Study of Energy Loss Characteristics of a Shaft Tubular Pump Device Based on the Entropy Production Method

The unstable flow of a shaft tubular pump device (STPD) leads to energy loss, thereby reducing its efficiency. The aim of this study is to investigate the distribution pattern of energy loss in STPDs. This paper reveals that the two components with the highest proportion of energy loss are the impeller and the outlet passage. Furthermore, turbulent entropy production is the primary cause of energy loss. Due to the wall effect, the energy loss in the impeller mainly occurs near the hub and shroud. Additionally, the presence of a tip leakage vortex near the shroud further contributes to the energy loss in the region near the shroud. This results in the energy loss proportion exceeding 40% in the region with a volume fraction of 14% near the shroud. In the outlet passage, the energy loss mainly occurs in the front region, with a volume fraction of 30%, and the energy loss in this part accounts for more than 65%. Finally, this study reveals the locations of the vortex in the STPD under different flow-rate conditions, and when the distribution of energy loss is visualized, it is found that the energy loss occurs high in the vortex regions.


Introduction
Water resources are indispensable for the production and life of human beings; however, with the occurrence of floods and droughts caused by global climate variation, many new pumping stations have been built worldwide to meet the continuously increasing needs of water resource allocation [1,2]. In China, water resources are severely unevenly distributed in space and time, and to improve the present situation, the government has invested in the construction of the famous South-to-North Water Transfer Project, of which the east line project is composed of numerous large-scale pumping stations. These largescale pumping stations provide huge amounts of flow and generally operate for more than 3000 h annually [3]. Based on such operating conditions, the pumping device used in these pumping stations must have excellent efficiency and stability. After being developed for decades, the shaft tubular pump device (STPD) has become the most widely used type of pump device in coastal areas of China for its excellent hydraulic performance, and up to now, the STPD has been adopted in about 100 pumping stations. In the past, the hydraulic performance and optimized design of the STPD have already been studied [4][5][6]. Xu et al. [7] optimized the design of the STPD by numerical simulation method and tested its performance. The results showed that the optimal efficiency of the device exceeded 83%. However, with the acceleration of abnormal climate variation induced by increasing carbon emissions, China has proposed a national strategy for the peaking of CO 2 emissions and carbon neutrality, which requires that pump devices as energy consumers must operate more efficiently. Therefore, conducting a more in-depth study on the energy loss mechanism of STPDs is of great significance.
In the past, hydraulic loss in most studies was typically calculated based on the pressure drop across two sections [8]. This method can only provide the value of the

Grid Division and Scheme Selection
In this study, the parts of an STPD are divided into structured meshes. ICEM software is used for the mesh generation of two passages, and Turbo-Grid software is used for the grid division of the impeller and guide vanes. The grid diagram is shown in Figure 2.

Inlet Passage
Outlet passage Impeller Guide vanes

Grid Division and Scheme Selection
In this study, the parts of an STPD are divided into structured meshes. ICEM software is used for the mesh generation of two passages, and Turbo-Grid software is used for the grid division of the impeller and guide vanes. The grid diagram is shown in Figure 2.

Inlet Passage
Outlet passage Impeller Guide vanes  Numerous studies have shown that the grid division of the simulation domains has an important influence on the results [26,27]. Therefore, in this study, the efficiency of STPDs is used as the control index for grid independence analysis. Five grid schemes are simulated in this study. Table 3 indicates that the simulation results are stable when the number of grid cells reaches 8.6 million, with an efficiency change of only 0.01%, which can be considered negligible. The grid convergence index (GCI) proposed by Roache is a commonly applied method to verify the reliability of the mesh, and the validity of the GCI has been verified by many scholars [28][29][30]. The procedure of GCI calculation in this study is shown in Table 4. As shown in the table, the GCI is less than 2%, which indicates that the dispersion error of the simulation is small. Considering the grid independence and GCI, the 8.6 million grid scheme is used for numerical simulation. The distribution of Yplus in the four components of the STPD is shown in Figure 3, and the average Yplus values are 29.33, 16.61, 23.37 and 24.87, respectively. According to the Yplus values in the numerical simulations in refs. [15,31], the mesh scheme in this study can be used.

