# Numerical Simulation of a Three-Stage Electrical Submersible Pump under Stall Conditions

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## Abstract

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## 1. Introduction

_{des}. At the same time, a positive slope segment appears on the performance curve. Pan et al. [17] investigated the external characteristic curve of a mixed-flow pump and observed a saddle-shaped curve. Their findings suggested that a vortex formed at the shroud of the pump, which reduced the effective outside diameter of the impeller outlet. Feng et al. [18] analyzed the internal flow pattern of a centrifugal pump under stall conditions and found that stall vortex frequently leads to the formation of a vortex at the outlet downstream. This phenomenon disrupts the flow pattern inside the pump and can cause a decrease in the pump’s head and efficiency. Li et al. [19] studied the impeller inlet prerotation in centrifugal pumps. Their research revealed that the presence of prerotation does not impact the occurrence of a positive slope in the external characteristic curve of the pump. However, prerotation may cause a sudden head drop, negatively affecting the pump performance. Hu et al. [20] conducted research on mixed-flow water jet propulsion pumps operating under low flow conditions. Their findings indicated that the presence of a diffuser did not impact the stall effect within the impeller.

_{des}conditions, vortices in the impeller gathered at the suction surface, and the range and intensity of the vortices at all stages were slightly different.

## 2. Numerical Methods

#### 2.1. Turbulence Model

#### 2.2. Entropy Production

^{3}; ω represents the turbulent eddy frequency, s

^{–1}; k represents the turbulent energy, m

^{2}/s

^{2}; and T represents Kelvin temperature, 298.15 K.

#### 2.3. Computational Model

_{des}= 104.2 m

^{3}/h, the single-stage head H

_{des}= 8 m, rated speed n

_{des}= 2900 r/min, and the specific speed n

_{s}= 375. To model the pump’s performance and behavior, the UG NX10.0 software was used to create a computational model of the pump’s calculation area, as shown in Figure 1. The model used a closed impeller design and consisted of detailed geometric parameters for the impeller and diffuser of the pump, as provided in Table 2.

#### 2.4. Mesh and Boundary Condition

^{−5}. The interface between the stationary and rotational domain is selected as the frozen rotor in the steady state simulation, while the transient rotor stator is used in the transient simulation. In the transient simulation, the time step is set to the time taken for the impeller to rotate 5°, which is 2.8736 × 10

^{−4}s. The rotor stator interfaces adopt the frozen rotor mode. Due to the complex flow and large pressure gradient near the blade wall, the mesh near the impeller blade and diffuser vane are encrypted to ensure the accuracy of numerical simulation. The SST k-ω turbulence model gives the most accurate results when y+ is less than 5, and the accuracy is also excellent when y+ is less than 100 [38]. The y+ values in the impeller domain are kept at less than 100 to meet the SST k-ω turbulence model requirements. The distribution of y+ values near the blades is shown in Figure 2. To minimize the impact of boundary conditions on the simulation results, the inlet section is set to 5 times the pipe diameter, and the outlet section is set to 10 times the pipe diameter. The fluid medium used in the simulation is clear water at a temperature of 25 °C. The boundary conditions are set to pressure inlet and mass flow outlet, while the wall boundary condition is set to the no-slip wall and automatic wall function.

## 3. Results and Discussions

#### 3.1. External Characteristics Comparison

_{des}for the critical stall and at 0.66 Q

_{des}for the deep stall.

_{a}represents the average head of a single stage, H

_{1}represents the difference in head between the inlet of the stage 1 impeller and the outlet of the diffuser, and H

_{2}and H

_{3}represent the head of stage 2 and stage 3, respectively. The head of stage 1 is higher than that of the other two stages, and the head of a single stage gradually decreases as the number of stages increases. The heads created by the different stages of a multistage pump may not be identical (even if the impellers and diffusers have the same geometry). This is because the boundary conditions change as the fluid flows out of stage 1 and passes through the diffuser. Therefore, each stage affects the other and changes the overall hydraulic performance of the pump.

#### 3.2. Analysis of Flow Field under Design Conditions

#### 3.3. Analysis of Flow Field under Critical Stall Conditions

#### 3.4. Analysis of Flow Field under Deep Stall Conditions

#### 3.5. Vortex Evolution Process in the Impeller and Diffuser under Stall Conditions

_{0}, the inner channel of impeller 1 contains vortex A near the outlet and vortex B in the middle of the flow channel. At 0.25T, vortex A disappears, and vortex B migrates towards the outlet, shedding at the front edge of the suction force to form vortex C. At 0.5T, the energy of vortex B weakens and further migrates towards the outlet, while the intensity of vortex C increases and moves towards the middle of the flow channel. At 0.75T, the high kinetic energy fluid brought by the leakage flow rapidly dissipates vortex B, and vortex C migrates to the middle of the flow channel. At moment T, vortex C tends to collapse and disappear, and the flow separation at the leading edge of the blade generates vortex D. The internal vortices of impellers 2 and 3 do not exhibit obvious evolution patterns.

