# Influence of Thermoelectric Properties and Parasitic Effects on the Electrical Power of Thermoelectric Micro-Generators

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Modelling

_{H}= 305 K and T

_{C}= 300 K, respectively. In the second case, one end of the thermocouple is connected to a hot thermostat (T

_{H}= 305 K), while a convective heat transfer is assumed between the cold side of the thermocouple and the surrounding environment supposed to remain at 300 K. This second regime is the so-called mixed boundary conditions, which are closer to those encountered in real applications. This general approach allows for the study of the influence of several key parameters, including the physical properties of the materials, the geometrical parameters of the µ-TEG, the load resistance, and the electrical and thermal contact resistances, on the electrical power delivered at the load resistance.

_{2}Te

_{3}-based materials, polyamide, and copper) used in these calculations are listed in Table 1. Considering the small temperature difference applied between the hot and cold thermostat (T

_{H}− T

_{C}= $\mathsf{\Delta}T$ = 5 K), the physical properties were assumed constant.

^{−4}Ω cm

^{2}. This upper limit is representative of a poor electrical contact. The values of the heat exchange coefficient, ${h}_{ex}$, examined in this study cover the range from 5 W m

^{−2}K

^{−1}, corresponding to natural air convection, up to 1000 W m

^{−2}K

^{−1}that corresponds to forced water convection.

## 3. Results and Discussion

#### 3.1. Fixed Temperature Boundary Conditions

^{−5}Ω cm

^{2}, ${P}_{max}$ dramatically decreases to a level that could limit the use of such device.

#### 3.2. Mixed Temperature Boundary Conditions

#### 3.2.1. Influence of Parasitic Effects

^{−2}K

^{−1}), the degradation of $\mathsf{\Delta}{T}_{TE}$ is moderate, still leading to an appreciable value of ${P}_{max}$ when ${\rho}_{c}$ is not too high. However, when the cooling is ensured by air (${h}_{ex}=$ 5 W m

^{−2}K

^{−1}), $\mathsf{\Delta}{T}_{TE}$ amounts only to a few mK for ${\rho}_{c}=$10

^{−4}$\mathsf{\Omega}$ cm

^{2}and 10

^{−6}$\mathsf{\Omega}$ cm

^{2}that is less than 0.6% and 0.4% of the applied temperature difference between the hot and cold thermostat. These values, which are extremely low, strongly limit ${P}_{max}$ that drops from 5 µW in the ideal case (fixed temperature boundary conditions) to only 10

^{−3}–10

^{−4}µW. As already discussed elsewhere [35,36], the TE materials operate in a regime limited by the thermal sink, which is reached when the thermal resistance of the heat sink is much larger than the thermal resistance of the TE element. A direct consequence of this regime is that, for a fixed ${h}_{ex}$, the maximum temperature that can be expected on the TE materials, when ${\rho}_{c}$ is null, is $\mathsf{\Delta}{T}_{TE}=\frac{\mathsf{\Delta}T}{2}$ for an optimized thickness of the TE elements [37].

_{c}, which would correspond to an equivalent heat transfer coefficient of 400 W m

^{−2}K

^{−1}for ${S}_{c}$ alone. Under this condition, $\mathsf{\Delta}{T}_{TE}\approx $ 2.52 K regardless of the value of ${\rho}_{c}$, leading to a maximum output power ${P}_{max}$ ranging from 0.48 to 5.9 µW for ${\rho}_{c}=$10

^{−4}$\mathsf{\Omega}$ cm

^{2}and ${\rho}_{c}=$10

^{−6}$\mathsf{\Omega}$ cm

^{2}, respectively. These values are 6.7% higher than those achieved with our previous design, and are among the best values reported in the literature [35].

#### 3.2.2. Influence of Thermoelectric Properties

^{−5}$\mathsf{\Omega}$ cm

^{2}(whatever the choice of ${\rho}_{c}$ the following conclusions will be the same). When ${h}_{ex}$ ≤ 50 W m

^{−2}K

^{−1}, an improvement in $ZT$ is beneficial to improve ${P}_{max}$, even though the improvement is not strictly similar. When $\frac{Z}{{Z}_{int}}\le $ 2, it is more interesting to focus on lowering the thermal conductivity or increasing the Seebeck coefficient, while for $\frac{Z}{{Z}_{int}}>$ 2, lowering the thermal conductivity is the best strategy.