Model Test
The high-precision test rig is located at the China Water Resources Beifang Co., Ltd. located in Tianjing, China, as shown in Figure 4. The measurement uncertainty of the V15712-HD1A1D7D electromagnetic flow meter is ±0.2%. The LDG-500s differential pressure transmitter is used to test the head, and the measurement uncertainty is ±0.1%. The measurement uncertainty of the JCZL2-500 torque and speed sensor is ±0.1%. Based on the measurement uncertainty of the instruments mentioned above, the uncertainty of the system can be determined as ±0.24%. This model test meets the requirements of IEC standards. This model test tested the hydraulic performance of the STPD at multiple blade

Model Test
The high-precision test rig is located at the China Water Resources Beifang Co., Ltd. located in Tianjing, China, as shown in Figure 4. The measurement uncertainty of the V15712-HD1A1D7D electromagnetic flow meter is ±0.2%. The LDG-500s differential pressure transmitter is used to test the head, and the measurement uncertainty is ±0.1%. The measurement uncertainty of the JCZL2-500 torque and speed sensor is ±0.1%. Based on the measurement uncertainty of the instruments mentioned above, the uncertainty of the system can be determined as ±0.24%. This model test meets the requirements of IEC standards. This model test tested the hydraulic performance of the STPD at multiple blade angles. Table 5 shows the 10 times repeatability test results for the highest-efficiency point with blade angle of −2 degrees, from which the random uncertainty of 0.2% can be deduced; finally, the comprehensive uncertainty of this experiment is obtained as 0.5%.

Impeller
Guide vanes

Model Test
The high-precision test rig is located at the China Water Resources Beifang Co., Ltd. located in Tianjing, China, as shown in Figure 4. The measurement uncertainty of the V15712-HD1A1D7D electromagnetic flow meter is ±0.2%. The LDG-500s differential pressure transmitter is used to test the head, and the measurement uncertainty is ±0.1%. The measurement uncertainty of the JCZL2-500 torque and speed sensor is ±0.1%. Based on the measurement uncertainty of the instruments mentioned above, the uncertainty of the system can be determined as ±0.24%. This model test meets the requirements of IEC standards. This model test tested the hydraulic performance of the STPD at multiple blade angles. Table 5 shows the 10 times repeatability test results for the highest-efficiency point with blade angle of −2 degrees, from which the random uncertainty of 0.2% can be deduced; finally, the comprehensive uncertainty of this experiment is obtained as 0.5%.

Pump device
Impeller Guide vanes

Traditional Pressure Drop Method
In order to evaluate the energy conversion performance of each domain in the STPD, the hydraulic loss can be calculated using total pressure drop between the inlet and outlet of each part. In the past, the calculations of hydraulic loss in most studies were based on this method [32,33]. The hydraulic loss ∆h s in the stationary domain (inlet and outlet passage, guide vanes) and ∆h r in the rotating domain (impeller) can be calculated by Equations (1) and (2), respectively. The above hydraulic loss can be converted into the energy loss by Equation (3).
where p 1 and p 2 represent the total pressure of inlet and outlet sections, Pa; and P is the input shaft power, kW.

Entropy Production Method
Near the wall region, the viscous force leads the kinetic and pressure energy to be converted into internal energy, thus leading to the increase in entropy. In the high Reynolds number region, the unstable flow causes an increase in entropy production. Therefore, the entropy production method can be used to calculate the energy loss in the STPD, and the equations involved in the entropy production method can be referred to Ref. [21]. The turbulence kinetic energy equations for a Newtonian fluid in Cartesian coordinates could be expressed as follows: Using the velocity components u 1 , u 2 and u 3 , the Φ can be expanded to Equation (6): The incompressible fluid satisfies the following continuity equation: Therefore, Equation (6) can be simplified to Equation (8).
The local entropy production rate (LEPR) can be calculated by using Equation (9).
In the turbulent flow, the LEPR includes the direct entropy production rate induced by time-averaged motion and turbulent entropy production rate (TEPR) due to turbulent dissipation.
The direct entropy production rate can be obtained by using Equation (11).
Mathieu and Scott proposed a method to obtain the TEPR in the k-ω model. The formula is as follows [34]: The local entropy production (LEP) includes direct entropy production and TEP: Entropy 2023, 25, 995 7 of 20 In this study, the EPW can be calculated by using Equation (14).
The total entropy production can be calculated by using Equation (15).

