Numerical simulation of the effect of installation height on self-priming performance of a prototype self-priming pump

Abstract

To investigate the impact of installation height on the self-priming performance of a self-priming pump, this study established a circulating pipeline system incorporating the self-priming pump, a water tank, and other components. Clear water was placed in the lower part of the water tank and air in the upper part. The user-defined function (UDF) was employed to simulate the speed increase upon pump startup and subsequent constant-speed operation. Based on the Volume of Fluid (VOF) multiphase flow model, numerical simulations of the complete self-priming process were conducted for two different installation heights, focusing on gas-liquid two-phase flow. Additionally, vortex identification methods and entropy production theory were utilized to analyze the internal unsteady flow characteristics. The study revealed that as the installation height increases from 0.5 to 1.0 m, the self-priming time significantly prolongs by 6.260s, representing an 82% relative increase. During the oscillatory air discharge phase, the gas volume fraction within the volute remains consistently lower than that inside the impeller chamber. Throughout the self-priming process, the regions of significant hydraulic loss are primarily located at the outer edge of the impeller and the impeller inlet. This study is of great significance because it not only complements the understanding of how installation height affects pump performance but also provides practical guidance for optimizing the design and operational parameters of pumps, thereby improving efficiency and reducing energy consumption in real-world applications.

Introduction

Self-priming pumps, as essential liquid transfer devices, have been widely applied in various fields including chemical processing, oil and gas, firefighting, and agricultural irrigation^1–4. Given their broad application scope, mobile self-priming pumps must not only meet the demands for efficient and stable liquid transfer but also adapt to complex and variable working environments, such as different altitudes, temperature conditions, and road conditions. The self-priming phase of a self-priming pump primarily involves the processes of gas-liquid mixing, gas-liquid separation, liquid reflux, and gas discharge. After completing the self-priming phase, the pump operates as a conventional centrifugal pump. Currently, Computational Fluid Dynamics (CFD) technology is extensively utilized in pump research^5–7, and numerical simulation has become the predominant method for studying self-priming pumps. This approach enables detailed analysis of the pump’s internal flow characteristics and performance under various operating conditions, facilitating the optimization of design and operational parameters to enhance the pump’s efficiency and reliability.

To enhance the work efficiency and self-priming height of self-priming pumps, researchers have conducted extensive studies. Qian et al. constructed a visualization test bench and used a high-speed camera to capture the gas-liquid two-phase flow patterns during the self-priming process for different reflux hole structures. They found that the reflux holes affect the pump’s self-priming performance by influencing the reflux rate of the gas-liquid two-phase flow during the self-priming process, with the diameter of the reflux hole having a particularly significant impact on the duration of the mid-self-priming phase⁸. Zhou et al. analyzed the pressure differences, reflux flow rates, and transient flow characteristics near the reflux holes to investigate the effect of the reflux hole area on the pump’s pressure fluctuation characteristics and performance⁹. Wang et al. discovered through numerical simulations that the position of the reflux hole significantly affects the gas phase mass fraction within the impeller, thereby markedly reducing the time required for the mass fraction to exceed 80%¹⁰. Chang et al. proposed a novel water-free start-up self-priming pump designed to reduce energy consumption and the time required for initial operation. Through numerical simulations, they found that viscous entropy production (VEP) and turbulent entropy production (TEP) are the primary factors affecting energy conversion within the pump¹¹.

Cheng et al., using Response Surface Methodology (RSM) and CFD, investigated the internal pressure distribution characteristics and performance parameters of the pump chamber. They discovered that adjusting the blade curvature radius and outlet angle can alter the area and uniformity of low-pressure zones within the fluid, thereby enhancing pump performance¹². Wang et al. found through numerical simulations that the self-priming process of multi-stage self-priming pumps can be divided into three stages, with the middle stage, characterized by gas-water mixing and gas-water separation, being the primary determinant of the self-priming time. They also observed the ejection state of gas and water at the pump outlet using a high-speed camera^13,14. Zhang et al. employed an orthogonal experimental design method to study the effects of impeller diameter, blade outlet width, blade wrap angle, and the number of blades on the hydraulic performance of self-priming pumps. They ultimately determined the optimal parameter combination¹⁵.

Zhang et al. investigated the erosion characteristics within the flow domain of self-priming pumps conveying particulate-laden media using FLUENT with the Discrete Phase Model (DPM) and the Oka erosion model. They found that the volute and the leading edges of the blades are most susceptible to particle erosion, and the suction side of the blades and the near-axis regions also experience some particle impact¹⁶. Hu et al. numerically simulated the characteristics of short jet self-priming pumps under both steady and unsteady flow conditions using the standard k-ε turbulence model. They discovered that under high-flow conditions, the asymmetry of the low-pressure zones within the impeller increases, preventing the formation of stable vortex structures at specific locations. This asymmetry is a significant cause of impeller vibration and noise¹⁷. Dolzan et al. designed a novel self-priming pump and optimized its structure. They found that the maximum flow rate and backpressure performance are achieved when the pump chamber, piezoelectric actuator, and inlet rectification elements are centrally aligned¹⁸. Sun et al. proposed a new type of vertical self-priming pump by optimizing the hydraulic components and the matching structure between the volute and the gas-liquid separation chamber, which reduces hydraulic and mechanical losses. They performed a three-dimensional simulation of the internal flow and found that there are considerable additional losses in the gas-liquid separation chamber and the auxiliary impeller¹⁹. Ibra et al. conducted numerical simulations of the self-priming process in self-priming pumps and discovered that the gas-liquid mixing and separation phenomena primarily occur during the early stages of self-priming. Gas-liquid mixtures appear at the outer edge of the impeller, and the instantaneous gas content at the impeller inlet and outlet significantly decreases²⁰. Liu et al. investigated the effect of the special volute structure on the self-priming performance of a swirl self-priming pump using two-dimensional Particle Image Velocimetry (PIV) measurements and three-dimensional CFX software²¹. M. Namazizadeh et al. showed that adding a splitter can increase the total head by approximately 10.6%. By modifying the geometry using the Design of Experiment (DoE) technique, a further improvement of 4.4% can be achieved, with minimal impact on the overall efficiency of the pump²².

