# A Novel Energy Performance Prediction Approach towards Parametric Modeling of a Centrifugal Pump in the Design Process

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## Abstract

**:**

_{mean}, root mean square error, sum of squares due to error, and mean absolute error for centrifugal pump energy performance. The research revealed that the HLCNN model achieves accurate energy performance prediction in the design of centrifugal pumps, reducing the development time and costs.

## 1. Introduction

## 2. Methodology and model

#### 2.1. Modeling of Hydraulic Loss and Energy Performance of Centrifugal Pump

_{j}), impeller outlet diameter (D

_{2}), blade outlet width (b

_{2}), the number of blades (z), blade outlet angle (β

_{2}), hub diameter (d

_{h}), blade inlet angle (β

_{1}), volute base circle diameter (D

_{3}), and volute inlet width (b

_{3}), initial angle of volute tongue (φ

_{0}), and throat area (F

_{t}). The specific speed (n

_{s}), flow rate (Q), and impeller rotational speed (n) are significant operating parameters of the pump. The design and operating parameters affecting the centrifugal pump energy performance (head (H) and efficiency (ղ)) have been widely researched [41,42,43].

_{iis}), (b) impeller surface friction loss (l

_{isf}), (c) impeller flow passage diffusion loss (l

_{ifd}), (d) volute inlet shock loss (l

_{vis}), (e) volute friction loss (l

_{vfri}), and (f) volute diffusion loss (l

_{vdif}).

#### 2.1.1. Impeller Inlet Shock Loss

_{iis}depends on the relative velocity of the blade inlet, which is defined as:

_{iis}is the impeller inlet shock loss coefficient, u

_{1}is the circumferential velocity of the impeller inlet, and Q

_{d}represents the flow rate of the centrifugal pump under design conditions.

#### 2.1.2. Impeller Surface Friction Loss

_{isf}is as follows:

_{isf}are presented in Table 1.

#### 2.1.3. Impeller Flow Passage Diffusion Loss

#### 2.1.4. Volute Inlet Shock Loss

_{2}and v

_{m2}are the circumferential and axial velocities at the impeller outlet, and ψ

_{2}is the extrusion coefficient of the blade outlet.

#### 2.1.5. Volute Friction Loss

_{vfri}can be estimated as:

#### 2.1.6. Volute Diffusion Loss

_{v}is the volute loss coefficient. v

_{3d}is the velocity component in a tangential direction to the impeller.

_{t}, ղ

_{h}, ղ

_{v}, and ղ

_{m}are the theoretical head, hydraulic efficiency, volumetric efficiency, and mechanical efficiency of the centrifugal pump [47]. The theoretical calculation of H

_{t}(as shown in Appendix A) can be referred to the literature [48].

#### 2.2. Hydraulic Loss–Convolution Neural Network (HLCNN)

_{i}), which represents the design and operation parameters of the centrifugal pump. The expression of the centrifugal pump feature information

**g**(

**z**) extracted from the first convolution layer (Conv1) through p channels is shown in Equation (21), where ${w}_{\mathbf{1},\mathbf{y},\mathbf{z}}^{\mathbf{z}}$ is the convolution kernel and

**b**is the bias of the Conv1.

^{z}_{1}represent the number of neurons of FCL1. Similarly, the feature information of the (n + l)th layer is inputted to the regression layer, and the regression function is shown in Equation (26), where w

^{T}and b are regression coefficients.

**m**represents all hydraulic loss of the centrifugal pump.

_{pl}**y**and

_{i}**ŷ**represent the experimental and predicted values of the i-th sample, and n is the number of samples.

_{i}## 3. Experimental Research and Data Sources

_{j}, D

_{2}, b

_{2}, z, β

_{2}, d

_{h}, β

_{1}, D

_{3}, b

_{3}, φ

_{0}, F

_{t}, and n

_{s}, Q, and n are considered as the input variables of the HLCNN model. The six hydraulic losses, namely l

_{iis}, l

_{isf}, l

_{ifd}, l

_{vis}, l

_{vfri}and l

_{vdif}, were selected as output variables. In the design of the centrifugal pump, the variations in these parameters were related to each other along with the working condition and the regulation mode. In this paper, the HLCNN model is proposed to predict the head and efficiency of the centrifugal pump. Consequently, the modeling samples of the centrifugal pump can be represented as

**M**= {

**X, Y**}, where

**X**= [

**x**]

_{1}, …, x_{N}^{T}∈R

^{n}

^{×14},

**Y**= [

**y**]

_{1}, …, y_{N}^{T}∈R

^{n}

^{×6}, and n is the number of the sampling set. The i-th sample set can be further described as

**m**= {(

_{i}**x**= [D

_{i}_{j}(i), D

_{2}(i), b

_{2}(i), z(i), β

_{2}(i), d

_{h}(i), β

_{1}(i), D

_{3}(i), b

_{3}(i), φ

_{0}(i), F

_{t}(i), n

_{s}(i), Q(i), n(i)]

^{T},

**y**= [l

_{i}_{iis}(i), l

_{isf}(i), l

_{ifd}(i), l

_{vis}(i), l

_{vfri}(i), l

_{vdif}(i)]

^{T})}.