Validation of Simulation and Comparison of Energy Analysis Methods
The test results showed that the variation pattern of hydraulic performance under different angles is basically the same, so the STPD with a blade angle of 0 degrees is selected for detailed analysis. The energy performance curves of the STPD obtained by the model test and numerical simulation are shown in Figure 5. At 0.8 Q d~1 .2 Q d , the head and the shaft power curves of the STPD decrease with the increase in flow rate. The efficiency curve of the STPD increases with the increasing flow rate, reaching a maximum at 1.0 Q d and then decreasing at 1.0 Q d . The relative errors of head, shaft power and efficiency between the model test and numerical simulation are 2.49%, 2.69% and 0.62%, respectively. The maximum error of the shaft power and efficiency occurs at 0.8 Q d , and the relative errors are 4.95% and 1.80%, respectively. The relative error at low-flow-rate conditions is mainly due to the flow separation at the blades. The maximum error of the head occurs at 1.2 Q d with a relative error of 5.2%, which is caused by the low absolute value of the head at high-flow-rate conditions. Although there are still some uncertainties in numerical simulations that may cause slight deviations from experimental results, overall, the results are reliable. As shown in Figure 6, although the energy loss obtained by the entropy productio method is slightly smaller than that obtained by the pressure drop method at eac condition, the two methods still exhibit good uniformity. The variation trend of energ loss obtained by the two methods is the same; the energy loss decreases first and the increases in the flow range of 0.8 Qd~1.2 Qd. The results obtained by the entrop production method are proved to be reliable according to the traditional pressure dro method; hence, this study can be conducted based on the entropy production method. As shown in Figure 6, although the energy loss obtained by the entropy production method is slightly smaller than that obtained by the pressure drop method at each condition, the two methods still exhibit good uniformity. The variation trend of energy loss obtained by the two methods is the same; the energy loss decreases first and then increases in the flow range of 0.8 Q d~1 .2 Q d . The results obtained by the entropy production method are proved to be reliable according to the traditional pressure drop method; hence, this study can be conducted based on the entropy production method. As shown in Figure 6, although the energy loss obtained by the entropy production method is slightly smaller than that obtained by the pressure drop method at each condition, the two methods still exhibit good uniformity. The variation trend of energy loss obtained by the two methods is the same; the energy loss decreases first and then increases in the flow range of 0.8 Qd~1.2 Qd. The results obtained by the entropy production method are proved to be reliable according to the traditional pressure drop method; hence, this study can be conducted based on the entropy production method.   Figure 7, the variation trend of energy loss in each component obtained by the two methods is consistent. In the inlet passage, the energy loss is very small and increases monotonically, which is due to the increasing flow velocity. In the remaining three parts, the energy loss decreases first and then increases, and the energy loss is minimal at the design flow condition. In the range of 0.8 Q d~1 .1 Q d , the loss proportion of each component from the largest to the smallest is impeller, outlet passage, guide vanes and inlet passage; however, the loss proportion of the outlet passage at 1.2 Q d is more than the impeller. According to Figure 7a, in the flow range of 0.8 Q d~1 .2 Q d , the loss in the inlet passage accounts for 3.0~8.2% of the loss in the STPD, the loss in the impeller accounts for 30.9~47.7%, the loss in the guide vanes accounts for 10.2~26.2% and the loss in the outlet passage accounts for 31.5~41.4%. As shown in Figure 7b, the percentage of energy loss in each component is basically consistent with the results obtained by the traditional pressure drop method. In addition, the proportions of different types of entropy production are shown in Figure 7b. It is obvious that the DEP in each component is small and can be neglected, while the TEP is the main contributor to energy loss in STPDs, which is consistent with the conclusions of previous research [31,35]. passage accounts for 3.0~8.2% of the loss in the STPD, the loss in the impeller accounts for 30.9~47.7%, the loss in the guide vanes accounts for 10.2~26.2% and the loss in the outlet passage accounts for 31.5~41.4%. As shown in Figure 7b, the percentage of energy loss in each component is basically consistent with the results obtained by the traditional pressure drop method. In addition, the proportions of different types of entropy production are shown in Figure 7b. It is obvious that the DEP in each component is small and can be neglected, while the TEP is the main contributor to energy loss in STPDs, which is consistent with the conclusions of previous research [31,35].

Analysis of TEP Distribution
According to the analysis in Section 5.2, the energy loss in STPDs is mainly concentrated in the impeller, guide vane and outlet passage, and the main source of energy loss is TEP. This section will further analyze the distribution of TEP in these three components. The impeller and the guide vanes are evenly divided into 10 subdomains along the axial and radial directions. The subdomains of the impeller along the axial direction are named IA1~IA10, the subdomains along the radial direction are named IR1~IR10, the subdomains of the guide vanes along the axial direction are named GA1~IA10 and the subdomains along the radial direction are named GR1~IR10, as shown in Figure 8.

