# Multiobjective Combination Optimization of an Impeller and Diffuser in a Reversible Axial-Flow Pump Based on a Two-Layer Artificial Neural Network

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Method

#### 2.1. Computational Model and Mesh

^{6}. Figure 3c,d shows the tendency of ${\mathit{\eta}}_{1}$ and ${\mathit{\eta}}_{2}$ with a grid number of the diffuser increasing. The maximum relative deviation was smaller than 0.16% when the grid number was greater than 1.47 × 10

^{6}. Hence, the number of nodes of the inlet section, impeller, diffuser and outlet section were finally chosen as approximately 0.74 × 10

^{6}, 1.25 × 10

^{6}, 1.47 × 10

^{6}and 0.95 × 10

^{6}respectively. Y+ is a dimensionless number and it can define the distance from the wall to the first node. The average Y+ of any component can be obtained by calculating the mean Y+ values from each node on all walls. The average Y+ of the straight pipe without the inlet section, impeller, diffuser and elbow pipe without the outlet section were 21.8, 24.3, 13.52 and 45.4 respectively.

#### 2.2. Numerical Setup

## 3. Optimization Procedure

#### 3.1. Optimization Functions

^{3}/s and the reverse design flow rate was chosen as 0.3 m

^{3}/s. The optimization objectives can be described as follow:

#### 3.2. Geometry Variables

#### 3.3. Orthogonal Test Design of Diffuser

#### 3.3.1. Orthogonal Test Scheme

#### 3.3.2. Range Analysis

#### 3.4. Latin Hypercube Sampling

#### 3.5. Multilayer Artificial Neural Networks

#### 3.6. NSGA for Multiobjective Optimization

## 4. Results and Analysis

#### 4.1. Comparison of Pump Performance

^{3}/s). The pump performance under the forward condition and reverse condition could not be improved at the same time. In order to take ${\eta}_{2}$ and ${H}_{2}$ into consideration during the optimization design, forward pump performance under all flowrates slightly dropped after optimization. The ${\eta}_{1}$ and ${H}_{1}$ of the optimized design were 77.61% and 4.21 m. However, under the reverse condition, the improvement of pump efficiency and the pump head were so obvious and the high efficiency range was also widened. The ${\eta}_{2}$ and ${H}_{2}$ of the optimized design were 61.58% and 3.8 m.

#### 4.2. Analysis of Inner Flow

#### 4.3. Validation of the Optimization Results

_{1}and η

_{2}were about 1.23% and 3.56%. In addition, the main design parameters of this optimized reversible axial-flow pump is shown in Table 9. The results shows that the reverse pump performance was improved actually, at the same time of maintaining the forward pump performance.

## 5. Conclusions

- (1)
- ${\alpha}_{5}$ and ${\phi}_{1}$ had the most important effect on the pump performance, according to the orthogonal experimental design. ${L}_{1}$, ${L}_{5}$, ${\theta}_{1}$, ${\theta}_{5}$, ${\alpha}_{5}$ and ${\phi}_{1}$ were set as the final design variables for the combination optimization of the impeller and diffuser.
- (2)
- Compared with other approximate models, the two-layer ANN could construct the functions with highest accuracy between the design variables and optimization objectives, based on 120 groups of sample points generated from LHS.
- (3)
- 300 Pareto-optimal solutions were obtained by NSGA and one representative Pareto-optimal solution was selected as the basis of the optimized design. The ${\eta}_{1}$ and ${H}_{1}$ of the optimized design decreased by 0.50% and 10.06%, respectively, compared with the original design. The axial velocity gradient near the suction side of the optimized diffuser was larger than that of the original diffuser. The circulation of the impeller outlet had decreased after optimization. However, the ${\eta}_{2}$ and ${H}_{2}$ of the optimized design increased by 18.62% and 60.40%, compared with the original design. In addition, the curve slope of the axial velocity and circulation in the impeller outlet of the optimized design decreased and increased after optimization.
- (4)
- The external characteristic test for the optimized design was completed to prove the validity of this optimization method. In addition, the test highest efficiency values under the forward condition and reverse condition were 78.73% and 63.85% respectively, which could meet engineering needs.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$Q$ (m^{3}/s) | Flow rate |

