Study on the Performance of Collaborative Production Mode for Gas Wave Ejector

Abstract

Gas wave ejector (GWE) is an efficient ejection equipment using pressure waves to extract and transfer energy. However, at present, GWE is designed only for single condition, not fully utilizing the production capacity. The collaborative production mode using one equipment to work simultaneously under two different conditions was proposed to resolve this issue in this study, and was analyzed by combining numerical simulation and experimental test. The research results show that the collaborative mode almost has no effects on average total efficiency compared to single mode. In the range of tests, the efficiency difference between two modes is within 4.4%. The state parameters of the stable-pressure region (where channels are closed at both ends) on one condition are the initial parameters of the functional region on the other condition in collaborative mode, accounting for the difference between single and collaborative mode. The variations of performance parameters (ejection rate and the isentropic efficiency) with the medium-port pressure in collaborative mode was similar to that of the single mode. Thus, the performance parameters difference between two modes can be predicted by the relative relationship between the medium-port pressure and the average pressure in stable-pressure region of GWE in single mode. In conclusion, the collaborative mode improves the utilization of equipment while maintaining total efficiency, which can promote the popularization and application of GWE.

1. Introduction

The employment of pressure energy transfer devices for comprehensive pressure energy recycling and extraction can significantly reduce the waste of energy and resource in industrial production [1,2]. In the field of natural gas exploitation and utilization, for example, the surplus pressure energy of high pressure gas wells over the gathering pressure can be used to supercharge low-pressure gas wells. It can avoid the waste of energy caused by throttling and depressurizing high-pressure gas, while also reducing the external power consumption required to pressurize and extract low-pressure gas [3,4]. In industry, the expansion-compressor units and static ejectors are commonly used for energy transfer. As shown in Figure 1a, the expansion-compressor units whose core components is impeller are made up of coaxially coupled expansion and compression. It is highly efficient and has a wide range of working conditions. However, the high rotational speed of impeller results in high costs of manufacturing and operating. It also has strict purity criteria of the fluid, limiting its usage in industrial production like natural gas extraction [5]. In contrast, as shown in Figure 1b, the static ejector whose core component is nozzle has the advantages of simple structure, low costs, high stability, and the ability to operate with impurities. However, the isentropic efficiency of static ejector is rather low due to the substantial viscous loss caused by turbulent mixing [6,7,8,9].

Due to the issues of both devices mentioned above, the gas wave ejector (GWE) [10] which combines the benefits of expansion-compressor units and static ejector has received more and more attention. Similar to other wave rotor technologies [11,12,13,14,15], the core components of GWE, as shown in Figure 2, are a wave rotor made up of uniformly distributed channels and the end plates with pressure ports on both sides of the rotor. And the wave rotor rotates between the intake and exhaust end plates when the device is running. As different pressure gases enter through the ports, shock and expansion waves are created to transfer energy between gases with various pressures in the channels. GWE mainly transfer energy through pressure waves rather than mechanical components like impellers. Therefore, GWE has advantages of high isentropic efficiency, low rotational speed, low manufacturing and operating costs, easy disassembly and maintenance, and a strong capacity to run with liquid and solid. As a result, it has been considered for application in a variety of fields, such as the exploitation of natural gas, refrigeration cycles, and so on [16,17,18,19].

In 1960s, Spalding [20] first proposed the concept of equalizer (the initial name of GWE) and conducted preliminary theoretical researches. The first experiment platform for GWE was built by Kentfield [21,22] at Imperial College London. The testing results of Kentfield proved that GWE could achieve high isentropic efficiency equivalent to turbo-machinery under certain operating conditions while operating at a much lower rotation speed. In 1990s, NASA built a novel three-port gas wave experiment platform [23], which was used to investigate the effects of fluid viscosity, gaps between channels and ports, and vortices on device performance. It was suggested that lowering leakage and flow loss was the key to improve the performance of wave rotor equipment. In 2000s, Kharazi et al. [24,25] at the University of Michigan expanded the application of GWE technology to refrigeration cycle, and they found that using GWE for utilizing the energy of high-pressure liquid to eject low-pressure vapor could improve the efficiency of the R718 refrigeration cycle. Then, in 2007, Okamoto et al. [26] from the University of Tokyo conducted a pressure monitoring in the wave rotor channels and acquired the actual pressure fluctuation characteristics, furthering the mechanistic research of wave rotor. Since 2010, Hu. et al. [10,27] from Dalian University of Technology have been conducting further theoretical and experimental research on GWE technology. The group of Hu. not only proposed several methods to improve the performance of GWE, but also obtained the laws of the equipment performance with changing of rotational speed and working condition, which provided theoretical support and data for the design of GWE.

