# Modeling and Optimization of Gaseous Thermal Slip Flow in Rectangular Microducts Using a Particle Swarm Optimization Algorithm

^{1}

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## Abstract

**:**

## 1. Introduction

#### 1.1. Rarefied Gas Flows

#### 1.2. Artificial Neural Networks

#### 1.3. Particle Swarm Optimization Algorithm

_{1}and r

_{2}are the random numbers in the interval 0 to 1.

_{1}= c

_{2}= 0.05 [30].

## 2. Mathematical Model

#### 2.1. Hydrodynamic Analysis

#### 2.2. Thermal Analysis

#### 2.3. Solution Procedure

## 3. Results and Discussion

## 4. Conclusions

- The small tolerance values for the mean squared error function values and a very close correlation coefficient of 1 in the NN models proved that the NN models were suitable to use for predicting the Poiseuille Po and Nusselt Nu numbers.
- Without going back to the numerical data, the proposed models were used with the PSO algorithm to find the optimal values of Po and Nu.
- The optimal values of Po and Nu are found when $\epsilon =1$ and $Kn=0.001$.
- The correlation coefficient of the models was greater than 0.95.
- The optimal values of the curves were determined when the curves became stable.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**Best Validation Performance is 0.00020768 at iteration 187 of using MATLAB to generate the NN model of Nu.

**Figure 5.**Best Validation Performance is 0.00022112 at iteration 452 using MATLAB to generate the neural network (NN) model of Po.

**Figure 6.**Correlation coefficient and the regression of the NN model values of Nu and the desired values.

**Figure 7.**Correlation coefficient and the regression of the NN model values of Po and the desired values.

$\mathit{\epsilon}$ | Morini et al. [36] | Sadeghi et al. [37] | Present Results |
---|---|---|---|

0.2 | 9.46 | 9.464 | 9.519 |

0.4 | 8.65 | 8.654 | 8.721 |

0.6 | 8.25 | 8.248 | 8.331 |

0.8 | 8.08 | 8.076 | 8.178 |

1 | 8.04 | 8.033 | 8.159 |

$\mathit{\epsilon}$ | Numerical | Exact [38] | % Difference |
---|---|---|---|

0.001 | 24.031 | 23.7 | 1.3774 |

0.01 | 23.743 | 23.41 | 1.4025 |

0.02 | 23.432 | 22.24 | 5.0871 |

0.1 | 21.259 | 20.95 | 1.4535 |

0.2 | 19.182 | 18.89 | 1.5223 |

0.3 | 17.641 | 17.36 | 1.5929 |

0.4 | 16.514 | 16.24 | 1.6592 |

0.5 | 15.712 | 15.43 | 1.7948 |

0.6 | 15.164 | 14.87 | 1.9388 |

0.7 | 14.811 | 14.5 | 2.0998 |

0.8 | 14.608 | 14.28 | 2.2453 |

0.9 | 14.519 | 14.17 | 2.4037 |

1 | 14.514 | 14.14 | 2.5768 |

$\mathit{\epsilon}$ | Numerical | Exact [38] | % Difference |
---|---|---|---|

0.001 | 10.944 | 10.9 | 0.402 |

0.01 | 10.862 | 10.82 | 0.3867 |

0.02 | 10.773 | 10.47 | 2.8126 |

0.1 | 10.139 | 10.09 | 0.4833 |

0.2 | 9.519 | 9.46 | 0.6198 |

0.3 | 9.056 | 9 | 0.6184 |

0.4 | 8.721 | 8.66 | 0.6995 |

0.5 | 8.487 | 8.41 | 0.9073 |

0.6 | 8.331 | 8.25 | 0.9723 |

0.7 | 8.233 | 8.14 | 1.1296 |

0.8 | 8.178 | 8.08 | 1.1983 |

0.9 | 8.157 | 8.04 | 1.4344 |

1 | 8.159 | 8.04 | 1.4585 |

$\mathit{\epsilon}$ | $\mathit{K}\mathit{n}\text{}=\text{}0.001$ | $\mathit{K}\mathit{n}\text{}=\text{}0.05$ | $\mathit{K}\mathit{n}\text{}=\text{}0.1$ |
---|---|---|---|

0.001 | 6.1658 | 1.4213 | 0.7164 |

0.25 | 5.1047 | 1.7625 | 1.0819 |

0.35 | 4.6786 | 1.8996 | 1.2287 |

0.5 | 4.0394 | 2.1051 | 1.4488 |

0.65 | 3.8048 | 2.1830 | 1.5590 |

0.75 | 3.6485 | 2.2348 | 1.6325 |

0.85 | 3.6156 | 2.2745 | 1.6816 |

1 | 3.5662 | 2.3341 | 1.7552 |

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**MDPI and ACS Style**

Hamadneh, N.N.; Khan, W.A.; Khan, I.; Alsagri, A.S.
Modeling and Optimization of Gaseous Thermal Slip Flow in Rectangular Microducts Using a Particle Swarm Optimization Algorithm. *Symmetry* **2019**, *11*, 488.
https://doi.org/10.3390/sym11040488

**AMA Style**

Hamadneh NN, Khan WA, Khan I, Alsagri AS.
Modeling and Optimization of Gaseous Thermal Slip Flow in Rectangular Microducts Using a Particle Swarm Optimization Algorithm. *Symmetry*. 2019; 11(4):488.
https://doi.org/10.3390/sym11040488

**Chicago/Turabian Style**

Hamadneh, Nawaf N., Waqar A. Khan, Ilyas Khan, and Ali S. Alsagri.
2019. "Modeling and Optimization of Gaseous Thermal Slip Flow in Rectangular Microducts Using a Particle Swarm Optimization Algorithm" *Symmetry* 11, no. 4: 488.
https://doi.org/10.3390/sym11040488