## 3 Numerical Simulation Results  
### Impact of the Internal Flow on External Characteristics  
To analyze the effect of internal flow on external characteristics of the pump, three calculation models with different internal leakage channels were built for numerical simulations, and the test-bed described in [10] was used for experiments under different flow conditions. As with all open and semi-open impeller high-speed centrifugal pumps, tip leakage is inevitable. The numerical simulation of a pump considering tip leakage can be carried out by using a model whose impeller contains a fixed tip clearance, so tip leakage is considered in most studies. Hub leakage refers to the flow of fluid into the hub clearance from the gap between the impeller outlet and the diffuser inlet. The sealing structure is  
Figure 3: Coupled steady-state calculation of the impeller and the bearing of the high-speed centrifugal pump.  
set in the hub clearance, which can reduce hub leakage to a certain extent, resulting in the omission of hub leakage in most studies. However, the hub leakage in a pump with a sealed structure is comprised of both the internal leakage and the internal flow. It flows through the internal flow channel and rejoins the main stream, during which time it lubricates the sliding bearing and cools the motor. As a consequence of hub leakage, the flow field becomes more complicated and the performance of the pump is affected. Therefore, in pumps with a fully sealed structure, hub leakage cannot be ignored, and a model that considers the complete internal flow channel is required to accurately study the influence of hub leakage on the pump.  
The first model, the tip leakage model, contains the main flow channel and the tip clearance, based on which the second model, the tip-hub leakage model, considers the hub leakage. The third model, the internal flow model, takes into account all flow channels, including the tip clearance channel and the internal flow channel. A comparison between the calculated results of each model and the experimental values is shown in Table 2. With the increase in the leakage channel, the following changes occur simultaneously: the tip leakage flow continues to increase, the pump head first increases and then decreases, and the efficiency shows a decreasing trend, which becomes increasingly similar to the experimental values. This variation is consistent with [17]. Between the tip leakage model and the tip-hub leakage model, the relative error of the head is 0.5% and the relative error of the efficiency is 10%, indicating that hub leakage greatly affects the efficiency. Between the tip-hub leakage model and the internal flow model, the relative error of the head is 2.9% and the relative error of the efficiency is 2.1%, indicating that the influence of leakage from the bearing clearance on the external characteristics of the pump cannot be ignored. Between the tip leakage model and the internal flow model, the relative error of the head is 2.4% and the deviation in the efficiency is 12.1%. Moreover, compared with the tip leakage model, the tip leakage of the internal flow model increases by 52%, indicating that the internal flow greatly increases the tip leakage and significantly affects the external characteristics of the pump. Since the internal flow lubricates the bearing, cools the motor, and has a great impact on the external characteristics of the pump, the complete internal flow channel must be adopted in order to build a model of a centrifugal pump with a fully sealed structure.