# Design and Verification of a Single-Channel Pump Model based on a Hybrid Optimization Technique

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Methods

#### 2.1. Single-Channel Pump Model

#### 2.2. Numerical Analysis

^{+}was kept below 2 to use the low Reynolds SST model, so that the flow near the wall was accurately analyzed. To eliminate the grid dependency of the numerical solutions, grid dependency test was performed at the design mass flow rate for the hydraulic performance of the pump. Based on the results in the previous work [23], the optimal grids system (a total of 2,500,000 nodes; 1,300,000 nodes in the impeller domain and 1,200,000 nodes in the volute domain) was selected, as shown in Figure 3.

^{−5}. The calculation for the steady RANS analysis were performed using an Intel Xeon CPU (2.70 GHz) processor and the CPU running time for each analysis was about 4 h.

## 3. Optimization Techniques

#### 3.1. Optimization Goal

_{s}represents the signed area of the polygon, considered as the sweep area of the radial force during one revolution. The centroid of a non-self-intersecting closed polygon defined by n vertices (x

_{0}, y

_{0}), (x

_{1}, y

_{1}), …, (x

_{(n-1)}, y

_{(n-1)}) is defined as the point (C

_{x}, C

_{y}), as follows:

_{η}and F

_{radial}as objective functions.

_{1}and CP

_{2}) for the impeller and three control points (CP

_{3}, CP

_{4}and CP

_{5}) for the volute were selected. Figure 5 shows the defined design variables and ranges are listed in Table 2.

#### 3.2. Surrogate Modeling

_{j}is the weight and Φ

_{j}is radial basis function. Machine learning finds the input vector with the largest error in the network and adjusts the weight or neurons of this vector to minimize the error in the network. This process is repeated until the network’s mean squared error falls below the set goal. The construction process can be accessed through graphical user interface (GUI) environment using the application ‘neural net fitting’, which is supported within MATLAB [27].

_{1}and SC

_{2}are constants that are utilized to configure the surrogate model with respect to the efficiency and radial force, respectively. The SC values were chosen using a k-fold cross-validation test [28], which is a validation method that estimates how the results of statistical analysis are generalized to independent datasets. The process of the k-fold cross-validation test used in the present work is described as follows:

- Step 1
- Construct a surrogate model using the 53 experimental points, except for one point of the 54 experimental points.
- Step 2
- Compare the value of the objective function at the location of the experimental point excluded from Step 1 (between CFD simulation value and predicted value by the surrogate model).
- Step 3
- This process is carried out at all experimental points. Then, evaluate the sum of the errors between predicted and CFD simulation values.

_{1}and SC

_{2}were 0.2 and 1.0, respectively. Neural networks with various numbers of neurons were trained 10 times and the averaged adjusted R

^{2}values are compared (Figure 8). The same quantity of neurons was used in each layer; when using 12 neurons, the statistically most accurate models with adjusted R

^{2}equals to 1.00 and 0.78 could be achieved.

#### 3.3. Searching Algorithm

^{−6}, and initial velocity of particles = 2.5. The pseudo-code of the hybrid PSO-GA used in this study is shown in Table A2 (Appendix B).

#### 3.4. Optimization Results

## 4. Results and Discussion

#### 4.1. Unsteady Analyses of Internal Flow Field

_{x}and F

_{y}values of the radial force were normalized based on the maximum radial force of the reference design. As shown in Figure 15, the sweep area distribution in the reference design leaned a little toward the fourth quadrant, whereas it was closer to the origin for AOD. Moreover, the sweep area in AOD reduced significantly compared with the reference model. As discussed previously in Section 3.4, the radial force in AOD decreased by 14.73, compared with the reference design.

#### 4.2. Performance Verification of the Optimized Prototype Model

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

AOD | Arbitrary optimum design |

BPF | Blade passing frequency |

CFD | Computational fluid dynamics |

CP | Control point |

D | Diameter of impeller |

DOE | Design of experiments |

FSI | Fluid-structure interaction |

g | Gravity acceleration |

GA | Genetic algorithm |

H | Total head |

LDV | Laser Doppler velocimetry |

LHS | Latin hypercube sampling |

N | Rotational speed |

P | Power |

PSO | Particle swarm optimization |

Q | Volume flow rate |

RANS | Reynolds-averaged Navier-Stokes |

RBNN | Radial basis neural network |

SC | Spread constant |

SST | Shear stress transport |

(U)RANS | Unsteady Reynolds-averaged Navier-Stokes |

ρ | Density |

Ф | Flow coefficient |

Ψ | Head coefficient |

## Appendix A

CP 1 | CP 2 | CP 3 | CP 4 | CP 5 | F_{η}/F_{η_Ref.} | F_{radial}/F_{radial_Ref} | |
---|---|---|---|---|---|---|---|

