During the operation of thermal power plant systems, severe challenges to the quality and safety of the heat supply arise owing to the effects of peak regulation. Steam ejectors, widely employed in the district heating sector, often exhibit poorer performance under off-design conditions as a result of their fixed structure. This study analyzes the effects of varying plugging rate, primary pressure, and induced pressure on the performance of adjustable steam ejectors. As the plugging rate increases, the mass flow rate of the primary fluid decrease, potentially falling below 32%, and that of the induced fluid initially experiences a minor increase, followed by a rapid decrease. Concurrently, the critical back pressure also decreases with an increase in the plugging rate, potentially falling by up to 65%. By contrast, the entrainment ratio surges with increasing plugging rate, with the maximum increase reaching up to 4.75 times. Under the same primary steam pressure and induced steam pressure, there exists an optimal entrainment ratio. Furthermore, both the critical back pressure and the optimal entrainment ratio increase as the primary steam pressure and induced steam pressure increase.

## NOMENCLATURE

### Greek

### Subscripts

## I. INTRODUCTION

An ejector is a device designed to merge two streams of fluids characterized by distinct pressures and temperatures, utilizing the turbulent diffusion effect within fluid microgroups to facilitate the transfer of mass and energy, which results in the formation of a mixed fluid reflecting the central pressure and temperature. The ejector’s appeal lies in its simple structure, cost-effectiveness, high reliability, and ease of maintenance. These attributes have led to the widespread application of ejectors in sectors as diverse as seawater desalination, aerospace, and refrigeration systems.

Ever since the first ejector was fabricated by Flugel in Germany,^{1} these devices have been widely studied. Zeuner contributed to the theoretical foundations of jet design through a momentum conservation theorem, while Keenan conducted extensive research on ejectors, calculating their properties in the case of air as the medium, and proposed a constant-pressure mixing theory.^{2,3} However, Keenan’s method struggled to predict the constant capacity of the ejector, leading Otis to propose plugging as the cause of internal congestion.^{4} Chou *et al.*^{5} examined potential locations of congestion locations within an ejector, including the inlet of the mixing section, the outlet of the isotropic mixing section, and an intermediate section. There have been a number of studies aimed at optimization of ejectors. Pereira *et al.*,^{6} Aphornratana and Eames,^{7} Lin *et al.*,^{8} and Hu *et al.*^{9} conducted studies optimizing the traditional fixed ejector, utilizing R600a, water, R410a, and R134a, respectively, as primary fluids.

Much of the research to date has been on fixed ejectors. However, traditional fixed ejectors suffer from inherent problems, such as significant irreversible loss and susceptibility to deviation from the design condition due to shock waves, and these remain unresolved. Consequently, adjustable steam ejectors have emerged as a structural enhancement to maintain efficient operation under fluctuating operating conditions. This type of ejector has attracted considerable interest. Kim *et al.*^{10} investigated the circulation rate of an adjustable ejector for refrigeration cycles under a fixed working pressure. Yapici *et al.*^{11} conducted an experimental study to understand the impact of operating vapor temperature on ejector performance at different area ratios. Jia and Wenjian^{12} examined the cycle rate of a tuned ejector for refrigeration cycles and evaluated the effect of operating conditions on the entrainment ratio. Lin *et al.*^{13} undertook a study focusing on the internal flow characteristics of an adjustable ejector. Chen *et al.*^{14} employed a one-dimensional semi-empirical model to analyze the variations in parameters such as the entrainment ratio of an adjustable flat ejector in response to changes in the equivalent diameter of the nozzle throat.

In addition to one-dimensional theories and experimental approaches, rapid advances in computer technology have fostered the widespread use of numerical methods. Attempts have been made to obtain more accurate depictions of the fluid mixing process within adjustable steam ejectors. A simulation-based study by Wang *et al.*^{15} suggested that an adjustable steam ejector could exhibit significantly enhanced performance. Varga *et al.*^{16} conducted a simulation study on the nozzle area ratio, demonstrating that adjusting the spindle position could effectively control the primary fluid flow rate, with the error relative to an experimental study being less than 5%. Through numerical simulation, Sharifi *et al.*^{17} focused on and optimized the influence of the geometry of the supersonic nozzle on the ejector compression and entrainment ratios, thereby improving the performance of the ejector. A study by Omidvar *et al.*^{18} revealed that entropy loss, a major factor in the mixing process, resulted in the ejector’s exit temperature exceeding the critical temperature. Dennis and Garzoli^{19} performed a simulation study of a solar ejection refrigeration system, finding that the refrigeration ratio increased by 8%–13% in comparison with a conventional ejector. Chong *et al.*^{20} conducted a study of a supersonic air ejector, analyzing the performance changes in critical mode and subcritical mode. Despite the improved performance of adjustable steam ejectors, current research into these devices suffers from several limitations, such as a lack of studies comparing supersonic and subsonic ejectors, and a lack of investigations of the medium-pressure range.^{21}