Combining the aforementioned research findings, it is evident that current numerical simulation studies exhibit certain limitations: Firstly, most computational models either focus solely on the self-priming pump itself or extend only slightly beyond the pump’s inlet and outlet. Secondly, initial boundary conditions for simulations are often artificially set by specifying a certain gas-liquid distribution in the inlet pipe. Thirdly, the transient phase during which the impeller accelerates from a stationary state to a steady rotational speed is frequently overlooked. Such approaches inevitably lead to significant discrepancies between simulation results and actual experimental processes. To address these issues, this paper establishes a computational model of a circulation pipeline system that includes the self-priming pump, valves, water tank, and piping components. Air is positioned above the water in the tank, replicating real-world conditions. Furthermore, the transient process of impeller acceleration following startup is taken into account, enabling a comprehensive numerical simulation of the entire self-priming process, from the increase in pump speed to steady-state operation. This approach aims to provide a more accurate representation of the self-priming pump’s behavior under realistic operating conditions.

Computational models and methods

Computational model

The computational model is an external-mixing self-priming pump, with the main design parameters listed in Table 1. The physical prototype of the self-priming pump and the internal computational domain are illustrated in Fig. 1. The computational domain encompasses the S-pipe, impeller, volute, gas-liquid separation chamber, front and rear pump chambers, reflux hole, as well as the inlet and outlet pipes. Since the content of this paper is a continuation of previous research, the research model used in this study is consistent with that in reference²³.

Table 1.

Self-priming pump design parameters.

Parameter	Unit	Value
Rotational speed	nd / (r/min)	2900
Flow rate	Qd / (m3/h)	15
Head	Hd / (m)	32
Specific speed	n s	51

Fig. 1.

Self-priming pump²³ (a) physical prototype (b) computational domain.

The reflux hole is located at a position 204° to 216° along the rotational direction of the impeller from the volute tongue, with a radius of 3.25 mm. Specific parameters and locations are shown in Fig. 2.

Fig. 2.

Reflux hole parameters²³.

The computational model presented in this paper is a circulating piping system that includes a self-priming pump, as depicted in Fig. 3a. This model is a full-scale replica of the constructed visualization experimental platform, with specific dimensions provided in Table 2. The mesh generation is illustrated in Fig. 3b.

Fig. 3.

Circulating piping system²³ (a) computation model (b) computation grid.

Table 2.

Geometric dimensions.

System component	Value
Inlet pipe diameter	40 mm
Inlet pipe horizontal length	1665 mm
Inlet pipe vertical length	1050 mm
Outlet pipe diameter	35 mm
Outlet pipe horizontal length	2300 mm
Outlet pipe vertical length	1388 mm
Tank	960 mm×700 mm×650 mm

The grid independence verification is shown in Fig. 4, and the final determination was made that the total number of grid cells for the entire system is 3,914,245.

Fig. 4.

System grid independence verification²³.

Governing equation

VOF model

The VOF two-phase flow model was developed by Hirt et al.²⁴ in 1981. This model exhibits high applicability and efficiency in tracking the free surface of incompressible fluids, while also being simple to operate and requiring minimal computational resources. Subsequent research by Kan and Parikh et al.^25,26 has confirmed the good applicability of the VOF model in tracking the free surface of centrifugal pumps and calculating the volume fraction of the water phase. Numerical simulation results have been found to be highly accurate when compared with experimental results. Therefore, the VOF model is selected in this paper for simulating the gas-liquid two-phase flow inside the pump during the self-priming process. The basic equations of the VOF model include the volume fraction continuity equation, the continuity equation, and the momentum equation, which are formulated as follows:

The volume fraction continuity equation:

1

2

The continuity equation:

3

The momentum equation:

4

where, and represent the volume fractions of the water phase and the gas phase, respectively, with ; u is the velocity; t is time; p is the static pressure; is the Hamiltonian operator; is the dynamic viscosity of the mixture, with ​, and and are the dynamic viscosities of the water phase and the gas phase, respectively; is the density of the mixture, with , and are the densities of the water phase and the gas phase, respectively; F is the gravitational acceleration; g represents the equivalent volumetric force due to surface tension.

Turbulence model

The Realizable k-ε two-equation model employs a new formula for turbulent viscosity that satisfies the constraints of Reynolds stress, thereby maintaining consistency with real turbulence in terms of Reynolds stress. He et al. conducted numerical simulations of gas-liquid two-phase flow within pumps based on the Realizable k-ε turbulence model^27–29. The simulation results showed high consistency with experimental results, confirming that this model is suitable for simulating turbulence in the gas-liquid two-phase flow process inside pumps. Therefore, this paper also adopts the Realizable k-ε turbulence model to close the system of control equations.

The transport constraint equations in this turbulence model are as follows:

5

6

where, the coefficient , the turbulent eddy viscosity coefficient . In the formula for calculating the turbulent eddy viscosity, , , , , , . The values of the constants in the above equation are: σ_ε = 1.2, C₂ = 1.92, A₀ = 4.0.

