# Performance of a Composite Thermoelectric Generator with Different Arrangements of SiGe, BiTe and PbTe under Different Configurations

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## Abstract

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## 1. Introduction

## 2. Composite Thermoelectric Generator

#### 2.1. Configuration of Thermal and Electrical Connections

**Figure 2.**(

**a**) Thermal-electrical circuit of series connected thermoelectric generator (SC-TEG) , which is composed of two stages, which are thermally and electrically connected in series; (

**b**) SC-composite TEG (CTEG) system.

**Figure 3.**(

**a**) Thermal-electrical circuit of a parallel series connected (PSC)-CTEG composed of segmented-conventional TEMs, which are thermally and electrically connected in parallel; (

**b**) PSC-CTEG composite system.

**Figure 4.**(

**a**) Thermal-electrical circuit of thermally and electrically in parallel (TEP)-CTEG composed of three conventional TEMs, which are thermally and electrically connected in parallel; (

**b**) TEP-CTEG composite system.

#### 2.2. Equivalent Figure of Merit for Different Configurations

## 3. Arrangements for Thermoelectric Materials in TEGS

#### 3.1. Case I: Homogeneous Thermoelectric Properties, Configuration Effect

#### 3.2. Case II: Heterogeneous Thermoelectric Properties, Arrangements of Thermoelectric Materials

## 4. Results and Discussion

#### 4.1. Homogeneous TEGS: Effect of the Configuration

**Table 1.**Numerical values of ${Z}_{eq}^{h}$, for each of the three configurations using different materials.

Material | ${Z}_{eq-SC}^{h}$ | ${Z}_{eq-PSC}^{h}$ | ${Z}_{eq-TEP}^{h}$ |
---|---|---|---|

BiTe | 0.00212133 | 0.00305269 | 0.00195898 |

PbTe | 0.00055109 | 0.000657238 | 0.000586714 |

SiGe | 0.000287562 | 0.00033337 | 0.000314212 |

#### 4.2. Heterogeneous TEGS: Different Arrangements of Thermoelectric Materials

**Figure 5.**The equivalent figure of merit corresponding to heterogeneous SC CTEG, under the condition $TE{M}_{2}=TE{M}_{3}\ne TE{M}_{1}$. The highest numerical value corresponding to $TE{M}_{2}=TE{M}_{3}=BiTe\ne TE{M}_{1}=PbTe$.

**Figure 6.**The equivalent figure of merit corresponding to heterogeneous PSC TEGS under the condition $TE{M}_{1}=TE{M}_{2}\ne TE{M}_{3}$. The highest numerical value corresponding to $TE{M}_{1}=TE{M}_{2}=PbTe\ne TE{M}_{3}=BiTe$.

**Figure 7.**The equivalent figure of merit corresponding to the TEP TEGS under the condition $TE{M}_{2}=TE{M}_{3}\ne TE{M}_{1}$. The highest numerical value corresponding to $TE{M}_{2}=TE{M}_{3}=PbTe\ne TE{M}_{1}=BiTe$.

**Table 2.**Numerical values of ${Z}_{eq-SC-max}^{Inh}$ for SC TEGS with the arrangement $TE{M}_{i}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}TE{M}_{j}\ne TE{M}_{l}$.

${TEM}_{1}$ | ${TEM}_{2}={TEM}_{3}$ | ${Z}_{eq-\mathrm{SC}-max}^{Inh}$ |
---|---|---|

BiTe | PbTe | 0.00168734 |

SiGe | 0.0012388 | |

PbTe | BiTe | 0.00273649 |

SiGe | 0.00118802 | |

SiGe | BiTe | 0.00150947 |

PbTe | 0.000994534 |

**Table 3.**Numerical values of ${Z}_{eq-PSC-max}^{Inh}$ for PSC TEGS with the arrangement $TE{M}_{i}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}TE{M}_{j}\ne TE{M}_{l}$.