${H}_{1}$ (m) | Head under forward design condition |

${H}_{2}$ (m) | Head under reverse design condition |

${H}_{3}$ (m) | Sum of ${H}_{1}$ and ${H}_{2}$ |

${\eta}_{1}$ (%) | Efficiency under forward design condition |

${\eta}_{2}$ (%) | Efficiency under forward design condition |

$N$ (r/min) | Rotating speed |

$\theta $ (°) | Blade angle of impeller |

$L$ (mm) | Chord length of impeller |

$\varnothing $ (°) | Included angle between camber line and chord |

$\mathsf{\sigma}$ (°) | theoretical lift coefficient |

$\lambda $ (°) | Outlet angle of skeleton line |

$\epsilon $ (°) | Inlet angle of skeleton line |

$\alpha $ (°) | Inlet angle of diffuser |

$\beta $ (°) | Outlet angle of diffuser |

$\phi $ (°) | Wrap angle of diffuser |

${C}_{\mathrm{u}}$ (m^{2}/s) | Circulation |

${V}_{\mathrm{z}}$ (m/s) | Axial velocity |

Abbreviations | |

3D | Three-Dimensional |

RANS | Reynolds-Averaged Navier–Stokes |

LHS | Latin Hypercube Sampling |

ANN | Artificial Neural Network |

NSGA | Non-domination based Genetic Algorithm |

CFD | Computational Fluid Dynamics |

DOE | Design of Experiment |

RSM | Response Surface Model |

EXP | Experiment |

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**Figure 3.**Grid independence of the original design for the (

**a**) impeller under a forward design flowrate, (

**b**) impeller under a reverse design flow rate, (

**c**) diffuser under a forward design flowrate and (

**d**) diffuser under a reverse design flow rate.

**Figure 6.**The definition of the main geometry variables of the impeller: (

**a**) definition of the airfoil section, (

**b**) main geometry variables of the airfoil and (

**c**) the rest of the geometry variables.

**Figure 7.**The definition of the main geometry variables of the diffuser: (

**a**) definition of the airfoil section and (

**b**) main geometry variables of the airfoil.

**Figure 10.**Pareto-optimal solutions in 2D function-space: (

**a**) ${\eta}_{1}\u2013{H}_{3},$, (

**b**) ${\eta}_{1}\u2013{\eta}_{2}\mathrm{and}$ (

**c**) ${\eta}_{2}\u2013{H}_{3}$.

**Figure 11.**Comparison of pump performance between the original and optimized design: (

**a**) under the forward condition and (

**b**) under the reverse condition.

**Figure 13.**Comparison of the velocity axial distribution in the impeller between the original design and optimized design under the forward design flow rate (span = 0.5).

**Figure 14.**Comparison in the impeller outlet between the original and optimized design for the (

**a**) velocity axial distribution and (

**b**) circulation distribution (under the forward design flow rate).

**Figure 15.**Comparison of the velocity axial distribution in the diffuser between the original and optimized design under the forward design flow rate (span = 0.5).

**Figure 16.**Comparison of the velocity distribution in the impeller between the original design and optimized design under the reverse design flow rate (span = 0.5).

