# Optimization of the Micro Channel Heat Sink by Combing Genetic Algorithm with the Finite Element Method

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Governing Equations

## 3. Methods

- (1)
- Generation of an initial population: The initial population is usually chosen randomly. These initial guesses are held as binary encoding of the true variable in which a binary representation is needed to describe each individual in the population of interest, although an increasing number of genetic algorithms use real valued encoding or encoding that has been chosen to mimic, in some manner, the natural data structure of the problems.
- (2)
- Calculation of the fitness for each individual: A fitness function needs to be defined in order to evaluate each individual. Once this is done, a fitness value is assigned to each individual reflecting its quality.
- (3)
- Selection and elitism: Selection is the method of choosing individuals from the population to be parents for the succeeding generation. Elitism is associated with the selection step that copies or preserves the best individual of each generation to prevent losing the best qualities.
- (4)
- Crossover: In this work, single point crossover is used to generate children from two parents by combining the information extracted from parents. The crossover probability is taken as 0.6 by random selection.
- (5)
- Mutation: The purpose of mutation is to provide new, random bits of information during the genetic search and to keep the genetic algorithm from converging too fast before sampling the entire cost surface. With each new generation the whole population is swept, with every bit position in every string visited, and occasionally, a 1 is flipped to a 0 or vice versa with a mutation probability of 0.01.

## 4. Results

#### 4.1. Comparison with the Inverse Method

#### 4.2. Sensitivity Analysis

#### 4.3. Optimal Study

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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**Figure 5.**Sensitivity analyses of the non-dimensional design variables for the micro channel heat sink.

**Figure 6.**Sensitivity analyses of the non-dimensional design variables for the micro channel heat sink (initial ${\mathrm{W}}_{\mathrm{c}}=50{\%\mathrm{W}}_{\mathrm{p}}$).

**Figure 9.**The optimal result of the micro multi-channel heat sink in Case 1 with a channel number of 60.

Propertities | Coolant | Solid |
---|---|---|

Material | Water | silicon |

Thermal conductivity ($\mathrm{W}/\mathrm{m}\xb7\mathrm{K}$) | 0.613 | 148 |

Viscosity ($\mathrm{kg}/\mathrm{m}\xb7\mathrm{s}$) | 0.000855 | |

Density $\left(\mathrm{kg}/{\mathrm{m}}^{3}\right)$ | 997 | 2329 |

Specific heat $\left(\mathrm{J}/\mathrm{kg}\xb7\mathrm{K}\right)$ | 4179 |

**Table 2.**The differences between the present model and that of [3].

Channel Number | 50 | 63 | 67 | 71 | |
---|---|---|---|---|---|

${\mathrm{R}}_{\mathrm{T}}(\frac{\mathrm{K}{\mathrm{m}}^{2}}{\mathrm{W}})$ | Ref. [3] | 0.142 | 0.138 | 0.138 | 0.144 |

Present | 0.145 | 0.142 | 0.142 | 0.149 | |

$\mathrm{Error}$ | 2% | 3% | 3% | 3.5% |

Results | Result 1 | Result 2 | Result 3 | Result 4 |
---|---|---|---|---|

${\mathrm{R}}_{\mathrm{T}}\left(\mathrm{W}/\mathrm{K}\right)$ | 0.149 | 0.149 | 0.150 | 0.152 |

Temperature (K) | 307.90 | 307.88 | 308.03 | 308.22 |

Channel number | 60 | 62 | 70 | 74 |

${\mathrm{W}}_{\mathrm{c}}\left(\mathsf{\mu}\mathrm{m}\right)$ | 106 | 105 | 99 | 94 |

${\mathrm{H}}_{\mathrm{c}}\left(\mathsf{\mu}\mathrm{m}\right)$ | 646 | 667 | 668 | 635 |

Results | Initial | Case 2 | |||
---|---|---|---|---|---|

Result 1 | Result 2 | Result 3 | Result 4 | ||

${\mathrm{R}}_{\mathrm{T}}\left(\mathrm{W}/\mathrm{K}\right)$ | 0.402 | 0.148 | 0.144 | 0.147 | 0.148 |

Temperature (K) | 333.16 | 307.80 | 307.45 | 307.71 | 307.85 |

Channel number | 56 | 54 | 55 | 61 | 63 |

${\mathrm{W}}_{\mathrm{c}}$ ($\mathsf{\mu}\mathrm{m}$) | 71 | 111 | 107 | 103 | 100 |

