reach up to 12% of that in the design frequency



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*Source: 7520456-2023-02-11.xml*