${TEM}_{3}$ | ${TEM}_{1}={TEM}_{2}$ | ${Z}_{eq-PSC-max}^{Inh}$ |
---|---|---|

BiTe | PbTe | 0.0055567 |

SiGe | 0.00325841 | |

PbTe | BiTe | 0.00445846 |

SiGe | 0.0011157 | |

SiGe | BiTe | 0.00392902 |

PbTe | 0.00172358 |

**Table 4.**Numerical values of ${Z}_{eq-TEP-max}^{Inh}$ from the TEP arrangement under condition $TE{M}_{i}=TE{M}_{j}\ne TEM-l$.

${TEM}_{1}$ | ${TEM}_{2}={TEM}_{3}$ | ${Z}_{eq-TEP-max}^{Inh}$ |
---|---|---|

BiTe | PbTe | 0.00405227 |

SiGe | 0.00108556 | |

PbTe | BiTe | 0.00137141 |

SiGe | 0.000871679 | |

SiGe | BiTe | 0.00368808 |

PbTe | 0.00140252 |

$System$ | $Arrangement$ |
---|---|

SC | $TE{M}_{2}=TE{M}_{3}=BiTe\ne TE{M}_{1}=PbTe$ |

PSC | $TE{M}_{1}=TE{M}_{2}=PbTe\ne TE{M}_{3}=BiTe$ |

TEP | $TE{M}_{2}=TE{M}_{3}=PbTe\ne TE{M}_{1}=BiTe$ |

**Figure 8.**The optimal configuration corresponds to the PSC TEGS under the condition $TE{M}_{1}=TE{M}_{2}=PbTe\ne TE{M}_{3}=BiTe$.

#### 4.3. Maximum Power and Efficiency

#### 4.3.1. Maximum Power

**Figure 9.**Maximum power of the system PSC under the condition $TEM1=TEM2\ne TE{M}_{3}$, the highest numerical value corresponding to case $TEM1=TEM2=PbTe\ne TE{M}_{3}=BiTe$.

#### 4.3.2. Efficiency

**Figure 11.**Contour plot: efficiency for the PSC-system assuming the condition $TE{M}_{1}=TE{M}_{2}\ne TE{M}_{3}$, assuming the maximum value of ${Z}_{eq-PSC}^{Inh}$ ($TE{M}_{1}=TE{M}_{2}=PbTe\ne TE{M}_{3}=BiTe$).

## 5. The Case of a Three-TEM Chain Thermally and Electrically Connected in Series

**Figure 13.**Equivalent figure of merit for the series-system as a function of ${T}_{i-(1,2)}$, assuming three different materials.

## 6. Corollary: Maximum ${Z}_{eq}$ for a CTEG

- (i)
- There exists a thermal-electrical connection between TEMs, which have the maximum value of ${Z}_{eq}$ when the thermoelectric material is the same for all TEMs.
- (ii)
- If different thermoelectric materials are used for each TEM, under the condition $TE{M}_{i}=TE{M}_{j}\ne TE{M}_{l}$ where $i;j;l$ can be $1,2\phantom{\rule{0.166667em}{0ex}}or\phantom{\rule{0.166667em}{0ex}}3$, then for a given thermal-electrical connection, there exists an optimal arrangement of the thermoelectric material in which the value of ${Z}_{eq}$ is maximum.

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

Symbol | Name |

CTEG | $\text{Composite}\phantom{\rule{4.pt}{0ex}}\text{thermoelectric}\phantom{\rule{4.pt}{0ex}}\text{generator}$ |

Z | $\text{Thermoelectric}\phantom{\rule{4.pt}{0ex}}\text{figure}\phantom{\rule{4.pt}{0ex}}\text{of}\phantom{\rule{4.pt}{0ex}}\text{merit}$ |

α | $\text{Seebeck}\phantom{\rule{4.pt}{0ex}}\text{coefficient}$ |

κ | $\text{Thermal}\phantom{\rule{4.pt}{0ex}}\text{conductivity}$ |

ρ | $\text{Electrical}\phantom{\rule{4.pt}{0ex}}\text{resistivity}$ |

${T}_{hot}$ | $\text{Hot}\phantom{\rule{4.pt}{0ex}}\text{side}\phantom{\rule{4.pt}{0ex}}\text{temperature}$ |