Previous research has either amalgamated theoretical and experimental approaches or lacked comprehensive simulations of adjustable steam ejectors. This paper focuses on an adjustable steam ejector in a steam unit, conducting a three-dimensional numerical simulation of the device. It scrutinizes the influences of a variety of parameters, such as the plugging rate and steam pressure, on the operating conditions of the unit. The aim of this study is to supplement research on adjustable steam ejectors within the medium-pressure range, extending numerical simulations to adjustable steam ejectors, and thereby provide data support and strategies for the regulation of medium-pressure cylinder pumping applications in power plants and other industrial operations.

## II. NUMERICAL SIMULATION MODEL

### A. Ejector structure

The structure of the adjustable steam ejector under consideration is depicted in Fig. 1. Its principal components include an adjusting device, Laval nozzle, mixing chamber, mixing tube, and diffuser chamber. The high-energy primary fluid accelerates through the tapering Laval nozzle, forming a supersonic shock wave chain and a low-pressure region at the nozzle exit, where pressure is transformed into kinetic energy. Under the influence of the pressure differential, the low-energy induced fluid is drawn into the mixing chamber, where it is accelerated to supersonic speed by the primary fluid. Subsequently, the two fluids engage in mass, energy, and momentum exchange within the mixing section of uniform cross-section, resulting in a homogeneous mixed fluid. The flow velocity of the mixed fluid is simultaneously reduced to subsonic speed. Finally, the mixed fluid passes through the diffuser chamber, where its kinetic energy is reconverted into pressure potential energy, and it is pressurized and then discharged from the ejector. This process completes the compression work of high-pressure steam on low-pressure steam, as well as the cooling process of low-temperature steam on high-temperature steam.

*ω*. This is defined as the ratio of the mass flow rate of the induced fluid to that of the primary fluid:

*q*

_{s}and

*q*

_{p}are the mass flow rates of the induced fluid and primary fluid, respectively (kg/s).

The critical back pressure represents a significant parameter of an ejector. As the outlet pressure (or back pressure) escalates, the primary shock wave may oscillate within the ejector, thereby generating a reflected shock wave, regarded as a secondary shock wave. This typically occurs when the back pressure attains a critical value. The critical back pressure is defined as the minimum back pressure that permits a shock wave to fully rebound within the ejector. When the back pressure exceeds this critical value, a primary shock wave oscillates within the ejector, potentially giving rise to one or more secondary shock waves.

The adjusting device serves primarily as an adjusting cone. Altering the position of the adjusting cone within the device modifies the ratio of the throat’s cross-sectional area to that of the Laval nozzle’s throat. This, in turn, facilitates adjustment of the critical pressure, the entrainment ratio, and the outlet flow rate of the adjustable steam ejector.

The geometrical parameters of the adjustable steam ejector considered in the present study are listed in Table I.