#### 3.6. Energy Loss Analysis

## 4. Conclusions

- (1)
- Under critical stall conditions, impeller 1 generates inlet and multiple channel vortices, which when combined with inherent secondary flow and wake, result in turbulence within the impeller’s flow field. The internal vortex evolution period of impeller 1 is approximately 0.75T; however, the scale of internal vortices in impellers 2 and 3 is small with no apparent evolution period.
- (2)
- Under deep stall conditions, large-scale vortex structures manifest within all stages of the impeller, resulting in severe channel blockage. Flow separation commences at the leading edge of the guide vanes in the diffuser, with vortices developing and expanding inside the channel to cause a significant impact on outlet performance. The scales and evolution patterns of these vortices vary across different channels within impeller 1.
- (3)
- The turbulent entropy production power within the impeller and diffuser constitutes the primary component of total entropy production power. The entropy production power loss inside the impeller increases with the increase in stages, and the power loss inside the diffuser is the same in each stage. Under deep stall conditions, frequent vortex shedding inside these components results in significant energy loss.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Numerical simulation model. (

**a**) Calculation domain of electrical submersible pump. (

**b**) 2D illustration of the ESP stage.

**Figure 8.**Streamline and velocity distributions on different span surfaces under design conditions. (

**a**) Schematic diagram of different blade-to-blade surfaces. (

**b**) 0.1 Span. (

**c**) 0.2 Span. (

**d**) 0.5 Span. (

**e**) 0.8 Span. (

**f**) 0.9 Span.

**Figure 9.**Velocity streamlines and low-speed zone distribution diagram under critical stall conditions.

**Figure 10.**Streamline and velocity distributions on different span surfaces under critical stall conditions. (

**a**) 0.1 Span. (

**b**) 0.2 Span. (

**c**) 0.5 Span. (

**d**) 0.8 Span. (

**e**) 0.9 Span.

**Figure 11.**Velocity streamlines and low-speed zone distribution diagram under deep stall conditions.

**Figure 12.**Streamline and velocity distributions on different span surfaces under deep stall conditions. (

**a**) 0.1 Span. (

**b**) 0.2 Span. (

**c**) 0.5 Span. (

**d**) 0.8 Span. (

**e)**0.9 Span.

**Figure 13.**Vortex evolution process in the impeller and the diffuser under critical stall conditions. (

**a**) 0.9 Span of the impeller and 0.2 Span of the diffuser. (

**b**) 0.5 Span of impeller and diffuser.

**Figure 14.**Vortex evolution process in the impeller and the diffuser under deep stall conditions. (

**a**) 0.9 Span of the impeller and 0.2 Span of the diffuser. (

**b**) 0.5 Span of impeller and diffuser.

**Figure 15.**Power loss of impeller and diffusers. (

**a**) Total entropy production power. (

**b**) Turbulent entropy production power. (

**c**) Wall entropy production power. (

**d**) Direct entropy production power.

Constants | Value |
---|---|

${a}_{1}$ | 0.31 |

${\alpha}_{1}$ | 5/9 |

${\alpha}_{2}$ | 0.44 |

${\beta}_{1}$ | 3/40 |

${\beta}_{2}$ | 0.0828 |

$\beta $ | 9/100 |

${\sigma}_{k1}$ | 0.85 |

${\sigma}_{k2}$ | 1 |

${\sigma}_{\omega 1}$ | 0.5 |

${\sigma}_{\omega 2}$ | 0.856 |

Impeller Hydraulic Geometry Parameters | Hydraulic Geometry Parameters of Diffuser | ||
---|---|---|---|

Inlet inner diameter D_{imph1}/mm | 36 | Inlet inner diameter D_{difh1}/mm | 76 |

Inlet outer diameter D_{imps1}/mm | 86.5 | Inlet outer diameter D_{difs1}/mm | 117.5 |

Outlet inner diameter D_{imph2}/mm | 82 | Outlet inner diameter D_{difh2}/mm | 36.5 |

Outlet outer diameter D_{imps2}/mm | 113.5 | Outlet outer diameter D_{difs2}/mm | 86.5 |

Number of blades Z_{imp}/vanes | 7 | Number of blades Z_{dif}/ vanes | 11 |

Inlet blade angle β_{imp1}/° | 25 | Inlet blade angle β_{dif1}/° | 55 |

Outlet blade angle β_{imp2}/° | 36 | Outlet blade angle β_{dif2}/° | 87 |

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**MDPI and ACS Style**

Wang, Y.; Wang, Z.; Song, X.; Bai, L.; El-Emam, M.A.; Zhou, L.
Numerical Simulation of a Three-Stage Electrical Submersible Pump under Stall Conditions. *Water* **2023**, *15*, 2619.
https://doi.org/10.3390/w15142619

**AMA Style**

Wang Y, Wang Z, Song X, Bai L, El-Emam MA, Zhou L.
Numerical Simulation of a Three-Stage Electrical Submersible Pump under Stall Conditions. *Water*. 2023; 15(14):2619.
https://doi.org/10.3390/w15142619

**Chicago/Turabian Style**

Wang, Yuqiang, Zhe Wang, Xiangyu Song, Ling Bai, Mahmoud A. El-Emam, and Ling Zhou.
2023. "Numerical Simulation of a Three-Stage Electrical Submersible Pump under Stall Conditions" *Water* 15, no. 14: 2619.
https://doi.org/10.3390/w15142619