Numerical Calculation of the Influence of Reflux Hole Area on the Self‐Priming Performance of a Prototype Self‐Priming Pump

Abstract

ABSTRACT

To investigate the impact of the reflux hole area on the self‐priming performance of a self‐priming pump, this study innovatively established a circulating pipeline system that includes the self‐priming pump, water tank, and other components. Additionally, the influence of increasing rotational speed was taken into account to ultimately simulate accurate self‐priming performance. The upper parts of the water tank, the self‐priming pump, and the initial part of the pipeline were set to contain air, while the rest was filled with clear water. Additionally, the acceleration phase of the rotational speed was also included in the calculation through user‐defined functions. Therefore, the computational physical model established closely matches the actual situation, effectively reflecting the real and complete self‐priming process. Based on the VOF (volume of fluid) multiphase flow model, the numerical study analyzed the effects of three different initial water storage volumes on the self‐priming performance of the pump. The study found that the largest reflux hole area does not result in the shortest time for a complete self‐priming process; instead, there exists an optimal value. When the reflux hole area is small, it takes longer during the oscillatory gas discharge stage, with an extended time of up to 4 s, indicating that the size of the reflux hole mainly affects this stage of the self‐priming process. The influence of the reflux hole area on the head and the time required to reach a stable value is relatively minor, with differences within approximately 1 s and a head difference of around 0.1 m. The “internal leakage” phenomenon during the early stages of the self‐priming process results in a slightly lower stable flow rate at the pump outlet compared to the stable flow rate at the pump inlet. The article reveals the impact of the size of the reflux hole area on the efficiency and stability of the self‐priming process in a self‐priming pump, which provides crucial guidance for optimizing the design and operating conditions of the pump.

Introduction

A self-priming pump is a type of centrifugal pump with the capability to prime itself, which means it can start up with only the impeller partially filled with liquid and then automatically draw in the liquid after starting. It is widely used in fields such as agricultural irrigation, industrial production, and urban water supply. The working principle of a self-priming pump relies on the use of special internal structures or auxiliary devices to expel air from the pump chamber at startup, creating a vacuum that subsequently draws in the liquid. This not only requires the pump body to have excellent sealing properties but also necessitates a rapid and efficient self-priming capability. To this end, numerous scholars have conducted in-depth studies on the self-priming performance of self-priming pumps [1].

Research on pumps has mainly focused on the optimization of pump body structure, blade design, and improvements to auxiliary devices to enhance self-priming efficiency and reduce self-priming time. For example, Wu et al. designed a dual-blade centrifugal self-priming pump. The study found that rotational speed and the position of the reflux hole significantly affect the movement path and size distribution of bubbles, thus affecting the self-priming speed of the pump. By optimizing these two parameters, it is possible to effectively shorten the self-priming time and improve the operational efficiency of the pump [2]. Zhao et al. studied the impact of adding non-connected miniature blades with different parameters to the outer casing of a centrifugal pump's impeller on cavitation performance, showing that the presence of small blades slightly increased the turbulent kinetic energy in the low-turbulent kinetic energy region near the impeller inlet while significantly decreasing the turbulent kinetic energy in the high-turbulent kinetic energy region near the outlet, and also reduced the overall amplitude of the main frequency of pressure pulsation during pump operation, making it more stable [3]. Chang et al. proposed a new type of self-priming pump that can start without pre-filling with water, aiming to reduce start-up time and energy consumption, and used CFD software for numerical calculations and comparative experiments for validation [4].

There has also been research on the cavitation phenomenon in pumps, indicating that cavitation can lead to adverse effects such as the deterioration of hydraulic performance in centrifugal pumps [5]. Yang et al. combined numerical simulation with experimental validation to address the cavitation issue of self-priming pumps under overload conditions [6]. The design of the internal flow passage of the pump, the clearance between the impeller and the pump casing, and the layout and dimensions of the reflux hole are all considered key factors affecting pump performance [1]. Allali et al. explored the impact of volute shape and fluid properties on the flow characteristics within the pump [7]. Jin et al. studied the effect of the clearance between the impeller and the volute tongue on pump performance and pressure fluctuation [8]. Cheremushkin et al. demonstrated, taking viscosity into account, that an impeller with specially shaped blades having a smaller angle and height operates more efficiently than a disc-type impeller [9]. Mohammedali et al. optimized the impeller blades and the volute tongue, thereby improving the efficiency of the pump [10].

At the same time, with the development of technology and the increase in application demands, researchers have begun to pay more attention to the internal flow characteristics of pumps and their specific impact on performance. Chalghoum et al. used numerical calculation methods with various turbulence models to predict the performance of fluids within the complex geometries of pumps [11]. Shim and Kim studied the flow instability in volute centrifugal pumps and its relationship with performance characteristics [12] and investigated the influence of the number of impeller blades on the interaction between the impeller and volute as well as on flow instability [13]. They successfully suppressed the flow instability in centrifugal pumps by optimizing the impeller and volute [14]. Many scholars have also improved gas–liquid two-phase flow characteristics by adjusting blade geometric parameters, reflux hole position, and area. For example, Cheng et al. aimed to improve the performance of self-priming pumps by optimizing blade geometric parameters, targeting head and efficiency. Through analysis of the internal pressure field of the pump, they concluded that adjusting the blade curvature radius and outlet angle can change the area and uniformity of the low-pressure region inside the fluid, thereby enhancing pump performance [15]. Alemi et al. studied the effect of the position of the blade relative to the volute tongue on the instantaneous pump characteristics [16]. Wang et al. researched the impact of the reflux hole position on the performance of self-priming pumps [17]. Zhou et al. conducted three-dimensional transient flow field numerical simulations for the studied pump under different operating conditions, analyzing the pressure difference, reflux volume, and transient flow characteristics near the reflux hole, and further studied the impact of the reflux hole area on the pressure fluctuation characteristics and performance of the pump [18]. Some scholars have also analyzed the pressure pulsation, radial force, and other characteristics through energy characteristic experiments and self-priming experiments on pumps with different structures and parameters [19, 20]. Achour et al. used the entropy generation method to study the hydraulic loss and performance degradation mechanism of a centrifugal volute pump when handling non-Newtonian emulsions [21]. Al-Obaidi performed transient numerical calculations of the flow field in a centrifugal pump under single-phase and cavitation conditions, conducting both qualitative and quantitative analyses of all results to better understand the flow structure under single-phase and cavitation conditions in centrifugal pumps [22]. Sakran et al. studied the effect of the blade wrap angle on the flow field characteristics and energy distribution of low specific speed centrifugal pumps [23]. Zhao et al. studied the pressure pulsation characteristics of self-priming pumps, and the results showed that under rated and high flow conditions, the monitoring points in the volute passage exhibited a distinct periodic pattern, with peak pressure pulsations occurring at the eighth harmonic band of the blade frequency [24].