Analysis of TEP Distribution
According to the analysis in Section 5.2, the energy loss in STPDs is mainly concentrated in the impeller, guide vane and outlet passage, and the main source of energy loss is TEP. This section will further analyze the distribution of TEP in these three components. The impeller and the guide vanes are evenly divided into 10 subdomains along the axial and radial directions. The subdomains of the impeller along the axial direction are named IA1~IA10, the subdomains along the radial direction are named IR1~IR10, the subdomains of the guide vanes along the axial direction are named GA1~IA10 and the subdomains along the radial direction are named GR1~IR10, as shown in Figure 8. 30.9~47.7%, the loss in the guide vanes accounts for 10.2~26.2% and the loss in the outlet passage accounts for 31.5~41.4%. As shown in Figure 7b, the percentage of energy loss in each component is basically consistent with the results obtained by the traditional pressure drop method. In addition, the proportions of different types of entropy production are shown in Figure 7b. It is obvious that the DEP in each component is small and can be neglected, while the TEP is the main contributor to energy loss in STPDs, which is consistent with the conclusions of previous research [31,35].

Analysis of TEP Distribution
According to the analysis in Section 5.2, the energy loss in STPDs is mainly concentrated in the impeller, guide vane and outlet passage, and the main source of energy loss is TEP. This section will further analyze the distribution of TEP in these three components. The impeller and the guide vanes are evenly divided into 10 subdomains along the axial and radial directions. The subdomains of the impeller along the axial direction are named IA1~IA10, the subdomains along the radial direction are named IR1~IR10, the subdomains of the guide vanes along the axial direction are named GA1~IA10 and the subdomains along the radial direction are named GR1~IR10, as shown in Figure 8.  Figure 9 shows the TEP distribution of each subdo Qd, the TEP gradually decreases, while in the flow increases, and the trend of variation of the energy lo shown in Figure 9a, it can be found that the TEP in the decreases along the axial direction at each condition. leading edge of the impeller leads to a surge of TEP. At which is due to the decrease in the attack angle; the flo the formation of backflow and vortex [36]. It can be fou the middle subdomains of the impeller is relatively st state in this region is stable. As it is influenced by the f the viscous resistance of the liquid, thus the flow in weakened and the loss increases [37]. Therefore, the TE than that in the adjacent subdomains. Additionally, the near the shroud further promotes the TEP in IR10. T exceeding 40% in IR10 with a volume fraction of 14% increases as the flow velocity decreases [38], hence the rate.  Figure 9 shows the TEP distribution of each subdomain of the impeller. At 0.8 Q d~1 .0 Q d , the TEP gradually decreases, while in the flow range of 1.0 Q d~1 .2 Q d , the TEP increases, and the trend of variation of the energy loss is consistent with Figure 7. As shown in Figure 9a, it can be found that the TEP in the impeller firstly increases and then decreases along the axial direction at each condition. In IA3, the flow separation at the leading edge of the impeller leads to a surge of TEP. At 0.8 Q d , the TEP in IA3 is the largest, which is due to the decrease in the attack angle; the flow separation is more serious, as is the formation of backflow and vortex [36]. It can be found from Figure 9b that the TEP in the middle subdomains of the impeller is relatively stable, which indicates that the flow state in this region is stable. As it is influenced by the frictional resistance of the wall and the viscous resistance of the liquid, thus the flow in the axial direction near the wall is weakened and the loss increases [37]. Therefore, the TEP in IR1 and IR10 are clearly higher than that in the adjacent subdomains. Additionally, the presence of the tip leakage vortex near the shroud further promotes the TEP in IR10. This results in the TEP proportion exceeding 40% in IR10 with a volume fraction of 14%. Furthermore, the range of TLV increases as the flow velocity decreases [38], hence the TEP in IR10 is large at a small flow rate.
the middle subdomains of the impeller is relatively stable, which indicates that the flow state in this region is stable. As it is influenced by the frictional resistance of the wall and the viscous resistance of the liquid, thus the flow in the axial direction near the wall is weakened and the loss increases [37]. Therefore, the TEP in IR1 and IR10 are clearly higher than that in the adjacent subdomains. Additionally, the presence of the tip leakage vortex near the shroud further promotes the TEP in IR10. This results in the TEP proportion exceeding 40% in IR10 with a volume fraction of 14%. Furthermore, the range of TLV increases as the flow velocity decreases [38], hence the TEP in IR10 is large at a small flow rate.
(a) (b) The guide vanes are positioned downstream of the impeller to reduce the circulation of the flow out of the impeller, and they enable the water to enter the outlet channel more smoothly. The distribution of TEP in the guide vanes is shown in Figure 10. According to Figure 10a, it can be found that at 0.9 Qd~1.2 Qd, the LEP in subdomains decreases along the axial direction, which indicates that the guide vanes reduce the circulation velocity of the flow out of the impeller, and finally the flow in the subdomains at the exit is stable and the TEP is lower. At 0.8 Qd, there is a vortex at the back of the guide vanes near the outlet The guide vanes are positioned downstream of the impeller to reduce the circulation of the flow out of the impeller, and they enable the water to enter the outlet channel more smoothly. The distribution of TEP in the guide vanes is shown in Figure 10. According to Figure 10a, it can be found that at 0.9 Q d~1 .2 Q d , the LEP in subdomains decreases along the axial direction, which indicates that the guide vanes reduce the circulation velocity of the flow out of the impeller, and finally the flow in the subdomains at the exit is stable and the TEP is lower. At 0.8 Q d , there is a vortex at the back of the guide vanes near the outlet section, therefore the TEP in GA10 increases. According to Figure 10b, along the radial direction, the TEP in GR1 and GR10 is clearly higher than that in the adjacent subdomains due to the wall effect. At 0.9 Q d~1 .1 Q d , the TEP in the guide vanes is stable, which indicates that the flow is stable. At 0.8 Q d and 1.2 Q d , the distribution of TEP has no clear regularity due to the large deviation from the design flow rate, which also indicates the internal flow of guide vanes is chaotic. section, therefore the TEP in GA10 increases. According to Figure 10b, along the radial direction, the TEP in GR1 and GR10 is clearly higher than that in the adjacent subdomains due to the wall effect. At 0.9 Qd~1.1 Qd, the TEP in the guide vanes is stable, which indicates that the flow is stable. At 0.8 Qd and 1.2 Qd, the distribution of TEP has no clear regularity due to the large deviation from the design flow rate, which also indicates the internal flow of guide vanes is chaotic.
(a) (b) Considering that the outlet passage is a leafless region, the distribution characteristics of TEP will be discussed only along the axial direction. According to Figure 11, the outlet passage is divided into 10 subdomains and named OA1~OA10 along the axial direction. Considering that the outlet passage is a leafless region, the distribution characteristics of TEP will be discussed only along the axial direction. According to Figure 11, the outlet passage is divided into 10 subdomains and named OA1~OA10 along the axial direction. Considering that the outlet passage is a leafless region, the distribution characteristics of TEP will be discussed only along the axial direction. According to Figure 11, the outlet passage is divided into 10 subdomains and named OA1~OA10 along the axial direction. The TEP in each subdomain of the outlet passage is shown in Figure 12. At 0.9 Qd~1.1 Qd, the distribution characteristics of TEP along the axial direction are consistent. The TEP increases first, with the maximum in the OA3 and OA4, and then decreases gradually. The volume of OA1~OA4 accounts for 30%; however, the TEP accounts for more than 65%. In the rear part, the energy loss is very small because the velocity and circulation have decreased due to the completion of the flow diffusion. At 0.8 Qd, the TEP value in the OA1 is the largest, which is due to the effect of the velocity circulation and guide vanes' wake vortex (GWV). At 1.2 Qd, due to the larger flow velocity, the TEP values in OA2~OA10 are larger than those at other flow-rate conditions. The TEP in each subdomain of the outlet passage is shown in Figure 12. At 0.9 Q d~1 .1 Q d , the distribution characteristics of TEP along the axial direction are consistent. The TEP increases first, with the maximum in the OA3 and OA4, and then decreases gradually. The volume of OA1~OA4 accounts for 30%; however, the TEP accounts for more than 65%. In the rear part, the energy loss is very small because the velocity and circulation have decreased due to the completion of the flow diffusion. At 0.8 Q d , the TEP value in the OA1 is the largest, which is due to the effect of the velocity circulation and guide vanes' wake vortex (GWV). At 1.2 Q d , due to the larger flow velocity, the TEP values in OA2~OA10 are larger than those at other flow-rate conditions.