## 4. Results and Discussions

#### 4.1. Influence of Convolutional Layer on Prediction Model

#### 4.2. Performance Prediction for the Test Samples

_{means}of the head and efficiency were both less than 9%, which meets the requirements of theoretical research and engineering practice [52]. The change in ARE indicates that it is feasible to use the HLCNN model to predict the energy performance of the centrifugal pump.

#### 4.3. Comparison with the Other Machine Learning Models

_{max}, ARE

_{min}, ARE

_{mean}, RMSE, SSE, and MAE values than the BPNN and LSSVR models. This indicates that the predicted head and efficiency of the HLCNN model and the experimental values have a great agreement, which means that it is more suitable for approximating the nonlinear mapping relationship between design, operation parameters, and energy performance of the centrifugal pump. At the same time, it is worth noting that the HLCNN model reduces the complexity of the prediction model with its special structure of local weight sharing, which is embodied in the data reconstruction in the process of centrifugal pump feature extraction.

#### 4.4. Comparison with the CFD Method

_{means}of the head and efficiency obtained by the CFD model were 17.30% and 7.16%. On the contrary, the ARE

_{means}of the HLCNN model were 4.36% and 4.00%, respectively, proving the strong generalization ability of the HLCNN model within a wide flow rate range.

^{3}/h). Therefore, the HLCNN model built in this paper can build a CPPP model to meet the needs of design, production, and operation quickly and accurately.

## 5. Conclusions

_{means}of head and efficiency were both less than 9%, which is consistent with the error range required by both theoretical research and engineering practice.

_{mean}of the HLCNN model was lower in predicting the head and efficiency compared with the BPNN and the LSSVR model.

_{means}of the HLCNN model were lower than the CFD method in predicting head and efficiency, and the whole prediction process only took 6.48 min. It indicates that the HLCNN model can predict the energy performance of centrifugal pumps in a wide flow rate range quickly and efficiently.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Abbreviations | Variables | ||

CPPP | Centrifugal pump performance prediction | D_{j} | Impeller inlet diameter |

HLM | Hydraulic loss method | D_{2} | Impeller outlet diameter |

CFD | Computational fluid dynamic | b_{2} | Blade outlet width |

ANN | Artificial neural network | β_{2} | Blade outlet angle |

BPNN | Back propagation neural network | z | The number of blades |

LSSVR | Least squares support vector regression | d_{h} | Impeller hub diameter |

DL | Deep learning | β_{1} | Blade inlet angle |

CNN | Convolution neural network | Φ | Blade cape angle |

HLCNN | HLM and CNN | D_{3} | Volute base circle diameter |

FCL | Fully connected layer | b_{3} | Volute inlet width |

ReLU | Rectified linear unit | φ_{0} | Initial angle of volute tongue |

ARE | Absolute relative error | F_{t} | Volute throat area |

ARE_{max} | Maximum absolute relative error | n_{s} | Centrifugal pump specific speed |

ARE_{min} | Minimum absolute relative error | n | Impeller rotational speed |

ARE_{mean} | Mean absolute relative error | Q | Centrifugal pump flow rate |

RMSE | Root mean square error | H | Centrifugal pump head |

SSE | Sum of squares due to error | ƞ | Centrifugal pump efficiency |

MAE | Mean absolute error | l_{iis} | Impeller inlet shock loss |

l_{isf} | Impeller surface friction loss | ||

Parameters | l_{ifd} | Impeller flow passage diffusion loss | |

CNN3 | Three convolution layers | l_{vis} | Volute inlet shock loss |

CNN4 | Four convolution layers | l_{vfri} | Volute friction loss |

CNN5 | Five convolution layers | l_{vdif} | Volute diffusion loss |

CNN6 | Six convolution layers | ||

Conv1 | First convolution layer | ||

Convn | n-th convolution layer | ||

FCL1 | First fully connected layer |

## Appendix A

_{t})

_{t}) represents the energy transmitted by the impeller to the liquid per unit weight. Its calculation formula can be expressed [53] as:

_{1}and u

_{2}are circumferential velocities at the inlet and outlet of the impeller, respectively. v

_{u1}and v

_{u2}are the circumferential components of absolute velocity at inlet and outlet, respectively (Figure A1).

_{2}), the theoretical head is written [45] as:

_{1}and A

_{2}are cross-section areas at the inlet and outlet of the impeller, respectively. ${D}_{1\mathrm{m}}^{*}$ is the meridional component of the diameter at the impeller inlet [48].

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**Figure 4.**Centrifugal pump energy performance test rig. (1) Motor. (2) Centrifugal pump. (3) Suction sump. (4) Electric valve. (5) Pressure sensor. (6) Electromagnetic flowmeter. (7) Photoelectric sensor. (8) Data acquisition system.