The aforementioned researches discovered essential operating principles of GWE, such as the ideal functional wave system, and demonstrated that GWE operated efficiently at low pressure ratios. In addition, a complete design and analysis method for GWE has been developed and verified. However, all current GWE applications and studies have been conducted in a single production condition. In order to meet the requirements of equipment structure and rotation line speed, as shown in Figure 2, the total number of channels generally far exceeds the minimum number of channels required for functional region. The majority of the channels are closed at both ends (stable-pressure region) and are not utilized efficiently during operation, limiting the production capacity of wave rotors. The use of a single rotor in collaborative production mode (a mode of simultaneous production on two different working conditions) allows the equipment to adapt to multiple industrial production conditions in the same area. Thus, the utilization rate of GWE can be improved while equipment costs can be reduced. However, no research on the analysis of GWE’s collaborative production mode has been conducted. By combining experiments and numerical simulations, this paper aims to analyze the mechanism of collaborative production mode and investigate its differences in equipment performance compared to the traditional single production mode.

2. Methodology

2.1. Performance Parameters

Based on the working mechanism and application background of GWE, its working conditions can be mainly expressed by the expansion ratio α and compression ratio β, which are calculated as Equations (1) and (2).

$$\alpha =\frac{p_{ht}}{p_{lt}}$$

(1)

$$\beta =\frac{p_{mt}}{p_{lt}}$$

(2)

where p_ht, p_mt and p_lt denote the total gas pressure in high, middle and low-pressure ports, respectively.

In general, the main parameters to evaluate the performance of ejection device are the ejection rate ξ and the isentropic efficiency η. The ejection rate ξ characterizes the actual ability of the equipment to eject low-pressure gas, which can be calculated by Equation (3).

$$\xi =\frac{m_{lp}}{m_{hp}}$$

(3)

where m_hp and m_lp represent the high and low-pressure gas mass flow rate, respectively [6]. The isentropic efficiency η is used to indicate the efficiency of energy transfer during device operation. The energy transfer between high and low-pressure gases in GWE can be divided into two processes, including the expansion work output of high-pressure gas and the process of supercharging low-pressure gas.

In detail, the isentropic expansion output work W_eps and the actual output work W_ep of high-pressure gas are expressed as Equations (4) and (5), respectively.

$$W_{eps}=c_p{m}_{hp}{T}_{ht}\left(1-{\left(\frac{p_{mt}}{p_{ht}}\right)}^{\frac{\gamma -1}{\gamma }}\right)$$

(4)

$$W_{ep}=W_{eps}{\eta }_{ep}$$

(5)

where c_p, γ, T_ht denote the specific heat, specific heat ratio, total temperature of the high-pressure gas, respectively. And η_ep is the isentropic efficiency of the high-pressure gas expansion process.

Similarly, the isentropic and actual work consumption for supercharging the low-pressure gas to middle-pressure gas are defined as W_coms and W_com. There expressions are as Equations (6) and (7).

$$W_{coms}=c_p{m}_{lp}{T}_{lt}\left({\left(\frac{p_{mt}}{p_{lt}}\right)}^{\frac{\gamma -1}{\gamma }}-1\right)$$

(6)

$$W_{com}=\frac{W_{coms}}{{\eta }_{com}}$$

(7)

where T_lt denote the total temperature of the low-pressure gas, and η_com is the isentropic efficiency of the low-pressure gas compression process.

The isentropic efficiency of GWE is the multiplication of the isentropic compression efficiency η_com and isentropic expansion efficiency η_ep. Since the actual output expansion work W_ep of the high-pressure gas is equal to the actual expansion work W_com consumed by the low-pressure gas. Thus, the isentropic efficiency of the equipment η can be expressed as Equation (8).

$$\eta ={\eta }_{ep}{\eta }_{com}=\frac{W_{coms}}{W_{eps}}=\frac{m_{lp}{T}_{lt}\left({\left(\frac{p_{mt}}{p_{lt}}\right)}^{\frac{\gamma -1}{\gamma }}-1\right)}{m_{hp}{T}_{ht}\left(1-{\left(\frac{p_{mt}}{p_{ht}}\right)}^{\frac{\gamma -1}{\gamma }}\right)}$$

(8)

2.2. Numerical Simulation Methods

The numerical simulation can acquire the distribution laws of internal state parameters in rotor channels, such as pressure, which cannot be measured experimentally, contributing to the study of the mechanism of GWE technology. In this paper, we used the CFD software FLUENT to conduct the numerical simulations. Besides, the three-dimensional model [28] was used to replace the typical two-dimensional and quasi-two-dimensional models [3,27] in order to obtain more precise numerical results.

2.2.1. Numerical Model Building

Figure 3 depicts the numerical model. The model consisted of high, middle and low-pressure fixed ports (one group in single operating mode, and two groups in collaborative mode), as well as several moving wave rotor channels. In addition, two gap calculation fields were added between the ports and the channels to improve calculation accuracy.

