## 2 Model and Methods
### Calculation Method  
During the operation of the high-speed centrifugal pump, the axial displacement of the rotor is 4 orders of magnitude different from the minimum scale of the centrifugal pump, so the influence of the axial displacement on the flow field can be ignored. Therefore, it is unnecessary to carry out a dynamic grid calculation considering the axial force balance. As for a pump with an internal flow channel, its rotor is floating during operation. When the working conditions change, both the balance position of the axial force and the bearing capacity change continuously, and the rotor moves back and forth in the axial direction. In addition, the liquid film thickness directly affects the flow field as well as the flow rate of the internal flow. Therefore, a MATLAB-CFX coupling calculation method was adopted.  
The clearance of the high-speed centrifugal pump with a fully sealed structure is composed of four main parts, namely the front liquid film thickness, the tail liquid film thickness, the tip clearance, and the hub clearance. When the pump operates in a stable state, the axial resultant force is zero and the rotor is in the balance position. When the working condition changes, the distribution of the flow field at the inlet of the sliding bearing changes, and the axial force of the impeller becomes non-zero. Moreover, the change in the working condition will also cause the motor to move, which will affect the size of the tip clearance and the hub clearance in turn. As a result, the flow field distribution of the impeller will change again. A flow chart of the calculation is shown in Figure 3. The coupled steady-state calculation of the impeller and the bearing was simulated by MATLAB and the ANSYS-CFX platform. In MATLAB, the axial force of the impeller and the bearing capacity of the bearing are both calculated through the empirical formula, and an approximate liquid film thickness is given in advance. Then, CFX is called through the batch file to write the flow, speed, and axial displacement of the rotor into the intermediate data file. In CFX, according to the displacement given by the data file, the grid of the flow field is updated by using the moving grid method, and the information on the axial force is transmitted to MATLAB after the calculation converges. Next, in MATLAB, the liquid film  
Figure 2: Diagram of the centrifugal pump with a fully sealed structure. (**a**) Model diagram of the pump (1, Inducer; 2, Impeller; 3, Diffuser; 4, Main axle; 5, Motor stator; 6, Tail end fixed bearing; 7, Front sliding bearing; 8, Front fixed bearing; 9, Tail end sliding bearing); and (**b**) flow passage diagram of the pump.  
thickness is modified again according to the imbalance in the axial force, after which CFX is called to perform a recalculation. The above-described steps repeat until the resultant force reaches equilibrium.  
Since the working medium in the pump, which is water, is an incompressible fluid, governing equations of incompressible flow were adopted that ignore the change in temperature. The equations for the pump head and efficiency are shown below.  
\[H=\frac{P_{\text{out}}}{\rho g}-\frac{P_{\text{in}}}{\rho g}+\Delta z \tag{1}\]  
\[\eta=\frac{P_{\text{u}}}{P_{\text{u}}+P_{\text{h}}+P_{\text{v}}+P_{\text{m}}} \tag{2}\]  
where \(P_{\text{out}}\) is the outlet pressure of the pump, \(P_{\text{in}}\) is the inlet pressure of the pump, \(\rho\) is the density of water, \(g\) is a constant term, and \(\Delta z\) is the height difference between the outlet and the inlet. In Equation (2), \(P_{\text{u}}\) is the useful power, \(P_{\text{h}}\) is the hydraulic loss power, \(P_{\text{v}}\) is the volume loss power of the internal flow and tip leakage, and \(P_{\text{m}}\) is the mechanical loss power, which is mainly composed of the disc friction loss power.