# Anti-Cavitation Design of the Symmetric Leading-Edge Shape of Mixed-Flow Pump Impeller Blades

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{d}flow rate range, especially within the 0.62–0.78 Q

_{d}and 1.08–1.20 Q

_{d}ranges, was improved. The head and hydraulic efficiency was numerically checked without obvious change. This provided a good reference for optimizing the cavitation or other performances of bladed pumps.

## 1. Introduction

## 2. Mixed-Flow Pump Object

## 3. Mathematical Methods

#### 3.1. Brief Introduction of the Diffusion-Angle Integral Method

^{*}is set to divide the LE ellipse arc and the t diffusion part. Five steps can be executed in sequence: (a) Giving the long–short axis ratio R

_{ab}= a

_{LE}/b

_{LE}of the LE ellipse arc, (b) scaling the leading-edge elliptical-arc to the arc based on R

_{ab}, (c) giving the straight thickness diffusion angle γ

_{s}(unit: degree) and calculating the leading-edge arc (scaled) radius r

_{LE}, (d) giving the coefficient B of the thickness integration expression and integrating out the thickness distribution in the thickness diffusion part, and (e) rescaling the leading-edge arc and the integrated thickness back to the elliptical-arc-scale based on R

_{ab}.

_{A}is the thickness at point A and m

_{A}is the m position at point A. Then, the increasing of t between A

^{*}and A can be integrated by

_{A}

^{*}is the m position at point A

^{*}, C

_{s}is the scale factor, and γ(m) is the thickness integral expression which can be expressed in the following form:

^{*}can be simply and well controlled by the coefficient B. The number of design parameters was simplified to three by keeping an accurate description of t along m.

#### 3.2. Genetic Algorithm and Setup

_{ref}and v

_{ref}are the reference pressure and velocity at the impeller inlet, respectively. p

_{v}is the saturation pressure. ρ is the density of fluid medium. To apply predictions for σ, the pressure coefficient C

_{p}was defined as:

_{v}. Therefore, σ = −C

_{p}at the time that cavitation inception occurs. As a result, the negative value of the minimum pressure coefficient −C

_{pmin}can be used instead of the cavitation inception number σ

_{i}as the basis of the fitness function f

_{fit}in GA. Three different flow rate conditions were considered so that f

_{fit}could be written as a weighted value:

_{i}is the weight value for different conditions. The weight value should be larger in off-design conditions and smaller in design conditions. The more the condition is different from design flow rate, the larger the weight value is. For a 0.7 Q

_{d}condition, w

_{1}= 0.5. For a 1.0 Q

_{d}condition, w

_{2}= 0.2. For a 1.2 Q

_{d}condition, w

_{3}= 0.3.

_{ab}, γ

_{s}, and B) because all the spanwise positions used the same thickness distribution. Eight-digit binary code was used for coding each parameter. Thus, 24-digit binary code was used for each sample. The probabilities of genetic operations are listed in Table 2. In total, 10 individual samples were set for each generation. The convergence criterion was set as the residual being less than 0.1% in 10 continuous generations. In the optimization, the design parameters can vary in the ranges shown in Table 3.

#### 3.3. CFD Simulation Setup

_{fit}in optimization. Commercial software ANSYS CFX 12.0 was used for numerical simulation. The SST-DES method [22,23], which hybrids RANS with LES by zonal division, was used to solve the turbulent flow. The equation of the SST k-ω turbulence model proposed by Menter is defined as:

_{t}is the turbulent eddy viscosity; σ

_{k}, σ

_{ω}, and β

_{k}are model constants; C

_{ω}is the coefficient of the production term; F

_{1}is the mixture function; l

_{k-ω}is the turbulence scale; k is the intensity of turbulence kinetic energy; t is the time; u

_{i}is the velocity; x

_{i}refers to the unit coordinates; and ω is the turbulence dissipation rate.