Numerical method

The assumptions are as follows: the physical properties of both the gas and liquid phases are constants, the liquid phase is incompressible, and the gas phase is a continuous medium. When the average volume fraction of the gas phase within the entire pump cavity is below 2%, it is considered that the self-priming process is complete. This paper employs ANSYS FLUENT 21.1 to perform unsteady numerical calculations for the entire self-priming process, using the SIMPLEC algorithm for solving the pressure-coupled equation system. No-slip boundary conditions are applied at solid walls, and standard wall functions are used near the walls to address flow computation issues in the near-wall region and at low Reynolds numbers. The convergence criteria for velocity components in all directions, turbulent kinetic energy k, and turbulent dissipation rate ε are set to 1 × 10^− 4.

The initial computational conditions of the model are shown in Fig. 5. To maintain consistency between the initial conditions of the computational model and the actual scenario, air is set above the water tank with a pressure value of one standard atmosphere. Data transfer between various flow components is achieved through internal coupling calculations. The vertical distance between the water level in the tank and the initial water level inside the pump is 0.75 m. Both the upper part of the self-priming pump and the upper part of the water tank are filled with air, with a gas phase volume fraction of 1; the lower parts are filled with clear water, with a liquid phase volume fraction of 1.

Fig. 5.

Gas-liquid two-phase system initial state²³.

After the self-priming pump is started, the pump speed rapidly increases from a stationary state to the rated speed and remains at this speed for continuous operation. Compared to the self-priming time required to complete the entire self-priming process, the time needed for the impeller to reach its rated speed is very short. However, during this period, the pump exhibits particularly complex unsteady gas-liquid two-phase flow characteristics. In previous studies, this brief acceleration phase has often been neglected. In this paper, to align with actual operating conditions, the control of the impeller speed is divided into two stages: an acceleration stage and a constant rotation stage. The duration of the acceleration stage is 0.2 s, and the constant speed is the rated speed of 2900r/min. During the computation, the pump speed control is achieved using a UDF, as shown in Eq. (7):

7

Numerical method verification

External characteristic verification

Under normal operating conditions with clear water, the comparison between the experimental results of the self-priming pump’s external characteristics and the numerical predictions of hydraulic performance is shown in Fig. 6. It can be observed that the maximum error between the experimental head and the calculated head is approximately 2.372%. The maximum error between the simulated shaft power and the experimental value is about 9.778%. The simulation results for hydraulic efficiency also show good agreement with the experimental results, with a maximum error of approximately 5.719%. Overall, the results demonstrate a good match, indicating that the adopted numerical calculation method is accurate and reliable.

Fig. 6.

External characteristics comparison²³.

Visual verification

A visualization experimental platform for a self-priming pump was established in this study, as shown in Fig. 7. The accuracy of the electromagnetic flowmeter is 0.5%, with a measurement range of 8–40 m³/h. The model of the three-phase asynchronous motor is YE4-100 L-2.

Fig. 7.

Visual experimental platform.

Visual experiments were conducted on the self-priming process of the pump to further validate the reliability of the numerical simulation method for gas-liquid two-phase flow. The inlet and outlet pipes of the self-priming pump are made of transparent acrylic material to facilitate the observation of changes in the water level during the self-priming process after startup. Before startup (the initial state of the experiment), the vertical height between the water level in the tank and the bottom of the inlet pipe (installation height) is 0.75 m, and the initial water storage in the self-priming pump is filled to the level of the bottom surface of the S-pipe.

Figure 8 shows the comparison between the simulation results and experimental results of the water level height in the vertical inlet pipe during the self-priming process of the self-priming pump³⁰. It can be observed that at four key time points during the self-priming process, the water levels from both methods are in good agreement, further confirming the high accuracy and reliability of the numerical calculation method used in this paper. In subsequent numerical calculations, this method will be employed to further investigate the hydraulic and self-priming characteristics of the self-priming pump.

Fig. 8.

Comparison between experimental and simulated liquid levels during self-priming process³¹. (a) t = 0.00s (b) t = 0.16s (c) t = 0.20s (d) t = 0.28s.

Model simplification

Due to the excessive length of the inlet pipe leading to a large volume of gas that needs to be discharged by the self-priming pump (as shown in Fig. 5), the computational effort required to complete the entire self-priming process is significantly increased. Considering computational efficiency, the pipe lengths in Fig. 3 have been appropriately adjusted in this paper to reduce the computational time. The inlet and outlet pipes have been shortened, and the installation height of the self-priming pump has been set to 1.0 m. The simplified dimensions of the circulating piping system are shown in Fig. 9a, and the system grid is illustrated in Fig. 9b. The total number of grid cells in the simplified model is only slightly different from the original, totaling 3,903,026 cells. The mesh for the self-priming pump remains completely unchanged. The number of grid cells for the water tank, inlet pipe, outlet pipe, and valve are 1,021,209, 223,856, 330,812, and 465,328, respectively. After simplification, the previously validated numerical calculation method was used to perform unsteady flow numerical calculations for the complete self-priming process of the self-priming pump. The initial state of the gas-liquid two-phase flow in the simplified computational model is shown in Fig. 9c.

Fig. 9.

Simplified circulation pipeline system (a) computation model (b) computational grid (c) gas-liquid two-phase initial state.

To meet the needs of various emergency rescue scenarios, self-priming pumps often need to perform rapid drainage operations under different operating conditions, such as at different installation heights. This paper will conduct self-priming process calculations for two different installation heights to determine the impact of installation height variations on the self-priming performance.

