# Analysis of Inner Flow in a Multi-Stage Double-Suction Centrifugal Pump Using the Detached Eddy Simulation Method

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Model and Computational Domain

#### 2.1. Three-Dimensional Model and Design Parameters

^{3}/h and 132 m, respectively, to meet the demand of water carrying and lifting.

#### 2.2. Computional Grid and Grid Convergence Analysis

_{r}/u was selected as an important indicator for grid convergence. The locations of the points on the impeller were labeled and ranged from the pressure side to the suction side, corresponding to 0° to 45°. The GCI value corresponding to each key variable was calculated and the maximum value is 4.02%, which is lower than the recommended value of 5%. Additionally, the comparison between the values of key variables corresponding to the fine-grid scheme and the extrapolated values shows that the two curves are almost identical.

## 3. Numerical Methodology

#### 3.1. Turbulence Model

- (1)
- The continuity equation, according to the continuity hypothesis of fluid, is expressed by the differential equation as follows:

_{i}denotes the fluid velocity component in the coordinate x

_{i}direction, and t stands for the time.

- (2)
- The momentum equation is as follows:

_{mi}is the generalized source term of the momentum equation, including gravity, and the force between multiphase flow, etc. Where τ

_{ij}belongs to the Reynolds stress terms.

_{t}is as follows:

_{k}and P

_{ω}are the turbulence generation terms. Both F

_{1}and F

_{2}are mixed functions.

_{RANS}is defined as follows:

_{RANS}is replaced by the DES length dimension l

_{DES}.

_{LES}is the filtering size of the LES sub-lattice model; ∆ is the maximum grid scale of the cell in three directions; and C

_{DES}is the model coefficient.

_{DES}= l

_{RANS}, the dissipation term k is the same as that of conventional RANS, which means DES is solved by the RANS model. On the other hand, if l

_{DES}= l

_{LES}, the dissipative term of the transport equation is defined as $\frac{\rho {k}^{3/2}}{{C}_{DES}\Delta}$, which means DES is solved by the LES model [20,21].

#### 3.2. Boundary Conditions

^{−5}and the transient time step was set to 0.00033557 s, which was equal to an angle of 3 degrees of rotation of the impeller. The total time was input at 0.604027 s, with the aim of studying the internal flow change of the impeller after 15 revolutions.

#### 3.3. Validation of Numerical Simulation

## 4. Results and Discussion

#### 4.1. Velocity Distribution

_{d}to 1.4 Q

_{d}, are selected to investigate the first-stage impeller due to the obvious phenomena, such as a large-scale vortex at 0.4 Q

_{d}and a high-speed zone at 1.4 Q

_{d}, which have a great impact on the efficiency of the pump. In total, the pressure side velocity is less than the suction side velocity, especially under the large flow condition of 1.4 Q

_{d}. Figure 5d shows that the low-speed zone covers nearly half of the length of the blade pressure side, resulting in flow separation. As the flow decreases to 0.6 Q

_{d}, the low-speed zone almost expands to the whole impeller channel. In addition, small-scale vortices begin to appear in impeller channels at random. The velocity changes suddenly near the pressure side, which forms an unstable flow structure comparable to a jet wake, as shown in red square. Furthermore, flow separation becomes more serious when the flow is reduced to the minimum working condition, which evolves into large-scale vortex structures near the impeller outlet, as shown in the red circles. In a word, the phenomenon mentioned above is the reason why efficiency decreases rapidly at low flow rates. In order to solve the above problems, multi-objective optimization is considered to achieve efficient operation of the modified pump under low flow conditions. Considering that optimizing individual components has certain limitations on improving pump performance, matching optimization of the impeller and other stationary components can also be considered to ultimately improve the efficiency of this type of pump.

_{d}and 1.2 Q

_{d}, there is an obvious vortex structure near the hub, indicating that the flow here is irregular. The average pre-whirl velocities corresponding to the four different conditions are −3.28 m/s, −4.99 m/s, −5.36 m/s and −15.84 m/s, respectively, which suggests that as the flow increases, the absolute value of the circumferential velocity at the inlet of the second-stage impeller also increases, and the influence of pre-whirl on the head of the second-stage impeller will also be greater. Overall, the inter-stage flow channel worsens the inflow conditions at the inlet of the second-stage impeller, leading to a large number of unstable structures in the pump, affecting the efficiency and stable operation of the pump.

_{d}conditions (as shown in the red squares), which undoubtedly cause great energy loss. However, the blockage in the second-stage impeller channel is alleviated along with the increased flow, and the streamline gradually becomes smooth. Overall, providing the original double-volute design is not reasonable. Consideration is given to subsequent improvements in the placement of the separator and the cross-sectional shape of the volute.