_{k-ω}will be replaced by min(l

_{k-ω}, C

_{DES}Δ). C

_{DES}is the model constant. Δ is the grid scale. For non-uniform grids, there is Δ = max(Δx, Δy, Δz), which is the maximum side length of the grid element. When l

_{k-ω}≤ C

_{DES}Δ, the DES method is solved by the SST k-ω turbulence model. When l

_{k-ω}≥ C

_{DES}Δ, the LES is used to solve the problem.

^{6}, as plotted in Figure 3. The y

^{+}value near the wall was guaranteed within 30–300 for applying the wall functions. The mass–flow inlet boundary was set at the impeller inlet. A static pressure outlet was set at the impeller outlet. The wall boundaries were set as no-slip. Rotational periodic boundaries were set to simplify the problem into a “single passage”. A general grid interface (GGI) was set between the impeller domain and the guide vane domain, based on the multiple reference frame (MRF) model. Steady-state simulations were conducted for predicting the −C

_{pmin}of the single-passage domain at 0.7, 1.0, and 1.2 Q

_{d}, with a maximum iteration number of 600 and a convergence criterion of 1 × 10

^{−5}. The rotor-stator interface was in the “frozen rotor” type in the steady-state simulation. The final verifications by full-passage domain were based on the transient-state simulations, with 0.41 s in total. The time step was set as 1.134 × 10

^{−4}s, with the iteration number up to 10 for each time step. The rotor-stator interface was in the “transient rotor-stator” type in the steady-state simulation.

#### 3.4. Model Test for Verification

_{sft}was tested by the power meter, which acquired the rotating speed and torque, respectively. The efficiency η, including mechanical, volumetric, and hydraulic efficiencies, was calculated using the flow rate, shaft power, and head by η = ρgQH/P

_{sft}, where g is the acceleration of gravity.

## 4. Experimental Verification of Computation

## 5. Results of Optimization Design

#### 5.1. Optimization Process

_{fit}changing with iteration steps. The f

_{fit}value of the initial impeller sample was −1.9945. After 60 iterative steps of optimization, the f

_{fit}value increased to about −0.9615, and converged when the residual was continually less than 0.1% for 10 steps. The comparison of blade thickness around the leading-edge is shown in Figure 7. The design parameter R

_{ab}changed from 2.00 to 3.96, γ

_{s}changed from 5.97° to 4.03°, and B changed from 1.50 to 3.00. The −C

_{pmin}values varied from −3.06 to −1.025 at 0.7 Q

_{d}, from −1.19 to −1.363 at 1.0 Q

_{d}, and from −0.755 to −0.588 at 1.2 Q

_{d}. The main improvements were under the partial-load 0.7 Q

_{d}and the over-load 1.2 Q

_{d}. At the design load, the cavitation performance was somehow deteriorated.

#### 5.2. Analysis of Cavitation Performance

_{p}distributions on the initial and optimal impeller blades. Sudden pressure drops can be found on the blade leading-edge because of the local flow separation. After optimization, the leading-edge pressure drops became much gentler at 0.7 and 1.2 Q

_{d}, especially on the spanwise 0.9 surface (spanwise 0 is the hub and 1 is the shroud). At 0.7 Q

_{d}, a C

_{pmin}value of about −1.5 occurred on the mid-span of the initial impeller. After optimization, the C

_{pmin}value became about −1.0 near the hub. At 1.0 Q

_{d}, a C

_{pmin}value of about −1.28 occurred near the shroud of the initial impeller. After optimization, the C

_{pmin}value was still around −1.0 near the shroud without any improvement. At 1.2 Q

_{d}, a C

_{pmin}value of about −1.0 occurred on the mid-span and near the shroud of the initial impeller. After optimization, the C

_{pmin}value became about −0.8, which was located on the mid-span.

_{pmin}law at different flow rate conditions. Before optimization, the condition-minimum C

_{pmin}value of about −3.06 was at 0.7 Q

_{d}. After optimization, the condition-minimum C

_{pmin}value increased to about −2.37. Its flow rate condition became 0.8 Q

_{d}. The cavitation inception performance was improved in the two ranges of 0.62–0.78 Q

_{d}and 1.08–1.20 Q

_{d}, as illustrated in Figure 9. The head and efficiency changes were less than 0.5% after optimization in the 0.5–1.2 Q

_{d}flow rate range according to the single-passage CFD prediction.

