*2.3. Grid Information*

The calculation domain should be discretized before grid-based simulation, and grid quality affects calculation accuracy and time. Generally, dividing a calculation model with a complex geometry and boundary by using hybrid grids is reasonable. The calculation domain was divided into hybrid grids by Gambit software in this study because the numerical model of the self-priming pump and the boundary conditions of the self-priming calculation were complex. Given that performing a grid-independence analysis for the unsteady self-priming calculation is inconvenient due to the large amount of calculation time, the number of grids was made as large as possible but still within the computing capability of the workstation. The main grid information is shown in Figure 4 and Table 2, and the total number of grids is more than four million.

**Figure 4.** *Cont.*

**Figure 4.** Grid of the self-priming centrifugal pump by using Gambit software (Gambit 2.4.6, ANSYS Corporation, Pittsburgh, PA, US.). (**a**) Total pump; (**b**) Impeller; (**c**) Diffuser; (**d**) Gas-water mixture cavity; (**e**) Gas-water separation cavity.

**Table 2.** Grid information of the self-priming centrifugal pump including the size, number, quality and type of the grid.


#### *2.4. Time Step Independence*

In general, in the periodic numerical simulation, the time step needs to satisfy the Courant number criterion [30], which is expressed as:

$$C\_o = v \Delta t / l < 100\tag{1}$$

In the formula, *v* is the absolute value of the estimated mean velocity, m/s; *l* is the smallest size of the grid, m; *t* is the time step, s; *Co* is the Courant number criterion and required no more than 100. When the numerical convergence is not good, it is appropriate to take smaller values.

If the time step size is too large, the value of Courant number will also be large; however, too small a time step will also lead to a significant increase in computing time [31,32]. Therefore, considering the computer configuration, the time step *<sup>t</sup>* was chosen as 5 <sup>×</sup> <sup>10</sup>−<sup>3</sup> s. Moreover, the value of *<sup>v</sup>* is within 10 m/s, and the value of *<sup>l</sup>* is more than 5 <sup>×</sup> <sup>10</sup>−<sup>3</sup> m (according to Table 1); therefore the value of Courant number is within 10.

#### *2.5. Setting of the Boundary Condition*

ANSYS Computational Fluid X software (ANSYS CFX 14.5, ANSYS Corporation, Pittsburgh, PA, USA) was used to perform numerical calculation in the self-priming process of the self-priming centrifugal pump. The impeller and shroud in the pump cavity were based on the rotating reference frame, whereas the other sub-domains were based on the stationary reference frame. Moreover, the pressure inlet and the opening in the outlet were selected as inlet and outlet boundaries in order to approximate the actual self-priming condition as much as possible. The gas-water volume fraction contour of the multistage self-priming pump in the initial state is shown in Figure 5. The height of the inlet elbow was 1.5 m, and the height of the outlet pipe was 1 m. The inlet elbow was placed in the

water. The elbow above the water surface was filled with gas (1 m in height), the elbow below the water surface was filled with water (0.5 m in height), and the outlet pipe was filled with gas (1 m in height). A check valve was installed in the pump inlet, and the entire pump was also filled with water. Given that the length of the inlet elbow below the water surface was 0.5 m, the initial pressure at the inlet was set to 5 kPa (gage pressure), and the pressure was reduced progressively to 0 kPa until at the water surface. The initial pressure at the inlet and outlet gas sections was 0 kPa. The initial pressure distribution of the entire self-priming pump is shown in Figure 6.

**Figure 6.** Static pressure contour of the self-priming pump in the initial state.