**Figure 17.**Comparison in the impeller outlet between the original and optimized design for the (

**a**) axial velocity distribution and (

**b**) circumferential velocity distribution (under the reverse design flow rate).

**Figure 19.**The comparison of pump performance between the simulated data and experiment data under the (

**a**) forward operation and (

**b**) reverse operation.

Design Parameters | |||

Rated flow rate (m^{3}/s) | 0.4 | Rotational speed (r/min) | 1450 |

Head on rated flow rate (m) | 3.87 | Specific speed | 1213.1 |

Geometry Parameters | |||

Impeller blade number | 3 | Diffuser blade number | 5 |

Impeller diameter (mm) | 300 | Outlet diameter of diffuser (mm) | 312.8 |

Optimization Parameters | |||

Forward design flow rate (m^{3}/s) | 0.36 | Efficiency on forward design flow rate (%) | 78.0 |

Reverse design flow rate (m^{3}/s) | 0.3 | Efficiency on reverse design flow rate (%) | 51.91 |

Level | Factor | |||
---|---|---|---|---|

A ${\mathit{\alpha}}_{5}$ | B ${\mathit{\alpha}}_{1}$ | C ${\mathit{\phi}}_{5}$ | D ${\mathit{\phi}}_{1}$ | |

Level 1 | 70 | 85 | 26 | 4.5 |

Level 2 | 65 | 80 | 30 | 6.5 |

Level 3 | 60 | 75 | 34 | 8.5 |

Level 4 | 55 | 70 | 38 | 10.5 |

Level 5 | 50 | 65 | 42 | 12.5 |

Test Number | Factor | Corresponding Parameter | Optimization Objective | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

A | B | C | D | ${\mathit{\alpha}}_{5}$ | ${\mathit{\alpha}}_{1}$ | ${\mathit{\phi}}_{5}$ | ${\mathit{\phi}}_{1}$ | ${\mathit{\eta}}_{1}/(\%)$ | ${\mathit{\eta}}_{2}/(\%)$ | ${\mathit{H}}_{3}/\left(\mathbf{m}\right)$ | |

