# Analysis of the Energy Loss Mechanism of Pump-Turbines with Splitter Blades under Different Characteristic Heads

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory of the Simulation

#### 2.1. Turbulence Model and Entropy Production Theory

^{–3}·K

^{–1}; ${\overline{u}}_{1}$, ${\overline{u}}_{2}$, and ${\overline{u}}_{3}$ denote the time-averaged velocity components in m/s; ${u}_{1}^{\prime}$, ${u}_{2}^{\prime}$, and ${u}_{3}^{\prime}$ denote the pulsating velocity components in m/s; T indicates temperature in Kelvin; μ

_{eff}is the effective dynamic viscosity of fluid in Pa·s; β is an empirical constant, approximated as 0.09; k denotes the turbulent kinetic energy in m

^{2}/s

^{2}; and ω denotes the turbulent vortex frequency in s

^{–1}.

^{−2}·K

^{−1}; ${\overrightarrow{\tau}}_{w}$ and ${\overrightarrow{v}}_{w}$ indicate the shear stress and velocity near the wall, respectively.

#### 2.2. Calculation Domain and Mesh Generation

#### 2.3. Grid Independence Verification

#### 2.4. Boundary Conditions and Case Information

^{–5}[28]. During the simulation, the interface between the rotating domain and the static domain is dealt with by means of the frozen rotor. This method is usually used for steady calculations, which reduce computational requirements by assuming that the flow field does not change significantly during one rotor revolution. The characteristic operating points during the actual operation of the power plant were selected for the study. The case information is shown in Table 4.

## 3. Experimental Test and Validation

#### 3.1. Test Rig

#### 3.2. Computational Validation

_{u}denotes the tangential velocity coefficient; C

_{m}denotes the axial velocity coefficient; V

_{u}and V

_{m}denote the velocity obtained by the test or numerical simulation, respectively, in m/s; n is the rotational speed in rev/min; and D

_{1}is the RN inlet diameter of the pump turbine in m.

_{M}of the model test is determined from Equation (17):

_{M}is the model test head, m; n

_{M}is the rotational speed of model pump-turbine, rev/min; D

_{M}is the RN inlet diameter of model pump-turbine, m.

## 4. Results and Analysis

#### 4.1. Total Energy Loss

_{1}, γ

_{2,}and γ

_{3}represent the TEP values in Cases 1, 2, and 3, respectively.

#### 4.2. Analysis of Flow Characteristics in Inlet Components

_{p}can be written as

_{0}means the average static pressure at the inlet of the SC, pa.

_{p}value. This indicates that the number of SV and GV was set correctly, effectively balancing the water pressure created by high heads. The presence of a distinct high C

_{p}region at the leading edge of SV and GV indicated a higher static pressure in this region. Compared to Case 1, the region with low C

_{p}values (dark blue) near the interface between GV and RN was larger in Case 3, indicating a faster pressure drop in the inlet component under high head conditions.

#### 4.3. Analysis of the Flow Characteristics in RN

_{p}value distribution at different span-wise surfaces of RN in the three cases. Here, span represents the dimensionless distance from hub to shroud, indicating the position of the blade-to-blade surface. For example, span = 0 indicates the blade-to-blade surface at the hub, and span = 1 indicates the blade-to-blade surface at the shroud. It can be clearly seen that the pressure inside the RN gradually decreased from the inlet to the outlet. At span = 0.05, the Cp value of the pressure side (PS) of the rotor blade was much higher than that of the suction side (SS), indicating that the pressure difference between the PS side and the SS side of the rotor blade drives the impeller to rotate. Moreover, the pressure difference on both sides of the blades was greater than that of the splitters, indicating that the ability of the splitter to drive the rotation of the impeller is weaker than that of the blades. At span = 0.5 and 0.95, the Cp distribution in the RN was consistent with that at span = 0.05, indicating that the pressure distribution of different spans is basically the same. Under the three different cases, the distribution of Cp inside the RN was basically the same, indicating that the design of the impeller of the unit is reasonable, and the pressure distribution inside the RN is basically unchanged within the operating range of the characteristic head.