Parameter . | Value (mm) . |
---|---|

Nozzle inlet diameter | 130 |

Nozzle throat diameter | 42.22 |

Nozzle outlet diameter | 46 |

Mixing chamber inlet diameter | 150 |

Mixing chamber throat diameter | 87.5 |

Outlet diameter of diffuser | 164 |

Maximum diameter of spindle | 30 |

Length of nozzle taper section | 190 |

Length of nozzle expansion section | 80 |

Diameter of ejector secondary fluid inlet | 250 |

Length of mixing chamber | 400 |

Length of mixing chamber throat | 500 |

Length of diffuser | 600 |

Length of spindle cone section | 78 |

Parameter . | Value (mm) . |
---|---|

Nozzle inlet diameter | 130 |

Nozzle throat diameter | 42.22 |

Nozzle outlet diameter | 46 |

Mixing chamber inlet diameter | 150 |

Mixing chamber throat diameter | 87.5 |

Outlet diameter of diffuser | 164 |

Maximum diameter of spindle | 30 |

Length of nozzle taper section | 190 |

Length of nozzle expansion section | 80 |

Diameter of ejector secondary fluid inlet | 250 |

Length of mixing chamber | 400 |

Length of mixing chamber throat | 500 |

Length of diffuser | 600 |

Length of spindle cone section | 78 |

### B. Model parameter setting

*ρ*and

*u*are the fluid density and velocity, respectively,

*p*is pressure,

*t*is time,

*E*is the total energy,

*α*

_{eff}is the effective thermal conductivity, and

*T*is the thermodynamic temperature.

*τ*

_{ij}is the turbulent shear stress and is given by

*μ*

_{eff}is the effective viscosity.

In an effort to improve computational efficiency and the reliability of the results, the discrete nonlinear governing equation is solved utilizing a pressure base in the settings. The standard *k*-*ε* model is selected as the turbulence model, and an implicit coupled solution method is employed to solve for velocity and pressure in the momentum equation. The solution method involves SIMPLE iteration and the use of least-squares-cell-based gradient interpolation. The working medium is designated as water vapor and treated as a compressible ideal gas. To ensure reliable numerical solutions, it is imperative to establish reasonable boundary conditions. As such, the boundary conditions are set as follows: pressure boundary conditions are applied at the inlet for both the primary and induced fluids, with the stagnation pressure and temperature provided. For the mixed fluid outlet, a pressure outlet boundary condition is implemented, specifying the static pressure; turbulence conditions are defined by turbulence intensity and hydrodynamic diameter. An adiabatic boundary with no slip and no seepage conditions is employed at the solid wall. Convergence of the iteration is assumed when the residuals of the variables (velocity, pressure, turbulent kinetic energy, etc.) fall below 10^{−4} and the mass flow rate at the monitoring point remains stable, as well as when the entrainment ratio value stabilizes.

The design parameters of the adjustable steam ejector are selected as shown in Table II.

Parameter . | Value . |
---|---|

Primary fluid pressure | 3.236–4.33 MPa |

Induced fluid pressure | 0.87–1.152 MPa |

Mixed fluid pressure | 1.3 MPa |

Fluid temperature | 623.15–639.15 K |

Parameter . | Value . |
---|---|

Primary fluid pressure | 3.236–4.33 MPa |

Induced fluid pressure | 0.87–1.152 MPa |

Mixed fluid pressure | 1.3 MPa |

Fluid temperature | 623.15–639.15 K |

### C. Grid-independence verification and model validation

To ensure the reliability of the numerical simulation results, the grid was continuously refined until the calculation results became independent of the grid number. When the error in the ratio of the mass flow rate of the induced fluid to that of the primary fluid fell below 0.5%, it could be inferred that the calculation results at this juncture were independent of the grid. The results of this grid-independence verification are shown in Table III.

Grid number . | q_{p} (kg/s)
. | Error (%) . | q_{s} (kg/s)
. | Error (%) . | ω
. | Error (%) . |
---|---|---|---|---|---|---|

2 000 000 | 5.5611 | … | 3.4302 | … | 0.6168 | … |

2 460 000 | 5.5644 | 1.490 | 3.5485 | 3.449 | 0.6377 | 3.388 |

3 120 000 | 5.5648 | 0.007 | 3.5871 | 1.088 | 0.6446 | 1.082 |

4 030 000 | 5.5737 | 0.160 | 3.6228 | 0.995 | 0.6500 | 0.838 |

5 500 000 | 5.5701 | −0.065 | 3.6227 | 0.003 | 0.6504 | 0.062 |

Grid number . | q_{p} (kg/s)
. | Error (%) . | q_{s} (kg/s)
. | Error (%) . | ω
. | Error (%) . |
---|---|---|---|---|---|---|

2 000 000 | 5.5611 | … | 3.4302 | … | 0.6168 | … |

2 460 000 | 5.5644 | 1.490 | 3.5485 | 3.449 | 0.6377 | 3.388 |

3 120 000 | 5.5648 | 0.007 | 3.5871 | 1.088 | 0.6446 | 1.082 |

4 030 000 | 5.5737 | 0.160 | 3.6228 | 0.995 | 0.6500 | 0.838 |

5 500 000 | 5.5701 | −0.065 | 3.6227 | 0.003 | 0.6504 | 0.062 |

To validate the accuracy of the simulation calculation models, such as the standard *k-ε* turbulence model, Table IV presents a comparison between the experimental and simulation results. The comparative analysis indicates that the simulated results from the model align well with the experimental findings, indicating the feasibility of the model.