Visualization Analysis
Three typical operation conditions are selected for visualization and analysis. Figures  13 and 14 show the distribution of TEPR and the flow characteristics in the impeller at the typical span of 0.03, 0.5 and 0.97. The location of the high TEPR region can be verified by the flow structure, and the TEPR is higher in the region of poor flow characteristics. At all conditions, there is an impeller wake vortex (IWV) at the trailing edge of the blade resulting in a small high TEPR region. Near the hub (span = 0.03), there is a high TEPR region at the leading edge of the impeller due to the vortex at the leading edge of impeller (ILV) at 0.8 Qd. The ILV is the flow separation caused by the large attack angle at a small flow rate. There is also a high TEPR region in the impeller passage, which is caused by the

Visualization Analysis
Three typical operation conditions are selected for visualization and analysis. Figures 13 and 14 show the distribution of TEPR and the flow characteristics in the impeller at the typical span of 0.03, 0.5 and 0.97. The location of the high TEPR region can be verified by the flow structure, and the TEPR is higher in the region of poor flow characteristics. At all conditions, there is an impeller wake vortex (IWV) at the trailing edge of the blade resulting in a small high TEPR region. Near the hub (span = 0.03), there is a high TEPR region at the leading edge of the impeller due to the vortex at the leading edge of impeller (ILV) at 0.8 Q d . The ILV is the flow separation caused by the large attack angle at a small flow rate. There is also a high TEPR region in the impeller passage, which is caused by the hub vortex of the impeller (IHV). At 1.0 Q d and 1.2 Q d , the attack angle decreases and the ILV disappears, hence the TEPR is significantly lower. In the middle passage (span = 0.5), the flow moves along the blade airfoil, hence the TEPR is obviously lower than that near the hub and shroud. Near the shroud region (span = 0.97), the TLV leads to the high TEPR region, and as the flow rate increases; the head and the pressure difference between the pressure side (PS) and suction side (SS) decrease, which leads to a reduction of the area of the TLV and high TEPR region.

