# Cooling and Mechanical Performance Analysis of a Trapezoidal Thermoelectric Cooler with Variable Cross-Section

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Modelling

#### 2.1. Physical Model

_{2}Te

_{3}) was selected as the thermoelectric material because it has the best thermoelectric properties at low temperature ranges. As a commercial material, it is widely used in thermoelectric coolers. Al

_{2}O

_{3}was chosen as the substrate layer material for heat conduction and electrical insulation. Copper slices were used as conductive layers which connect the thermoelectric legs. The Sn-Pb solder was used as a welding layer in this study. Ceramic materials and thermoelectric materials are considered as brittle materials. Copper slices and solders were identified as elastoplastic materials for reducing thermal stress in the trapezoidal TEC [39]. The temperature dependent properties of thermoelectric leg materials used in this simulation are shown in Figure 3 and residual material properties are shown in Table 2.

#### 2.2. Thermo-Physical Properties

_{2}Te

_{3}materials. Because of its bipolar diffusion effect, the Seebeck coefficient peaks around 400 K, and the thermal conductivity starts to rise at 350 K, making the extreme value of ZT appear at around 350 K. The electrical resistivity increases monotonously with temperature.

^{2}σT/κ. The ZT is proportional to the square of the Seebeck coefficient (S) and electrical conductivity (σ). It is inversely proportional to the thermal conductivity (κ). The ZT can effectively reflect the cooling performance of thermoelectric coolers.

#### 2.3. Mathematical Model

#### 2.4. Numerical Model

#### 2.4.1. Mathematical Equations of Thermoelectric Analysis

#### 2.4.2. Mathematical Equations of Thermal Stress Analysis

#### 2.4.3. Boundary Conditions

- Steady state conditions are assumed in this study;
- Apart from the cold side and hot side for the trapezoidal TEC, all lateral surfaces are considered to be adiabatic;
- Electrical and thermal contact resistance are neglected;
- The thermal power of the power chip on the cold side of trapezoidal TEC is set to 5 W;
- The heat sink connected to the hot end of the trapezoidal TEC is assumed to be the equivalent convection heat transfer coefficient, its value is set to 1000 W/(m
^{2}K) to meet the range of forced convection heat conduction for water; - The cold side and hot side of the trapezoidal TEC are fixed by clamps during thermal stress analysis.

#### 2.4.4. Model Validation

## 3. Results and Discussion

_{A}values are shown in Table 4.

#### 3.1. Influence of Electrical Current

_{A}= 2.5), and the optimal electrical current decreases along with increasing leg height owing to the higher electrical resistance in the trapezoidal TEC with increasing leg height. Furthermore, the optimal electrical current first increases and then decreases with increasing R

_{A}for the leg height of the trapezoidal TEC at a value of 1.6 mm, as shown in Figure 5b, and the optimal electrical current reaches the maximum value when R

_{A}= 1 (the traditional rectangular TEC). For any two R

_{A}values that are reciprocal to each other, their corresponding optimal electrical currents are equal, because they have the same electrical resistance and thermal resistance. The optimal electrical current defined in this section is used in the remaining sections of this study.

#### 3.2. Influence of Leg Height

_{A}values. The minimum chip temperature has the same decreasing trend as increasing leg height under different R

_{A}values. It can be found that a higher leg height reduces the minimum chip temperature, which suggests that higher leg height is beneficial to enhance the cooling performance of the trapezoidal TECs. For example, the minimum chip temperature decreased by 4.5% for R

_{A}= 1.8 when the leg height increased from 0.6 mm to 2.0 mm. The minimum chip temperature reached the minimum value of 284.085 K when H = 2.0 mm and R

_{A}= 1.8, as shown in Figure 6.

#### 3.3. Influence of Cross-Sectional Area Ratio

_{A}= 1 and 2.5 when H = 1.6 mm. The temperature distribution of the trapezoidal-shaped leg along the leg height is different from that of the rectangular-shaped leg. As shown in Figure 3, the thermal conductivity, electrical resistivity, Seebeck coefficient and ZT for thermoelectric materials are all temperature-dependent. The variable cross-sectional area design can adjust the temperature distribution along the leg height compared with a constant cross-sectional area design. Therefore, compared with the rectangular-shaped thermoelectric leg, these thermo-physical properties of the material in the trapezoidal-shaped thermoelectric leg vary with the change in spatial temperature distribution, which affects the performance of the trapezoidal thermoelectric cooler.