1 | 0.000 | 0.000 | 1.000 | 0.000 | 0.000 | −1.021 | 3.064 |

2 | 0.000 | 0.000 | 0.000 | 0.000 | 1.000 | −1.033 | 1.759 |

3 | 0.000 | 0.000 | 1.000 | 1.000 | 0.000 | −1.017 | 2.144 |

4 | 1.000 | 1.000 | 0.000 | 0.000 | 1.000 | −0.982 | 1.527 |

5 | 0.265 | 0.306 | 0.429 | 0.980 | 0.837 | −1.026 | 1.403 |

6 | 1.000 | 0.469 | 0.286 | 0.592 | 0.571 | −0.994 | 1.133 |

7 | 0.551 | 0.735 | 0.327 | 0.490 | 0.000 | −1.012 | 1.355 |

8 | 0.959 | 0.184 | 0.857 | 0.531 | 0.327 | −1.007 | 2.109 |

9 | 0.020 | 0.857 | 0.224 | 0.449 | 0.265 | −1.014 | 1.161 |

10 | 0.939 | 0.653 | 0.796 | 0.673 | 0.878 | −0.988 | 1.344 |

11 | 0.449 | 0.510 | 0.735 | 0.000 | 0.469 | −1.016 | 1.362 |

12 | 0.347 | 0.408 | 0.776 | 0.918 | 0.163 | −1.015 | 1.958 |

13 | 0.694 | 0.041 | 0.510 | 0.694 | 0.694 | −1.024 | 1.441 |

14 | 0.633 | 0.878 | 0.122 | 0.735 | 0.633 | −1.003 | 1.377 |

15 | 0.796 | 0.714 | 0.592 | 0.061 | 0.082 | −1.000 | 1.861 |

16 | 0.571 | 0.265 | 1.000 | 0.388 | 0.714 | −1.020 | 1.778 |

17 | 0.857 | 0.245 | 0.061 | 0.184 | 0.898 | −1.013 | 0.628 |

18 | 0.918 | 0.694 | 0.449 | 0.122 | 0.776 | −1.013 | 0.347 |

19 | 0.388 | 0.959 | 0.551 | 0.265 | 0.408 | −1.011 | 1.429 |

20 | 0.245 | 0.918 | 0.980 | 0.755 | 0.245 | −1.014 | 1.495 |

21 | 0.184 | 0.000 | 0.184 | 0.551 | 0.755 | −1.036 | 0.738 |

22 | 0.306 | 0.020 | 0.878 | 0.857 | 0.551 | −1.023 | 1.630 |

23 | 0.837 | 0.061 | 0.633 | 0.143 | 0.673 | −1.013 | 1.011 |

24 | 0.327 | 0.429 | 0.020 | 0.510 | 0.306 | −1.020 | 0.569 |

25 | 0.878 | 0.980 | 0.245 | 0.327 | 0.367 | −0.990 | 0.122 |

26 | 0.469 | 0.939 | 0.673 | 0.837 | 0.816 | −1.003 | 1.557 |

27 | 0.429 | 0.796 | 0.000 | 0.224 | 0.612 | −1.029 | 0.244 |

28 | 0.898 | 0.571 | 0.694 | 0.939 | 0.347 | −0.997 | 1.591 |

29 | 0.816 | 0.837 | 0.918 | 0.306 | 0.490 | −0.989 | 1.920 |

30 | 0.714 | 0.143 | 0.367 | 0.653 | 0.143 | −1.013 | 1.559 |

31 | 0.143 | 0.082 | 0.653 | 0.082 | 0.592 | −1.031 | 1.271 |

32 | 0.735 | 0.592 | 0.898 | 0.612 | 0.020 | −1.008 | 1.950 |

33 | 0.286 | 0.776 | 0.959 | 0.286 | 0.796 | −1.007 | 1.275 |

34 | 0.224 | 0.327 | 0.265 | 0.041 | 0.102 | −1.025 | 1.103 |

35 | 0.061 | 0.286 | 0.306 | 0.878 | 0.204 | −1.022 | 1.207 |

36 | 0.755 | 0.490 | 0.041 | 0.163 | 0.184 | −1.007 | 0.738 |

37 | 0.102 | 0.633 | 0.143 | 0.633 | 0.857 | −1.019 | 0.373 |

38 | 0.673 | 1.000 | 0.612 | 0.776 | 0.286 | −0.995 | 1.152 |

39 | 0.980 | 0.755 | 0.102 | 0.816 | 0.122 | −0.989 | 1.095 |

40 | 0.776 | 0.122 | 0.490 | 0.102 | 0.061 | −1.018 | 1.457 |

41 | 0.612 | 0.347 | 0.082 | 0.714 | 0.918 | −1.019 | 0.821 |

42 | 0.000 | 0.388 | 0.469 | 0.429 | 0.449 | −1.022 | 1.374 |

43 | 0.592 | 0.449 | 0.163 | 1.000 | 0.388 | −1.012 | 1.328 |

44 | 0.490 | 0.367 | 0.531 | 0.245 | 0.939 | −1.017 | 0.387 |

45 | 0.367 | 0.163 | 0.755 | 0.408 | 0.224 | −1.019 | 1.892 |

46 | 0.510 | 0.531 | 0.571 | 0.571 | 0.510 | −1.013 | 1.373 |

47 | 0.408 | 0.898 | 0.408 | 0.347 | 1.000 | −1.005 | 0.605 |

48 | 0.204 | 0.816 | 0.347 | 0.959 | 0.429 | −1.005 | 1.218 |

49 | 0.653 | 0.224 | 0.939 | 0.898 | 0.980 | −1.013 | 1.769 |

