# Prediction of Performance and Geometrical Parameters of Single-Phase Ejectors Using Artificial Neural Networks

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Artificial Neural Networks

#### 2.2. Database Description

- Only experimental data from HDRC, including single-phase ejectors, were collected;
- The working fluids are R134a, R245fa, R141b, R1234ze(E) and R1233zd(E). No mixture is considered.

#### 2.3. Construction of the ANN Models

#### 2.4. Training

#### 2.5. Model Validation

## 3. Results

#### 3.1. Generalization Capacity

#### 3.2. Operating Conditions

#### 3.3. Importance of Parameters

## 4. Conclusions

- Both approaches (Cases 1 and 2) show good accuracy. The maximum relative error for both models on the training set was less than 10%. Moreover, Case 1 absolute fractions of variance on the training and test dataset were, respectively, R = 0.9972 and R = 0.9968. For Case 2, the corresponding coefficients were, respectively, R = 0.9714 and R = 0.9166.
- When presented with a dataset of 159 new instances, models show absolute fractions of variance R = 0.9868 (Case 1) and R = 0.9262 (Case 2). This shows the high capacity of the models to predict newly presented data within their training range.
- Comparison between on-design and off-design with 159 data indicates that the two models agree very well with the experimental data. Otherwise, the critical mode offers better results than the sub-critical mode for both operating conditions and geometrical characteristics.
- The most important parameters in the prediction of the ejector performance are the operating conditions for the secondary inlet and outlet, followed by NXP and the primary inlet conditions. Whereas the primary and secondary inlet conditions are the most influential parameters for the estimation of geometrical parameters (${D}_{col}$, ${D}_{prim,out}$, NXP and ${D}_{cas}$).
- Comparisons with experimental and numerical data show that the ANN can provide better accuracy over a wide range of data.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) General schematics of the HDRC, along with (

**b**) its corresponding pressure–enthalpy diagram.

**Figure 5.**Comparison between the ANN predictions versus the target values (experimental data) for Case 1: prediction of $\omega $ and ${P}_{CR}$.

**Figure 6.**Comparison between the ANN predictions and the target values (experimental data) for Case 2: prediction of ${D}_{col}$, ${D}_{prim,out}$, NXP and ${D}_{cas}$.

**Figure 7.**Comparison between the ANN predictions and the target values (experimental data) for the two models using the 159 data points from the database initially set aside: (

**a**) Case 1 and (

**b**) Case 2.

**Figure 8.**Comparison between the on-design (

**a**,

**b**) and off-design (

**c**,

**d**) modes in terms of $\omega $ (

**a**,

**c**) and ${P}_{CR}$ (

**b**,

**d**).

**Figure 9.**Predictions of the ANN model in terms of: (

**a**) ${D}_{col}$, (

**b**) ${D}_{prim,out}$, and (

**c**) ${D}_{cas}$ for Case 2.

**Figure 10.**Relative importance of the input parameters on the model predictions for Cases (

**a**) 1 and (

**b**) 2.

**Figure 11.**Scatter plot matrix of the input data regarding operating conditions. Rectangles and lines show the range of optimum parameters.

**Figure 12.**Scatter plot matrix of input data concerning geometric characteristics. Rectangles and lines show the range of optimum parameters.

**Figure 13.**Comparison between the experimental data (□), thermodynamic (⋄) and ANN (△) models in terms of (

**a**) critical pressure ${P}_{out}$ and (

**b**) entrainment ratio $\omega $. Results obtained for a single-phase supersonic ejector working with R245fa and considered by Bencharif et al. [47].

**Table 1.**Experimental studies on ejector refrigeration systems gathered for the construction of the database.

Working Fluid | Reference | Number of Data Points |
---|---|---|

R134a | Selvaraju and Mani [34] | 360 |

García del Valle et al. [12] | ||

Yan et al. [35] | ||

Li et al. [36] | ||

Poirier et al. [37] | ||

Falat et al. [38] | ||

R245fa | Haghparast et al. [39,40] | 247 |

Hamzaoui et al. [5] | ||

Shestopalov et al. [41] | ||

Mazzelli and Milazzo [42] | ||

Scott et al. [43] | ||

Eames et al. [44,45] | ||

Narimani et al. [46] | ||

Bencharif et al. [47] | ||

R141b | Huang et al. [14] | 264 |

Thongtip and Aphornratana [6,48] | ||

Ruangtrakoon and Thongtip [49] | ||

R1234ze(E)-R1233zd(E) | Gagan et al. [50] | 88 |

Mahmoudian et al. [51] |

**Table 2.**Range of inputs and outputs with their minimum, maximum and average values, as well as their distribution percentage.