_{w}) is calculated as follows [33]:

_{a}is the axial velocity, U

_{t}is tangential velocity, and R is the hydraulic radius, representing the impeller radius.

_{w}values on different horizontal planes of RN in the three cases. According to the physical structure of the RN, it could be divided into three regions: R1 is the region where the water flows in tangentially, R2 is the transition region, and R3 is the region where the water flows out axially. It can be clearly seen that the S

_{w}value in the R1 domain was much greater than 1.0, indicating that the tangential velocity of the water flow was much greater than the axial velocity. Moreover, the S

_{w}value dropped sharply along the axial direction, which showed that the kinetic energy of the water flow was quickly converted into the rotational mechanical energy of RN, which was also related to the greater pressure difference between the two sides of the blade in the R1 domain. In the R2 domain, the S

_{w}value was less than 1.0, indicating that the main flow direction of the water flow was axial. At the same time, the S

_{w}value decreased slowly along the axial direction, and it could be seen that the energy transferred by the water flow to RN was reduced. In the R3 domain, the S

_{w}value was maintained at a low level, far below 1.0, and the velocity circulation still existed in the outlet domain. Comparing the three cases, it could be seen that on the same level, the S

_{w}value of case 1 was the highest, followed by case 2, and that of case 3 was the least. Especially in the R3 domain, the average S

_{w}values of the three cases were 0.23, 0.22, and 0.19, respectively. It could be seen that the velocity circulation of the water flow in Case 3 was the smallest, which was also the performance of Case 3 with less energy loss.

#### 4.4. Analysis of the Flow Characteristics in DT

## 5. Conclusions

- (1)
- Among the three terms of entropy production, EPTD dominates, accounting for over 98% of the TEP. Within the five flow components, RN and DT play a dominant role, and TEP increases significantly along the flow direction. In all three cases, the growth rate of TEP decreases with increasing head, suggesting that the high-efficiency region of the turbine is at high-head operating conditions.
- (2)
- In the inlet components, the presence of a large velocity gradient at the trailing edge of the GV leads to a significant EPR. Furthermore, under high head conditions, the higher flow rate increases the velocity in the GV area, increasing the velocity gradient at the trailing edge and causing more energy loss. The average energy loss growth rate of inlet components in case 2 and case 3 are 14.87% and 42.69%, respectively.
- (3)
- In the RN domain, at spans of 0.05 and 0.95, the high spiralization and high EPR are observed at the leading edge of the splitter on SS where the flow direction deviates from the center line of the blade profile. This deviation, together with the high flow velocity, leads to significant spiraling and significant energy losses. At span = 0.5, the high EPR and high flow velocity are observed at the trailing edge of the splitter, with no appreciable spiralization occurring. This indicates that the energy loss in this region is mainly due to the large velocity gradient and not to the presence of vortices. In the RN domain, the S
_{w}value continues to decrease along the axial direction, indicating that the water kinetic energy is continuously converted into the rotational mechanical energy of the RN. Especially in the R3 domain, the average S_{w}values of the three cases are 0.23, 0.22, and 0.19, respectively. - (4)
- In Case 1, the region close to the wall in the inflow section of the DT exhibits a high tangential velocity, which leads to significant spiralization. Additionally, the central region of the inflow section with large velocity gradients generates higher TKE and EPR. Case 3 shows significantly lower TKE and EPR values compared to Case 1, indicating that the improved flow properties in the RN domain reduce turbulence and energy losses in the DT.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 6.**Comparison between numerical simulation and experimental results: (

**a**) efficiency; (

**b**) velocity distribution of DT (C

_{uSim}and C

_{mSim}denote the velocity coefficient obtained in CFD, and C

_{uLDV}and C

_{mLDV}denote the velocity coefficient obtained in LDV).

**Figure 7.**Entropy production of different entropy terms in three cases and the growth rate relative to case 1: (

**a**) Entropy production; (

**b**) growth rate.

**Figure 8.**TEP of different unit components for three investigated cases, and the growth rate relative to Case 1: (

**a**) TEP; (

**b**) growth rate.

**Figure 9.**C

_{p}value distributions on the horizontal plane of the inflow part of the pump turbine in three cases.