50 | 0.041 | 0.551 | 0.837 | 0.796 | 0.653 | −1.010 | 1.476 |

51 | 0.122 | 0.612 | 0.388 | 0.020 | 0.735 | −1.017 | 0.936 |

52 | 0.163 | 0.673 | 0.714 | 0.367 | 0.041 | −1.014 | 1.613 |

53 | 0.082 | 0.204 | 0.816 | 0.469 | 0.959 | −1.028 | 1.234 |

54 | 0.531 | 0.102 | 0.204 | 0.204 | 0.531 | −1.030 | 0.696 |

## Appendix B

g = 0 * g: generation number |

fori = 1 to M do * M: population (particles) size |

Initialize particles of PSO x_{i} to random values |

x_{i}^{b} = x_{i} * x^{b}: initial information of particle |

F_{i} = f(x_{i}) * f: fitness assignment |

end for |

x_{g}^{b} = best{x_{i}^{b}; i = 1, …, M} * x_{g}: initial global best particle |

Pop = {x_{1}, x_{2}, …, x_{M}} |

F = {F_{1}, F_{2}, …, F_{M}} |

<Main Loop> |

while do |

{Evaluation Loop 1} |

for i = 1 to M do |

if f(x_{i}) is better than f(x_{i}^{b}) then |

x_{i}^{b} = x_{i} |

end if |

iff(x_{i}^{b}) is better than f(x_{g}^{b}) then |

x_{g}^{b} = x_{i}^{b} |

end if |

end for |

{Genetic Operators – Update particles’ position} |

Pop ← Selection(Pop, F) |

Pop ← Crossover(Pop, C) * C: crossover rate |

Pop ← Mutation(Pop, M) * M: mutation rate |

{Evaluation Loop 2} |

for i = 1 to M do |

F_{i} = f(x_{i}) |

end for |

F = {F_{1}, F_{2}, …, F_{M}} |

g = g+1 |

end while |

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**Figure 17.**Instantaneously changed unsteady pressure contours during one rotation at exit surface of impeller.

**Table 1.**Design major specifications of the single-channel pump [23].

Flow coefficient (Ф) | 0.019 |

Head coefficient (ψ) | 0.074 |

Rotational speed (RPM) | 1760 |

Impeller inlet-outlet diameter ratio | 1.9 |

LB | Ref. | UB | |
---|---|---|---|

CP 1 | 0.03 | 0.32 | 0.61 |

CP 2 | 0.42 | 0.71 | 1.00 |

CP 3 | 0.00 | 0.25 | 0.50 |

CP 4 | 0.00 | 0.50 | 1.00 |

CP 5 | 0.50 | 0.75 | 1.00 |

Design Variables | Predicted Values | (U)RANS | Relative Error (%) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

AOD | CP 1 | CP 2 | CP 3 | CP 4 | CP 5 | F_{η}/Fη_{_Ref}. | F_{radial}/F_{radial_Ref} | F_{η}/F_{η_Ref.} | F_{radial}/F_{radial_Ref} | F_{η}/F_{η_Ref.} | F_{radial}/F_{radial_Ref} |

0.600 | 0.004 | 0.003 | 0.214 | 0.837 | −1.038 | 0.5520 | −1.040 | 0.6050 | 0.16 | 8.77 |

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**MDPI and ACS Style**

Kim, J.-H.; Ma, S.-B.; Kim, S.; Choi, Y.-S.; Kim, K.-Y.
Design and Verification of a Single-Channel Pump Model based on a Hybrid Optimization Technique. *Processes* **2019**, *7*, 747.
https://doi.org/10.3390/pr7100747

**AMA Style**

Kim J-H, Ma S-B, Kim S, Choi Y-S, Kim K-Y.
Design and Verification of a Single-Channel Pump Model based on a Hybrid Optimization Technique. *Processes*. 2019; 7(10):747.
https://doi.org/10.3390/pr7100747

**Chicago/Turabian Style**

Kim, Jin-Hyuk, Sang-Bum Ma, Sung Kim, Young-Seok Choi, and Kwang-Yong Kim.
2019. "Design and Verification of a Single-Channel Pump Model based on a Hybrid Optimization Technique" *Processes* 7, no. 10: 747.
https://doi.org/10.3390/pr7100747