# Comprehensive Improvement of Mixed-Flow Pump Impeller Based on Multi-Objective Optimization

^{*}

## Abstract

**:**

_{des}, 1.0Q

_{des}, and 1.2Q

_{des}improved by 0.63%, 3.39%, and 3.77% respectively. The low-pressure region on the blade surface reduced by 96.92% while the pump head difference was less than 1.84% at the design point. In addition, a comparison of the flow field of the preferred impeller and the original impeller revealed the effect of SDIEC on mixed-flow pump performance improvement and flow mechanism.

## 1. Introduction

## 2. Mixed-Flow Pump Model

## 3. Optimization Design System

#### 3.1. CFD Analyses

#### 3.2. 3D Inverse Design Method

- Design specifications and fluid properties.
- Meridional shape.
- Spanwise distribution of $r\overline{{V}_{\theta}}$ at the impeller exit.
- The meridional derivative of the circulation $\partial \left(r\overline{{V}_{\theta}}\right)/\partial m$ (blade loading) at the hub and shroud.
- Stacking condition.

#### 3.3. Optimization Process

## 4. Optimization Settings

#### 4.1. Design Parameters

#### 4.2. Optimization Objectives and Constraints

_{des}and 1.2Q

_{des}were selected as the optimization objectives to enlarge the operating range of the high efficiency area. To make the optimized pump have similar specific speed, higher pump efficiency and better cavitation performance at the design point, the pump head ${H}_{1.0}$, efficiency ${\eta}_{1.0}$ and the normalized area of low-pressure area ${A}^{*}$ (the area on the blade surface where pressure is lower than water vaporization pressure 3169 Pa) at 1.0Q

_{des}were selected as the constrains. The optimization objective can be calculated by Equation (7), the constrains can be described as Equations (8)–(10):

#### 4.3. Algorithm Settings

## 5. Results and Discussions

#### 5.1. Experimental Validation

#### 5.2. Analysis of the Main Effect

_{des}is in the following order: $DRV{T}_{s}>{K}_{h}>r{V}_{s}>r{V}_{h}>DRV{T}_{h}>{K}_{s}>N{D}_{s}>N{D}_{h}>\beta $. The contribution of each design parameter to the pump efficiency at 0.8Q

_{des}is in the following order:$r{V}_{s}r{V}_{h}DRV{T}_{s}{K}_{s}N{D}_{s}DRV{T}_{h}\beta N{D}_{h}{K}_{h}$. Therefore, it is necessary to consider $r{V}_{s}$ and $r{V}_{h}$ in the optimization process after comprehensively considering their effect on the pump performance.

#### 5.3. Optimization Results

_{des}and 1.2Q

_{des}, respectively. The red points indicate that the performance of the corresponding mixed-flow pump does not meet the requirement of the constraints. To make the optimization results more intuitive, all the optimal solutions that satisfy the constraints were selected as shown in Figure 10b. It can be seen that the pump efficiency at 0.8Q

_{des}and 1.2Q

_{des}has a competitive relationship, with the increasing pump efficiency at 0.8Q

_{des}, the pump efficiency at 1.2Q

_{des}inevitably decreases. According to different selection criteria, three impellers (I1~I3) with different configurations were selected for further study, the selection criteria can be described as Equation (11), where ${\eta}_{0.8}{}_{\mathrm{ori}}$ and ${\eta}_{1.2}{}_{\mathrm{ori}}$ are the pump efficiency of the original model at 0.8Q

_{des}and 1.2Q

_{des}respectively.

_{des}was within the acceptable range. Impeller I2 was selected as the preferred impeller when the efficiency, head and cavitation were considered. The SDIEC, and blade loading distribution along meridional shape at hub and shroud of the optimized impeller I2 are shown in Figure 11.