. | Primary fluid . | Induced fluid . | Mixed fluid . | Entrainment ratio . | . | ||||
---|---|---|---|---|---|---|---|---|---|

Working condition . | P (MPa)
. | T (K)
. | P (MPa)
. | T (K)
. | P (MPa)
. | T (K)
. | Experiment . | Simulation . | Error (%) . |

1 | 11.7 | 773 | 2 | 495 | 3.7 | 653 | 0.51 | 0.522 | 2.32 |

2 | 11.7 | 773 | 2.4 | 515 | 3.8 | 653 | 0.71 | 0.689 | 2.99 |

3 | 11.7 | 773 | 2.8 | 565 | 3.9 | 653 | 0.97 | 0.969 | 0.12 |

4 | 11.7 | 783 | 2 | 753 | 3.5 | 723 | 0.57 | 0.577 | 1.26 |

5 | 11.7 | 783 | 2.25 | 753 | 3.8 | 723 | 0.56 | 0.559 | 0.26 |

6 | 11.7 | 783 | 2.5 | 753 | 3.8 | 723 | 0.97 | 0.975 | 0.47 |

. | Primary fluid . | Induced fluid . | Mixed fluid . | Entrainment ratio . | . | ||||
---|---|---|---|---|---|---|---|---|---|

Working condition . | P (MPa)
. | T (K)
. | P (MPa)
. | T (K)
. | P (MPa)
. | T (K)
. | Experiment . | Simulation . | Error (%) . |

1 | 11.7 | 773 | 2 | 495 | 3.7 | 653 | 0.51 | 0.522 | 2.32 |

2 | 11.7 | 773 | 2.4 | 515 | 3.8 | 653 | 0.71 | 0.689 | 2.99 |

3 | 11.7 | 773 | 2.8 | 565 | 3.9 | 653 | 0.97 | 0.969 | 0.12 |

4 | 11.7 | 783 | 2 | 753 | 3.5 | 723 | 0.57 | 0.577 | 1.26 |

5 | 11.7 | 783 | 2.25 | 753 | 3.8 | 723 | 0.56 | 0.559 | 0.26 |

6 | 11.7 | 783 | 2.5 | 753 | 3.8 | 723 | 0.97 | 0.975 | 0.47 |

## III. SIMULATION RESULTS AND ANALYSIS

The performance of an adjustable steam ejector in practice is mainly influenced by operating and structural parameters. The main operational parameters include the back pressure *P*_{b}, primary pressure *P*_{p}, and induced pressure *P*_{s}. The main structural parameters are the plugging rate *γ* of the nozzle throat, which can be varied by changing the position of the adjusting cone, which is the basis for the adjustment of the structural parameters of the adjustable steam ejector. In this paper, the influence of operational and structural parameters on ejector performance is analyzed by means of control variables, with the aim of providing guidance for the production of practical adjustable steam ejectors.

*γ*is defined as the ratio of the area of the adjusting cone on the throat section to the area of the nozzle throat section:

*y*is the diameter of the adjusting cone at the throat section of the Laval nozzle (m) and

*D*is the diameter of the throat section of the Laval nozzle (m). By adjusting the position of the adjusting cone along the axis, it is possible to change the value of

*y*in relation to the cross-sectional area of the Laval nozzle, thereby effectively controlling the plugging rate, as shown in Fig. 2.

### A. Effect of plugging rate on primary fluid

With reference to the ejector design parameters listed in Table II, the back pressure *P*_{b} is held constant while the plugging rate *γ* is adjusted to explore its impact on the primary fluid. Within the designated parameter range, the primary pressure *P*_{p} and the priming steam pressure *P*_{s} are controlled within the ranges 3.236–4.33 MPa and 0.87–1.152 MPa, respectively. Concurrently, the plugging rate *γ* varies from 0% to 70%.