The end faces of each pressure port away from the channel was set as high-pressure inlet, low-pressure inlet, and middle-pressure outlet, respectively. In addition, the pressure and temperature values for these pressure boundaries were set according to the experimental conditions. Since the entire wave rotor was calculated with a large number of flow paths operating in the stable-pressure region, which led to a waste of computational resources. Therefore, only a portion of the flow channels were included in the numerical model. And the channels were reciprocated between the pressure ports by periodic boundary conditions. As the transmission speed of pressure waves is much larger than the rotation speed of channels, thus the pressure exchange between the different pressure gases can be regarded as instantaneous. In addition, the temperature difference of the gas in channel was relatively small. For the reasons stated above, the walls of simulation model were set as adiabatic boundary conditions. The specific calculation settings are shown in

Table 1. Solver settings of numerical simulation in Fluent.

Parameter	Setting
Solver type	Density based
Time	Transient
Fluid	Air
Density	Real gas
Energy model	Activated
Species transport model	Activated
Turbulence model	SST-SAS
Convective flux type	AUSM
Computing fradients	Least squares cell based
Spatial discretization
Flow parameters	Second-order upwind
Turbulent kinetic energy	Second-order upwind
Specific dissipation rate	Second-order upwind
Transient formulation	Second-order implicit
Time step	Minimum = 1 × 10−7 s
Grid size	Minimum = 3 × 10−5 m
Convergence criteria	1 × 10−4

below.

2.2.2. Governing Equations

The air with pressures within 0.3 MPa and temperatures around 298 K was used as the fluid middle in this paper, which can be assumed to be ideal gases. As a result, the gas satisfied the Equation (9) as

$$p=\rho RT$$

(9)

where ρ is density of gas, and R denotes specific gas constant.

The gas conducts a high-speed compressible viscous flow in rotor channels when the GWE is running. The flow processes satisfy the basic conservation laws of fluid mechanics such as conservation of mass, momentum and energy. The governing equations under the coordinate system are as follows [29,30].

Mass conservation Equation (10).

$$\frac{\partial \rho }{\partial t}+\Delta \cdot (\rho \overrightarrow{u})=0$$

(10)

where t denotes the time and $\overrightarrow{u}$ is the vector of velocity.

Momentum conservation Equation (11).

$$\frac{\partial \left(\rho \overrightarrow{u}\right)}{\partial t}+\nabla \cdot (\rho \overrightarrow{u})=-\nabla p+\nabla \cdot (\tau )+\overrightarrow{F}$$

(11)

where $\overrightarrow{F}$ represents the external volume forces and τ is the viscous force tensor.

Energy conservation Equation (12).

$$\frac{\partial }{\partial t}\left(\rho E\right)+\nabla ·\left(\overrightarrow{V}\left(\rho E+p\right)\right)=-V·\left(k_{eff}\nabla T-{\displaystyle \sum _{j}hj\overrightarrow{J}j}+\left({\tau }_{eff}\cdot \overrightarrow{V}\right)\right)$$

(12)

where E and h are the total energy and enthalpy of the gas, respectively, while $\overrightarrow{J}$ represents the diffusion flux. And k_eff, τ_eff respectively denote the effective thermal conductivity and the effective viscous force tensor.

As the gas flow inside GWE is complex and turbulent, the selection of a suitable turbulence model according to the flow characteristics is of great value to ensure the accuracy of the calculation. In this paper, a new scale-adaptive turbulence model SST-SAS [31] was selected by considering the calculation accuracy and efficiency. This model is a new development form of the traditional SST-k-ω model. The governing equations of turbulent kinetic energy k and specific dissipation rate ω can be formulated as Equations (13) and (14).

$$\frac{\partial \rho k}{\partial t}+\frac{\partial }{\partial x_i}\left(\rho u_{i}k\right)=G_k-\rho {\beta }_{k}k\omega +\frac{\partial }{\partial x_j}\left[\left(\mu +\frac{{\mu }_t}{{\sigma }_k}\right)\frac{\partial k}{\partial x_j}\right]$$

(13)

$$\frac{\partial \rho \omega }{\partial t}+\frac{\partial }{\partial x_i}\left(\rho u_i\omega \right)={\alpha }_k\frac{\omega }{k}{G}_k-\rho {\beta }_{\omega }{\omega }^2+Q_{SAS}+\frac{\partial }{\partial x_j}\left[\left(\mu +\frac{{\mu }_t}{{\sigma }_{\omega }}\right)\frac{\partial \omega }{\partial x_j}\right]+\left(1-F_1\right)\frac{2\rho }{{\sigma }_{\omega 2}}\frac{1}{\omega }\frac{\partial k}{\partial x_j}\frac{\partial \omega }{\partial x_j}$$