Result analysis

Computation model

To investigate the impact of different installation heights (the vertical distance between the water level in the storage tank and the water level inside the pump) on the self-priming performance of the self-priming pump, two different installation configurations are set up in this paper. The models, meshes, and initial states of the different circulating piping systems are shown in Figs. 10 and 11. The change in installation height is achieved by modifying the vertical sections of the inlet and outlet pipes.

Fig. 10.

The circulatory piping system with ∆h = 0.5 m (a) model (b) grid (c) initial state.

Fig. 11.

The circulatory piping system with ∆h = 0.5 m (a) model (b) grid (c) initial state.

External characteristic

To study the impact of different installation heights on the self-priming characteristics, the volume fraction of the gas phase within various flow components during the self-priming process was monitored, as shown in Fig. 12. Figure 12a presents the gas phase volume fraction curve within the S-pipe. It can be observed that before t = 0.270s, the influence of the installation height on the curve variation is relatively small. However, during the oscillatory exhaust phase and the water intake exhaust phase, the differences become more pronounced. For Δh = 0.5 m, the initial value of the gas phase volume fraction within the S-pipe is consistent with that of Δh = 1.0 m, both being 0.613. As the initial liquid is drawn into the impeller, the curve begins to rise initially at an accelerating rate and then slows down. At t = 0.265s, it reaches its maximum value of 0.978. Subsequently, as the liquid enters the S-pipe, the curve drops sharply, and by t = 0.875s, the gas volume fraction has almost reached zero. For Δh = 1.0 m, the curve reaches its maximum value at t = 0.360s, with a maximum value of approximately 0.991. The time point at which the curve drops to near zero is t = 1.470s. Comparing the two cases, it can be seen that as the installation height increases, the time it takes for the liquid in the storage tank to reach the entrance of the S-pipe and the time it takes to expel the gas from the S-pipe both increase. Additionally, the maximum value of the gas phase volume fraction within the S-pipe for Δh = 1.0 m is slightly higher than that for Δh = 0.5 m.

Fig. 12.

The gas-phase volume fraction in flowing components. (a) S-pipe (b) impeller (c) volute (d) gas-liquid separation chamber.

Figure 12b shows the gas phase volume fraction curves within the impeller flow field under different installation heights. It can be observed that at the beginning of the start-up, the initial values of the curves for all schemes are 0. Before t = 0.210s, the overall differences between the two curves are minimal, after which the distinctions gradually become evident. For Δh = 0.5 m, the time point when the gas from the S-pipe begins to enter the impeller flow field is around t = 0.152s. Subsequently, the curve gradually rises, with the rate of increase first slowing, then accelerating, and finally decelerating again. Around t = 0.440s, the curve reaches its maximum value of 0.967. Following this, the liquid gradually enters the impeller, causing the curve to decrease at a decreasing rate. Eventually, around t = 2.510s, the gas within the impeller chamber is almost completely expelled. For Δh = 1.0 m, the gas begins to enter the impeller around t = 0.163s, leading to a rapid increase in the curve, reaching its peak value of 0.989 around t = 0.525s. Afterward, the impeller is gradually occupied by the liquid phase, and by t = 3.020s, the curve essentially decreases to 0. Comparing the two scenarios, it is evident that as the installation height increases, the timing of gas entry into the impeller, the peak value of the gas phase volume fraction curve within the impeller, and the time point when the gas is completely expelled from the impeller all experience a delay, with the peak value of the curve also increasing.

Figure 12c shows the variation of the gas phase volume fraction curve within the volute flow field over time. Similarly, at the beginning of the start-up, the initial values of the curves are both 0. The trends of the two curves are generally consistent, with only the timing differing. For Δh = 0.5 m, the curve begins to rise gradually and accelerates from t = 0.184s, with a consistently high rate of increase. By t = 0.510s, the curve reaches its maximum value of 0.824. Subsequently, as the liquid thrown out by the impeller gradually enters the volute and the continuous supply of liquid through the reflux hole, the curve starts to decrease at a decreasing rate. By t = 2.205s, the volute is almost completely filled with liquid, and the gas is largely expelled. For Δh = 1.0 m, the moment when gas begins to enter the volute is around t = 0.193s, and the curve starts to rise. By t = 0.595s, the curve reaches its peak value of approximately 0.840. Following this, the entry of liquid causes the curve to drop rapidly, and by t = 2.230s, the gas in the volute is almost completely expelled. Comparing the two scenarios, it can be seen that as the installation height increases, the time points for gas entry into the volute, the onset of liquid entering the volute flow field, and the time when the gas in the volute is largely expelled all experience a delay. Additionally, the maximum value of the gas phase volume fraction within the volute also increases slightly.

Figure 12d shows the gas phase volume fraction curve within the gas-liquid separation chamber. As shown in the figure, under both installation heights, the curves rapidly decrease from an initial value of 0.318. For ∆h = 0.5 m, the curve stops decreasing around t = 0.280s, with an instantaneous value of approximately 0.241. At this point, most of the discharge from the volute outlet is in the gas phase, causing the curve to begin increasing. By t = 0.840s, the curve reaches its peak value of 0.277. Subsequently, as the liquid phase gradually enters the pump and the amount of liquid entering the gas-liquid separation chamber increases significantly, the curve begins to decrease again, with the rate of decrease gradually slowing down. Ultimately, by t = 7.880s, the gas in the gas-liquid separation chamber is almost completely expelled. For ∆h = 1.0 m, the curve also reaches its minimum value of approximately 0.249 around t = 0.280s. The curve then begins to rise, reaching its peak value of 0.284 around t = 1.080s. Under this installation height, it takes until t = 10.310s for the gas in the gas-liquid separation chamber to be almost completely expelled. Comparing the two scenarios, it can be observed that as the installation height increases, the time points at which the curve reaches its local maximum and the time when the gas in the gas-liquid separation chamber is almost completely expelled are both delayed. Additionally, the minimum and maximum values of the gas phase volume fraction curve within the gas-liquid separation chamber during the self-priming process are both slightly higher.