**Figure 10.**Comparison of performance prediction of centrifugal pump with different models. (

**a**) Head. (

**b**) Efficiency.

Parameter | Expression | |
---|---|---|

k_{3}: impeller friction loss correction coefficient | ${k}_{3}=1/4.68{n}_{\mathrm{s}}{}^{0.0185}-4.84$ | (3) |

λ_{1}: impeller linear friction coefficient | ${\lambda}_{1}={\left[1.74+2\mathrm{lg}({D}_{\mathrm{a}}/2\delta )\right]}^{-2}$ | (4) |

l_{a}: hydraulic length of the flow passage | ${l}_{\mathrm{a}}=({D}_{2}-{D}_{\mathrm{j}})/(\mathrm{sin}{\beta}_{2}+\mathrm{sin}{\beta}_{1})$ | (5) |

D_{a}: average diameter of the flow passage | ${D}_{\mathrm{a}}=({D}_{2}+{D}_{\mathrm{j}})/2$ | (6) |

w_{a}: average relative velocity | ${w}_{\mathrm{a}}=0.5({w}_{1}+{w}_{2})$ | (7) |

_{1}and w

_{2}are the relative velocity of the impeller inlet and outlet.

Parameter | Expression | |
---|---|---|

k_{7}: volute friction loss correction coefficient | ${k}_{7}=-0.071+6.3\frac{{n}_{\mathrm{s}}}{100}-2{(\frac{{n}_{\mathrm{s}}}{100})}^{2}-3.44{(\frac{{n}_{\mathrm{s}}}{100})}^{3}+1.7{(\frac{{n}_{\mathrm{s}}}{100})}^{4}$ | (11) |

λ_{2}: volute linear friction coefficient | ${\lambda}_{2}={\left[1.2+2\mathrm{lg}(D/2\delta )\right]}^{-2}$ | (12) |

l: equivalent tube length of volute | $l=\mathsf{\pi}\left(1-{\phi}_{0}/360\right)\left({D}_{3}+D\right)$ | (13) |

D: equivalent tube diameter of volute | $D=\sqrt{\frac{2{F}_{\mathrm{t}}}{\mathsf{\pi}}}$ | (14) |

_{a}is the average velocity in the volute.

Network Layer | Type | Parameters | ||
---|---|---|---|---|

Convolutional Kernel Size | Feature Maps | Neurons | ||

Input | Input layer | - | - | 14 |

Conv1 | Convolutional layer | 1 × 3 | 16 | - |

Conv2 | Convolutional layer | 1 × 3 | 24 | - |

Conv3 | Convolutional layer | 1 × 3 | 24 | - |

Conv4 | Convolutional layer | 1 × 3 | 16 | - |

FCL1 | Fully connected layer | 1 × 1 | - | 32 |

FCL2 | Fully connected layer | 1 × 1 | - | 12 |

FCL3 | Fully connected layer | 1 × 1 | - | 6 |

Output | Output layer | - | - | 6 |

Model | ARE_{max} | ARE_{min} | ARE_{mean} | RMSE | SSE | MAE |
---|---|---|---|---|---|---|

HLCNN | 10.943% | 0.112% | 4.866% | 2.774 m | 600.331 m^{2} | 0.783 m |

BPNN | 20.268% | 0.252% | 7.587% | 3.607 m | 1014.695 m^{2} | 1.053 m |

LSSVR | 15.318% | 0.118% | 5.718% | 3.608 m | 1015.125 m^{2} | 0.980 m |

Model | ARE_{max} | ARE_{min} | ARE_{mean} | RMSE | SSE | MAE |
---|---|---|---|---|---|---|

HLCNN | 10.583% | 0.072% | 4.769% | 4.011% | 0.125 | 0.443% |

BPNN | 16.724% | 2.319% | 8.019% | 6.263% | 0.306 | 1.217% |

LSSVR | 16.693% | 0.316% | 6.538% | 5.154% | 0.207 | 0.463% |

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

D_{j} (mm) | 76 | D_{3} (mm) | 15 |

D_{2} (mm) | 137 | b_{3} (mm) | 30 |

b_{2} (mm) | 14 | φ_{0} (°) | 22 |

z | 6 | F_{t} (mm^{2}) | 1477 |

β_{2} (°) | 30 | n_{s} | 129 |

d_{h} (mm) | 0 | Q_{d} (m^{3}/h) | 50 |

β_{1} (°) | 17 | n (r/min) | 2900 |

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## Share and Cite

**MDPI and ACS Style**

Nan, L.; Wang, Y.; Chen, D.; Huang, W.; Zhu, Z.; Liu, F.
A Novel Energy Performance Prediction Approach towards Parametric Modeling of a Centrifugal Pump in the Design Process. *Water* **2023**, *15*, 1951.
https://doi.org/10.3390/w15101951

**AMA Style**

Nan L, Wang Y, Chen D, Huang W, Zhu Z, Liu F.
A Novel Energy Performance Prediction Approach towards Parametric Modeling of a Centrifugal Pump in the Design Process. *Water*. 2023; 15(10):1951.
https://doi.org/10.3390/w15101951

**Chicago/Turabian Style**

Nan, Lingbo, Yumeng Wang, Diyi Chen, Weining Huang, Zuchao Zhu, and Fusheng Liu.
2023. "A Novel Energy Performance Prediction Approach towards Parametric Modeling of a Centrifugal Pump in the Design Process" *Water* 15, no. 10: 1951.
https://doi.org/10.3390/w15101951