(14)

where u_i denotes the x_i-directional velocity, and μ_t is the vortex viscosity coefficient. G_k represents the generating terms of a partial differential equation. α_k, β_k, β_ω, σ_k, σ_ω and F₁ are all constants or specific model parameters [32,33]. What is more, a scale-adaptive source term Q_SAS is added into the SST-SAS model, which is the main difference with the traditional SST-RANS model. The expression of Q_SAS is Equation (15).

$$Q_{SAS}=\max \left[\rho {\eta }_2\kappa S^2\left(\frac{L}{L_{v\kappa }}\right)-C\frac{2\rho k}{{\sigma }_{\Phi }}\max \left(\frac{1}{{\omega }^2}\frac{\partial \omega }{\partial x_j}\frac{\partial \omega }{\partial x_j},\frac{1}{k^2}\frac{\partial k}{\partial x_j}\frac{\partial k}{\partial x_j}\right),0\right]$$

(15)

where η₂, σ_Φ, κ and C are constants of model parameters, while S is a scalarized invariant of the strain rate tensor S_ij. L and L_νκ respectively denote the turbulence scale and the von Karmen scale.

The source term Q_SAS is almost equal to 0 in stable flow, which makes this equation results similar to that of RANS model. While in an unstable flow, Q_SAS is constantly larger than 0, which makes this model capture more turbulent characteristics and obtain results similar to the results of LES model.

Zacharzewski et al. [34] proved that SST-SAS model has excellent computational capability for the simulation of strong cyclonic and interacting flows by the comparison of simulation and experiment. And the flows mentioned above are just the main unsteady flow phenomena in the wave rotor device. In addition, Kurec et al. [35] used Particle Image Velocimetry (PIV) to experimentally validate the calculation results of the internal flow in rotor channels by SST-SAS turbulence model. Their research results revealed that the distributions of velocity and pressure, the shape and movement laws of the contact surface between different pressure gases, and the propagation speed of pressure waves in channels were basically identical in experiments and simulations with SST-SAS model. It proves that this model can not only calculate the distribution of state parameters in GWE channels accurately, but also can precisely reflect the contact and mixing between different pressure gases. Thus it is perfectly suitable for the theoretical study and analysis of GWE.

2.2.3. Grid Independence Verification

Due to the regular shape of the geometric model in this paper, a hexahedral mesh was used to partition the model. Because the size of gap is small and the pressure parameters during the connection between the channels and ports change rapidly, the mesh near the interfaces of different flow domains are refined. In order to accurately simulate the unsteady flow, the boundary layer mesh was added to each wall in this model. The grid independence verification was performed with varying grid quantities models in order to save as many computational resources as possible while maintaining computation accuracy. In detail, the specific settings of the model for verification are shown in

Table 2. Initial condition settings of model used for grid independence verification.

	High Pressure Inlet	Channels
Pressure (kPa)	151.5	101
Temperature (K)	283	293
Mass fraction of HP air	1	0
Mass fraction of LP air	0	1
Rotation linear velocity (m/s)	0	36.96

.

Figure 4 shows the pressure distribution and mass fraction curves at 0.6 ms after the channel starting to connect with the high-pressure port. When the number of grids in a single channel reaches 4.2 × 10⁴, the numerical model can accurately capture the low-pressure area caused by vortices at the entrance of channels (around 0.015 m away from the channel inlet end) and the small pressure fluctuations that occurred after the action of multiple pressure waves, as illustrated in Figure 4a. The distortion of contact surface during propagation can be accurately captured at this grid scale, resulting in a significantly different mass fraction distribution in Figure 4b compared to the models with less grids in a single channel. It can also be found that the calculation results of each parameter basically no longer change when the number of single channel grids exceed 4.2 × 10⁴. Thus further mesh refinement has no significant impact on the calculation results. Therefore, the grid number of single channel was chosen as 4.2 × 10⁴, and the total grid number of the model is around 1.22 × 10⁶.

2.2.4. Model Validation

The accuracy of simulation model was verified according to the test results from the wave rotor internal pressure by Okamoto et al. [26] at the University of Tokyo. Figure 5a shows the diagram of their testing device which consists of three stationary channels and two rotating pressure nozzles at both ends of channels. Considering the mutual influence between adjacent channels, the point P09 in the middle channel II was chosen for the pressure monitoring and comparing. As shown in Figure 5b, there are minor differences between the simulation and the experimental results with different sizes of gaps that ensure the relative movement between the channels and pressure ports. Furthermore, the shape and the peak position of the pressure curves are essentially identical. The above results proves that the simulation model in this study can accurately predict the formation and propagation laws of pressure waves, and it is applicable to calculate the flow inside the wave rotor channels.