Using Large Eddy Simulation technology, Kye et al. studied the flow field characteristics of a volute-type centrifugal pump under both design and off-design conditions, finding that separation bubbles occur on both the pressure and suction surfaces of the impeller [25]. Qian et al. established an experimental platform including a high-speed camera and a transparent pump body to investigate the gas–liquid two-phase flow patterns in self-priming pumps during the self-priming process [26, 27]. Chang et al. further revealed the gas–liquid two-phase flow characteristics of self-priming pumps during the self-priming process using numerical calculation methods [28]. These studies show that self-priming pumps experience a gas–liquid two-phase flow state during the start-up process, with the appearance of bubbles having a significant impact on internal flow characteristics. At the same time, this also confirms the effectiveness of numerical calculation methods in simulating such complex flow processes.

New intelligent fault diagnosis methods have also been proposed to address the issue of difficulty in accurately extracting key fault feature information with traditional methods. Some studies have utilized machine learning and artificial intelligence algorithms to optimize self-priming pump design. For example, artificial neural networks and particle swarm optimization algorithms were applied to globally optimize mathematical models, thereby improving the pump's efficiency [29]. Wang et al. proposed a new approach that uses artificial intelligence algorithms to enhance the pump's head, efficiency, and power by optimizing parameters such as the cross-sectional area of the centrifugal pump casing, impeller interference, volute tongue length, and angle. Experimental results verified the performance improvement [30].

Al-Obaidi and Alhamid conducted an in-depth investigation into the mixed-flow hydrodynamics of axial flow pumps under five different operating conditions [31], emphasizing the importance of studying the internal flow characteristics of axial pumps [32]. Their research revealed significant impacts of various flow conditions on the main flow characteristics and dynamics of axial flow pumps, particularly the high dependency of vortex structures and transport, including vortex extension, induced vortex formation, and tip clearance flow phenomena [33]. Considering changes in flow conditions, they found that the study of dynamic flow patterns within axial flow pumps is especially critical [34]. Moreover, they explored the internal flow characteristics and pressure fluctuation features of axial flow pumps under different water conditions, noting that clearance return flow phenomena cannot be ignored due to the large clearances present in axial flow pumps [35]. Regarding the impact of guide vanes, the results showed that adding guide vanes to the axial impeller can increase pressure, kinetic energy, shear stress, and velocity [36]. Meanwhile, tail blades play a crucial role under different flow rates due to mixing and separation effects [37]. Notably, Al-Obaidi also introduced the method of cavitation detection using acoustic technology, which can provide detailed information [38], and mentioned the application of vibration techniques for the detection and diagnosis of cavitation phenomena within centrifugal pumps [39].

The reflux hole is crucial in self-priming pumps, enabling gas–liquid separation during priming. The liquid, separated from gases expelled via the pump outlet, returns to the volute region through the reflux hole under gravity, aiding in continuous self-priming and gas discharge. Investigating the reflux hole's size is essential for optimizing pump design.

A test system with a self-priming pump, valves, water tank, and pipelines was set up, containing water and connected to atmospheric pressure. The study also accounted for the acceleration phase of rotational speed using user-defined functions, ensuring the physical model accurately represents real-world conditions. Numerical analysis explored how three different reflux hole sizes affect the pump's self-priming performance.

Calculation Model and Method

Self-Priming Pump Model

The physical self-priming pump and the internal fluid domain of the pump are shown in Figure 1. The main performance parameters of this pump are: flow rate Q = 120 m³/h, head H = 75 m, and rotational speed n = 2950 r/min. Other geometric parameters are listed in Table 1. The internal fluid computational domain of the self-priming pump includes the S-pipe, impeller domain, volute domain, gas–liquid separation chamber, front and rear cavities, wear ring clearance, and reflux hole.

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Table 1 Main geometric parameters of the self-priming pump.

Component	Parameter	Value
Pipeline	Pump inlet diameter/mm	157
Inlet pipe length/mm	500
Pump outlet diameter/mm	120
Outlet pipe length/mm	500
Gas–liquid separation chamber	Height/mm	680
Width/mm	480
Volute	Outlet size/mm	90 × 48
Base circle diameter/mm	282
Impeller	Rotational speed/r·min−1	2950
Outlet width/mm	14
Inlet diameter/mm	113.5
Outer edge diameter/mm	251
Number of blades	5

Circulating Pipeline System Model

The circulating pipeline system established in this paper is shown in Figure 2. The system consists of a water tank, a self-priming pump, inlet and outlet pipelines, and valves. Among these, the valves are used to regulate and control the flow rate within the pipeline system and thus have been simplified in the model. The overall geometric dimensions of the circulating system are as shown in Figure 2b. The inlet pipeline of the self-priming pump is composed of a vertical section, a horizontal section, and a bend, with the diameter of the inlet pipeline being 157 mm throughout; the outlet pipeline of the self-priming pump consists of two sections, named outlet pipeline 1 and outlet pipeline 2, with the diameter of the outlet pipelines being 120 mm each. The lengths of the vertical section of the inlet pipeline and the vertical section of outlet pipeline 2 are 600 and 1100 mm, respectively. The dimensions of the water tank are 1500 × 1200 × 900 mm. To prevent the fluctuations and surges at the outlet of the self-priming pump's outlet pipeline from affecting the inflow at the inlet of the inlet pipeline, a baffle was set up in the middle part of the water tank, with its dimensions being 50 × 600 × 900 mm.