In summary, during the acceleration period of the impeller, the impact of the two different installation heights on the gas phase volume fraction curves within various flow components of the pump is minimal. This is because, during this stage, the fluid being drawn into the pump is primarily in the gas phase, and under the same impeller speed variation, the distribution of gas and liquid within the pump remains largely similar. After the impeller acceleration stage, an increase in installation height means that the self-priming pump must expel a greater volume of gas, leading to a delay in the characteristic moments of curve changes and an extension of the self-priming time. This also results in a slight increase in the peak and trough values of the curves within each flow component. Additionally, the gas phase volume fraction within the volute remains consistently lower than that within the impeller chamber during the oscillatory exhaust phase. This is due to the presence of liquid residues on the inner walls of the volute and the entry of liquid from the bottom of the gas-liquid separation chamber through the reflux hole into the volute flow channel.

During the self-priming process, there exists a complex two-phase flow state involving both gas and liquid. Monitoring the gas mass flow rate (Q_m) at the inlet and outlet surfaces of the self-priming pump helps to better understand the process of gas entering and exiting the pump, as shown in Fig. 13. Figure 13a illustrates the temporal variation of the gas mass flow rate curve at the inlet of the self-priming pump under different installation heights. It can be observed that the variation patterns of the two curves are broadly similar across different installation heights: the curves rise from 0 to a maximum value, then gradually decline to a lower level with minor fluctuations, and eventually approach 0. For ∆h = 0.5 m, within the time interval 0.00s < t < 0.200s, as the impeller accelerates, the curve rises rapidly, with the rate of increase gradually increasing. By t = 0.191s, it reaches its maximum value of 8.61 × 10⁻³kg/s. Subsequently, due to the gas occupying most of the impeller area, the impeller’s ability to do work diminishes, and the vacuum degree at the impeller inlet significantly decreases, causing the curve to start declining. By t = 0.520s, the gas is mostly sucked into the pump. At t = 0.340s, the gas enters the S-pipe in a flow state entrained by the liquid, resulting in a noticeable reduction in the rate of decline of the curve and minor fluctuations; for ∆h = 1.0 m, the curve reaches its maximum value of approximately 8.40 × 10⁻³kg/s around t = 0.197s. Following the reduction in vacuum degree at the impeller inlet, the curve also begins to decline, approaching 0 by t = 0.815s. Similarly, at t = 0.485s, the rate of decline of the curve also noticeably slows down, for the same reason as described earlier. Comparing the two scenarios, it can be observed that after the impeller acceleration stage, an increase in installation height significantly reduces the rate of decline of the gas mass flow rate curve at the pump inlet and delays the time point at which the curve approaches 0.

Fig. 13.

The gas-phase mass flow rate of the pump at two installation heights (a) inlet (b) outlet.

Figure 13b shows the Q_m curves at the pump outlet cross-section under different installation heights. Similarly, during the rapid suction phase, the differences between the two curves are minimal. However, during the oscillatory exhaust and water intake exhaust phases, the differences between the two curves become increasingly apparent. For ∆h = 0.5 m, the curve reaches its peak value of 7.83 × 10⁻³kg/s around t = 0.179s. Subsequently, some of the stored liquid inside the pump is directly discharged to the pump outlet, causing the gas mass flow rate to decrease rapidly. By t = 0.225s, the curve drops to a minimum value of 3.37 × 10⁻⁴kg/s. Thereafter, the majority of the mixture discharged from the volute outlet is gas, causing the gas mass flow rate to gradually rise to a higher level. As the gas content in the impeller and volute decreases, the gas content in the mixture discharged from the volute outlet also decreases. Ultimately, by t = 1.750s, the curve declines to a lower level, with significant fluctuations. For ∆h = 1.0 m, the curve reaches its peak value of 7.75 × 10⁻³kg/s around t = 0.190s. The curve then drops to a minimum value of 4.58 × 10⁻⁴kg/s around t = 0.260s. Finally, by t = 1.950s, the curve declines to a lower level, also accompanied by fluctuations. Comparing the two scenarios, it can be observed that after the impeller acceleration stage, an increase in installation height significantly reduces the average gas mass flow rate within the time interval 0.200s < t < 2.000s, indicating a noticeable decrease in the efficiency of gas discharge from the pump.

In summary, during the impeller acceleration period, the gas mass flow rate curves at the pump inlet and outlet show minimal differences. However, during the oscillatory exhaust phase and the water intake exhaust phase, an increase in installation height significantly reduces the efficiency of gas intake and exhaust in the self-priming pump.

The most important criterion for evaluating the self-priming performance in this paper is the gas phase volume fraction curve within the entire pump cavity. Figure 14 shows the temporal variation of the gas phase volume fraction curve within the pump cavity under two different installation heights.

Fig. 14.

The gas-phase volume fraction in the pump at two installation heights.