#### 5.4. Comparisons between Original and Preferred Impellers

_{des}, 1.0Q

_{des}and 1.2Q

_{des}are increased by 0.63%, 3.39%, and 3.77%, respectively. The head deviation at 1.0Q

_{des}is less than 1.84%, which meets the design requirements, however, the head at 0.8Q

_{des}has decreased by 5.38%.

_{des}. Compared with the original model, the low-pressure area of the working surface and suction surface of the preferred impeller blade almost completely disappeared, which means the cavitation performance of the preferred impeller has been improved. In addition, the pressure distribution on the blade surface of the preferred impeller is also more uniform, and the radial adverse pressure gradient is greatly reduced, which helps reduce the radial secondary flow and improve the efficiency of the impeller.

_{des}and 0.8Q

_{des}, these data are extracted at the ${r}^{*}=0.1$, 0.5 and 0.9 after considering the viscous effect caused by the friction between the fluid and the wall. It can be seen from Figure 14a that at 1.2Q

_{des}of the original impeller, the relative velocity distribution at the blade-to-blade channel is not uniform, especially at the leading edge of the blade where is close to the shroud. There is an obvious low-speed area accompanied by backflow phenomenon. After optimization, the non-uniformity of the relative velocity distribution among the blade-to-blade channel is reduced and the low-speed area is completely suppressed. The same phenomenon was observed in Figure 14b, the difference is that the low-speed area has changed from the shroud to the hub, and there is no backflow in the low-speed area.

_{des}and 1.2Q

_{des}. It can be seen that at 1.2Q

_{des}, the velocity distribution at the outlet of the preferred impeller becomes more uniform, especially the high-velocity area near the shroud side is eliminated. The same phenomenon was observed at 0.8Q

_{des}, the difference is that the high-velocity area is weakened at both the hub and shroud.

_{des}and 1.2Q

_{des}. It can be seen that compared with the original impeller, the total pressure on the hub side of the preferred impeller is increased, the total pressure at the shroud side is reduced, and the spanwise distribution of impeller exit total pressure becomes more uniform. This means that the mixing loss due to uneven total pressure distribution in the impeller will be reduced, and the possibility of cavitation at shroud is also reduced since the total pressure at the shroud is reduced.

## 6. Conclusions

_{des}and 1.2Q

_{des}were selected as the optimization objectives while the head, efficiency and area of low-pressure area at 1.0Q

_{des}were selected as constraints. The results of this study can be summarized as follows:

- (1)
- The CFD calculations accurately simulated the flow in the pump, and the performance curves agreed well with the experimental curves. The maximum efficiency and head deviations did not exceed 2% and 4% respectively.
- (2)
- The optimization results show that the non-linear SDIEC is better than the constant and linear distributions. The analysis of the main effect also shows that $r{V}_{s}$ and $r{V}_{h}$ have a greater impact on the performance of the mixed-flow pump impeller than other design parameters. Therefore, it is necessary to take $r{V}_{s}$ and $r{V}_{h}$ as design parameters in the optimization process to further improve the performance of the mixed-flow pump impeller.
- (3)
- The pump efficiency with preferred impeller at 0.8Q
_{des}, 1.0Q_{des}, and 1.2Q_{des}are 81.11%, 88.60%, and 77.62%, respectively. Compared with the original model, these efficiencies increased by 0.63%, 3.39%, and 3.77%, respectively. At the same time, the cavitation performance of preferred impeller has been significantly improved. The area of low-pressure region further reduced by 96.92% and the pump head deviation at 1.0Q_{des}is less than 1.84%, which is within an acceptable range.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${n}_{s}$ | specific speed |

H | head |

N | rotational speed |

Q | volume flow rate |

$\overline{{V}_{\theta}}$ | tangentially velocity |

r | radius or radial direction |

B | blade numbers |

${p}^{+}$ | static pressure at blade work surface |

${p}^{-}$ | static pressure at blade suction surface |

${W}_{m}$ | relative velocity on the blade surface |

m | streamline in the meridional shape or meridional distance |

$\overline{{V}_{z}}$ | axial components of the circumferential average absolute velocity |

