**1. Introduction**

A pump is a kind of general machine with tremendous variety and extensive application fields, and it can be said that pumps serve in all places with liquid flow [1–4]. According to statistics, the power consumption of pumps account for 22% of the power generation and the oil consumption accounts for about 5% of the total oil consumption [5–10]. Self-priming centrifugal pumps, or self-priming pumps, have no bottom valve in the inlet pipe. The pump structure is changed such that it can store some water after stopping and exhaust the air in the inlet pipe at the start by the mixture and separation of air and water. The water returns to the appropriate position in the pump via the backflow channel, and the above process is repeated, thereby realizing the self-priming. The process requires a water supply only at the initial stage; no water supply is necessary in subsequent start-ups. After a short operating period, the pump itself can suck up the water and be put into normal operation. A self-priming pump is easy to operate and has stronger adaptability than an ordinary centrifugal pump. Such a pump is extremely suitable for situations with frequent start-ups or difficult liquid irrigations [11–16]. Self-priming is an important parameter in order to evaluate the performance of self-priming pumps, and it determines the normal operation of pumps. According to the Chinese standard JB/T6664-2007, the self-priming time of a 5 m vertical pipe should be controlled within 100 s. Given that the self-priming process of a self-priming pumps is a complicated, unsteady gas–liquid flow phenomenon, studying the self-priming performance of pumps is difficult. Moreover, many challenges emerge in the study of the influencing factors of the self-priming time of self-priming pumps.

Many studies on the self-priming process of self-priming centrifugal pumps have been conducted by theoretical calculation, numerical calculation, and test measurement; certain research results have also been obtained. Using theoretical calculations, Zhao et al. [17] deduced the formulas of self-priming time and exhaustion rate according to fluid mechanics, thermodynamics, air dynamic equation, and energy invariant equation cited in their studies of vertical self-priming centrifugal pumps. Yi [18] summarized the data about self-priming time and the specific speed of 31 modes of external-mixture self-priming centrifugal pumps and explored the corresponding rules and trends. However, the self-priming time gained from the relevant calculation formula had a great range and only had statistical significance. Although the proposed theoretical calculation of self-priming time is not highly accurate and has narrow applicability, the theoretical calculation process indicates that moving bubbles are the relevant results of the main media accomplished by self-priming. Using numerical calculation, Wang et al. [19,20] adopted an inlet void fraction of 15% for the self-priming process of rotational flow of a self-priming pump using Fluent software. They found that the liquid phase drove the gas phase flow by phase interaction during the self-priming process and finally obtained the self-priming time. Using the Fluent software, Liu et al. [21,22] made a numerical calculation of the gas-water two-phase flow in the self-priming process of a single-stage self-priming centrifugal pump, and the pressure, velocity and gas distribution of the flow field under different assumed void fraction conditions were obtained. However, the exact void fraction in the pump inlet was unknown. Li et al. [23,24] simulated the gas–liquid states in a pump at different moments (initial, middle, and late stages) of the self-priming process using the quasi-steady method with a decreased void fraction, and then estimated the time required for the entire self-priming process. However, such a quasi-steady method is markedly different from the real self-priming process. For the numerical calculation of the self-priming time of self-priming pumps, scholars hypothesized that either the inlet void fraction of a pump comprised several fixed numerical values (5%, 10%, and 15%) and the velocity inlet was set or gas filled the entire pump inlet and the gas inlet velocity was a mean value. The former hypothesis was not based on the self-priming numerical calculation of self-priming pumps but on the numerical simulation of ordinary gas–liquid flow pumps. The latter hypothesis was close to the real self-priming situation, but it assumes that the velocity inlet was an average, which obviously contradicted the inversely V-shaped variation of self-priming speed in the self-priming process. The study of Huang et al. [25] was the closest to the real simulation of the self-priming time of a single-stage self-priming pump because they did not set the velocity inlet or mass outlet in the simulation process. However, the relevant calculation model only hypothesized that the self-priming height was 0.25 m. This result was inconsistent with the vertical self-priming height of 3 or 5 m in the real self-priming process, thereby failing to reflect the flow rule of the whole self-priming process. With regard to experimental measurements, existing studies mainly focused on the influences of structural improvement [26], volume of fluid reservoir [27], area and position of backflow hole [28], and tongue gap [29] of self-priming pumps on self-priming time.

In summary, the existing numerical calculations of the self-priming process in self-priming pumps are not precise, and relevant experimental studies on the self-priming process are scarce. Thus, it's very necessary to study the unsteady flow of the self-priming process through numerical methods. In the current study, a typical self-priming centrifugal pump was designed, and a numerical calculation of the gas-water two-phase flow in the self-priming process was performed using the ANSYS CFX.

#### **2. Methodology**

## *2.1. Three-Dimensional Model of the Impeller and Di*ff*user*

In this study, a motor direct connection mode in the self-priming pump is used with a compact structure and easy installation and operation. The entire pump is composed of inlet and outlet pipes, gas–liquid mixture cavity, self-priming cover plate, impeller, diffuser, outer casing, shaft, gas–liquid separation cavity, and a motor, among others (Figure 1). Moreover, the core components of the

self-priming pump include the impeller and diffuser, whose geometric parameters are calculated by using the velocity coefficient method, which is shown in Table 1. The three-dimensional models and practical pictures of the impeller and diffuser are shown in Figure 2, respectively.

**Figure 1.** Assembly diagram of the self-priming centrifugal pump. 1. Inlet section; 2. Gas-water mixture cavity; 3. Self-priming cover plate; 4. Impeller; 5. Diffuser; 6. Outer casing; 7. Gas-water separation cavity; 8. Outlet section.

**Table 1.** Basic geometrical parameters of the pump by using the velocity coefficient method.


**Figure 2.** Three-dimensional models and practical pictures of the impeller and diffuser. (**a**) Plane and axial projection of the impeller; (**b**) Outward and return vane of the diffuser; (**c**) Practical impeller; (**d**) Practical diffuser.