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Grid Generation and Independence Verification

The entire fluid domain was meshed using ICEM CFD 19.2, with the grid for the circulating system and the self-priming pump shown in Figure 3. To eliminate the influence of the number of grid cells on the calculation results, an independence verification for the grid of the circulating system was performed, with the results as shown in Figure 4. After the grid independence verification, it was found that when the number of grid cells in the circulating system reached 4,431,931, the static pressure difference at the inlet and outlet of the self-priming pump had basically stabilized, corresponding to a grid count of 220,561 for the computational domain of the self-priming pump. The numbers of grid cells for the water tank, inlet pipeline, valve, outlet pipeline 1, and outlet pipeline 2 were 1,344,340, 364,996, 135,688, 208,852, and 172,424, respectively. This number of grid cells is slightly insufficient for simulating fine flow within the boundary layer but is sufficient for predicting external characteristics and capturing the macroscopic flow structures inside. Through the mesh quality inspection, it was found that the mesh quality is good, with the minimum orthogonality quality being greater than 0.2. Additionally, except for the gas–liquid separation chamber in the self-priming pump, which used tetrahedral unstructured grids, all other flow components adopted hexahedral structured grids.

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Governing Equations

During the self-priming process of the self-priming pump, there is a distinct interface characteristic between the gas and liquid phases. Therefore, the VOF (Volume of Fluid) multiphase flow model, which is most suitable for describing the gas–liquid interface [41], was selected for the unsteady simulation of the self-priming stage. The VOF model is a surface tracking method based on a fixed Eulerian grid, where the phases do not mix with each other, and the VOF model can be used to observe the interface between the phases. In transient calculations, the basic equations of the VOF model include the volume fraction continuity equation, the continuity equation, and the momentum equation, as follows:

Volume fraction continuity equation [41]: 1$\frac{\partial {\alpha }_1}{\partial t}+u\cdot \nabla {\alpha }_1=0,$ 2$\frac{\partial {\alpha }_2}{\partial t}+u\cdot \nabla {\alpha }_2=0.$

Continuity Equation [41]: 3$\frac{\partial \rho }{\partial t}+\nabla \cdot \left(\rho u\right)=0.$

Momentum Equation [41]: 4$\frac{\partial \rho u}{\partial t}+(\rho u\cdot \nabla )u=-\nabla p+\nabla \cdot \left[\mu (\nabla u+\nabla u^T)\right]+\rho g+F.$

In the equations: ${\alpha }_1$ and ${\alpha }_2$ represent the volume fractions of the water phase and the gas phase, respectively, and ${\alpha }_1+{\alpha }_2=1$; u is the velocity; t is time; p is the static pressure; $\nabla$ is the Hamiltonian operator; $\mu$ is the mixture dynamic viscosity coefficient, $\mu ={\alpha }_1{\mu }_1+{\alpha }_2{\mu }_2$, where ${\mu }_1$ and ${\mu }_2$ are the dynamic viscosity coefficients of the water phase and the gas phase, respectively; $\rho$ is the mixture density, $\rho ={\alpha }_1{\rho }_1+{\alpha }_2{\rho }_2$, where ${\rho }_1$ and ${\rho }_2$ are the densities of the water phase and the gas phase, respectively; g is the gravitational acceleration; F is the equivalent volumetric force form of surface tension.

Existing literature has confirmed that the RNG k-ε model can achieve good computational results in pump studies [42–44]. Therefore, this paper selects the RNG k-ε two-equation model to close the Reynolds-averaged equations for unsteady turbulence calculations. The RNG k-ε model is an improvement over the Standard k-ε model; compared to the standard k-ε model, the RNG k-ε model introduces the mean strain rate of the main flow, increasing the influence of the average strain rate [45]. Additionally, the turbulence generation and turbulent kinetic energy dissipation equations in this model are the same as those in the standard k-ε model, with only the constant terms being modified. Thus, for unsteady, incompressible flows, the turbulent kinetic energy dissipation equation is transformed into [45]: 5$\frac{\partial \left(\rho U_j\epsilon \right)}{\partial x_j}=\frac{\partial }{\partial x_j}\left[\left(\mu +\frac{{\mu }_t}{{\sigma }_{\epsilon \text{RNG}}}\right)\frac{\partial \epsilon }{\partial x_j}\right]+\frac{\epsilon }{\kappa }(C_{\epsilon 1\text{RNG}}{P}_{\kappa }-C_{\epsilon 2\text{RNG}}\rho \epsilon +C_{\epsilon 1\text{RNG}}{P}_{\epsilon b}).$

In the equation: ${\sigma }_{\mathrm{\epsilon RNG}}=0.7179$; $C_{\epsilon 1\mathrm{RNG}}=1.42-f_{\eta }$; $C_{\epsilon 2\mathrm{RNG}}=1.68$; The expression for the coefficient $f_{\eta }$ is [45]: 6$f_{\eta }=\frac{\eta \left(1-\frac{\eta }{4.38}\right)}{(1+{\beta }_{\text{RNG}}{\eta }^3)}.$

In the equation: ${\beta }_{\text{RNG}}=0.012$; The expression for the coefficient η is [45]: 7$\eta =\sqrt{\frac{P_{\kappa }}{\rho C_{\mu \text{RNG}}\epsilon }}.$

Considering the viscous effects, no-slip boundary conditions are applied at the wall, and the coupling of velocity and pressure is achieved using the SIMPLEC algorithm.