Visualization Analysis
Three typical operation conditions are selected for visualization and analysis. Figures  13 and 14 show the distribution of TEPR and the flow characteristics in the impeller at the typical span of 0.03, 0.5 and 0.97. The location of the high TEPR region can be verified by the flow structure, and the TEPR is higher in the region of poor flow characteristics. At all conditions, there is an impeller wake vortex (IWV) at the trailing edge of the blade resulting in a small high TEPR region. Near the hub (span = 0.03), there is a high TEPR region at the leading edge of the impeller due to the vortex at the leading edge of impeller (ILV) at 0.8 Qd. The ILV is the flow separation caused by the large attack angle at a small flow rate. There is also a high TEPR region in the impeller passage, which is caused by the hub vortex of the impeller (IHV). At 1.0 Qd and 1.2 Qd, the attack angle decreases and the ILV disappears, hence the TEPR is significantly lower. In the middle passage (span = 0.5), the flow moves along the blade airfoil, hence the TEPR is obviously lower than that near the hub and shroud. Near the shroud region (span = 0.97), the TLV leads to the high TEPR region, and as the flow rate increases; the head and the pressure difference between the pressure side (PS) and suction side (SS) decrease, which leads to a reduction of the area of the TLV and high TEPR region.   Figures 15 and 16 show the distribution of TEPR and the flow characteristics in the guide vanes at a typical span of 0.03, 0.5 and 0.97. Due to the presence of GWV at three operating conditions, there is a small high TEPR region at the trailing edge of the guide vanes. In addition, there is the guide vanes hub vortex (GHV) near the hub, which results in high TEPR at corresponding locations. As the flow rate increases, the GHV first decreases and then increases, and the high TEPR region shows the same variation. The matching relationship between the flow direction and the guide vanes placement angle is shown in Figure 17. At 0.8 Q d , because of the misfit of flow direction and the guide vanes placement angle, there is a guide vanes separation vortex (GSV) in the SS side near the outlet, and the TEPR is high in this region. At 1.0 Q d , the flow in the guide vanes is stable without flow separation, hence the TEPR is low. At 1.2 Q d , the location of GSV shifts to the PS of the guide vanes near the inlet and results in a high TEPR region.  vanes. In addition, there is the guide vanes hub vortex (GHV) near the hub, which results in high TEPR at corresponding locations. As the flow rate increases, the GHV first decreases and then increases, and the high TEPR region shows the same variation. The matching relationship between the flow direction and the guide vanes placement angle is shown in Figure 17.  In order to further investigate the distribution of the TEPR in the impeller and guide vanes, six monitoring sections are set up along the axial direction, as shown in Figure 18. Figure 19 shows the distribution of TEPR and flow characteristics in the monitoring sections of the impeller; from left to right, they are S1, S2 and S3. The distribution of TEPR and flow characteristics can verify each other, and the TEPR is high in the region with disordered flow. At each of the flow-rate conditions, the high TEPR regions are mainly distributed on the blade surfaces and near the hub and shroud regions due to the influence of IHV and TLV. As the IHV gradually increases along the axial direction, the area of the high TEPR region near the hub also increases. In order to further investigate the distribution of the TEPR in the impeller and g vanes, six monitoring sections are set up along the axial direction, as shown in Figur Figure 19 shows the distribution of TEPR and flow characteristics in the monito sections of the impeller; from left to right, they are S1, S2 and S3. The distribution of T and flow characteristics can verify each other, and the TEPR is high in the region disordered flow. At each of the flow-rate conditions, the high TEPR regions are ma distributed on the blade surfaces and near the hub and shroud regions due to the influ of IHV and TLV. As the IHV gradually increases along the axial direction, the area o high TEPR region near the hub also increases. In order to further investigate the distribution of the TEPR in the impeller and guide vanes, six monitoring sections are set up along the axial direction, as shown in Figure 18. Figure 19 shows the distribution of TEPR and flow characteristics in the monitoring sections of the impeller; from left to right, they are S1, S2 and S3. The distribution of TEPR and flow characteristics can verify each other, and the TEPR is high in the region with disordered flow. At each of the flow-rate conditions, the high TEPR regions are mainly distributed on the blade surfaces and near the hub and shroud regions due to the influence of IHV and TLV. As the IHV gradually increases along the axial direction, the area of the high TEPR region near the hub also increases.  