As shown in the figure, before t = 0.215s, the differences between the two curves are minimal, but they become more pronounced thereafter. For ∆h = 0.5 m, within the time interval t < 0.140s, the curve decreases slightly from 0.294 to 0.293. The slight decrease in the curve can be attributed to the fact that, at the beginning of the start-up, the exhaust rate at the self-priming pump outlet is slightly higher than the gas intake rate at the pump inlet. Subsequently, as the stored liquid is discharged from the pump and gas continues to be drawn into the S-pipe, the curve rises rapidly, reaching a peak value of 0.373 around t = 0.325s. As liquid gradually enters the S-pipe, the curve begins to decline, with the rate of decline gradually slowing down. Ultimately, by t = 7.625s, the gas phase volume fraction decreases to below 3%, completing the self-priming process. For ∆h = 1.0 m, the curve decreases from 0.294 to 0.292 within the time interval 0.000s < t < 0.153s. The curve then rises to a peak value of 0.429 around t = 0.470s. Finally, by t = 13.885s, the curve decreases to below 3%, marking the end of the self-priming process. Comparing the two scenarios, it can be seen that increasing the installation height from 0.5 m to 1.0 m extends the self-priming time by 6.260s, a relative increase of 82.098%. During the impeller acceleration stage, the change in installation height has a minimal impact on the curve variation pattern. However, as the installation height increases, the time point at which the gas phase volume fraction curve reaches its peak is slightly delayed, and the peak value also increases.

Gas liquid two-phase distribution

The process of gas-liquid two-phase flow in the meridian plane (x = 0) can illustrate the air intake and water intake processes of the self-priming pump, thereby exploring the effects of different installation heights on the self-priming process, as shown in Fig. 15. From Fig. 15a–c, it can be observed that within the time frame of 0.0s < t ≤ 0.30s, the overall distribution of the gas-liquid two-phase in the meridian plane shows minor differences under different installation heights. By t = 0.30s, the impeller is almost entirely occupied by gas. However, at t = 0.60s, it becomes evident that significant differences exist; under ∆h = 0.5 m, the liquid from the storage tank has almost completely filled the S-pipe and started entering the impeller flow area, whereas under ∆h = 1.0 m, only a small amount of liquid has reached the impeller inlet. From Fig. 15d–f, it can be seen that as the liquid gradually enters the impeller, by t = 1.50s, the regions with a higher gas volume fraction in the impeller chamber are significantly smaller under ∆h = 0.5 m compared to those under ∆h = 1.0 m, where a large area of the impeller chamber still contains a considerable amount of gas. Figure 15g–i indicate that after t = 2.80s, most of the gas in the impeller and volute flow areas has been expelled, with the main process being the discharge of gas dispersed in the gas-liquid separation chamber. It is also evident that during the later stages of the self-priming process, due to the lack of an outlet in the rear pump chamber and the relatively weak centrifugal force acting on the gas, it is challenging to expel the gas, leading to some gas accumulation in the rear pump chamber across all schemes.

Fig. 15.

Evolution of gas-liquid two-phase distribution in cross-section of S-pipe at two installation heights. (a) t = 0.17s (b) t = 0.19s (c) t = 0.30s (d) t = 0.60s (e) t = 1.00s (f) t = 1.50s (g) t = 2.80s (h) t = 5.00s (i) t = 10.0s.

Figure 16 illustrates the evolution of the gas-liquid two-phase distribution in the mid-plane of the impeller (z = 0) for different schemes. From Fig. 16a and b, it can be observed that at t = 0.170s, gas has already entered the impeller flow passages under ∆h = 0.5 m, while only a very small amount of gas has entered the impeller inlet under ∆h = 1.0 m. By t = 0.20s, under the effect of the impeller’s centrifugal force, the liquid is thrown to the outer edge of the impeller and into the volute. A comparison of the two scenarios shows that under ∆h = 1.0 m, a small amount of liquid still remains at the trailing edge of the impeller, whereas under ∆h = 0.5 m, the liquid has been largely expelled from the impeller. Although the timing of these characteristic phenomena differs, the overall differences during this phase are relatively minor. By t = 0.30s, most of the initial stored liquid in the impeller and volute has been thrown into the gas-liquid separation chamber and outside the pump along the volute flow path. At t = 0.60s, the liquid has already entered the impeller under ∆h = 0.5 m, while very little liquid has entered the impeller chamber under ∆h = 1.0 m. Thereafter, the differences between the two scenarios become more pronounced, with the self-priming process progressing faster under ∆h = 0.5 m. By t = 3.40s, the gas in the impeller and volute has been largely expelled. At t = 10.0s, the gas content in the gas-liquid separation chamber under ∆h = 0.5 m is already minimal, while a certain amount of gas still remains under ∆h = 1.0 m.

Fig. 16.

Evolution of gas-liquid two-phase distribution in the impeller cross-section at two installation heights (a) t = 0.17s (b) t = 0.20s (c) t = 0.30s (d) t = 0.60s (e) t = 1.00s (f) t = 1.50s (g) t = 2.20s (h) t = 3.40s (i) t = 10.0s.