_{A}= 1. It can be seen that the variation tendency of the minimum chip temperature when H = 1.6 mm is different from the other leg heights, as is the increase in R

_{A}. The minimum temperature of a chip cooled by a trapezoidal TEC is lower than the minimum temperature of a chip cooled by a rectangular TEC. For instance, the minimum chip temperature corresponding to the trapezoidal TEC is 1.2 K lower than the minimum chip temperature corresponding to the rectangular TEC when H = 1.6 mm. The minimum chip temperature decreases more significantly at R

_{A}> 1 compared to R

_{A}< 1. This indicates that the area of the cold side for the trapezoidal TEC is larger than that of its hot side, which is more conducive to improving the cooling performance of the trapezoidal TEC. The minimum value of the minimum chip temperature always occurs at R

_{A}= 2.5 in this figure.

_{A}makes the heat conduction effect dominate over the Joule heating effect at this stage. Afterwards, when R

_{A}reaches a critical value, the Joule heating effect begins to dominate and the chip temperature begins to decrease. This occurs up until the point when R

_{A}value is too large (R

_{A}= 2.5), when the heat conduction effect once again dominates, making the minimum chip temperature rises again. When the leg height value is 1.6 mm, there is no first stage because the higher leg height weakens the heat conduction effect, a smaller cross-sectional area ratio is needed to make the heat conduction effect dominate.

#### 3.4. Thermal Stress Analysis

_{A}< 1, the maximum von Mises stress is generated on the cold side of the thermoelectric legs. When R

_{A}> 1, the maximum von Mises stress is generated on the hot side of the thermoelectric legs. Furthermore, the maximum von Mises stress is generated at the edge of thermoelectric legs. These positions are easily destroyed first. From this figure, it can also be seen that the hot side and cold side of the trapezoidal-shaped thermoelectric legs generate a non-uniform thermal stress distribution, and the thermal stress for the thermoelectric legs at the edge of the chip is greater. This is due to the inhomogeneous temperature distribution resulting in a large local thermal stress gradient.

_{A}. The maximum von Mises stress for the thermoelectric leg reduces as leg height is increased due to the decreasing temperature gradient, which indicates that the reliability of the trapezoidal TEC with a higher leg height is better. In addition, we observed that the maximum von Mises stress for the trapezoidal-shaped leg with variable cross-sectional area decreases more obviously as leg height increases, compared with the rectangular-shaped leg with a constant cross-sectional area. Taking R

_{A}= 1.8 as an example, the maximum von Mises stress decreases by 46.7% as the leg height increases from 0.6 mm to 2.0 mm.

_{A}increases, the maximum von Mises stresses have the same changing trend at different leg heights. It can be seen that the maximum von Mises decreases first, then increases, and finally decreases as the R

_{A}value increases. The minimum value for the maximum von Mises stress always occurs at R

_{A}= 1, which indicates that the mechanical reliability of the rectangular-shaped leg with constant cross-sectional area compared with that of the trapezoidal-shaped leg with non-constant cross-sectional area is better. Meanwhile, when the R

_{A}value exceeds 2.5, the maximum von Mises stress begins to decrease. From Figure 11 and Figure 12, it can be seen that the minimum value for the maximum thermal stress of the thermoelectric leg is reached when H = 2 mm and R

_{A}= 1. The maximum von Mises stress of the leg for the trapezoidal thermoelectric cooler with optimal cooling performance has increased by 40.1% compared to the original rectangular thermoelectric cooler.

## 4. Conclusions

- 1.
- The optimal electrical current corresponding to the trapezoidal-shaped thermoelectric leg under different geometric parameters is different. The optimal electrical current should be used when analyzing the performance and thermal stress of the trapezoidal TEC.
- 2.
- Increasing the thermoelectric leg height simultaneously improves the cooling performance and mechanical reliability of the trapezoidal TEC. For R
_{A}= 1.8, the minimum chip temperature decreased by 4.5% and the maximum von Mises stress of the leg decreased by 46.7% as the leg height increased from 0.6 mm to 2.0 mm. - 3.
- Compared to the rectangular TEC, the variable cross-sectional design for the trapezoidal TEC improves the cooling performance. The minimum chip temperature was reduced by 0.87% under the trapezoidal thermoelectric cooler with optimized geometry.
- 4.
- The maximum von Mises stress for the trapezoidal-shaped leg was greater than that of the rectangular-shaped leg. The maximum von Mises stress of the leg for the trapezoidal thermoelectric cooler with optimal cooling performance increased by 40.1% compared to the original rectangular thermoelectric cooler. Therefore, both the cooling performance and reliability need to be considered at the same time when designing a trapezoidal TEC.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 3.**Temperature dependent thermoelectric material properties (

**a**) Seebeck coefficient (

**b**) electrical resistivity (

**c**) thermal conductivity and (

**d**) figure of merit (ZT) [40].

**Figure 4.**Comparison of simulation results and experimental results (

**a**) performance validation and (

**b**) thermal stress validation.

**Figure 5.**Variation of the chip temperature with the electrical current for (

**a**) different leg heights when R

_{A}= 2.5 and (

**b**) different cross-sectional area ratios of the cold side to hot side for the legs when H = 1.6 mm.

**Figure 7.**Temperature distributions of thermoelectric legs when H = 1.6 mm: (

**a**) R

_{A}= 1; (

**b**) R

_{A}= 2.5.