Computational Settings and Boundary Conditions

The discrete equations are solved using the SIMPLEC method to achieve the coupling of pressure and velocity, with the working pressure in all defined domains set to 0 Pa. The two-phase flow consists of air and water at room temperature, with densities of 1.225 kg/m³ and 998.2 kg/m³ and viscosities of 1.7894 × 10⁻⁵kg/(m·s⁻¹) and 1.003 × 10⁻³kg/(m·s⁻¹), respectively. The convergence accuracy for all parameters is set to 10⁻⁴, and the blade, as well as the front and rear cover walls, rotate without slip along with the impeller domain. Considering the actual situation, the top of the water tank is connected to the air. Therefore, the top of the water tank is set to atmospheric pressure. Since the entire self-priming system is a closed loop, no other boundary conditions need to be set. The initial gas–liquid distribution within the self-priming pump's circulating pipeline system under rated conditions is shown in Figures 5 and 6. At this valve opening, the stable flow rate of the circulating pipeline system is the rated flow rate of 120 m³/h. The liquid surface is 400 mm from the top of the water tank, with air above and water below. The initial liquid level inside the self-priming pump is level with the bottom of the inlet pipe, with the liquid phase below and air above. In addition, the remaining parts of the circulating pipeline system are initially filled with air. The self-priming height of this computational model is 1.0 m. Initially, the initial liquid level of the stored water in the self-priming pump is the same height as the bottom of the inlet pipe. Within the first 0.2 s after startup, the impeller speed increases linearly from rest to a stable 2950 r/min through the application of user-defined functions. Afterward, the impeller rotates at a constant speed of 2950 r/min. The specific speed change can be expressed as follows: 8$n=\left\{\begin{array}{l}14,750t0\le t\le 0.2\\ 2950t>0.2\end{array}\right..$

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Results Analysis

Verification of Numerical Method

To verify the reliability of the numerical calculation method, the numerical results were compared with the experimental results, as shown in Figure 7 [40]. It can be seen that the calculation results are in good agreement with the experimental results, with the absolute value of relative deviation within 5%, indicating that the numerical calculation method is reliable and the calculation results are accurate.

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Scheme Design

The change in the reflux hole area affects the pressure difference on both sides, thereby influencing the reflux volume. Studying the impact of changes in the reflux hole on the self-priming characteristics of a self-priming pump is of great significance. The area of the reflux hole is typically determined by an empirical formula [18], as shown in the following equation: 9$S=\frac{\pi }{4}{(1.1~1.8)}^2{(Q/n)}^{2/3}.$

In the equation: S is the area of the reflux hole, m²; Q is the design flow rate, m³/s; n is the design rotational speed, r/min.

By substituting the pump's design parameters, the optimal range for the reflux hole area is determined to be: S = 478.49~1281.31 mm². According to experience, the shape of the reflux hole for the self-priming pump is designed as an oblong hole, with an initial area of 715 mm², located at the bottom of the volute, starting from the self-sealing tongue and extending approximately 130° in the direction of the impeller rotation. When studying the influence of the reflux hole on the self-priming characteristics of the pump, the position of the reflux hole remains unchanged, and two additional area schemes are added based on the initial reflux hole. The corresponding three reflux hole areas S are 521.06, 715.16, and 938.39 mm², with a ratio of about 0.73:1:1.31. These are sequentially named small, medium, and large reflux hole areas, as shown in the schematic diagram in Figure 8.

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Grid Generation and Boundary Conditions

The computational domain for the fluid was discretized using structured grids with ICEM CFD 21.1 software, where the reflux hole grid and the volute grid were treated as a single component. The number of grids in the volute flow field and within the self-priming pump under different reflux hole schemes is shown in Table 2, and the grid generation is illustrated in Figure 9.

Table 2 Grid number of the volute.

Scheme	Area S/(mm2)	Volute grid count/Pieces	System grid count/pieces
Small reflux hole	521.06	364,978	4,431,793
Medium reflux hole	715.16	365,116	4,431,931
Large reflux hole	938.39	365,048	4,431,863

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Self-Priming Performance

Figure 10 shows the variation of liquid phase mass flow rate at the inlet of the self-priming pump under different reflux hole areas. It can be seen from the figure that, under the three reflux hole schemes, the liquid flow rate at the inlet of the self-priming pump generally follows a similar evolution trend, initially remaining constant before fluctuating and rising to a stable value. Among these, with medium and large reflux hole areas, the instantaneous flow rate values at the pump inlet at various time points are relatively close, whereas with a small reflux hole area, there is a noticeable delay.

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Before 2.0 s, the gas–liquid mixing and separation occurring within the self-priming pump only involves the originally stored liquid, and the liquid from the water tank has not yet entered the pump. At t = 2.0–2.5 s, under the medium and large reflux hole area schemes, the liquid gradually enters the pump through the pump inlet, reaching a peak of 10.163 and 11.524 kg/s at 2.5 s, respectively; whereas, with the small reflux hole area, due to the smaller area of the reflux hole, the exhaust inside the pump is relatively slower, and the liquid from the water tank has still not entered the pump. During the period of t = 2.5–2.9 s, for the medium and large reflux hole areas, the liquid impacts the S-pipe wall, with some of the liquid flowing in the opposite direction, causing the flow rate value at the pump inlet to decrease rapidly; with the small reflux hole area, only a very small amount of liquid enters the pump. After 2.9 s, for the medium and large reflux hole areas, the liquid phase flow rate at the pump inlet shows a fluctuating upward trend, reaching relatively stable values of 33.06 and 32.28 kg/s at 6.7 and 6.3 s, respectively. After this, the flow rate at the pump inlet exhibits an up-and-down fluctuating trend. For the small reflux hole area scheme, it is only after 3.4 s that there is a noticeable increase in the liquid phase flow rate at the pump inlet, reaching a relatively stable value of 34.67 kg/s at 7.6 s. It can be observed that, with the small reflux hole area, the time for the liquid from the water tank to enter the self-priming pump is significantly delayed compared to the other two schemes; however, the final stable flow rates achieved by all three are not much different, being around 33.8 kg/s.

The variation of instantaneous liquid phase mass flow rate at the pump outlet under different reflux hole area schemes is shown in Figure 11. It can be seen that the evolution trend of the instantaneous liquid phase flow rate under the three reflux hole area schemes is relatively similar, all initially remaining constant before rapidly increasing, then rapidly decreasing, and finally fluctuating to a stable value. However, with the small reflux hole area, the time for the rapid increase at the pump outlet is noticeably faster than the other two, and the duration during which the liquid phase flow rate at the pump outlet is negative is also longer than that of the other two schemes.