1 | A_{1} | B_{1} | C_{1} | D_{1} | 70 | 85 | 26 | 4.5 | 73.62 | 54.04 | 7.30 |

2 | A_{1} | B_{2} | C_{2} | D_{2} | 70 | 80 | 30 | 6.5 | 75.43 | 53.30 | 7.21 |

3 | A_{1} | B_{3} | C_{3} | D_{3} | 70 | 75 | 34 | 8.5 | 76.86 | 52.30 | 7.12 |

4 | A_{1} | B_{4} | C_{4} | D_{4} | 70 | 70 | 38 | 10.5 | 77.85 | 51.12 | 6.99 |

5 | A_{1} | B_{5} | C_{5} | D_{5} | 70 | 65 | 42 | 12.5 | 78.90 | 49.69 | 6.88 |

6 | A_{2} | B_{1} | C_{2} | D_{3} | 65 | 85 | 30 | 8.5 | 75.98 | 52.52 | 7.17 |

7 | A_{2} | B_{2} | C_{3} | D_{4} | 65 | 80 | 34 | 10.5 | 77.37 | 51.52 | 7.06 |

8 | A_{2} | B_{3} | C_{4} | D_{5} | 65 | 75 | 38 | 12.5 | 78.79 | 50.31 | 6.98 |

9 | A_{2} | B_{4} | C_{5} | D_{1} | 65 | 70 | 42 | 4.5 | 77.75 | 51.23 | 6.99 |

10 | A_{2} | B_{5} | C_{1} | D_{2} | 65 | 65 | 26 | 6.5 | 77.32 | 51.96 | 7.03 |

11 | A_{3} | B_{1} | C_{3} | D_{5} | 60 | 85 | 34 | 12.5 | 77.74 | 50.73 | 7.00 |

12 | A_{3} | B_{2} | C_{4} | D_{1} | 60 | 80 | 38 | 4.5 | 76.43 | 51.75 | 7.03 |

13 | A_{3} | B_{3} | C_{5} | D_{2} | 60 | 75 | 42 | 6.5 | 78.22 | 50.53 | 6.94 |

14 | A_{3} | B_{4} | C_{1} | D_{3} | 60 | 70 | 26 | 8.5 | 77.92 | 51.34 | 7.01 |

15 | A_{3} | B_{5} | C_{2} | D_{4} | 60 | 65 | 30 | 10.5 | 79.38 | 49.85 | 6.92 |

16 | A_{4} | B_{1} | C_{4} | D_{2} | 55 | 85 | 38 | 6.5 | 76.96 | 50.95 | 6.98 |

17 | A_{4} | B_{2} | C_{5} | D_{3} | 55 | 80 | 42 | 8.5 | 78.45 | 49.66 | 6.87 |

18 | A_{4} | B_{3} | C_{1} | D_{4} | 55 | 75 | 26 | 10.5 | 78.54 | 50.45 | 6.96 |

19 | A_{4} | B_{4} | C_{2} | D_{5} | 55 | 70 | 30 | 12.5 | 79.82 | 48.90 | 6.85 |

20 | A_{4} | B_{5} | C_{3} | D_{1} | 55 | 65 | 34 | 4.5 | 77.64 | 49.81 | 6.79 |

21 | A_{5} | B_{1} | C_{5} | D_{4} | 50 | 85 | 42 | 10.5 | 78.77 | 48.56 | 6.80 |

22 | A_{5} | B_{2} | C_{1} | D_{5} | 50 | 80 | 26 | 12.5 | 79.06 | 49.67 | 6.91 |

23 | A_{5} | B_{3} | C_{2} | D_{1} | 50 | 75 | 30 | 4.5 | 77.19 | 50.54 | 6.87 |

24 | A_{5} | B_{4} | C_{3} | D_{2} | 50 | 70 | 34 | 6.5 | 78.47 | 49.07 | 6.76 |

25 | A_{5} | B_{5} | C_{4} | D_{3} | 50 | 65 | 38 | 8.5 | 80.34 | 47.07 | 6.69 |

Test Indices | D | A | B | C |
---|---|---|---|---|

K_{1} | 382.63 | 382.66 | 383.08 | 386.45 |

K_{2} | 386.41 | 387.21 | 386.73 | 387.81 |

K_{3} | 389.54 | 389.70 | 389.60 | 388.08 |

K_{4} | 391.90 | 391.41 | 391.81 | 390.37 |

K_{5} | 394.32 | 393.83 | 393.59 | 392.10 |

${\overline{K}}_{1}$ | 76.53 | 76.53 | 76.62 | 77.29 |

${\overline{K}}_{2}$ | 77.28 | 77.44 | 77.35 | 77.56 |

${\overline{K}}_{3}$ | 77.91 | 77.94 | 77.92 | 77.62 |

${\overline{K}}_{4}$ | 78.38 | 78.28 | 78.36 | 78.07 |

${\overline{K}}_{5}$ | 78.86 | 78.77 | 78.72 | 78.42 |

R | 2.34 | 2.23 | 2.10 | 1.13 |

Test Indices | A | B | D | C |
---|---|---|---|---|

K_{1} | 260.45 | 256.80 | 257.38 | 257.45 |

K_{2} | 257.55 | 255.90 | 255.81 | 255.10 |

K_{3} | 254.20 | 254.14 | 252.88 | 253.44 |

K_{4} | 249.76 | 251.65 | 251.50 | 251.21 |

K_{5} | 244.91 | 248.38 | 249.30 | 249.68 |

${\overline{K}}_{1}$ | 52.09 | 51.36 | 51.48 | 51.49 |

${\overline{K}}_{2}$ | 51.51 | 51.18 | 51.16 | 51.02 |