**Figure 8.**Variation of the minimum chip temperature with cross-sectional area ratios of the cold to hot ends of the legs for different H.

**Figure 9.**Variation of the minimum chip temperature with cross-sectional area ratios of the cold to hot ends of the legs and leg heights.

**Figure 10.**The von Mises stress nephogram of full-scale trapezoidal-shaped thermoelectric legs when H = 1.6 mm, I = 4 A (

**a**) R

_{A}= 0.56 and (

**b**) R

_{A}= 1.8.

**Figure 12.**Change of the maximum von Mises stress with cross-sectional area ratio of the cold side to hot side of the leg for different H.

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

Leg height (mm) | 1.6 |

Electrode thickness (mm) | 0.2 |

Solder thickness (mm) | 0.1 |

Substrate thickness (mm) | 0.8 |

Cross-section area of the legs (mm^{2}) | 1.4 × 1.4 |

pitch (mm) | 1.0 |

Power chip thickness (mm) | 0.4 |

Thermal grease thickness (mm) | 0.1 |

Cross-section area of power chip (mm^{2}) | 10 × 10 |

Material | Thermal Conductivity (W/(m∙K)) | Electrical Conductivity (S/m) | Seebeck Coefficient (V/K) | Density (kg/m ^{3}) | Specific Heat Capacity (J/kg·K) | Young’s Modulus (GPa) | Coefficient of Thermal Expansion (1/K) | Poisson’s Ratio |
---|---|---|---|---|---|---|---|---|

Ceramic | 25 | 1 × 10^{−12} | 0 | 3970 | 800 | 380 | 6.5 × 10^{−6} | 0.22 |

Copper | 385 | 5.9 × 10^{7} | 6.5 × 10^{−6} | 8930 | 386 | 115 | 1.7 × 10^{−5} | 0.31 |

Solder | 55 | 2 × 10^{7} | 0 | 7240 | 210 | 44.5 | 2.7 × 10^{−5} | 0.33 |

P-Bi_{2}Te_{3} | - | - | - | 7740 | 154.4 | 65~59 | 0.8 × 10^{−5}~1.32 × 10^{−5} | 0.23 |

N-Bi_{2}Te_{3} | - | - | - | 7740 | 154.4 | 65~59 | 0.8 × 10^{−5}~1.32 × 10^{−5} | 0.23 |

Power chip | 420 | 1 × 10^{−14} | 0 | 3100 | 800 | 410 | 4 × 10^{−6} | 0.14 |

Thermal grease | 1 | 2.5 × 10^{−12} | 0 | 1630 | 1450 | 60 | 1.8 × 10^{−5} | 0.19 |

Grid Number | Chip Temperature for I = 2 A (K) | Chip Temperature for I = 3 A (K) | Chip Temperature for I = 4 A (K) | Chip Temperature for I = 5 A (K) |
---|---|---|---|---|

171040 | 332.183 | 322.049 | 318.728 | 321.935 |

209655 | 332.224 | 322.091 | 318.762 | 321.961 |

266264 | 332.215 | 322.081 | 318.752 | 321.950 |

R_{A} | 0.27 | 0.4 | 0.56 | 0.75 | 1 | 1.33 | 1.8 | 2.5 | 3.67 |
---|---|---|---|---|---|---|---|---|---|

A_{c} (mm^{2}) | 0.84 | 1.12 | 1.4 | 1.68 | 1.96 | 2.24 | 2.52 | 2.8 | 3.08 |

A_{h} (mm^{2}) | 3.08 | 2.8 | 2.52 | 2.24 | 1.96 | 1.68 | 1.4 | 1.12 | 0.84 |

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## Share and Cite

**MDPI and ACS Style**

Lu, T.; Li, Y.; Zhang, J.; Ning, P.; Niu, P.
Cooling and Mechanical Performance Analysis of a Trapezoidal Thermoelectric Cooler with Variable Cross-Section. *Energies* **2020**, *13*, 6070.
https://doi.org/10.3390/en13226070

**AMA Style**

Lu T, Li Y, Zhang J, Ning P, Niu P.
Cooling and Mechanical Performance Analysis of a Trapezoidal Thermoelectric Cooler with Variable Cross-Section. *Energies*. 2020; 13(22):6070.
https://doi.org/10.3390/en13226070

**Chicago/Turabian Style**

Lu, Tianbo, Yuqiang Li, Jianxin Zhang, Pingfan Ning, and Pingjuan Niu.
2020. "Cooling and Mechanical Performance Analysis of a Trapezoidal Thermoelectric Cooler with Variable Cross-Section" *Energies* 13, no. 22: 6070.
https://doi.org/10.3390/en13226070