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For the medium and large reflux hole area schemes, before 0.9 s, there is no liquid flow at the pump outlet, and the liquid phase flow rate remains at 0. At t = 0.9–1.61 s, under the effect of impeller rotation, the liquid originally stored in the pump is thrown toward the pump outlet, causing the liquid phase flow rate at the pump outlet to rise, reaching maximum values of 51.59 and 45.22 kg/s at 1.52 and 1.61 s, respectively. After reaching the maximum flow rate, due to the lack of liquid phase distribution within the impeller domain, the centrifugal force generated by the rotating impeller rapidly decreases, and the liquid flowing out from the pump outlet cannot continue to flow out. Under the effect of inertia, the liquid phase at the pump outlet still shows a state of outflow, but the value continuously decreases, reaching minimum values of 0.359 and 0.563 kg/s at 2.93 and 2.43 s, respectively. After this, the liquid from the water tank enters the impeller, and the centrifugal force generated by the impeller increases again, leading to a fluctuating increase in the liquid phase flow rate at the pump outlet, reaching a stable value of 33.23 kg/s at 10.0 s. For the small reflux hole scheme, the flow rate at the pump outlet quickly rises at 0.19 s, reaching a maximum value of 50.38 kg/s at 0.22 s. It then rapidly decreases, reaching a minimum value of −3.08 kg/s at 1.22 s. The reason for the negative value of the liquid phase flow rate at the pump outlet may be that the liquid cannot be continuously thrown out from the volute domain, and at this time, the liquid outside the pump outlet flows back into the pump under the action of gravity. During the period of t = 1.22−3.91 s, the liquid from the water tank does not enter the impeller domain, and there is no continuous liquid discharge, resulting in severe fluctuations in the liquid phase flow rate at the pump outlet. After 3.91 s, the liquid from the water tank enters the pump, and the liquid gradually flows out from the pump outlet, corresponding to a fluctuating increase in the liquid phase flow rate at the pump outlet, reaching a stable value of 33.01 kg/s at 11.9 s.

It can be observed that during the self-priming process, under different reflux hole area schemes, the stable flow rate values reached at the pump outlet are basically consistent, but the time it takes for the small reflux hole area to reach a stable value lags behind the other two conditions. Numerically, the stable flow rate at the pump outlet is slightly lower than the stable flow rate at the pump inlet, indicating that an “internal leakage” of flow occurs during the self-priming process.

During the self-priming process, gas and liquid continuously mix and separate, and the proportion of gas phase in each flow passage component within the pump keeps changing. The proportions of the gas phase in each flow passage component of the self-priming pump under different reflux hole area schemes are shown in Figure 12. It can be clearly seen from Figure 12a–e that compared with the other two schemes, there is a significant difference in the proportion of the gas phase within the flow passage components for the small reflux hole area scheme.

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During the first stage of self-priming (the rapid suction stage), under the action of impeller rotation, a vacuum is generated at the impeller inlet, and the gas in the inlet pipeline is quickly drawn into the pump. The originally stored liquid is thrown into the gas–liquid separation chamber. For the medium and large reflux hole area schemes, from t = 0–1.2 s, the impeller starts from rest and accelerates uniformly, reaching the rated speed of 2950 r/min at 0.2 s, and then maintains the rated speed. Under the effect of the impeller, a vacuum is generated at the impeller inlet, and the gas in the inlet pipeline is drawn into the pump. The liquid originally stored in the S-pipe, impeller, and volute domains is thrown into the gas–liquid separation chamber and outside the pump. During this stage, the proportion of the gas phase in the S-pipe domain rapidly increases from 0.0293 to 1.0. In the impeller domain, there is no gas phase distribution before 0.71 s, and it rises from 0 to 0.93 after 0.71 s. In the volute domain, there is no gas phase distribution before 1.1 s, and it rises to 0.114 at 1.2 s. In the pump cavity domain, there is no gas phase distribution before 1.08 s, and it reaches 0.084 at 1.2 s. In the gas–liquid separation chamber domain, the proportion of the gas phase decreases rapidly from 0.356 to 0.074. For the small reflux hole area scheme, from t = 0–1.07 s, the gas in the inlet pipeline enters the pump. Due to the smaller reflux hole area, the flow leakage at the reflux hole is less, so its rise rate is much faster than the other two schemes. In the S-pipe domain, the proportion of the gas phase rises extremely fast, reaching 0.959 at 0.75 s. In the impeller domain, the proportion of the gas phase remains constant initially and then rises rapidly to its maximum value, reaching 0.835 at 0.3 s and 0.991 at 1.07 s. In the volute domain, the proportion of the gas phase remains constant initially and then rises to its maximum value, reaching 0.823 at 0.95 s. In the pump cavity domain, the proportion of the gas phase remains constant initially and then rises rapidly; there is no gas phase before 0.2 s, and it rises to reach its maximum value of 0.954 at 1.02 s. In the gas–liquid separation chamber domain, the trend is for the proportion of the gas phase to decrease rapidly at first and then increase slowly, reaching a minimum value of 0.217 at 0.32 s and then rising to 0.258 at 1.07 s.

During the second stage of self-priming (the oscillatory gas discharge stage), the liquid in the gas–liquid separation chamber enters the volute domain through the reflux hole and repeatedly participates in the self-priming exhaust, gradually expelling the gas from the inlet pipeline. The proportion of the gas phase in each flow passage component exhibits different characteristics under various reflux hole area schemes. For the medium and large reflux hole area schemes: from t = 1.2–2.0 s, for the S-pipe domain, since the liquid in the water tank has not yet entered the pump, the proportion of the gas phase remains at 1. For the impeller domain, the change in the two-phase flow within the domain is small, with the proportion of the gas phase rising from 0.93 to 0.98. For the volute domain, under the two reflux hole area schemes, the proportion of the gas phase rises from 0.114 to 0.765 and 0.679, respectively. For the pump cavity domain, the proportion of the gas phase rises from 0.084 to 0.804 and 0.746, respectively. For the gas–liquid separation chamber domain, as the volume of gas flowing out from the pump outlet decreases, and the gas from the inlet pipeline continuously enters the gas–liquid separation chamber through the volute outlet, the proportion of the gas phase in the chamber shows a trend of first decreasing and then increasing. Under the two reflux hole area schemes, it reaches a minimum value of 0.015 and 0.011 at 1.45 s, and 0.062 and 0.054 at 2.0 s, respectively. For the small reflux hole area scheme, the duration of the oscillatory gas discharge stage is significantly longer than that of the other two area schemes. From t = 1.2–3.4 s, in the S-pipe domain, the proportion of the gas phase remains around 0.97; in the impeller domain, the proportion of the gas phase also remains around 0.97, with almost no liquid phase distribution; in the volute and pump cavity domains, compared to the medium and large reflux hole area schemes, the fluctuations in the proportion of the gas phase are much more intense; in the gas–liquid separation chamber domain, the gas from the inlet pipeline flows into the gas–liquid separation chamber through the volute outlet, and the proportion of the gas phase slowly increases from 0.258 to 0.277.