Figure 20 shows the distribution of TEPR and flow characteristics in the monitoring sections of the guide vanes. In Figure 20, from left to right, they are S4, S5 and S6. The distribution of TEPR and flow characteristics can verify each other, and the TEPR is high in the region with disordered flow. At 0.8 Qd, the presence of GHV near the hub leads to a small high TEPR region. Along the axial direction, GSV appears in the SS of the guide   Figure 20 shows the distribution of TEPR and flow characteristics in the monitoring sections of the guide vanes. In Figure 20, from left to right, they are S4, S5 and S6. The distribution of TEPR and flow characteristics can verify each other, and the TEPR is high in the region with disordered flow. At 0.8 Qd, the presence of GHV near the hub leads to a small high TEPR region. Along the axial direction, GSV appears in the SS of the guide  Figure 20 shows the distribution of TEPR and flow characteristics in the monitoring sections of the guide vanes. In Figure 20, from left to right, they are S4, S5 and S6. The distribution of TEPR and flow characteristics can verify each other, and the TEPR is high in the region with disordered flow. At 0.8 Q d , the presence of GHV near the hub leads to a small high TEPR region. Along the axial direction, GSV appears in the SS of the guide vanes and gradually increases, hence the area of the high TEPR region also increases. When the flow rate is 1.0 Q d and the TEPR is small in each section of the guide vanes, there is only a small high TEPR region that appears near the hub caused by GHV and gradually increases with the axial direction. At 1.2 Q d , in each monitoring section, there are high TEPR regions near the hub and the PS of the guide vanes due to the influence of GSV and GHV. Along the axial direction, the high TEPR region gradually decreases because of the decrease in the GSV range.
vanes and gradually increases, hence the area of the high TEPR region also increases. When the flow rate is 1.0 Qd and the TEPR is small in each section of the guide vanes, there is only a small high TEPR region that appears near the hub caused by GHV and gradually increases with the axial direction. At 1.2 Qd, in each monitoring section, there are high TEPR regions near the hub and the PS of the guide vanes due to the influence of GSV and GHV. Along the axial direction, the high TEPR region gradually decreases because of the decrease in the GSV range.  Figure 21 shows the distribution of TERP in the horizontal and vertical sections of the outlet passage under each operation condition. According to Figure 21, it is found that the TEPR in the outlet passage has basically symmetrical distribution and is sensitive to the change of flow rate. The high TEPR region decreases and then increases with the increase in flow rate, which is because the energy loss in the outlet passage is influenced by both the circulation velocity and the flow velocity. The high TEPR region is completely concentrated in the front part of the outlet passage, which further validates the distribution law of energy loss in Figure 12. Six monitoring sections are set up along the axial direction; the location and numbers are shown in Figure 22.  Figure 21 shows the distribution of TERP in the horizontal and vertical sections of the outlet passage under each operation condition. According to Figure 21, it is found that the TEPR in the outlet passage has basically symmetrical distribution and is sensitive to the change of flow rate. The high TEPR region decreases and then increases with the increase in flow rate, which is because the energy loss in the outlet passage is influenced by both the circulation velocity and the flow velocity. The high TEPR region is completely concentrated in the front part of the outlet passage, which further validates the distribution law of energy loss in Figure 12. Six monitoring sections are set up along the axial direction; the location and numbers are shown in Figure 22.    Figure 23 shows the distribution of TEPR in the monitoring sections of the outflow passage. The common characteristics can be found that the high TEPR region in the sections decreases gradually along the axial direction. In S7~S9, the distribution of high TEPR is clearly influenced by the guide vanes, with six high TEPR regions appearing in each section, which is consistent with the number of guide vanes. The flow out of the guide vanes still has circulation, so the flow moves toward the outer wall by centrifugal force; therefore, there is a high TEPR region near the outer wall surface of each section, and the high TEPR region near the outer wall surface increases as the flow rate increases. In S11~S12, the flow has been adjusted by the outlet passage to complete the diffusion and kinetic energy recovery, so the flow is stable and therefore the TEPR is quite low, which is consistent with the law in Figure 12. Compared with Figure 20, it is found that the distribution characteristics of TEPR in S7 are basically the same as those in S6, but the intensity of TEPR is reduced; this indicates that the flow in the outlet passage is still influenced by GSV, GWV and GHV, and the influence gradually decreases along the axial direction.