Figure 3 illustrates the impact of the plugging rate on the mass flow of the primary fluid at different primary pressures. As the plugging rate *γ* increases, the mass flow rate of the primary fluid *q*_{p} decreases, and this trend becomes more pronounced with an increase in primary pressure *P*_{p}. It can be inferred from Fig. 3 that modifications of the plugging rate can directly influence the mass flow rate of the primary fluid. For instance, when *P*_{p} is set to 4.33 MPa and *P*_{s} to 1.152 MPa, the *q*_{p} corresponding to zero plugging rate can reach more than triple the *q*_{p} corresponding to a 70% plugging rate.

The larger the primary pressure *P*_{p}, the higher is the vapor density, leading to a larger mass flow through the section under an identical volume flow rate. By altering the position of the adjusting device to modify the plugging rate, the sectional area of the Laval nozzle’s throat is directly affected. As the primary fluid passes through the Laval nozzle, pressure is converted into kinetic energy, forming a supersonic flow, which means that the adjusting cone can directly regulate the primary fluid’s mass flow rate *q*_{p}. When the adjusting cone is fully opened, the inlet volume of the Laval nozzle is maximized. Conversely, a larger plugging rate corresponds to a smaller inlet volume. When the adjusting device approaches the nozzle throat from the far left along the axis, the increase in steam resistance through the throat is more substantial for a higher primary fluid pressure compared with a lower primary fluid pressure, thus resulting in larger variations in *q*_{p} with changes in the plugging rate.

Figure 4 illustrates the impact of the plugging rate on the primary fluid’s mass flow rate under various induced fluid pressures. In a similar vein, the mass flow rate of the primary fluid *q*_{p} decreases as the plugging rate increases. However, alterations in the induced fluid pressure *P*_{s} do not influence *q*_{p}. This is attributed to the formation of a low-pressure area in the mixing chamber after the primary fluid has been accelerated by the Laval nozzle. Consequently, the induced fluid is entrained by the low pressure in the mixing chamber and enters it in conjunction with the primary fluid. Given that the pressure of the induced fluid is lower than that within the mixing chamber, it does not affect the primary fluid’s pressure and flow.

### B. Effect of plugging rate on induced fluid

Referring again to the ejector design parameters outlined in Table II, the back pressure *P*_{b} is held steady while the plugging rate *γ* is adjusted to examine its influence on the induced fluid. Within the prescribed range of design parameters, the primary pressure *P*_{p} and induced pressure *P*_{s} are regulated within the ranges 3.236–4.33 MPa and 0.87–1.152 MPa, respectively. The plugging rate *γ* varies from 0% to 70%, with the back pressure *P*_{b} maintained at 1.3 MPa.

Figure 5 demonstrates the impact of the plugging rate on the induced fluid’s mass flow rate under different primary pressures. Initially, the mass flow rate *q*_{s} of the induced fluid marginally increases and then rapidly decreases with increasing plugging rate *γ*. The higher the primary fluid pressure *P*_{p}, the later the alteration in *q*_{s} occurs with *γ*, and the more pronounced the change becomes. As illustrated in Fig. 5, *q*_{s} starts at ∼5.5 kg/s and remains unaffected by changes in *P*_{p} and *γ*. When *P*_{p} is set to 3.236, 4, and 4.33 MPa, *q*_{s} decreases with the plugging rate by 20%, 40%, and 40%, respectively. In all cases, *q*_{s} falls to 0 kg/s when the plugging rate reaches around 50%–60%, indicating a backflow of the induced fluid. Since the a pressure boundary condition is adopted for the primary fluid and induced fluid inlet, a fixed pressure value is set. When the plugging rate reaches a certain value, the critical back pressure of the working condition drops to the value of the back pressure. The ejector enters a subcritical state, and the induced fluid flow rate begins to decline and backflow occurs. When the pressure of the mixing chamber is greater than the set pressure of the induced fluid inlet, the induced fluid flow rate begins to show a negative value.