During the third stage of self-priming (the accelerated gas discharge stage), the liquid in the water tank quickly enters the pump, accelerating the completion of gas discharge. Among these, under the medium and large reflux hole area schemes, each flow passage component exhibits the same evolution characteristics. In the S-pipe domain, the proportion of the gas phase first rapidly decreases, reaching a minimum value of 0.492 and 0.458 at 3.5 s, respectively, and then rapidly increases, reaching a maximum value of 0.697 and 0.808 at 3.96 and 4.10 s, respectively. The reason for this increase may be the occurrence of backflow in the impeller inlet pipeline during this period. After that, the gas fluctuation in the S-pipe decreases, and by 8.86 s, the gas phase in the S-pipe is basically expelled. In the impeller domain, the proportion of the gas phase shows a trend of fluctuating decrease, with the gas being expelled at 10.1 s, which is slightly later than the time required to expel the gas from the S-pipe. In the volute domain, the proportion of the gas phase also shows a trend of fluctuating decrease, with the gas being basically expelled by 9.02 s. In the pump cavity domain, the proportion of the gas phase first increases and then slowly decreases to a stable value, reaching a stable value of 0.277 around 10.0 s. In the gas–liquid separation chamber domain, the proportion of the gas phase within the chamber shows a trend of first increasing and then decreasing, with the proportion of the gas phase being 0.007 and 0.010 at 12.0 s. For the small reflux hole area scheme, the proportion of the gas phase in the S-pipe rapidly decreases, and the gas is expelled at 9.17 s, which is slightly later than the other two schemes. In the impeller domain, the time for gas expulsion is not much different from the other two schemes, also expelling the gas around 10.0 s. In the volute domain, the time required for gas expulsion is significantly longer than the previous two schemes, with the gas being basically expelled only by 10.5 s. In the pump cavity domain, there is always some gas that cannot be expelled. In the gas–liquid separation chamber domain, compared to the other two schemes, the rate of decrease in the proportion of the gas phase is significantly greater, with the proportion of the gas phase decreasing to 0.017 at 12.0 s.

It can be seen that during the self-priming process of a self-priming pump, the area of the reflux hole mainly affects the oscillatory gas discharge stage, with a relatively minor impact on the overall gas discharge time within the pump.

Figure 13 shows the real-time changes in the volume fraction of gas inside the self-priming pump under different reflux hole areas. It is evident that the change in the volume fraction of gas inside the pump for the small reflux hole area scheme is distinctly different from the other two reflux hole area schemes. For the small reflux hole area scheme, during the first stage of self-priming (t = 0–1.07 s), the curve of the volume fraction of gas inside the self-priming pump shows a rapid upward trend, reaching 0.425 at 1.07 s. During this stage, the volume of gas discharged from the pump outlet is less than the volume of gas entering the pump through the inlet pipeline. In the second stage of self-priming (t = 1.07–3.4 s), the volume fraction of gas inside the self-priming pump fluctuates within a certain range, indicating a dynamic equilibrium between the volumes entering and exiting the pump. In the third stage of self-priming (t > 3.4 s), the gas content inside the pump first fluctuates up and down, then rapidly decreases, completing self-priming at 12.36 s. For the medium and large reflux hole area schemes, during the first stage of self-priming (t = 0–1.2 s), the volume of gas entering the pump is less than the volume being discharged, and the volume fraction of gas shows a decreasing trend. In the second stage of self-priming (t = 1.2–2.0 s), the liquid in the gas–liquid separation chamber repeatedly enters the volute domain through the reflux hole, participating in the self-priming gas discharge, gradually expelling the gas from the inlet pipeline; however, the volume expelled is smaller compared to the volume of gas entering the pump, leading to an increasing trend in the volume fraction of gas. In the third stage of self-priming (t > 2.0 s), the liquid from the water tank enters the pump, accelerating the expulsion of gas inside the pump, completing self-priming at 10.88 and 11.43 s, respectively. It can be observed that the size of the reflux hole area slightly affects the time required for self-priming completion, with a too small reflux hole area leading to a reduced gas discharge rate and a longer self-priming time.

[IMAGE OMITTED. SEE PDF]