Vortex identification analysis

Figure 17 illustrates the evolution of vortex core distribution within the pump under two different installation heights over time. From Fig. 17a, it can be observed that a significant number of vortex cores are generated at the impeller inlet, due to the initial entry of gas into the impeller, which creates a relatively complex gas-liquid flow state. Additionally, the jet-wake structures at the outer edge of the impeller and the interference effects at the volute tongue, where the outer edge of the impeller and the volute are predominantly liquid, lead to substantial energy losses and the generation of numerous vortex cores. Figure 17a–c show that within the time interval t < 0.300s, as the gas mixes with the liquid at the outer edge of the impeller, the number of vortex cores within the impeller flow passages and near the outer edge gradually increases. During this phase, the impeller inlet is primarily occupied by gas, but the leakage through the clearance at the front pump chamber and the residual liquid at the bottom of the S-pipe also enter the impeller, causing a relatively chaotic two-phase flow and a high concentration of vortex cores at the impeller inlet. However, the overall velocity of the vortex cores is relatively low, gradually increasing along the impeller flow passages. The maximum velocity of the vortex cores is approximately 20 m/s, located at the trailing edge of the impeller. From Fig. 17d–f, it can be seen that the number of vortices within the impeller and volute gradually decreases, indicating the process of the impeller being gradually occupied by liquid. At t = 0.600s, the liquid distribution within the impeller and volute flow passages under both installation heights is relatively chaotic and dispersed, showing strong instability in the internal flow field of the pump. Both the impeller and volute generate a large number of vortex cores, with higher velocities at the trailing edge of the impeller. However, under ∆h = 1.0 m, the velocity of the vortex cores is even greater, with a maximum velocity of approximately 50 m/s, located near the pressure side of the trailing edge of the impeller. As the liquid gradually fills the impeller and volute flow areas, the distribution of vortices significantly decreases.

Fig. 17.

Evolution of the vortex cores distribution in the pump at two installation heights. (a) t = 0.17s (b) t = 0.20s (c) t = 0.30s (d) t = 0.60s (e) t = 1.00s (f) t = 1.50s (g) t = 2.20s (h) t = 3.40s (i) t = 10.0s.

From Fig. 17g, it can be observed that at t = 2.20s, the majority of the impeller and volute chambers are filled with liquid, with only a small amount of gas remaining on the suction side of the blades. Overall, the vortex core velocities are very low at this point, with the highest velocities occurring near the volute tongue. The maximum vortex core velocities for ∆h = 0.5 m and ∆h = 1.0 m are approximately 25 m/s and 20 m/s, respectively. Corresponding to Fig. 17g, it is evident that there are numerous vortices near the suction side of the blades, indicating a certain correlation between the gas-liquid two-phase flow state and the regions where vortices are generated. Figure 17h–i depict the later stages of the water intake exhaust phase, where the impeller and volute are almost entirely occupied by liquid, resulting in fewer losses associated with single-phase flow. A few vortex cores are distributed at the impeller inlet, the blade backside, the outer edge of the impeller, the bottom of the gas-liquid separation chamber, and near the volute tongue. A larger number of vortex cores are distributed along the flow path near the volute outlet, extending to the exit of the gas-liquid separation chamber. Overall, in the later stages of the self-priming process, the number and velocity of vortex cores are relatively low, with velocities below 30 m/s. The high-velocity regions are mainly concentrated near the volute tongue. In summary, under both installation heights, a significant number of vortices persist at the impeller inlet, the outer edge of the impeller, and near the volute tongue, indicating that these regions consistently experience notable energy losses. In contrast, although the gas distribution within the gas-liquid separation chamber is also scattered and chaotic, the overall velocities are lower, resulting in fewer energy losses and a relatively smaller number of vortices. Comparing the same time points under different installation heights, it is evident that the number and distribution of vortices within the pump are consistently higher for ∆h = 0.5 m than for ∆h = 1.0 m, especially in the later stages of the self-priming process. This is primarily due to the faster progression of the self-priming process at the lower installation height, resulting in higher velocities of the liquid being discharged from the volute.

Entropy production analysis

The entropy production rate (EPR) signifies the inevitable dissipative effects accompanying various energy conversion processes. The numerical simulation of the self-priming pump is based on the Reynolds-averaged Navier-Stokes (RANS) equations. For turbulence, the entropy production of particles can be calculated in two parts: one is the entropy production caused by time averaged motion, and the other is the entropy production caused by velocity fluctuations. The calculation formula is as follows:

8

where is the entropy production generated by average velocity; is the entropy production generated by pulsating velocity.

The entropy production generated by the average velocity can be calculated by Eq. (9):

9

The entropy production generated by the pulsating velocity can be calculated by Eq. (10):

10

where,is the kinematic viscosity;、、 are time averaged velocities; u′, v′, w′ are pulsating velocities; T is the temperature (set as a constant of 293 K); is the effective motion viscosity, where is the turbulent kinematic viscosity.

can be directly solved through numerical calculations and cannot be directly solved. According to the theory of local entropy production³², the local entropy production caused by pulsating velocity can be expressed as³³:

11

where, α = 0.09, ω is the frequency of turbulent eddies, s^− 1;k is the turbulence intensity, m²/s².

However, due to the strong wall effect of EPR and the obvious time averaged term, the formula for calculating entropy production near the wall is:

12

where, is the wall shear stress, Pa;S is the area, m²; v is the near wall velocity, m/s.

