# Numerical Analysis and Model Test Verification of Energy and Cavitation Characteristics of Axial Flow Pumps

^{1}

^{2}

^{*}

## Abstract

**:**

_{in}= 11 × 10

^{4}–4 × 10

^{4}Pa), the vacuole gradually increases under the same flow rate and the cavitation degree increases. The research results of this paper can provide a reference for the study of the energy and cavitation mechanism of the same type of axial flow pump.

## 1. Introduction

## 2. Numerical Calculation Models, Grids, and Methods

#### 2.1. Numerical Calculation Model

#### 2.2. Mesh Division

^{+}is a dimensionless quantity of distance from the wall, which is proportional to the height of the first grid layer of the wall. In numerical calculations using SST k-ω and RNG k-ε turbulence models, the rotational and shear flow y

^{+}is taken to be 30–100). After the mesh irrelevance analysis, the total mesh number is finally selected as 2114505 for numerical calculation.

#### 2.3. Control Equations and Boundary Conditions

^{3}); x

_{i}and x

_{j}are spatial coordinates; u

_{i}and u

_{j}are the velocity components of the fluid parallel to the corresponding axes x

_{i}and x

_{j}, respectively; F

_{i}is the volume force component in the i-direction; μ is the fluid dynamic viscosity coefficient; and p is the pressure (Pa).

_{k}, G

_{ω}is the generating term of the equation; Y

_{k}, Y

_{ω}is the generating term of the diffusive action; S

_{k}, S

_{ω}is the user-defined source term; D

_{ω}is the term generated by the orthogonal divergence; k is the turbulent kinetic energy; ω is the turbulent special dissipation; and μ

_{t}is turbulent dynamic viscosity coefficient.

_{v}is the vapor mass fraction, Γ is the diffusion coefficient, Re is the evaporation conversion of the gas-liquid phase, Rc is the condensation conversion of the gas-liquid phase, R

_{B}is the bubble radius, α

_{nuc}is the volume fraction of the nucleation site, F

_{vap}is the evaporation coefficient, F

_{cond}is the condensation coefficient, ρ

_{v}is the vapor density, α

_{v}is the volume fraction of the vapor phase, P

_{v}is the pressure inside the bubble, p is the pressure around the bubble in the liquid, ρ

_{l}is the liquid density, $\overrightarrow{{V}_{v}}$ is the mode of the relative velocity of liquid and vapor, and f

_{g}is the gas mass fraction.

^{4}Pa and the outlet condition is set to a flow rate of Q = 210–434 L/s when performing the energy characteristic calculation, and the inlet condition is set to a total pressure of 11 × 10

^{4}Pa, 10 × 10

^{4}Pa, 8 × 10

^{4}Pa, 7 × 10

^{4}Pa, 6 × 10

^{4}Pa, 5 × 10

^{4}Pa, and 4 × 10

^{4}Pa and the outlet condition is set to a flow rate of 411 L/s, 380 L/s, 348 L/s, and 234 L/s, respectively, when performing the cavitation calculation. The diffusion term and pressure gradient are represented by finite element functions, the convective term is represented by a high-resolution format (High-Resolution Scheme), and the velocities u, v, and w, in the pressure p, x, y, and z directions of the monitored flow field are calculated; the convergence conditions of turbulent kinetic energy k and dissipation rate ε are set to 10

^{−6}, and, in principle, the smaller the residuals, the better.

#### 2.4. Numerical Calculation Results Analysis Formula

_{net}, and efficiency, η, of the overflow components based on the flow velocity and pressure fields obtained from numerical calculations are:

_{net}is the head (m), S

_{1}, S

_{2}is the area of the inlet and outlet section of the axial flow pump (m

^{2}), P

_{1}, P

_{2}is the static pressure at each point of the inlet and outlet section of the axial flow pump (Pa), u

_{t}

_{1}, u

_{t}

_{2}is the normal component of the flow velocity at each point of the inlet and outlet section of the axial flow pump (m/s), ρ is the density (kg/m

^{3}), Q is the flow rate (m

^{3}/s), g is the acceleration of gravity (m/s

^{2}), H

_{1}, H

_{2}is the elevation of the inlet and outlet section of the axial flow pump (m), u

_{1}

_{, }u

_{2}is the flow velocity at each point of the inlet and outlet channel section of the axial flow pump (m/s), η is the efficiency (%), T

_{p}is the torque (N·m), and ω is the rotational angular velocity of the impeller (rad/s).