To better observe the gas–liquid distribution and streamline distribution at the reflux hole, a cross-section passing through the reflux hole was created as shown in Figure 14. The gas phase distribution on the cross-section of the reflux hole during the self-priming process under three different reflux hole area schemes is shown in Figure 15. From the figures, it can be seen that under the small reflux hole area scheme, the self-priming pump enters the oscillatory gas discharge stage much faster than the other two schemes, and this stage lasts for a longer duration. For the medium and large reflux hole area schemes, between t = 0.2 and 1.0 s, the liquid phase in the S-pipe, impeller, and volute domains is slowly discharged, with no gas phase distribution near the reflux hole; between t = 1.0 and 2.0 s, the impeller and volute domains are filled with gas, and under the effect of pressure difference, the liquid at the bottom of the gas–liquid separation chamber flows into the volute domain through the reflux hole, participating in the self-priming gas discharge, which oscillates to expel the gas from the inlet pipeline; after t > 2.0 s, the liquid from the water tank gradually enters the impeller domain through the S-pipe, participating in self-priming, and the gas–liquid mixture is discharged from the volute outlet, while part of the liquid returns to the gas–liquid separation chamber under the action of gravity and re-enters the volute domain through the reflux hole to participate in self-priming gas discharge. For the small reflux hole area scheme, by 0.5 s, the liquid in the impeller domain is already drained, and the liquid in the gas–liquid separation chamber flows toward the volute domain through the reflux hole, but due to the smaller reflux hole area, the reflux volume is correspondingly reduced, leading to a slower gas discharge rate. At 4.0 s, the liquid from the water tank just begins to enter the pump, entering the third stage of self-priming, accelerating the expulsion of gas inside the pump. As can be seen in Figure 15j,k, at 6.0 s, the gas content in the gas–liquid separation chamber under the small reflux hole area scheme is significantly higher than that of the other two schemes, and there is still unexpelled gas at 8.0 s, whereas for the medium and large reflux hole area schemes, except for the pump cavity, there is basically no gas phase distribution within the pump. In summary, when the reflux hole area is smaller, the reflux volume entering the volute domain through the reflux hole is smaller, and the duration of the oscillatory gas discharge stage is longer; for the medium and large reflux hole area schemes, the gas phase distribution during the self-priming process is not significantly different, and the self-priming times are also similar.

[IMAGE OMITTED. SEE PDF]

[IMAGE OMITTED. SEE PDF]

Hydraulic Characteristics

Figure 16 shows the instantaneous flow rate changes of the mixed phase at the pump outlet during the self-priming process under different reflux hole area schemes. It can be observed that in the initial start-up phase, there is a flow surge phenomenon for all three area schemes, but the time of occurrence is delayed as the area increases, with the maximum surge flow rates of 174.61, 251.25, and 160.68 m³/h reached at 0.22, 1.10, and 1.53 s, respectively. After this, with very little liquid in the impeller domain, the centrifugal force generated by the rotating impeller decreases, causing the instantaneous flow rate to drop rapidly until the liquid from the water tank enters the impeller domain, which occurs at 2.0, 2.0, and 4.0 s, respectively. It can be noted that during this stage, under the small reflux hole area scheme, there is a backflow phenomenon of the gas–liquid mixture at the pump outlet, with corresponding negative values for the instantaneous flow rate. After the gas from the water tank enters the impeller domain, the flow rate at the pump outlet begins to fluctuate and rise, reaching relatively stable values of 119.6 m³/h at 11.13, 6.59, and 10.21 s, respectively. In summary, both too small and too large reflux hole areas will delay the time it takes for the flow rate to reach a stable value.

[IMAGE OMITTED. SEE PDF]

The instantaneous head changes during the self-priming process under different reflux hole area schemes are shown in Figure 17. It can be seen that before 2.8 s, the gas–liquid distribution inside the pump is complex, with intense pressure fluctuations at the inlet and outlet, which is reflected in the head as significant fluctuations. After 2.8 s, the head begins to increase for all cases. For the medium and large reflux hole area schemes, the increase in head follows a similar pattern, with an initial rapid fluctuating growth; the growth rate decreases at 6.3 s and reaches a stable value of 75.7 m at 9.6 s. Under the small reflux hole area scheme, there is a slow initial increase; the growth rate increases at 5.2 s, decreases again at 8.0 s, and reaches a stable value of 75.6 m at 10.7 s. In summary, the reflux hole area has a minor impact on the numerical value of the head and the time required to reach a stable value.

[IMAGE OMITTED. SEE PDF]

The instantaneous shaft power curves of the self-priming pump during the self-priming process under different reflux hole area schemes are shown in Figure 18. As can be seen from the figure, the instantaneous shaft power curve under the small reflux hole area scheme has a significant difference compared to the other two. Before 0.2 s, the impeller and volute domain is filled with water, and as the rotational speed increases, the shaft power rises rapidly, increasing by 54.36 kW within 0.2 s. Afterward, as the liquid phase is continuously discharged, the shaft power of the pump drops quickly until 4.0 s, when the liquid from the water tank enters the impeller domain, and the instantaneous shaft power begins to increase, reaching a stable value of 75.50 kW at 10.5 s. For the small and medium reflux hole area schemes, similarly, the shaft power rises rapidly before 0.2 s; between 0.2 and 1.03 s, there is still liquid in the impeller domain, and the shaft power maintains a small fluctuation; between 1.03 and 1.37 s, the liquid in the impeller domain is emptied, and the shaft power decreases rapidly; until 2.8 s, when the liquid re-enters the impeller domain, the shaft power starts to increase, reaching a stable value of 75.1 kW at 10.0 s. It can be observed that a smaller reflux hole area prolongs the time required for the self-priming pump to exhaust and also delays the time it takes for the shaft power to reach a stable value.

[IMAGE OMITTED. SEE PDF]

The size of the reflux hole has a significant impact on the direction of the liquid phase flow near the reflux hole. To better investigate its influence, Figures 19 and 20, respectively, show the instantaneous changes in the liquid phase mass flow rate near the gas–liquid separation chamber side reflux hole during the self-priming process under different reflux hole schemes, as well as the velocity streamline distribution near the reflux hole.

[IMAGE OMITTED. SEE PDF]

[IMAGE OMITTED. SEE PDF]

During the early stage of self-priming, the impeller and volute passage are filled with liquid, and the liquid phase mass flow rate at the reflux hole is negative, indicating that the direction of liquid phase flow is from the volute domain through the reflux hole into the gas–liquid separation chamber. Combined with the flow at the inlet and outlet, an “internal leakage” phenomenon occurs within the self-priming pump during this stage. From Figure 19, it can be observed that under three different reflux hole area schemes, the minimum flow rates are reached at 0.186, 1.075, and 0.985 s, with values of −9.03, −15.85, and −19.99 m³/h, respectively. The larger the reflux hole area, the greater the leakage volume. Observing Figure 20a–e, one can clearly see that before reaching the minimum flow rate, the direction of liquid flow within the reflux hole is from the volute domain toward the bottom of the gas–liquid separation chamber, and the velocity values inside the reflux hole are relatively higher compared to its sides; moreover, the flow inside the chamber is significantly more complex under the small reflux hole area scheme, forming multiple vortices within the gas–liquid separation chamber.