Therefore, the formula for calculating the total entropy production within the entire system calculation domain is:

13

During the self-priming process of the self-priming pump, complex gas-liquid two-phase flow occurs, and the distribution of the gas-liquid phases and changes in the flow field velocity all contribute to an increase in entropy production. Figure 18 illustrates the variations in entropy production distribution on the mid-plane of the impeller under different installation heights. From Fig. 18a–b, it can be observed that during the impeller acceleration stage, the overall region of entropy production within the pump is relatively small, primarily located at the outer edge of the impeller, the volute tongue, and within the volute flow passages. Among these, the entropy production values within the volute flow passages are relatively low, while the values at the outer edge of the blade pressure side and near the volute tongue are higher. At this stage, the high-entropy production regions are very limited, with the maximum EPR located near the volute tongue, reaching approximately 50,000 W·m^− 3·K^− 1. Figure 18c–d show that within the time interval 0.300s < t < 0.600s, the distribution of entropy production within the pump is highly scattered, mainly concentrated at the trailing edges of some impeller flow passages, the outer edge of the impeller, the volute flow passages, and near the volute tongue. Combining this with Fig. 16d, it is evident that at t = 0.600s, the liquid distribution within the impeller and volute is very chaotic and disordered, leading to a significant increase in entropy production. Figure 18e–g depict the entropy production distribution within the pump during the water intake exhaust phase. As the liquid gradually fills the impeller, the flow within the pump becomes more stable, and the distribution of entropy production transitions from a chaotic state to one that is nearly symmetric about the center. The entropy production at the outer edge of the impeller gradually decreases, with high-entropy production regions primarily concentrated at the impeller inlet and the leading edge of the blade suction side. Meanwhile, the entropy production values near the volute tongue increase, and the distribution range also expands. Under both installation heights, the entropy production in the region near the impeller inlet has reached around 50,000 W·m^− 3·K^− 1.

Fig. 18.

Evolution of the entropy production distribution in the cross-section of impeller at two installation heights. (a) t = 0.17s (b) t = 0.20s (c) t = 0.30s (d) t = 0.60s (e) t = 1.00s (f) t = 1.50s (g) t = 2.20s (h) t = 3.40s (i) t = 10.0s.

Figure 18h–i show that in the later stages of the self-priming process, the distribution of entropy production within the pump gradually increases, presenting a centrally symmetric pattern near the impeller inlet. The values of entropy production are very high, significantly exceeding 50,000 W·m⁻³·K⁻¹, and decrease radially along the impeller. This phenomenon can be attributed to two main reasons: first, the sudden change in flow direction at the impeller inlet, where the flow changes from axial to radial along the impeller flow passages; second, the water impact on the trailing edge of the blade inlet due to the impeller inlet being fully occupied by liquid, which causes the liquid to strike the blade upon entry. Additionally, the entropy production near the volute tongue spreads from the near-wall region to the middle of the volute outlet section, extending all the way to the exit of the gas-liquid separation chamber. Comparing the entropy production distributions at the same time points under different installation heights, it can be observed that in most cases, the entropy production distribution on the mid-plane of the impeller under ∆h = 0.5 m is greater in both range and magnitude compared to ∆h = 1.0 m. Notably, throughout the entire self-priming process, entropy production persists near the outer edge of the impeller in both schemes. This is due to the jet-wake structure near the outer edge of the impeller, which consistently leads to entropy generation.

Conclusion

This paper conducts numerical calculations of the self-priming process of a self-priming pump under different operating conditions. By analyzing the gas phase volume fraction within various flow components of the self-priming pump and the gas-liquid two-phase distribution within the pump, the impacts of two different installation heights on the self-priming performance are compared. Additionally, the flow characteristics within the pump under different operating conditions are analyzed using vortex identification methods and entropy production theory. The conclusions are as follows:

1. As the installation height increases, the self-priming time significantly extends. When the installation height increases from 0.5 m to 1.0 m, the self-priming time increases by 6.260s, representing an 82% relative increase. This indicates that higher installation heights pose greater challenges for the pump’s ability to prime itself due to the increased effort required to remove air and replace it with liquid at greater heights.

2. During the impeller acceleration period, changes in installation height have a minimal impact on the gas phase volume fraction curves within the various flow components of the pump. After the impeller acceleration stage, as the installation height increases, the self-priming time also extends. This suggests that the critical factor affecting self-priming time is not the initial acceleration phase but rather the subsequent phases where the system must work harder to fully evacuate gases and achieve stable liquid flow at higher elevations.

3. The gas phase volume fraction within the volute remains consistently lower than that within the impeller chamber during the oscillatory exhaust phase. This is due to the presence of liquid residues on the inner walls of the volute and the entry of liquid from the bottom of the gas-liquid separation chamber through the reflux hole into the volute flow channel.

4. Before the impeller and volute computational domains are completely filled with liquid, a complex two-phase mixing flow state always exists at the outer edge of the impeller, characterized by turbulent flow and significant energy loss. During the water intake exhaust phase, sudden changes in flow direction and water impact cause a noticeable increase in entropy at the impeller inlet. The jet-wake structure near the outer edge of the impeller and the interference effects near the volute tongue consistently lead to energy loss, generating numerous vortices and significant entropy increase phenomena. These observations suggest that optimizing the design of the impeller and volute could potentially reduce turbulence and energy losses, thereby improving the overall efficiency and priming speed of the pump.

List of symbols

n

Rotational speed

Q

Flow rate

H

Head

n_s

Specific speed

α₁ and α₂

Volume fractions of the water phase and the gas phase

u

Velocity

t

Time

p

Pressure

Hamiltonian operator

Dynamic viscosity of the mixture

and

Dynamic viscosities of the water Phase and the gas phase

Density of the mixture

and

Densities of the water phase and the gas phase

F

Equivalent volumetric force due to surface tension

g

Gravitational acceleration

Wall shear stress

S

Area

v

Near wall velocity

Author contributions

Ying-Yu Ji carried out the numerical simulation and experiments. Shao-Han Zheng wrote the manuscript. Yu-Liang Zhang proposed the innovative idea and supervised this work. Kai-Yuan Zhang analyzed the numerical results. Zu-Chao Zhu revised the manuscript. All authors have read and agreed to the published version of the manuscript.

Data availability

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

Declarations

Competing interest

The authors declare no competing interests.