${\mathsf{\delta}}_{1}$ ($\mathsf{\mu}\mathrm{m}$) | 188 | 84 | 131 | 131 | |

${\mathsf{\delta}}_{2}$ ($\mathsf{\mu}\mathrm{m}$) | 69 | 53 | 85 | 85 | |

$\mathsf{\alpha}$ | 4.8 | 5.78 | 7.11 | 6.62 | 6.84 |

$\mathsf{\beta}$ | 0.4 | 0.60 | 0.59 | 0.63 | 0.63 |

Results | Initial Design | Case 1 | |||
---|---|---|---|---|---|

Single J | Multi-J | ||||

${\mathrm{R}}_{\mathrm{T}}\left(\mathrm{W}/\mathrm{K}\right)$ | 0.402 | 0.149 | 0.149 | 0.165 | 0.166 |

Temperature (K) | 333.16 | 307.90 | 307.88 | 309.45 | 309.66 |

Channel number | 56 | 60 | 62 | 53 | 60 |

${\mathrm{W}}_{\mathrm{c}}$ ($\mathsf{\mu}\mathrm{m}$) | 71 | 106 | 105 | 143 | 132 |

${\mathrm{H}}_{\mathrm{c}}$ ($\mathsf{\mu}\mathrm{m}$) | 342 | 646 | 667 | 680 | 647 |

$\mathsf{\alpha}$ | 4.8 | 6 | 6.3 | 4.7 | 4.9 |

$\mathsf{\beta}$ | 0.4 | 0.63 | 0.65 | 0.77 | 0.79 |

$\mathrm{Weight}\left(\mathrm{g}\right)$ | 0.178 | 0.113 | 0.108 | 0.088 | 0.089 |

${\mathrm{W}}_{\mathrm{i}}/\mathrm{W}$(%) | 1 | 63.48 | 60.67 | 49.44 | 50 |

${\mathrm{T}}_{\mathrm{i}}/\mathrm{T}$ (%) | 1 | 92.42 | 92.41 | 92.88 | 92.95 |

Results | Initial Design | Case | Case 2 | |||||
---|---|---|---|---|---|---|---|---|

Single J | Multi-J (${\mathbf{p}}_{1}:{\mathbf{p}}_{2}$ = 1:1) | Multi-J (${\mathbf{p}}_{1}:{\mathbf{p}}_{2}$ = 1:1.5) | ||||||

${\mathrm{R}}_{\mathrm{T}}\left(\mathrm{W}/\mathrm{K}\right)$ | 0.402 | 0.144 | 0.147 | 0.156 | 0.161 | 0.156 | 0.160 | |

Temperature (K) | 333.16 | 307.45 | 307.71 | 308.60 | 309.09 | 308.63 | 309.01 | |

Channel number | 56 | 55 | 61 | 77 | 77 | 77 | 79 | |

${\mathrm{W}}_{\mathrm{c}}$ ($\mathsf{\mu}\mathrm{m}$) | 71 | 106 | 105 | 87 | 81 | 88 | 82 | |

${\mathrm{H}}_{\mathrm{c}}$ ($\mathsf{\mu}\mathrm{m}$) | 342 | ${\mathsf{\delta}}_{1}$ ($\mathsf{\mu}\mathrm{m}$) | 84 | 131 | 177 | 152 | 115 | 152 |

${\mathsf{\delta}}_{2}$ ($\mathsf{\mu}\mathrm{m}$) | 53 | 85 | 114 | 108 | 94 | 108 | ||

$\mathsf{\alpha}$ | 4.8 | 7.11 | 6.62 | 6.99 | 7.95 | 7.82 | 8.45 | |

$\mathsf{\beta}$ | 0.4 | 0.59 | 0.63 | 0.67 | 0.62 | 0.68 | 0.65 | |

Weight $\left(\mathrm{g}\right)$ | 0.178 | 0.276 | 0.243 | 0.177 | 0.180 | 0.163 | 0.160 | |

${\mathrm{W}}_{\mathrm{i}}/\mathrm{W}$ (%) | 1 | 155.05 | 136.52 | 99.44 | 101.12 | 91.57 | 89.89 | |

${\mathrm{T}}_{\mathrm{i}}/\mathrm{T}$ (%) | 1 | 92.28 | 92.36 | 92.63 | 92.78 | 92.64 | 92.75 |

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## Share and Cite

**MDPI and ACS Style**

Lin, D.T.W.; Kang, C.-H.; Chen, S.-C.
Optimization of the Micro Channel Heat Sink by Combing Genetic Algorithm with the Finite Element Method. *Inventions* **2018**, *3*, 32.
https://doi.org/10.3390/inventions3020032

**AMA Style**

Lin DTW, Kang C-H, Chen S-C.
Optimization of the Micro Channel Heat Sink by Combing Genetic Algorithm with the Finite Element Method. *Inventions*. 2018; 3(2):32.
https://doi.org/10.3390/inventions3020032

**Chicago/Turabian Style**

Lin, David T. W., Chung-Hao Kang, and Sheng-Chung Chen.
2018. "Optimization of the Micro Channel Heat Sink by Combing Genetic Algorithm with the Finite Element Method" *Inventions* 3, no. 2: 32.
https://doi.org/10.3390/inventions3020032