During the middle stage of self-priming, the liquid phase content within the impeller and volute domain is extremely low, and the liquid phase mass flow rate at the reflux hole remains negative, but the magnitude of the flow rate continuously decreases. The direction of liquid phase flow at the reflux hole has not yet changed to flow from the volute domain toward the gas–liquid separation chamber. From the flow rate change graph, it can be seen that after reaching the minimum flow rate, the instantaneous flow rate value rapidly increases, with the flow direction changing at 0.25, 1.76, and 1.72 s, respectively, to flow from the gas–liquid separation chamber toward the volute domain. As shown in Figures 20f,g, at 1.5 s, under the small reflux hole area scheme, the liquid inside the gas–liquid separation chamber flows into the volute domain due to the pressure difference on both sides of the reflux hole; there are four vortices at the bottom of the gas–liquid separation chamber, and the flow is very complex. For the medium and large reflux hole area schemes, the liquid within the volute domain still flows toward the gas–liquid separation chamber; there is only one vortex inside the gas–liquid separation chamber, but the velocity values within the domain are relatively high, reaching a maximum of 20.0 m/s. At 2.0 s, the liquid from the water tank has entered the impeller and volute domain. Under all three reflux hole area schemes, the flow velocity within the volute domain is greater than that within the gas–liquid separation chamber, and under the influence of the pressure difference, the liquid phase flows from the gas–liquid separation chamber toward the volute domain.

In the later stage of self-priming, the liquid in the inlet pipeline rapidly enters the pump, accelerating the completion of gas discharge. During this stage, the direction of liquid phase flow at the reflux hole is from the gas–liquid separation chamber toward the volute domain. In this stage, the gas–liquid mixture inside the self-priming pump is thrown into the gas–liquid separation chamber by the impeller, where the liquid returns to the bottom of the volute and then back to the volute domain under the action of gravity, while the gas is discharged from the pump outlet. As can be seen from Figure 18, during this stage, the liquid phase flow rate at the reflux hole is positive and fluctuates within a certain range. Under the three different reflux hole area schemes, the stable flow rates are reached at 9.6, 9.9, and 9.5 s, with values of 1.62, 2.17, and 2.33 kg/s, respectively.

In summary, as the reflux hole area increases, the flow at the bottom of the gas–liquid separation chamber becomes more stable; the reflux volume from the gas–liquid separation chamber to the volute increases, and the phenomenon of flow leakage in the early stage of self-priming becomes more pronounced.

Conclusions

This paper innovatively established a circulating pipeline system including a self-priming pump, water tank, and other components, while also considering the impact of increasing rotational speed. The study focuses on the influence of the reflux hole area on the performance of the self-priming pump, ultimately achieving precise simulation results. The main conclusions are as follows:

• For large, medium, and small reflux hole areas, the self-priming times are approximately 11.43, 10.88, and 12.36 s, respectively. This indicates that a larger reflux hole does not guarantee the fastest priming.

• The pump with the smallest reflux hole shows a notably higher gas phase ratio compared to the other two. Both excessively small and large reflux holes can delay achieving a stable flow rate by up to 1.5 s. Under medium and large reflux holes, the head stabilizes at 75.7 m in 9.6 s, whereas the small reflux hole reaches 75.6 m in 10.7 s. Thus, reflux hole size has a minor impact on head stability and stabilization time.

• At 6.0 s, the small reflux hole scheme exhibits significantly more gas in the gas–liquid separation chamber than the others, with some gas remaining at 8.0 s. In contrast, medium and large schemes show almost no gas except inside the pump cavity. A smaller reflux hole results in less reflux flow into the volute, extending the oscillatory gas discharge phase.

• Increasing the reflux hole area leads to greater stability in the gas–liquid separation chamber's bottom flow, with reflux volumes to the volute increasing to 1.62, 2.17, and 2.33 kg/s for small, medium, and large holes, respectively. The “internal leakage” phenomenon becomes more evident as the reflux hole grows, slightly reducing the pump's outlet flow rate compared to its inlet.

These findings are of significant importance for optimizing the design of self-priming pumps and enhancing their efficiency. By adjusting the size of the recirculation hole, it is possible to improve the priming time and stability without compromising the overall performance of the pump. Moreover, understanding the impact of different sizes of recirculation holes on the gas–liquid flow conditions inside the pump facilitates the design of more efficient and stable self-priming pump systems.

In conclusion, future research can further explore the relationship between the optimal reflux hole area and pump performance. By adopting a more refined design and experimentation, the most optimized reflux hole size under various operating conditions can be determined to achieve faster self-priming times and higher efficiency. Future studies should focus on developing or improving existing gas–liquid separation technologies to reduce gas retention during the self-priming process, shorten the time of gas oscillation discharge phase, and enhance system stability.

Nomenclature

• α₁
• volume fraction of the water phase
• α₂
• volume fraction of the gas phase
• $u$
• velocity
• t
• time
• p
• static pressure
• $\nabla$
• Hamiltonian operator
• μ
• dynamic viscosity coefficient of the mixture
• ${\mu }_1$
• dynamic viscosity coefficient of the water phase
• ${\mu }_2$
• dynamic viscosity coefficient of the gas phase
• $\rho$
• density of the mixture
• ${\rho }_1$
• density of the water phase
• ${\rho }_2$
• density of the gas phase
• g
• acceleration due to gravity
• F
• equivalent volumetric force form of surface tension
• $f_{\eta }$
• equation coefficients
• η
• equation coefficients
• n
• rotational speed
• S
• reflux hole area
• Q
• design flow rate
• ε
• kinetic energy dissipation rate

Acknowledgments

The research was financially supported by the National Natural Science Foundation of China (No.52376037), Science and Technology Project of Quzhou (No.2023NC08), and Zhejiang Provincial Natural Science Foundation of China (No. LZY21E050001).

Conflicts of Interest

The authors declare no conflicts of interest.

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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