${T}_{cold}$ | $\text{Cold}\phantom{\rule{4.pt}{0ex}}\text{side}\phantom{\rule{4.pt}{0ex}}\text{temperature}$ |

$\Delta V$ | $\text{Potential}\phantom{\rule{4.pt}{0ex}}\text{difference}$ |

$\Delta T$ | $\text{Temperature}\phantom{\rule{4.pt}{0ex}}\text{difference}$ |

S | $\text{Entropy}\phantom{\rule{4.pt}{0ex}}\text{per}\phantom{\rule{4.pt}{0ex}}\text{carrier}$ |

e | $\text{Electron}\phantom{\rule{4.pt}{0ex}}\text{charge}$ |

s | $\text{Compatibility}\phantom{\rule{4.pt}{0ex}}\text{factor}$ |

SC | $\text{Series}\phantom{\rule{4.pt}{0ex}}\text{connection}$ |

${Z}_{eq-SC}$ | $\text{Equivalent}\phantom{\rule{4.pt}{0ex}}\text{figure}\phantom{\rule{4.pt}{0ex}}\text{of}\phantom{\rule{4.pt}{0ex}}\text{merit}\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{the}\phantom{\rule{4.pt}{0ex}}\text{SC}\phantom{\rule{4.pt}{0ex}}\text{system}\phantom{\rule{4.pt}{0ex}}$ |

PSC | $\text{Parallel}\phantom{\rule{4.pt}{0ex}}\text{segmented}\phantom{\rule{4.pt}{0ex}}\text{conventional-CTEG}$ |

${Z}_{eq-PSC}$ | $\text{Equivalent}\phantom{\rule{4.pt}{0ex}}\text{figure}\phantom{\rule{4.pt}{0ex}}\text{of}\phantom{\rule{4.pt}{0ex}}\text{merit}\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{the}\phantom{\rule{4.pt}{0ex}}\text{PSC}\phantom{\rule{4.pt}{0ex}}\text{system}\phantom{\rule{4.pt}{0ex}}$ |

TEP | $\text{Thermally}\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}\text{electrically}\phantom{\rule{4.pt}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}\text{parallel}$ |

${Z}_{eq-TEP}$ | $\text{Equivalent}\phantom{\rule{4.pt}{0ex}}\text{figure}\phantom{\rule{4.pt}{0ex}}\text{of}\phantom{\rule{4.pt}{0ex}}\text{merit}\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{the}\phantom{\rule{4.pt}{0ex}}\text{TEP}\phantom{\rule{4.pt}{0ex}}\text{system}$ |

R | $\text{Electrical}\phantom{\rule{4.pt}{0ex}}\text{resistance}$ |

K | $\text{Thermal}\phantom{\rule{4.pt}{0ex}}\text{conductance}$ |

${Z}_{eq-SC}^{h}$ | $\text{Equivalent}\phantom{\rule{4.pt}{0ex}}\text{figure}\phantom{\rule{4.pt}{0ex}}\text{of}\phantom{\rule{4.pt}{0ex}}\text{merit}\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{the}\phantom{\rule{4.pt}{0ex}}\text{SC}\phantom{\rule{4.pt}{0ex}}\text{system,}\phantom{\rule{4.pt}{0ex}}\text{homogeneous}\phantom{\rule{4.pt}{0ex}}$ |

${Z}_{eq-PSC}^{h}$ | $\text{Equivalent}\phantom{\rule{4.pt}{0ex}}\text{figure}\phantom{\rule{4.pt}{0ex}}\text{of}\phantom{\rule{4.pt}{0ex}}\text{merit}\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{the}\phantom{\rule{4.pt}{0ex}}\text{PSC}\phantom{\rule{4.pt}{0ex}}\text{system,}\phantom{\rule{4.pt}{0ex}}\text{homogeneous}\phantom{\rule{4.pt}{0ex}}$ |