2.3. Experimental Setup

The experimental system shown in Figure 6 was established to obtain the actual working performance of the GWE collaborative production mode. The GWE experimental equipment was driven by a motor whose speed was adjusted by a frequency converter. The high-pressure inlet gas was provided by a compressor, and the gas collector and buffer tank were used to ensure the stability of the air supply during the experiment. The valves were used to control the pressure of the high, low and middle-pressure ports. In addition, sensors and micromanometers were used to measure the temperature, pressure at each pressure port. Besides, the mass flow rate at each port was measured by the vortex flowmeter.

Table 3. The parameters of the control and measuring instruments.

Instrumentation	Brand	Measuring/Controlling Range	Nominal Accuracy
Frequency convertor	KEWO	0~7.5 kW	±0.01 Hz
Temperature sensor	Sino Measure	−50~200 °C	±0.3 + 0.005 T
Pressure sensor	Asmik	0~0.5 MPa	±0.08%
Vortex flowmeter	Gallop	10~100 m3/h	±1.5%
Screw compressor	Fusheng	0~1.0 MPa	±0.5%
Micromanometer	YIOU	0~0.1 MPa	±0.1%

displays the specific parameters of the control and measurement instruments.

Figure 7a shows an exploded schematic diagram of GWE used in experiment. The base of the equipment was equipped with four gas inlets, which can meet the inlet requirements for single and collaborative modes. The middle-pressure gas generated in the energy transfer process was discharged from the channels and entered the middle-pressure chamber through the nozzle. In collaborative production mode, the two groups of gas products converged in the equipment chamber and were discharged by the MP outlet of device together, so both working conditions had the identical static pressure of the middle-pressure gas. The total pressure of the two streams of middle-pressure gas was measured respectively at the corresponding nozzles, facilitating the comparative analysis of the total pressure efficiency of each working condition in the single and collaborative production modes. In addition, the equipment was equipped with angle adjusting panel to achieve accurate external adjustment of the equipment port declination, which was conducive to the installation and control of the equipment. The wave rotor used in the experiments and its structural dimensions are given in Figure 7b and

Table 4. Structural parameters of GWE.

Parameter	Value	Parameter	Value
Rotor mean diameter dm	256 mm	Channel length l	250 mm
Rotor channel internal diameter d	240 mm	Channel hight h	16 mm
Rotor channel outer diameter D	272 mm	Channel mean width wip	9.7 mm
Wall thickness of channel gap s	1.5 mm	Number of channel am	72

.

3. Results and Discussion

Experimental Results

The performance of the two modes was compared in experiments under each operating condition listed in

Table 5. Working conditions in experiment.

Working Conditions Group Code	Working Conditions Code	pht (kPa)	plt (kPa)
A	A1	135	101
	A2	135	105
B	B1	152	101
	B2	152	110
C	C1	177	110
	C2	177	120

, and the results are shown in Figure 8.

As shown in Figure 8a,c,e, the curves of η in different modes intersect on the same working conditions, and the static pressure of middle-pressure port p_ms corresponding to the intersection points in a group of working conditions are nearly identical (for example, p_ms of intersection point of both modes on condition A₁ is nearly equal to that on condition A₂). It indicates that when η of one working condition in collaborative mode is higher than that of the single mode, η of the collaborative mode on the other working condition in the same group became relatively worse. The variation laws of η with p_ms in both modes are exactly consistent at each condition, and the p_ms for achieving the maximum η are also basically identical. On the fixed gas intake conditions, though the collaborative mode results in a change in value of η when compared to the single mode, it may not change the variations of η with p_ms on each working condition.

Similar to the variations of η, as shown in Figure 8b,d,f the variations of ξ with the p_ms also remain consistent in both modes, and ξ of both modes decreases with increasing p_ms on the fixed gas intake conditions. Moreover, when ξ of one condition of the collaborative mode is higher than that of the single mode, the ξ of the other condition becomes relatively worse. The values of p_ms corresponding to the intersection points of ξ curves in different modes are approximately identical on two different p_lt conditions in a group, such as A₁ and A₂.

In conclusion, the device performance of one working condition in collaborative mode is improved compared with the single mode, the other condition will be relatively decreased. Therefore, we propose the GWE’s average total isentropic efficiency η_Total to evaluate the combined energy transfer efficiency in both modes. Because the total compression work actually consumed for compressing the two different low-pressure gases is the same as the total expansion work actually output by high-pressure gas, the expression of η_Total can be derived as Equation (16) according to the Equations (4)–(8).

$${\eta }_{Total}=\frac{m_{l1}{T}_{lt1}\left({\left(\frac{p_{mt1}}{p_{lt1}}\right)}^{\frac{\gamma -1}{\gamma }}-1\right)+m_{l2}{T}_{lt2}\left({\left(\frac{p_{mt2}}{p_{lt2}}\right)}^{\frac{\gamma -1}{\gamma }}-1\right)}{m_{h1}{T}_{ht}\left(1-{\left(\frac{p_{mt1}}{p_{ht}}\right)}^{\frac{\gamma -1}{\gamma }}\right)+m_{h2}{T}_{ht}\left(1-{\left(\frac{p_{mt1}}{p_{ht}}\right)}^{\frac{\gamma -1}{\gamma }}\right)}$$

(16)

The total pressure efficiency difference Δη_Total between the two modes can be calculated as Equation (17).

$$\Delta {\eta }_{Total}={\eta }_{Total-c}-{\eta }_{Total-s}$$

(17)

where η_Total−c and η_Total−s represent the average total efficiency in collaborative and single modes, respectively.