**Figure 10.**Velocity streamlines on the horizontal plane of the inflow part of the pump turbine in three cases.

**Figure 11.**EPR distributions on the horizontal plane of the inflow part of the pump turbine in three cases.

Domain | Parameter | Variable/Unit | Value |
---|---|---|---|

RN | Blades | Z_{b} | 5 |

Splitters | Z_{s} | 5 | |

GV | Guide vanes | Z_{G} | 16 |

Spiral casing (SC) | Wrap angle | Φ/(°) | 360 |

Stay Vane (SV) | Stay vanes | Z_{S} | 16 |

Domain | Grid Type | Number of Grid Cells | y+ |
---|---|---|---|

RN | Hexahedral | 3,011,474 | <15 |

GV | Tetrahedral | 2,988,878 | <20 |

SV | Hexahedral | 1,699,875 | <30 |

DT | Hexahedral | 1,665,874 | <30 |

SC | Hexahedral | 1,856,253 | <50 |

Total | / | 11,222,354 | / |

Parameters | φ_{1} | φ_{2} | φ_{3} | F_{S} | $\mathit{\zeta}$ | GCI |
---|---|---|---|---|---|---|

Q | 381.75 | 381.25 | 378.35 | 1.2 | 10.78 | 1.55% |

Efficiency | 87.3 | 87.2 | 86.9 | 1.2 | 3.89 | 1.35% |

**Table 4.**Simulation case selection (all data in the table is from test data are used as boundary conditions for simulation).

Case | H (m) | GVO (°) | n (rev/min) | Q (m^{3}/s) | N (MW) | η (%) | Δh (m) |
---|---|---|---|---|---|---|---|

Case 1 | 580 | 10 | 500 | 36.6 | 180.2 | 86.69 | 77.20 |

Case 2 | 600 | 10 | 500 | 38.4 | 198.9 | 88.10 | 71.40 |

Case 3 | 640 | 10 | 500 | 41.3 | 233.9 | 90.30 | 62.08 |

Parameters | Value |
---|---|

Maximum test flow rate (m³/s) | 1.5 |

Maximum test head (m) | 150 |

Maximum test speed (r/min) | 2500 |

Model RN diameter (mm) | 250–500 |

Dynamometer maximum power (kW) | 500 |

Rated power of pump motor (kW) | 2 × 850 |

DT pressure (kPa) | –85 to +250 |

Uncertainty of efficiency measurements | ≤±0.25% |

Model type | Reaction turbine |

Parameters | Value |
---|---|

Speed range | –150–1000 m/s |

Measurement error | 0.1% |

Sample frequency | 400–800 MHZ |

Maximum processing frequency | 175 MHz |

Minimum processing frequency | 300 Hz |

Bits | 8 |

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## Share and Cite

**MDPI and ACS Style**

Gui, Z.; Xu, Z.; Li, D.; Zhang, F.; Zhao, Y.; Xu, L.; Zheng, Y.; Kan, K.
Analysis of the Energy Loss Mechanism of Pump-Turbines with Splitter Blades under Different Characteristic Heads. *Water* **2023**, *15*, 2776.
https://doi.org/10.3390/w15152776

**AMA Style**

Gui Z, Xu Z, Li D, Zhang F, Zhao Y, Xu L, Zheng Y, Kan K.
Analysis of the Energy Loss Mechanism of Pump-Turbines with Splitter Blades under Different Characteristic Heads. *Water*. 2023; 15(15):2776.
https://doi.org/10.3390/w15152776

**Chicago/Turabian Style**

Gui, Zhonghua, Zhe Xu, Dongkuo Li, Fei Zhang, Yifeng Zhao, Lianchen Xu, Yuan Zheng, and Kan Kan.
2023. "Analysis of the Energy Loss Mechanism of Pump-Turbines with Splitter Blades under Different Characteristic Heads" *Water* 15, no. 15: 2776.
https://doi.org/10.3390/w15152776