#### 4.2. Pressure Fluctuation Characteristics

_{0}represents the initial time (s); j is the number of time steps.

_{d}to 1.2 Q

_{d}, the intensity in the impeller channel decreases gradually, but shows an upward trend near the inlet of the impeller. The main reason for this is because the energy exchange at the interface and the suction chamber becomes more intense under the increased flow rate.

_{d}.

#### 4.3. Internal Energy Loss and Flow Characteristics

^{2}; ${\mu}_{eff}$ is the effective dynamic viscosity, N·s/m

^{2}.

_{D}and S

_{W}respectively. The entropy production in the main flow area rises rapidly while the wall entropy production climbs gradually as the flow rate increases. More precisely, the maximum value of total entropy production is 136.6 W/K at 1.2 Q

_{d}, and the minimum value is 74.1 W/K at 0.6 Q

_{d}. The wall entropy production ranges from 18% at 0.6 Q

_{d}to 30% at 1.2 Q

_{d}, which occupies a considerable share.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Abbreviations | Description | Unit |
---|---|---|

H | Head | m |

nd | Roatation speed | rpm |

Qd | Design flow rate | m^{3}·h^{−1} |

P | Static pressure | Pa |

ρ | Water density | kg·m^{−3} |

g | Gravity acceleration factor | m·s^{−2} |

t | Time | s |

u | Circumferential velocity | m·s^{−1} |

Ψ | Head coefficient | - |

η | Efficiency | % |

µt | Eddy viscosity coefficient | - |

v | Relative velocity | m·s^{−1} |

$\overline{p}$ | The time-averaged pressure component | pa |

$\tilde{p}$ | The periodic pressure component | pa |

${C}_{\mathrm{p}}^{*}$ | Pressure fluctuation intensity coefficient | - |

S | Entropy production rate | W·K^{−1} |

${\dot{S}}_{D}^{\u2034}$ | Entropy production rate caused by time-averaged velocity | W·m^{−3}·K^{−1} |

${\dot{S}}_{{D}^{\prime}}^{\u2034}$ | Entropy production rate caused by pulsating velocity | W·m^{−3}·K^{−1} |

T | Temperature | K |

$\mu $ | Dynamic viscosity | N·s/m^{2} |

${\mu}_{eff}$ | Effective dynamic viscosity | N·s/m^{2} |

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**Figure 1.**Three-dimensional model of the pump, which includes (

**a**) Physical drawing, (

**b**) assembly drawing, (

**c**) inter-stage flow channel, and (

**d**) second-stage impeller.

**Figure 2.**Structured mesh of all domains, which includes (

**a**) suction chamber, (

**b**) inter-stage flow channel, (

**c**) impeller, and (

**d**) double volute.

**Figure 5.**Relative velocity distribution on the mid-span of the first-stage impeller under different flow rates.

**Figure 7.**Circumferential velocity distribution of the inlet for the second-stage impeller under different flow rates.

**Figure 9.**Blade to blade view of the pressure fluctuation intensity distribution of the first-stage impeller under different flow rates.

**Figure 10.**Pressure fluctuation intensity distribution on the mid-span of the forward channel under different flow rates.

**Figure 11.**Pressure fluctuation intensity distribution on the mid-span of the volute under different flow rates.

**Figure 13.**Different representations of entropy production under different flow rates include (

**a**) energy loss distribution and (

**b**) total entropy production rate distribution.

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**MDPI and ACS Style**

Peng, W.; Pei, J.; Yuan, S.; Wang, J.; Zhang, B.; Wang, W.; Lu, J. Analysis of Inner Flow in a Multi-Stage Double-Suction Centrifugal Pump Using the Detached Eddy Simulation Method. *Processes* **2023**, *11*, 1026.
https://doi.org/10.3390/pr11041026

**AMA Style**

Peng W, Pei J, Yuan S, Wang J, Zhang B, Wang W, Lu J. Analysis of Inner Flow in a Multi-Stage Double-Suction Centrifugal Pump Using the Detached Eddy Simulation Method. *Processes*. 2023; 11(4):1026.
https://doi.org/10.3390/pr11041026

**Chicago/Turabian Style**

Peng, Wenjie, Ji Pei, Shouqi Yuan, Jiabin Wang, Benying Zhang, Wenjie Wang, and Jiaxing Lu. 2023. "Analysis of Inner Flow in a Multi-Stage Double-Suction Centrifugal Pump Using the Detached Eddy Simulation Method" *Processes* 11, no. 4: 1026.
https://doi.org/10.3390/pr11041026