As shown in Figs. 6 and 7, *P*_{p} = 3.236 MPa and *P*_{s} = 1.152 MPa. Under the same back pressure, as the plugging rate decreases and the proportion of the adjusting cone area decreases, it becomes easier to generate secondary shock waves in the diffuser chamber. This makes it easier for the shock wave chain generated after the Laval nozzle to pass through the mixing tube with uniform cross-section. The ability of the primary fluid to entrain the induced fluid is enhanced, resulting in a larger critical back pressure. At a certain back pressure, when the plugging rate is small, the ejector is not in a subcritical state, with the induced fluid being less affected by changes in the plugging rate. When the plugging rate reaches a certain value, the critical back pressure decreases to the outlet back pressure value, and the ejector enters a subcritical state, at which point the secondary shock wave disappears, backflow occurs (as shown in Fig. 6), and the mass flow rate of the induced fluid and entraining capacity of the primary fluid start to drop abruptly (as shown in Fig. 5).

Figure 8 illustrates the impact of the plugging rate on the mass flow rate of the induced fluid under different induced fluid pressures. As can be seen, the flow rate of the induced fluid under several operating conditions decreases almost completely to zero at a plugging rate of 55%. Mirroring previous findings, *q*_{s} initially increases slightly with the rise in plugging rate *γ*, then experiences a rapid decline. Once *q*_{s} starts to alter, the rate of change progressively intensifies with the increase in plugging rate.

For a given primary pressure *P*_{s}, a higher primary pressure *P*_{p} results in a greater ability to resist back pressure and a stronger entrainment effect on the induced fluid in the mixing chamber. Therefore, with a higher *P*_{p}, the induced fluid flow rate *q*_{s} starts to change with the plugging rate later. Additionally, for a given *P*_{p}, a higher induced pressure *P*_{s} corresponds to a greater ability of the induced fluid to resist back pressure. This also explains why *q*_{s} begins to decrease later at *P*_{s} = 1.152 MPa compared with *P*_{s} = 0.87 MPa.

In the practical operation of a thermal power plant system, the primary fluid of the steam ejector is sourced from the superheated steam pipe, while the ejector steam originates from the reheated steam pipe. As the electrical load of the unit decreases, the heat load correspondingly diminishes, causing a concurrent reduction in the two steam parameters. Unlike a fixed steam ejector, an adjustable steam ejector is able to overcome the problem of a sudden drop in steam mass flow rate under fluctuating operating conditions. When the operational parameter, the back pressure *P*_{b}, remains constant and the unit load decreases, both the primary pressure *P*_{p} and the induced pressure *P*_{s} will decrease correspondingly. To ensure stability of the unit’s heating capacity, it is only necessary to adjust the adjusting device, positioning the adjusting cone axially away from the nozzle throat. Through reducing the plugging rate, this action increases both *q*_{s} and *q*_{p}, thereby ensuring heating stability. This approach facilitates the application of the traditional heating method of “fixing electricity by heat” and achieves thermal–electric decoupling.

### C. Effect of plugging rate on critical back pressure and entrainment ratio

The impact of the plugging rate *γ* on the critical back pressure $Pb*$ and entrainment ratio *ω* is investigated by adjusting *t γ* in accordance with the ejector design parameters outlined in Table II. Within the specified ranges of these parameters, the primary pressure *P*_{p} and induced pressure *P*_{s} are regulated between 3.236 and 4.33 MPa and between 0.87 and 1.152 MPa, respectively. The plugging rate *γ* is set to vary from 0% to 70%.

The critical back pressure $Pb*$ is the boundary value between the subcritical and critical states of the ejector. When *P*_{b} < $Pb*$ (Fig. 9), both the primary fluid and the induced fluid can reach sonic or supersonic speeds in the mixing chamber, resulting in plugging at two locations within the ejector. At this point, the back pressure *P*_{b} does not affect the flow rates of the primary fluid and the induced fluid, but the presence of secondary shock waves leads to irreversible energy losses, causing a slight decrease in ejector performance. When *P*_{b} = $Pb*$ (Fig. 10), the ejector is in a critical state and achieves optimal performance, with the fluid being pumped and discharged most efficiently. In this state, a small change in back pressure can lead to a sharp decline in performance. When *P*_{b} > $Pb*$ (Fig. 11), the ejector is in a subcritical state, and the induced fluid cannot reach a blocked state in the mixing chamber. The flow velocity at the end of the mixing chamber is less than the speed of sound, and increasing the back pressure in this case will decrease the flow rate of the induced fluid, leading to inadequate mixing of the primary fluid and the induced fluid, thereby reducing ejector performance.