${\overline{K}}_{3}$ | 50.84 | 50.83 | 50.58 | 50.69 |

${\overline{K}}_{4}$ | 49.95 | 50.33 | 50.30 | 50.24 |

${\overline{K}}_{5}$ | 48.98 | 49.68 | 49.86 | 49.94 |

R | 3.11 | 1.68 | 1.62 | 1.55 |

Test Indices | A | B | C | D |
---|---|---|---|---|

K_{1} | 35.51 | 35.25 | 35.22 | 34.98 |

K_{2} | 35.24 | 35.09 | 35.03 | 34.93 |

K_{3} | 34.90 | 34.87 | 34.72 | 34.85 |

K_{4} | 34.45 | 34.60 | 34.67 | 34.73 |

K_{5} | 34.03 | 34.31 | 34.49 | 34.62 |

${\overline{K}}_{1}$ | 7.10 | 7.05 | 7.04 | 7.00 |

${\overline{K}}_{2}$ | 7.05 | 7.02 | 7.01 | 6.99 |

${\overline{K}}_{3}$ | 6.98 | 6.97 | 6.94 | 6.97 |

${\overline{K}}_{4}$ | 6.89 | 6.92 | 6.93 | 6.95 |

${\overline{K}}_{5}$ | 6.81 | 6.86 | 6.90 | 6.92 |

R | 0.30 | 0.19 | 0.14 | 0.07 |

Approximation Model | ${\mathit{R}}^{2}$$\mathbf{of}\text{}{\mathit{\eta}}_{1}/\%$ | ${\mathit{R}}^{2}$$\mathbf{of}\text{}{\mathit{\eta}}_{2}/\%$ | ${\mathit{R}}^{2}$$\mathbf{of}\text{}{\mathit{H}}_{1}/\mathbf{m}$ | ${\mathit{R}}^{2}$$\mathbf{of}\text{}{\mathit{H}}_{2}/\mathbf{m}$ |
---|---|---|---|---|

Two-layer ANN Model | 0.982 | 0.990 | 0.993 | 0.996 |

RSM | 0.913 | 0.962 | 0.970 | 0.994 |

Kriging Model | 0.792 | 0.953 | 0.938 | 0.948 |

RBF Model | 0.905 | 0.983 | 0.972 | 0.993 |

Measurement Items | Equipment Name | Instrument Model | Measurement Range | Measurement Accuracy |
---|---|---|---|---|

Flow rate | Intelligent electromagnetic flowmeter | OPTIFLUX2000F | 0–1800 m^{3}/h | <±0.2% |

Head | Intelligent differential pressure transmitter | EJA | 0–10 m | <±0.1% |

Torque | Intelligent torque speed sensor | JCL1 | 0–200 N·m | <±0.1% |

Rotation speed |

Design Parameters | Value |
---|---|

Forward rated flowrate (m^{3}/s) | 0.368 |

Efficiency under forward rated condition (%) | 78.73 |

Head under forward rated condition (m) | 3.72 |

Reverse rated flowrate (m^{3}/s) | 0.298 |

Efficiency under reverse rated condition (%) | 63.85 |

Head under reverse rated condition (m) | 3.51 |

Rotation speed (rev/min) | 1450 |

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## Share and Cite

**MDPI and ACS Style**

Meng, F.; Li, Y.; Yuan, S.; Wang, W.; Zheng, Y.; Osman, M.K.
Multiobjective Combination Optimization of an Impeller and Diffuser in a Reversible Axial-Flow Pump Based on a Two-Layer Artificial Neural Network. *Processes* **2020**, *8*, 309.
https://doi.org/10.3390/pr8030309

**AMA Style**

Meng F, Li Y, Yuan S, Wang W, Zheng Y, Osman MK.
Multiobjective Combination Optimization of an Impeller and Diffuser in a Reversible Axial-Flow Pump Based on a Two-Layer Artificial Neural Network. *Processes*. 2020; 8(3):309.
https://doi.org/10.3390/pr8030309

**Chicago/Turabian Style**

Meng, Fan, Yanjun Li, Shouqi Yuan, Wenjie Wang, Yunhao Zheng, and Majeed Koranteng Osman.
2020. "Multiobjective Combination Optimization of an Impeller and Diffuser in a Reversible Axial-Flow Pump Based on a Two-Layer Artificial Neural Network" *Processes* 8, no. 3: 309.
https://doi.org/10.3390/pr8030309