Figure 9 depicts the experimental results of η_Total on three groups of conditions A, B, and C listed in Table 5. Interestingly, the η_Total of both modes show a trend of increasing and then decreasing with the increase of p_ms, and the p_ms corresponding to the maximum η_Total are identical. Furthermore, Δη_Total of both modes increases and then decreases as p_ms increases. The highest η_Total obtained for both modes on different conditions in the experimental range could exceed 42.1% and Δη_Total between the two modes is less than 4.64% at each p_ms. The above experimental results show that employing one device for collaborative mode instead of two devices for single mode devices simplifies the system and significantly reduces manufacturing costs while also having less effect on the total energy transfer efficiency. Moreover, it can even improve the equipment η_Total in some conditions, demonstrating the feasibility and superiority of the collaborative mode.

Applying the numerical model described in the previous section to analyze the mechanism and performance variations of the GWE collaborative mode shall give further insights into the gas dynamics within the rotor channels that the experiments cannot provide. Before this is done, it is necessary to establish how well the model results in terms of performance parameters correlate with the experimental data.

Figure 10 shows the variations of η in different modes obtained by numerical simulations and experiments on condition of p_ht = 152 kPa and p_lt = 101 kPa. Although the numerical η obtained in both modes are slightly higher within 4.4% than the actual experimental results, the variations of η with p_ms remains nearly consistent in both modes, proving the rationality of the calculation in this study.

The pressure contours for single mode and collaborative mode on conditions 1 and 2 in

Table 6. Working conditions employed in the numerical comparative analysis.

Working Conditions Code	pht (kPa)	plt (kPa)	pms (kPa)
1	177	120	125
2	177	110	125
3	177	120	140
4	177	110	140
5	152	120	125
6	152	110	125

are shown in Figure 11. In the single mode, the average pressure in the stable-pressure region of condition 1 (defined as p_s1) is higher than that of condition 2 (defined as p_s2). The parameters in the channels of the stable-pressure region on the condition 1 are the initial parameters before entering the functional region of condition 1, and such a parameters relationship is also satisfied on the condition 2. Similar to the single mode, p_s1 is still higher than p_s2 in the collaborative mode. Nonetheless, in the collaborative mode, the gas state parameters in channels of the stable-pressure region on condition 1 become the initial parameters of the functional region on condition 2, while the parameters of the stable-pressure region on the condition 2 become the initial parameters of the functional region on the condition 1. The p_s1 in collaborative mode is significantly higher than the p_s2 in single mode, which accounts for a higher average static pressure in the functional region of the condition 2 in collaborative mode. As a result, in the collaborative mode, the gas velocity in the functional region of condition 2 is reduced and the ejection capacity is increased compared to the single mode. According to the calculation results, the inlet mass flow of high and low pressure gas in condition 2 collaborative mode is reduced by 36.4% and 8.73%, respectively, compared to the single mode. Conversely, the static pressure in the functional region of condition 1 on the collaborative mode is lower than that of single mode. Therefore, the gas flow rate and exhausting velocity in the functional region are increased relative to the single mode, resulting in a reduction in working performance of condition 1 in the collaborative mode compared to the single mode. Compared to the single mode, the high and low pressure mass flow rates of collaborative mode under condition 1 are increased by 72.5% and 17.3%, respectively, while the ξ and η are decreased by 34.4% and 6.5% compared to that in the single mode. The difference in the parameter relationship between the stable-pressure and functional regions results in the average pressure in the stable-pressure region of collaborative mode is different from that of the single mode. When compared to the single mode, the p_s1 of condition 1 with a relatively higher p_lt increases, while the p_s2 of condition 2 decreases in the collaborative mode.