Parameter | Units | Minimum | Maximum | Mean | Missing Values (%) |
---|---|---|---|---|---|

Geometrical Parameters | |||||

Primary nozzle throat diameter (${D}_{col}$) | mm | 0.50 | 20.20 | 5.68 | 0% |

Primary nozzle outlet diameter (${D}_{prim,out}$) | mm | 0.80 | 26.32 | 9.13 | 0% |

Nozzle exit position (NXP) | mm | 0.00 | 69.93 | 18.55 | 0% |

Constant area section diameter (${D}_{cas}$) | mm | 0 | 34.07 | 10.46 | 0% |

Constant area section length (${L}_{cas}$) | mm | 0 | 223.77 | 69.21 | 4% |

Diffuser oulet diameter (${D}_{out}$) | mm | 2.60 | 108.30 | 34.91 | 4% |

Diffuser length (${L}_{diff}$) | mm | 11.50 | 950.00 | 212.39 | 4% |

Operating parameters | |||||

Primary flow temperature (${T}_{prim}$) | ^{∘}C | 48 | 120 | 88.91 | 0% |

Primary flow pressure (${P}_{prim}$) | kPa | 400 | 3907.93 | 1514.13 | 0% |

Secondary flow temperature (${T}_{sec}$) | ^{∘}C | −7 | 30.60 | 9.98 | 0% |

Secondary flow pressure (${P}_{sec}$) | kPa | 20.50 | 630.00 | 195.72 | 0% |

Condenser temperature (${T}_{cond}$) | ^{∘}C | 11.93 | 42.50 | 30.64 | 0% |

Performance parameters | |||||

Double-choke entrainment ratio ($\omega $) | - | 0.01 | 0.99 | 0.33 | 3.54% |

Limiting compression ratio (${P}_{CR}$) | - | 0.21 | 4.80 | 2.14 | 0% |

ANN Model | Input Parameters | Output Parameters |
---|---|---|

Case 1: Ejector performance prediction | ${D}_{col}$, ${D}_{prim,out}$, NXP | $\omega $, ${P}_{CR}$ |

${D}_{cas}$, ${L}_{cas}$, ${D}_{out}$, ${L}_{diff}$ | ||

${T}_{prim}$, ${P}_{prim}$, ${T}_{sec}$, ${P}_{sec}$, ${T}_{cond}$ | ||

Case 2: Geometry determination | $\omega $, ${P}_{CR}$, ${T}_{prim}$, ${P}_{prim}$ | ${D}_{col}$, ${D}_{prim,out}$, NXP, ${D}_{cas}$ |

${T}_{sec}$, ${P}_{sec}$, ${T}_{cond}$ |

Parameters/Functions | Value |
---|---|

Algorithm | Levenberg–Marquardt |

Transfer function | Tangent sigmoid |

Performance parameters | Mean squared error |

Number of hidden layers | 10 |

Number of input layers | Case 1 (12) |

Case 2 (7) | |

Number of output layers | Case 1 (2) |

Case 2 (4) | |

Kinds of samples | Training 70 %, validation 15% and testing 15% |

Train epoch | Dividerand |

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## Share and Cite

**MDPI and ACS Style**

Bencharif, M.; Croquer, S.; Fang, Y.; Poncet, S.; Nesreddine, H.; Zid, S.
Prediction of Performance and Geometrical Parameters of Single-Phase Ejectors Using Artificial Neural Networks. *Thermo* **2023**, *3*, 1-20.
https://doi.org/10.3390/thermo3010001

**AMA Style**

Bencharif M, Croquer S, Fang Y, Poncet S, Nesreddine H, Zid S.
Prediction of Performance and Geometrical Parameters of Single-Phase Ejectors Using Artificial Neural Networks. *Thermo*. 2023; 3(1):1-20.
https://doi.org/10.3390/thermo3010001

**Chicago/Turabian Style**

Bencharif, Mehdi, Sergio Croquer, Yu Fang, Sébastien Poncet, Hakim Nesreddine, and Said Zid.
2023. "Prediction of Performance and Geometrical Parameters of Single-Phase Ejectors Using Artificial Neural Networks" *Thermo* 3, no. 1: 1-20.
https://doi.org/10.3390/thermo3010001