## 3. Test Device and Test Method

#### 3.1. Test Device

#### 3.2. Test Methods

_{1}, P

_{2}is the static pressure at the inlet and outlet of the flow field (Pa), z

_{1}, z

_{2}is the height of the inlet and outlet of the flow field (m), u

_{1}, u

_{2}is the flow velocity of the inlet and outlet of the flow field (m/s), ρ is the density of the water in real-time of the test (kg/m

^{3}), g is the local acceleration of gravity (m/s

^{2}), N is the shaft power (w), $M$ is the input torque of the pump (N·m), $M\prime $ is the mechanical loss torque of the pump (N·m), and n is the test speed of the pump (r/min).

_{av}) corresponding to a 1% drop in efficiency is defined as the critical cavitation margin (NPSH

_{re}) when the flow rate is kept constant. The effective cavitation margin value for the pump at different inlet pressures, NPSH

_{av}, is calculated by the following Equation [31]

_{av}is the pump effective cavitation margin (m), p

_{av}is the pump into the water tank pressure measurement point of the absolute pressure, measured by the absolute pressure transmitter (Pa), ρ is the test of real-time water density (kg/m

^{3}), g is the local acceleration of gravity (m/s

^{2}), v is the pump into the tank pressure measurement section average flow rate (m/s), P

_{v}is the test water temperature of the water saturation vapor pressure (Pa), and h is the absolute pressure transmitter above the pump vane rotation centerline (pump shaft) height value (m).

^{3}/s), ρ is the test real-time water density (kg/m

^{3}), g is the local gravitational acceleration (m/s

^{2}), and N is the shaft power (w).

## 4. Numerical Calculations and Analysis of Experimental Results

#### 4.1. Numerical Calculations and Experimental Energy Analysis

_{d}= 5.56 m and the efficiency is η = 84.6% when the flow rate is Q

_{d}= 350.39 L/s. The head at the design point meets the design requirements and the efficiency is within the high-efficiency zone, which shows the reasonableness of the design. The highest operating head is H

_{m}= 8.80 m, 1.58 times the design head, indicating that the high specific speed axial flow pump designed in this paper can be operated in a wider range. When the head is greater than 8.80 m, the axial flow pump enters the saddle area (Q = 174.68–218.74 L/s), and the operation becomes unstable, accompanied by an increase in bad flow patterns, vibration, and a sharp increase in noise; operation in this area should be avoided.

#### 4.2. Numerical Calculations and Experimental Cavitation Properties

_{re}is required to be less than 5.5 m for the design condition (Q = 348 L/s), and the NPSH

_{re}is 4.6 m for both the numerical calculation and test results, which meets the design requirement of cavitation. Both the numerical calculation and test show that with the increase in flow rate, the critical cavitation margin first decreases, and the critical cavitation margin is the smallest near the high-efficiency zone. With the increase in flow rate, the critical cavitation margin continues to increase. The numerical calculation is close to the test critical cavitation margin, with small error, and shows essentially the same trend and high reliability of the numerical calculation.

## 5. Analysis of Cavitation Numerical Calculation Results

_{in}= 11 × 10

^{4}Pa are organized as shown in Figure 9.

_{in}= 11 × 10

^{4}Pa) shows that the cavitation occurs at the inlet of the pressure surface under the design condition and high flow rate (Q = 348–411 L/s), presenting a strip-like distribution, while the cavitation occurs at the suction surface under the low flow rate (Q = 234 L/s), near the inlet blade tip, presenting a sheet-like distribution. As the flow rate decreases, the cavitation area of the pressure surface gradually decreases, and the area of the cavitation at the hub toward the wheel rim also gradually decreases; no air bubble appears at the pressure surface at the flow rate of Q = 234 L/s, at the suction surface, no air bubble appears at the high flow rate and design conditions (Q = 348–411 L/s), and the area of the air bubble is larger at the small flow rate conditions (Q = 234 L/s) and exceeds the most intense occurrence of cavitation when at the pressure surface. At the top of the leaf, there are vacuoles in all operating conditions; the vacuole area is small in the high flow condition and design condition (Q = 348–411 L/s), and the cavitation at the top of the leaf is more intense in the small flow condition (Q = 234 L/s), having a larger vacuole area and showing a cloud-like distribution.

_{in}= 10 × 10

^{4}Pa and different flow rates are organized as shown in Figure 10.