As shown in Figure 12, when p_ms is raised to 140 kPa (conditions 3 and 4), the pressure variations of the stable-pressure region and functional region in the collaborative mode are similar to those of the condition with p_ms = 125 kPa (conditions 1 and 2). However, when the inlet conditions are fixed, the increment of static pressure in the functional region has a greater impact on the ejection capacity on condition 4 than on condition 2, since the expansion depth (lowest pressure in the channels of functional region) and ejection capacity of GWE may decrease with the increasing of p_ms. Therefore, in the collaborative mode, the low-pressure gas flow rate on the condition 4 is decreased by 38.4% compared to the single mode, and the decrement is significantly larger than that of condition 2. As a result, though the high-pressure gas flow rate in collaborative mode on condition 4 is reduced by 32.3% compared to the single mode, which is similar to that of condition 2, the ξ and η on condition 4 in collaborative mode are respectively decreased by 2.5% and 10.9% in comparison with the single mode. On the contrary, in the collaborative mode, the high and low pressure gas flow rates on condition 3 are respectively increased by 32.6% and 32.1% compared to the single mode, and it can be calculated that ξ and η are increased by 2.4% and 8.2%, respectively.

As shown in Figure 13, p_ms and p_lt on conditions 5 and 6 are identical with those on conditions 1 and 2, while the p_ht is reduced to 152 kPa. In the collaborative mode, the pressure variations of stable-pressure region and functional region on conditions 5 and 6 compared to single mode are similar to those on conditions 1 and 2. However, as p_ht decreases, the energy input is decreases, resulting in a diminished ability to pressurize the low-pressure gas. As a result, the low and high-pressure gas mass flow rates in the collaborative mode are significantly reduced on condition 6. In the collaborative mode, the mass flow rates of high and low-pressure gases on condition 6 are reduced by 52.5% and 44.9% compared to the single mode. It can be calculated that the ξ and η on condition 6 in collaborative mode are respectively 12.4% and 4.1% higher than those of the single mode. And the increase is obviously less than that on condition 2. As shown in Figure 11, although the average pressure of stable-pressure region on condition 2 in collaborative mode is lower than that in single mode, its average pressure is still higher than the p_ms on condition 2. However, the average pressure of stable-pressure region in collaborative mode on condition 6 is lower than the p_ms on condition 6. Thus a reversed compression wave is created in functional region on condition 5 when the channel starts to connect with the MP port, having a negative impact on sucking the low-pressure gas. Consequently, ξ and η on condition 5 in collaborative mode are respectively decreased by 37.6% and 6.6% compared to the single mode, and the decrease is higher than that on condition 1.

According to the analysis mentioned above, the difference of performance parameters between the single mode and collaborative mode is directly affected by the pressure in the stable-pressure region (defined as p_s). The simulations in two different modes on the conditions with different p_ms were conducted to clarify the specific relationship and difference between the two modes, and the results are shown in Figure 14.

The difference of efficiency η_m−d and the difference of ejection rate ξ_m−d are defined as Equations (18) and (19) for facilitate research.

$${\eta }_{m-d}={\eta }_m-{\eta }_d$$

(18)

$${\xi }_{m-d}={\xi }_m-{\xi }_d$$

(19)

where η_m and ξ_m denote the total pressure efficiency and ejection rate in collaborative mode, respectively, and η_d, ξ_d denote the corresponding values in the single mode.

As shown in Figure 14a,c, the curves of η_m−d on the conditions with different p_lt intersect at the point where η_m−d is roughly equal to 0. It is consistent with the experiment results, indicating that in the collaborative mode, if η on one operating condition in a group is higher than that of the single mode, the η on the other operating condition will be relatively smaller. Furthermore, on the working conditions as shown in Figure 8c and Figure 14a, the p_ms associated with the intersection point of the η_m−d curves obtained by simulations is essentially identical to p_ms of the intersection point found in experiments. It demonstrates that the numerical model employed in this study accurately reflects the relative performance relationship between the collaborative and the single modes, and the quantitative simulation results can be used to analyze the practical application effects of the collaborative mode. According to the simulation results, static pressure of stable-pressure region on lower operating condition in single mode (defined as p_sl) gradually decreases with the increase of p_ms, and the p_ms at p_sl = p_ms is close to the value of p_ms corresponding to the intersection of η_m−d curves. As a result, the relationship between p_sl and p_ms can be used as the judgment criteria for determining the relationship of relative magnitude between η of the two modes. In conclusion, when the condition with lower p_lt satisfies p_sl > p_ms in the collaborative mode, η on this condition increases compared to the single mode, whereas η on the condition with higher p_lt decreases relatively. When p_sl > p_ms, however, η on the condition with higher p_lt in collaborative mode may be improved compared to the single mode, while η on the condition of with lower p_lt may be relatively declined. Furthermore, the difference of efficiency |η_m−d| between collaborative and single mode is positively correlated with |p_sl − p_ms|.