$\overline{{V}_{r}}$ | radial components of the circumferential average absolute velocity |

${v}_{zbl}$ | axial component of periodic velocity |

${v}_{rbl}$ | radial component of periodic velocity |

${v}_{\theta bl}$ | circumferential component of periodic velocity |

$f$ | wrap angle |

${V}_{h}$ | tangentially velocity at hub |

${V}_{s}$ | tangentially velocity at shroud |

${r}^{*}$ | normalized spanwise distance |

${\overline{{V}_{\theta}}}^{*}$ | normalized tangentially velocity |

${m}^{*}$ | normalized meridional distance |

DRVT | blade loading at leading edge |

NC | fore connection points |

ND | aft fore connection points |

K | slope of linear line |

$\beta $ | stacking condition |

$\eta $ | pump efficiency |

${A}^{*}$ | normalized area of low-pressure area |

$P$ | pressure at inlet or outlet |

$M$ | torque on the impeller |

$\omega $ | angular velocity of the impeller |

$\rho $ | density of the fluid |

$g$ | gravitational acceleration |

## Abbreviations

SDIEC | spanwise distribution of impeller exit circulation |

CFD | computational fluid dynamics |

LHS | Latin hypercube sampling |

RSM | response surface model |

NSGA-Ⅱ | non-dominated sorting genetic algorithm |

DOE | design of experiment |

CCD | central composite design |

RANS | Reynolds average Navier–Stokes |

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**Figure 8.**Experimental validation (

**a**) Experimental bench and (

**b**) performance comparison between CFD and experimental.

**Figure 9.**Main effect of each design parameter on different optimization objectives (

**a**) 1.2Q

_{des}and (

**b**) 0.8Q

_{des}.

**Figure 11.**Circulation distribution and blade loading distribution of the optimized impeller I2 (

**a**) spanwise distribution of impeller exit circulation and (

**b**) blade loading distribution along meridional shape at hub and shroud.

**Figure 13.**Comparison of pressure distribution on the blade surface (

**a**) original impeller and (

**b**) preferred impeller.

**Figure 14.**Comparison of relative velocity distribution at different span (

**a**) 1.2Q

_{des}and (

**b**) 0.8Q

_{des}.

**Figure 16.**Comparison of total pressure distribution at impeller exit (

**a**) 1.2Q

_{des}and (

**b**) 0.8Q

_{des}.

Design Flow Rate (m^{3}/s) | 0.427 | Impeller Blade Number | 4 |

Head on Design Points (m) | 12.66 | Impeller Diameter (mm) | 320 |

Rotational Speed (r/min) | 1450 | Specific Speed | 511 |

Inflow Pipe $(\times {10}^{4})$ | Outflow Pipe $(\times {10}^{4})$ | Impeller $(\times {10}^{4})$ | Vane Diffuser $(\times {10}^{4})$ | Total Mesh $(\times {10}^{4})$ | Efficiency (%) | Head (m) |
---|---|---|---|---|---|---|

27 | 49 | 42 | 48 | 166 | 83.828 | 12.064 |

58 | 67 | 84 | 92 | 301 | 84.647 | 12.146 |

83 | 95 | 141 | 152 | 471 | 85.208 | 12.103 |

129 | 142 | 211 | 268 | 750 | 85.206 | 12.096 |

Variable | $\mathit{r}{\mathit{V}}_{\mathit{h}}$ | $\mathit{r}{\mathit{V}}_{\mathit{s}}$ | $\mathit{D}\mathit{R}\mathit{V}{\mathit{T}}_{\mathit{h}}$ | $\mathit{D}\mathit{R}\mathit{V}{\mathit{T}}_{\mathit{s}}$ | ${\mathit{K}}_{\mathit{h}}$ | ${\mathit{K}}_{\mathit{s}}$ | $\mathit{N}{\mathit{D}}_{\mathit{h}}$ | $\mathit{N}{\mathit{D}}_{\mathit{s}}$ | $\mathit{\beta}$ |
---|---|---|---|---|---|---|---|---|---|