Figure 12 shows the variations in the critical back pressure and the corresponding entrainment ratio *ω* as functions of the plugging rate at a primary pressure *P*_{p} = 3.236 MPa and an induced pressure *P*_{s} = 1.152 MPa. As the plugging rate increases, the critical back pressure decreases, while the corresponding entrainment ratio increases, exhibiting a larger growth trend at higher plugging rates. As can be seen in Fig. 12, the entrainment ratio at a 70% plugging rate can exceed four times that at a 0% plugging rate.

From Fig. 9, it can be observed that there are two locations where shock waves are formed inside the ejector: one is in the mixing chamber, and the other is at the entrance section of the expansion chamber. The critical back pressure and the corresponding entrainment ratio decrease and increase respectively with increasing plugging rate. This is because a decrease in the plugging rate leads to the generation of secondary shock waves in the diffuser chamber, causing backflow and a reduction in the entrainment ratio. At the same time, a decrease in the plugging rate makes it easier for the shock wave chain generated by the Laval nozzle to pass through the mixing tube, enhancing the ability of the primary steam to entrain the induced steam, thereby increasing the critical back pressure.

Figures 13 and 14 demonstrate the effect of the plugging rate *γ* on the critical back pressure $Pb*$ under different primary fluid pressures *P*_{p} and induced fluid pressures *P*_{s}. When *P*_{p} is held constant and *P*_{s} is varied, the critical back pressure $Pb*$ increases with a rise in *P*_{s}. Similarly, when *P*_{s} is constant and *P*_{p} is varied, the critical back pressure $Pb*$ also increases with increasing *P*_{p}. The larger the values of *P*_{p} and *P*_{s}, the higher is the pressure of the mixed fluid resulting from the combination of the primary and induced fluids. This leads to a higher outlet back pressure, greater back pressure resistance, and, correspondingly, a higher critical back pressure. At the same time, both Figs. 13 and 14 also demonstrate the trend of change of the critical back pressure with the plugging rate: the critical back pressure gradually decreases as the adjusting cone moves axially toward the throat, and the trend of decrease is relatively uniform. For every 10% increase in plugging rate, the critical back pressure decreases by around 0.085 MPa.

Figure 15 presents the impact of the plugging rate *γ* on the entrainment ratio *ω* under different primary fluid pressures *P*_{p}. Given the same *P*_{p} and *P*_{s}, there exists a maximum *ω* corresponding to a specific *γ*, known as the optimal entrainment ratio *ω**. For *P*_{p} = 4 MPa and *P*_{s} = 0.87 MPa, the optimal entrainment ratio *ω** is 26%, whereas for *P*_{p} = 4.33 MPa and *P*_{s} = 0.87 MPa, it is 30%. The value of the optimal entrainment ratio increases with increasing *P*_{p}. This can be attributed to the fact that when the back pressure is less than the critical back pressure, the ejector is in a supercritical state. The insertion of the adjusting cone elevates the plugging rate *γ*, which in turn can reduce the primary fluid flow, conserve energy, and increase the entrainment ratio. However, when the plugging rate *γ* increases to the point at which it disrupts the critical state, leading to *P*_{b} exceeding the critical back pressure $Pb*$, backflow is generated. This results in a rapid decline in the entrainment capacity of the induced fluid, and consequently a swift decrease in the entrainment ratio.

From Figs. 16 and 17, it can be observed that with a gradual increase in the plugging rate *γ*, the throat area of the nozzle decreases and the shock wave chain shortens. When *γ* exceeds 28% and 42% respectively, the secondary shock wave ceases to be generated at the throat diameter. The shock wave produced by the nozzle can just traverse the mixing chamber and the region of uniform diameter, allowing the ejector to achieve its optimal working state and obtain the maximum entrainment ratio *ω**.

As indicated in Fig. 3, within the range of *γ* = 20%–40%, the primary fluid flow rate *q*_{p} under a primary pressure *P*_{p} of 4 MPa is larger than that under *P*_{p} = 3.236 MPa. As shown in Fig. 5, the induced fluid flow rates *q*_{s} under both conditions are nearly identical, providing an explanation as to why a larger *P*_{p} corresponds to a greater plugging rate *γ*.