As shown in Figure 14b,d, the ξ_m−d curves on different p_lt conditions also intersect at the point corresponding to ξ_m−d = 0, and the variations of ξ_m−d are also consistent with the experimental results. The p_ms corresponding to the intersection point of the ξ_m−d curves obtained by simulations on each operating condition are virtually identical to the p_ms corresponding to the intersection point found in the experiments. In the single mode, there has been a steady decrease for static pressure of stable-pressure region on higher p_lt condition in single mode (defined as p_sh) with the increase of p_ms, and p_ms at p_sh = p_ms is nearly equal to the p_ms corresponding to the intersection point of ξ_m−d curves. And thus p_ms at p_sh = p_ms can be employed as judgment criteria for the relationship between ξ of the two modes. In the collaborative mode, when p_sh > p_ms, ξ on the condition with lower p_lt is increased compared to the single mode, while ξ on the condition with higher p_lt is relatively decreased. Conversely, when p_sh > p_ms, ξ on the condition with higher p_lt is increased in collaborative mode compared to the single mode, while ξ on the condition with lower p_lt is relatively decreased. In addition, the |ξ_m−d| is positively correlated with |p_sh − p_ms|.

Within the pressure range of 10 MPa, gas in the industrial production, such as natural gas, does not display nonclassical flow characteristics [36,37]. Although the propagation speed and strength of pressure waves may vary for different gas [38], the functional wave system within the GWE is identical. In addition, the pressure distribution characteristics of the functional and stable pressure region, as well as their pressure relationship with the pressure of pressure ports is also identical. As a result, the pertinent findings of this study have broad application prospects in industry. The collaborative mode can be designed in the actual application of GWE based on the specific working conditions. By combining the calculation results of the pressure distribution in the channels of stable-pressure region and the performance difference judgment criteria, the ejection capacity of GWE in single and collaborative mode can be compared and analyzed. Eventually, the production solution is chosen according to the actual demand and the optimal benefit.

4. Conclusions

The issue of excessive stable-pressure region that results in the underutilization of the rotor capacity is one of the apparent shortcomings of the gas wave ejector (GWE).

In this study, a novel collaborative mode of GWE was proposed, which may improve the equipment utilization and make single equipment satisfy the demand of multi-condition and co-production in industry. The performance characteristics of the collaborative mode were obtained experimentally, and the internal mechanism was investigated using numerical simulation. In conclusion, the main findings of this research can be listed as follows:

• The experimental results show that η_Total of collaborative mode can reach up to 48.8% and the difference of η_Total between two modes is less than 4.4% within the range of this study. It proves that the collaborative mode can improve the device utilization while having almost no effect on the total energy transfer efficiency of GWE when compared to the single mode. This is due to the fact that the collaborative mode allows for the synchronous production of two working conditions without changing the energy transfer mode, which is mediated by pressure waves inside the rotor channel, and there is no significant difference in gas flow losses between the two modes.
• Simulation findings demonstrate that the collaborative mode causes a difference in the pressure distribution in the rotor channels on each operating condition compared to the single mode. The state parameters in the stable-pressure region on one operating condition are the initial parameters of the functional region on the other condition of a same group, resulting in a performance between the collaborative and single modes.
• The variation laws of ejection rate ξ and efficiency η with the static pressure of middle-pressure port p_ms obtained by experiments are basically identical in both the single and collaborative modes. However, when the performance parameters values on one operating condition of a group (one working condition group includes two conditions with the same p_ht and p_ms, as well as different p_lt) in collaborative mode are higher than that of the single mode, the performance parameters of the collaborative mode on the other condition in the same group are relatively lower. As a result, on one working condition in a group, p_ms corresponding to the intersection point of the performance parameter curves in different modes are nearly equal to that on the other working condition.
• It is found through simulation and experiment that the relationship between the p_ms and p_s (p_sl/p_sh) in single mode can be used as a basis to predict the change trend of the performance parameters.

  (1).

  When the condition with lower p_lt in a group satisfies p_sl > p_ms, η of this condition in collaborative mode increases in comparison to the single mode, whereas η of the other condition with higher p_lt in the collaborative mode drops relatively. In contrast, compared to the single mode, η of the condition with higher p_lt in collaborative mode is improved when p_sl < p_ms, while η of the condition with lower p_lt is relatively reduced.

  (2).

  When the condition with higher p_lt in a group satisfies p_sh > p_ms, ξ of the condition with higher p_lt in collaborative mode decreases compared to the single mode, while ξ of the condition with lower p_lt increases relatively. In contrast, compared to the single mode, ξ of the condition with higher p_lt in collaborative mode is improved when p_sh < p_ms, while ξ ofthe condition with lower p_lt is relatively reduced.

  Furthermore, the research results demonstrate that the difference in ejection rate |ξ_m−d| and the difference in total pressure isentropic efficiency |η_m−d| between the collaborative mode and single mode are positively correlated with |p_sh − p_ms| and |p_sl − p_ms|, respectively.
• The working characteristics of a single GWE equipment for multi-condition production were preliminarily obtained for the first time in this study. Based on this, the application effect of the equipment under more than dual working conditions can be analyzed and studied in the future, so as to further improve the utilization rate and application scope of GWE.