Distribution of TEPR
(a) Figure 22. The location of monitoring sections of the outlet passage. Figure 23 shows the distribution of TEPR in the monitoring sections of the outflow passage. The common characteristics can be found that the high TEPR region in the sections decreases gradually along the axial direction. In S7~S9, the distribution of high TEPR is clearly influenced by the guide vanes, with six high TEPR regions appearing in each section, which is consistent with the number of guide vanes. The flow out of the guide vanes still has circulation, so the flow moves toward the outer wall by centrifugal force; therefore, there is a high TEPR region near the outer wall surface of each section, and the high TEPR region near the outer wall surface increases as the flow rate increases. In S11~S12, the flow has been adjusted by the outlet passage to complete the diffusion and kinetic energy recovery, so the flow is stable and therefore the TEPR is quite low, which is consistent with the law in Figure 12. Compared with Figure 20, it is found that the distribution characteristics of TEPR in S7 are basically the same as those in S6, but the intensity of TEPR is reduced; this indicates that the flow in the outlet passage is still influenced by GSV, GWV and GHV, and the influence gradually decreases along the axial direction.   Figure 23 shows the distribution of TEPR in the monitoring sections of the outflow passage. The common characteristics can be found that the high TEPR region in the sections decreases gradually along the axial direction. In S7~S9, the distribution of high TEPR is clearly influenced by the guide vanes, with six high TEPR regions appearing in each section, which is consistent with the number of guide vanes. The flow out of the guide vanes still has circulation, so the flow moves toward the outer wall by centrifugal force; therefore, there is a high TEPR region near the outer wall surface of each section, and the high TEPR region near the outer wall surface increases as the flow rate increases. In S11~S12, the flow has been adjusted by the outlet passage to complete the diffusion and kinetic energy recovery, so the flow is stable and therefore the TEPR is quite low, which is consistent with the law in Figure 12. Compared with Figure 20, it is found that the distribution characteristics of TEPR in S7 are basically the same as those in S6, but the intensity of TEPR is reduced; this indicates that the flow in the outlet passage is still influenced by GSV, GWV and GHV, and the influence gradually decreases along the axial direction.

Conclusions
In this study, the flow field and energy loss of an STPD are simulated, and the accuracy of the numerical simulation is verified by a model test. The energy loss characteristics at each working condition are analyzed based on the entropy production and pressure drop methods. The main achievements of this study are as follows: 1. The energy loss in each part of the STPD is analyzed, and the results show that the prediction of the energy loss by the above two methods is consistent. The TEP is the

Conclusions
In this study, the flow field and energy loss of an STPD are simulated, and the accuracy of the numerical simulation is verified by a model test. The energy loss characteristics at