_{in}= 10 × 10

^{4}Pa, compared with the inlet pressure p

_{in}= 11 × 10

^{4}Pa, shows the same law of change with the working conditions. In the same working condition with the reduction in inlet pressure (inlet pressure from p

_{in}= 11 × 10

^{4}Pa to p

_{in}= 10 × 10

^{4}Pa), the area of the cavitation bubble increased and appeared in the same position. Because the inlet pressure p

_{in}= 10 × 10

^{4}Pa and the inlet pressure p

_{in}= 11 × 10

^{4}Pa show only small changes in the inlet pressure, and are born in the cavitation stage, the development of the degree of cavitation compared to the pressure change is not significant.

_{in}= 8 × 10

^{4}Pa are organized as shown in Figure 11.

_{in}= 8 × 10

^{4}Pa, compared with the inlet pressures p

_{in}= 10 × 10

^{4}Pa and p

_{in}= 11 × 10

^{4}Pa, shows the law of change with the working condition is similar. In the same working condition, inlet pressure decreases (from p

_{in}= 10 × 10

^{4}Pa to p

_{in}= 8 × 10

^{4}Pa), the area of the air bubble further increases in comparison, and it can be seen that the area of the air bubble on the pressure surface increases significantly under the large flow rate and design working condition (Q = 348–411 L/s). Additionally, the cavitation area of the pressure surface increased noticeably, the cavitation area at the pressure surface still did not appear under the small flow condition (Q = 234 L/s), and the cavitation area at the suction surface still did not appear under the high flow and design conditions (Q = 348–411 L/s). The cavitation area at the blade tip increased obviously under the small flow condition (Q = 234 L/s), the cavitation area at the blade tip increased under each condition (Q = 234–411 L/s), and the cavitation area at the blade tip increased significantly under all operating conditions (Q = 234–411 L/s).

_{in}= 7 × 10

^{4}Pa are organized as shown in Figure 12.

_{in}= 7 × 10

^{4}Pa, compared with the inlet pressure p

_{in}= 8 × 10

^{4}Pa, p

_{in}= 10 × 10

^{4}Pa, and p

_{in}= 11 × 10

^{4}Pa, shows the change law with the working condition is similar. In the same working condition with the decrease in inlet pressure (inlet pressure decreased from p

_{in}= 8 × 10

^{4}Pa to p

_{in}= 7 × 10

^{4}Pa), the area of the vacuole further increased in comparison, and the pressure surface change law was similar to the previous one. The vacuole appeared at the hub at the suction surface at the high flow condition (Q = 411 L/s) and the design condition (Q = 348 L/s), and the vacuole appeared at the suction surface, distributed in the middle of the blade head region. The cavitation at the blade tip at each working condition (Q = 234–411 L/s) is more obvious and starts to break away from the blade surface and develop into the fluid.

_{in}= 6 × 10

^{4}Pa are organized as shown in Figure 13.

_{in}= 6 × 10

^{4}Pa, with the change of working conditions, the pressure surface of the air bubble area gradually reduced; the suction surface of the air bubble area was first reduced in the design working conditions (Q = 348 L/s) under the smallest air bubble area and then increased, but both the pressure surface or suction surface of each working conditions contained an air bubble.

_{in}= 7 × 10

^{4}Pa to p

_{in}= 6 × 10

^{4}Pa), the area of the vacuole further increases in comparison. The area of the vacuole at the hub further increases at the suction surface at the high flow condition (Q = 411 L/s), and the vacuole also exists at the suction surface at the design condition (Q = 348 L/s). The area of the vacuole distributed at the blade head in the middle region further increases, and the cavitation at the blade tip under each working condition (Q = 234–411 L/s) is more obvious as the vacuole produces obvious stripping and movement into the fluid.

_{in}= 5 × 10

^{4}Pa are organized as shown in Figure 14.

_{in}= 5 × 10

^{4}Pa, compared with the inlet pressure p

_{in}=6 × 10

^{4}Pa, shows the same law of changing with the working conditions. In the same working condition, as the inlet pressure decreases (inlet pressure decreases from p

_{in}= 6 × 10

^{4}Pa to p

_{in}= 5 × 10

^{4}Pa), the area of the air bubbles further increases in comparison to the pressure surface in the high flow condition (Q = 411 L/s), the impeller and guide vane domain are basically full of air bubbles and the guide vane is also surrounded by air bubbles. The suction surface in the high flow condition (Q = 411 L/s) shows there is a large number of air bubbles at the hub; the air bubbles essentially wrapped 2/3 of the blade surface, in the design condition (Q = 348 L/s), the suction surface existing air bubbles essentially wrapped 1/2 of the blade surface and the cavitation at the top of the blade under each condition (Q = 234–411 L/s) is more obvious.