Range | 0.3~0.34 | 0.3~0.34 | −0.2~0.2 | −0.2~0.2 | −1.5~1.5 | −1.5~1.5 | 0.1~0.9 | 0.4~0.9 | −15~15 |

Setting | Value |
---|---|

Population Size | 100 |

Number of Generations | 100 |

Crossover Probability | 0.9 |

Cross Distribution Index | 10 |

Mutation Distribution Index | 20 |

Initialization Mode | Random |

Variable | $\mathit{r}{\mathit{V}}_{\mathit{h}}$ | $\mathit{r}{\mathit{V}}_{\mathit{s}}$ | $\mathit{D}\mathit{R}\mathit{V}{\mathit{T}}_{\mathit{h}}$ | $\mathit{D}\mathit{R}\mathit{V}{\mathit{T}}_{\mathit{s}}$ | ${\mathit{K}}_{\mathit{h}}$ | ${\mathit{K}}_{\mathit{s}}$ | $\mathit{N}{\mathit{D}}_{\mathit{h}}$ | $\mathit{N}{\mathit{D}}_{\mathit{s}}$ | $\mathit{\beta}$ |
---|---|---|---|---|---|---|---|---|---|

I1 | 0.3363 | 0.3363 | 0.1790 | 0.1925 | −0.8935 | 1.0366 | 0.2116 | 0.6057 | −14.9394 |

I2 | 0.3364 | 0.3363 | 0.1956 | 0.0747 | −1.1088 | 0.6562 | 0.2601 | 0.5003 | −14.7681 |

I3 | 0.3364 | 0.3365 | 0.1363 | −0.0617 | −1.1141 | 0.3910 | 0.2937 | 0.7271 | −14.9837 |

Performance | RSM | CFD | ||||||
---|---|---|---|---|---|---|---|---|

Impeller | ${\mathit{\eta}}_{0.8}(\%)$ | ${\mathit{\eta}}_{1.2}(\%)$ | ${\mathit{\eta}}_{0.8}(\%)$ | ${\mathit{\eta}}_{1.2}(\%)$ | ${\mathit{H}}_{1.0}\left(\mathbf{m}\right)$ | ${\mathit{\eta}}_{1.0}(\%)$ | ${\mathit{A}}^{*}(\%)$ | |

I1 | 80.58 | 80.77 | 80.40 | 80.31 | 11.92 | 88.72 | 53.97 | |

I2 (Preferred) | 81.40 | 78.19 | 81.11 | 77.62 | 12.37 | 88.60 | 3.08 | |

I3 | 82.00 | 74.30 | 81.31 | 74.75 | 12.44 | 87.68 | 18.22 | |

Original model | 80.48 | 73.84 | 12.10 | 85.21 | 100 |

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## Share and Cite

**MDPI and ACS Style**

Wang, M.; Li, Y.; Yuan, J.; Meng, F.; Appiah, D.; Chen, J.
Comprehensive Improvement of Mixed-Flow Pump Impeller Based on Multi-Objective Optimization. *Processes* **2020**, *8*, 905.
https://doi.org/10.3390/pr8080905

**AMA Style**

Wang M, Li Y, Yuan J, Meng F, Appiah D, Chen J.
Comprehensive Improvement of Mixed-Flow Pump Impeller Based on Multi-Objective Optimization. *Processes*. 2020; 8(8):905.
https://doi.org/10.3390/pr8080905

**Chicago/Turabian Style**

Wang, Mengcheng, Yanjun Li, Jianpin Yuan, Fan Meng, Desmond Appiah, and Jiaqi Chen.
2020. "Comprehensive Improvement of Mixed-Flow Pump Impeller Based on Multi-Objective Optimization" *Processes* 8, no. 8: 905.
https://doi.org/10.3390/pr8080905