Figure 18 illustrates the influence of the plugging rate on the entrainment ratio under varying induced fluid pressures. Similar to previous observations, for a given primary fluid pressure *P*_{p} and induced fluid pressure *P*_{s}, a specific plugging rate *γ* corresponds to an optimal entrainment ratio. Altering the pressure of the ejector fluid not only has an impact on the optimal entrainment ratio, but also significantly affects the corresponding *γ*. Keeping *P*_{p} constant at 4 MPa and increasing *P*_{s} from 0.87 to 1.152 MPa results in the optimal entrainment ratio *ω** changing from 0.812 to 1.507, with the corresponding *γ* shifting from 26 to 35.

As Fig. 4 demonstrates, the primary fluid flow rate *q*_{p} remains constant under different induced fluid pressures *P*_{s}. As indicated in Fig. 8, within the *γ* range of 20%–30%, the induced fluid flow rate *q*_{s} changes only very slightly with increasing *γ*, but increases with increasing *P*_{s}. This is because a larger *P*_{p} creates a greater differential pressure between the induced fluid and the low-pressure area of the mixing chamber. When *γ* exceeds 30%, *q*_{s} begins to decline substantially. This explains why at *P*_{p} = 4 MPa and *P*_{s} = 1.152 MPa, the optimal *γ* is 35, and the optimal entrainment ratio decreases sharply with increasing *γ*.

The entrainment ratio, as the most critical performance indicator of an ejector, holds significant practical implications for production. In the operational dynamics of a thermal power plant system, amidst load fluctuations and changing working conditions, the adjustable steam ejector sustains a high *ω** by modulating the adjustment device according to the entrainment ratio. This adjustment process is not only convenient, but also ensures that heating stability is maintained.

## IV. CONCLUSION

In this study, we have employed a numerical simulation method to investigate the performance of an adjustable steam ejector under variable operating conditions, considering the operational parameters of primary pressure, induced pressure, and a structural parameter, namely, the plugging rate. Through an analysis of the results of a three-dimensional numerical simulation of the adjustable steam ejector, we arrive at the following conclusions:

The primary fluid mass flow rate decreases with increasing plugging rate and primary pressure. A reduction in the plugging rate results in an increase in primary pressure, while an increase in the primary pressure leads to a decrease in primary fluid mass flow rate.

As the plugging rate increases, the mass flow rate of the induced fluid initially experiences a minor increase, followed by a rapid decrease. The larger the primary pressure and induced pressure, the later the change in induced fluid mass flow with the plugging rate occurs. In a production environment, induced fluid mass flow stability can be achieved by adjusting the plugging rate within an appropriate range, as well as by modifying the primary pressure and induced pressure.

The critical back pressure decreases with increasing plugging rate, and, correspondingly, the entrainment ratio rises. When the back pressure reaches the critical back pressure, the steam ejector attains its optimal operating point. However, if the back pressure exceeds this value, the secondary shock wave ceases to be generated, and the shock wave chain shortens, leading to a degradation in the performance of the steam ejector.

Under identical primary pressure and induced pressure conditions, there exists a plugging rate corresponding to the maximum entrainment ratio, referred to as the optimal entrainment ratio. This optimal entrainment ratio increases with increasing induced pressure and primary pressure. In a practical production environment, adjustments can be made to both the critical back pressure and the entrainment ratio by modifying the induced pressure, primary pressure, and plugging rate, thereby achieving heating stability.

## ACKNOWLEDGMENTS

This work was supported by the National Natural Science Foundation of China (Grant Nos. U2241257 and 51876166) and the Fundamental Research Funds for the Central Universities and Innovative Scientific Program of CNNC.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Jianxiang Gao**: Formal analysis (equal); Data curation (equal); Investigation (equal); Methodology (equal); Writing – original draft (lead). **Zepeng Yuan**: Formal analysis (supporting); Investigation (supporting). **Yuyan Hou**: Data curation (equal); Investigation (supporting); Methodology (equal). Writing – review & editing (equal). **Weixiong Chen**: Project administration (lead); Funding acquisition (lead); Supervision (lead); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available within the article.