_{in}= 4 × 10

^{4}Pa are organized as shown in Figure 15.

_{in}= 4 × 10

^{4}Pa, compared with the inlet pressure p

_{in}= 5 × 10

^{4}Pa, shows the same law of changing with the working condition. The area of air bubbles further increases with the decrease in inlet pressure (from p

_{in}= 5 × 10

^{4}Pa to p

_{in}= 4 × 10

^{4}Pa) in the same working condition, and the pressure surface, suction surface, guide vane blade, and outlet bend are basically full of air bubbles under the high flow condition (Q = 411 L/s). At the design condition (Q = 348 L/s), the air bubbles existing at the suction surface basically wrap 4/5 of the blade surface. The cavitation at the top of the blade under each working condition (Q = 234–411 L/s) is more obvious.

## 6. Conclusions

_{m}= 8.80 m, which is 1.58 times the design head, giving the pump a wide operating range.

_{re}is less than 5.5 m at the design condition (Q = 348 L/s), and the numerical calculation and test result of NPSH

_{re}are 4.6 m, which meets the design requirement of cavitation. Both the numerical calculations and tests show that as the flow rate increases, the critical cavitation margin decreases first, and near the high-efficiency zone, the critical cavitation margin is minimal and continues to increase as the flow rate increases. The cavitation performance and energy of the high-ratio axial flow pump studied in this paper are excellent, which can provide a reference for the design and development of high-ratio axial flow pumps and the research of cavitation performance.

## 7. Suggestions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**High-precision plan of the hydraulic mechanical test bench. 1. Inlet tank, 2. Test pump, 3. Pressure outlet tank, 4. Bifurcation tank, 5–6. Flow rate in-situ calibration device, 7. Working condition regulating gate valve, 8. Pressure regulating rectifier, 9. Electromagnetic flow meter, 10. Operating control gate valve, and 11. Auxiliary pump unit.

Parameters | Numerical Value | Unit |
---|---|---|

Impeller speed/n | 1450 | rpm |

Design flow/Q_{d} | 350 | L/s |

Design head/H_{d} | 5.5 | m |

Specific speed/n_{s} | 872 | r/min |

Parts | Grid Division Method | Average y^{+} Value |
---|---|---|

Impeller | “J” topology | ≈50 |

Guide vane | “O” topology | ≈50 |

Impeller leaf top clearance | “H” topology and 8-layer grid arrangement | <10 |

Boundary Conditions | Setting Method |
---|---|

Impeller speed | 1450 r/min |

Inlet | Total pressure |

Outlet | Flow |

Static wall | Application of no-slip conditions |

Near Wall Area | Standard Wall Functions |

Dynamic and static interface | The “Stage” interface |

Interfaces | GGI grid stitching technology |

Q | NPSH_{re} of Numerical Calculation | NPSH_{re} of Test | Design Requirements | Calculation and Test Error |
---|---|---|---|---|

(L/s) | (m) | (m) | (m) | (m) |

411 | 7.1 | 7.5 | / | −0.4 |

380 | 5.7 | 4.9 | / | 0.8 |

348 | 4.6 | 4.6 | <5.5 | 0 |

234 | 9.5 | 9.5 | / | 0 |

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## Share and Cite

**MDPI and ACS Style**

Xie, C.; Zhang, C.; Fu, T.; Zhang, T.; Feng, A.; Jin, Y.
Numerical Analysis and Model Test Verification of Energy and Cavitation Characteristics of Axial Flow Pumps. *Water* **2022**, *14*, 2853.
https://doi.org/10.3390/w14182853

**AMA Style**

Xie C, Zhang C, Fu T, Zhang T, Feng A, Jin Y.
Numerical Analysis and Model Test Verification of Energy and Cavitation Characteristics of Axial Flow Pumps. *Water*. 2022; 14(18):2853.
https://doi.org/10.3390/w14182853

**Chicago/Turabian Style**

Xie, Chuanliu, Cheng Zhang, Tenglong Fu, Tao Zhang, Andong Feng, and Yan Jin.
2022. "Numerical Analysis and Model Test Verification of Energy and Cavitation Characteristics of Axial Flow Pumps" *Water* 14, no. 18: 2853.
https://doi.org/10.3390/w14182853