^{−2}K

^{−1}, that is, when the cooling becomes efficient, for which a gain is observed when $\alpha $ increases. Only an incremental improvement of ${P}_{max}$ is obtained when $\rho $ drops due to the contribution of the electrical contact resistance that remains constant. The benefit of a reduced thermal conductivity becomes apparent only when the ratio $\frac{Z}{{Z}_{int}}$ is higher than 3.5 for ${h}_{ex}=$ 400 W m

^{−2}K

^{−1}, and even higher for ${h}_{ex}=$ 1000 W m

^{−2}K

^{−1}. For the sake of completeness, the value of $s$ associated with each of the previous situation is indicated in Figure 8. Strong deviations from unity are observed for high Seebeck coefficient values when ${h}_{ex}\ge 400$ W m

^{−2}K

^{−1}. To compensate the high Peltier heat, consecutive to high Seebeck values, created at the cold side of the thermoelectric material, it is beneficial to reduce the current by making the load resistance higher than for the ideal case. These results are in agreement with a previous report [38].

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Design of the µ-TEG integrating n- and p-type Bi

_{2}Te

_{3}-based materials; (

**b**) typical dimensions of the µ-TEG are indicated.

**Figure 4.**Output power as a function of the load resistance, ${R}_{load}$, for various electrical contact resistances ${\rho}_{c}$.

**Figure 5.**Maximal output power ${P}_{max}$ as a function of ${\rho}_{c}$ for different values of the heat exchange coefficient.

**Figure 6.**Maximum output power ${P}_{max}$ as a function of s with varying magnitude of electrical contact resistance.

**Figure 7.**Maximum output power as a function of $\frac{Z}{{Z}_{int}}$ for different values of ${h}_{ex}$. All of these calculations were performed with ${\rho}_{c}=$ 10

^{−5}$\mathsf{\Omega}$ cm

^{2}.

n Type | p Type | Copper | Polyimide | |
---|---|---|---|---|

Electrical conductivity (S m^{−1}) | 105,000 | 76,000 | 5.99 × 10^{8} | - |

Thermal conductivity (W m^{−1} K^{−1}) | 0.75 | 0.75 | 400 | 0.15 |

Specific heat (J kg^{−1} K^{−1}) | 190 | 190 | 385 | 0.904 |

Density (g cm^{−3}) | 7.70 | 7.74 | 8.96 | 1.4 |

Seebeck coefficient (µV K^{−1}) | −130 | 210 | 6.5 | - |

Geometrical Parameter | L_{cop} | W_{cop} | h_{cop} | h_{poly} | b | l_{gap} | m | e | d |
---|---|---|---|---|---|---|---|---|---|

Value (mm) | 2 | 3 | 0.07 | 0.10 | 0.09 | 0.50 | 0.10 | 0.50 | 0.50 |

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**MDPI and ACS Style**

El Oualid, S.; Kosior, F.; Span, G.; Mehmedovic, E.; Paris, J.; Candolfi, C.; Lenoir, B.
Influence of Thermoelectric Properties and Parasitic Effects on the Electrical Power of Thermoelectric Micro-Generators. *Energies* **2022**, *15*, 3746.
https://doi.org/10.3390/en15103746

**AMA Style**

El Oualid S, Kosior F, Span G, Mehmedovic E, Paris J, Candolfi C, Lenoir B.
Influence of Thermoelectric Properties and Parasitic Effects on the Electrical Power of Thermoelectric Micro-Generators. *Energies*. 2022; 15(10):3746.
https://doi.org/10.3390/en15103746

**Chicago/Turabian Style**

El Oualid, Soufiane, Francis Kosior, Gerhard Span, Ervin Mehmedovic, Janina Paris, Christophe Candolfi, and Bertrand Lenoir.
2022. "Influence of Thermoelectric Properties and Parasitic Effects on the Electrical Power of Thermoelectric Micro-Generators" *Energies* 15, no. 10: 3746.
https://doi.org/10.3390/en15103746