#### 5.3. Comparison of Head and Efficiency

## 6. Conclusions

_{fit}increased from about −1.9945 to about −0.9615 after optimization design. The combination strategy used in this case provided a quick and reliable solution for the optimization of pump impellers.

_{pmin}value strongly increased within the 0.5–1.2 Q

_{d}flow rate range, and the cavitation performance in the 0.62–0.78 Q

_{d}, and 1.08–1.20 Q

_{d}ranges was obviously improved.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Parameter control. (

**a**) Traditional parameter control; (

**b**) Diffusion-Angle Integral (DI) method.

**Figure 3.**Schematic map of the flow domain, mesh, and boundaries. RP is the rotational periodic boundary, RSI is the rotor-stator interface, and other unmarked boundaries are no slip walls.

**Figure 5.**The experimental and Computational Fluid Dynamics (CFD)-predicted data of head and efficiency before optimization. (

**a**) Head curves; (

**b**) Efficiency curves.

**Figure 8.**C

_{p}distributions on initial and optimal impeller blades. (

**a1**) 0.7 Q

_{d}, Spanwise 0.1; (

**a2**) 0.7 Q

_{d}, Spanwise 0.5; (

**a3**) 0.7 Q

_{d}, Spanwise 0.9; (

**b1**) 1.0 Q

_{d}, Spanwise 0.1; (

**b2**) 1.0 Q

_{d}, Spanwise 0.5; (

**b3**) 1.0 Q

_{d}, Spanwise 0.9; (

**c1**) 1.2 Q

_{d}, Spanwise 0.1; (

**c2**) 1.2 Q

_{d}, Spanwise 0.5; (

**c3**) 1.2 Q

_{d}, Spanwise 0.9

**Figure 10.**The comparisons of head and efficiency between the initial and optimal impeller blades. (

**a**) Head; (

**b**) Efficiency.

Parameter | Value |
---|---|

Rotation speed n_{d} | 1470 [rpm] |

Design flow rate Q_{d} | 1 [m^{3}/s] |

Design head H_{d} | 55 [m] |

Specific speed n_{q} | 72.8 |

Impeller blade number | 6 |

Guide vane blade number | 7 |

Impeller inlet diameter D_{1} | 0.4 [m] |

Operation | Copy/Eliminate | Crossover | Mutation |
---|---|---|---|

Probability | 1.0 | 0.6 | 0.1 |

Parameter | R_{ab} | γ_{s} [°] | B |
---|---|---|---|

Range | 1–5 | 1–10 | 1–6 |

Number of Mesh Nodes [×10^{6}] | 0.3180 | 0.5594 | 0.8269 | 1.2056 | 1.7427 | 2.3546 |

Residual [%] | 100 | 1.1862 | 0.7266 | 0.0809 | 0.3019 | 0.3782 |

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**MDPI and ACS Style**

Zhu, D.; Tao, R.; Xiao, R.
Anti-Cavitation Design of the Symmetric Leading-Edge Shape of Mixed-Flow Pump Impeller Blades. *Symmetry* **2019**, *11*, 46.
https://doi.org/10.3390/sym11010046

**AMA Style**

Zhu D, Tao R, Xiao R.
Anti-Cavitation Design of the Symmetric Leading-Edge Shape of Mixed-Flow Pump Impeller Blades. *Symmetry*. 2019; 11(1):46.
https://doi.org/10.3390/sym11010046

**Chicago/Turabian Style**

Zhu, Di, Ran Tao, and Ruofu Xiao.
2019. "Anti-Cavitation Design of the Symmetric Leading-Edge Shape of Mixed-Flow Pump Impeller Blades" *Symmetry* 11, no. 1: 46.
https://doi.org/10.3390/sym11010046