${Z}_{eq-TEP}^{h}$ | $\text{Equivalent}\phantom{\rule{4.pt}{0ex}}\text{figure}\phantom{\rule{4.pt}{0ex}}\text{of}\phantom{\rule{4.pt}{0ex}}\text{merit}\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{the}\phantom{\rule{4.pt}{0ex}}\text{TEP}\phantom{\rule{4.pt}{0ex}}\text{system,}\phantom{\rule{4.pt}{0ex}}\text{homogeneous}\phantom{\rule{4.pt}{0ex}}$ |

${Z}_{eq-SC}^{Inh}$ | $\text{Equivalent}\phantom{\rule{4.pt}{0ex}}\text{figure}\phantom{\rule{4.pt}{0ex}}\text{of}\phantom{\rule{4.pt}{0ex}}\text{merit}\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{the}\phantom{\rule{4.pt}{0ex}}\text{SC}\phantom{\rule{4.pt}{0ex}}\text{system,}\phantom{\rule{4.pt}{0ex}}\text{heterogeneous}\phantom{\rule{4.pt}{0ex}}$ |

${Z}_{eq-PSC}^{Inh}$ | $\text{Equivalent}\phantom{\rule{4.pt}{0ex}}\text{figure}\phantom{\rule{4.pt}{0ex}}\text{of}\phantom{\rule{4.pt}{0ex}}\text{merit}\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{the}\phantom{\rule{4.pt}{0ex}}\text{PSC}\phantom{\rule{4.pt}{0ex}}\text{system,}\phantom{\rule{4.pt}{0ex}}\text{heterogeneous}\phantom{\rule{4.pt}{0ex}}$ |

${Z}_{eq-TEP}^{Inh}$ | $\text{Equivalent}\phantom{\rule{4.pt}{0ex}}\text{figure}\phantom{\rule{4.pt}{0ex}}\text{of}\phantom{\rule{4.pt}{0ex}}\text{merit}\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{the}\phantom{\rule{4.pt}{0ex}}\text{TEP}\phantom{\rule{4.pt}{0ex}}\text{system,}\phantom{\rule{4.pt}{0ex}}\text{heterogeneous}\phantom{\rule{4.pt}{0ex}}$ |

${P}_{max}$ | $\text{Maximum}\phantom{\rule{4.pt}{0ex}}\text{power}$ |

${P}_{max-PSC}$ | $\text{Maximum}\phantom{\rule{4.pt}{0ex}}\text{power}\phantom{\rule{4.pt}{0ex}}\text{of}\phantom{\rule{4.pt}{0ex}}\text{the}\phantom{\rule{4.pt}{0ex}}\text{PSC}\phantom{\rule{4.pt}{0ex}}\text{system}$ |

${K}_{c}$ | $\text{Contact}\phantom{\rule{4.pt}{0ex}}\text{thermal}\phantom{\rule{4.pt}{0ex}}\text{conductance}$ |

${K}_{I=0}$ | $\text{Thermal}\phantom{\rule{4.pt}{0ex}}\text{conductance}\phantom{\rule{4.pt}{0ex}}\text{at}\phantom{\rule{4.pt}{0ex}}\text{zero}\phantom{\rule{4.pt}{0ex}}\text{electrical}\phantom{\rule{4.pt}{0ex}}\text{current}$ |

η | $\text{Efficiency}\phantom{\rule{4.pt}{0ex}}\text{of}\phantom{\rule{4.pt}{0ex}}\text{the}\phantom{\rule{4.pt}{0ex}}\text{thermoelectric}\phantom{\rule{4.pt}{0ex}}\text{generator}$ |

${\eta}_{eq-PSC}^{Inh}$ | $\text{Efficiency}\phantom{\rule{4.pt}{0ex}}\text{of}\phantom{\rule{4.pt}{0ex}}\text{the}\phantom{\rule{4.pt}{0ex}}\text{PSC}\phantom{\rule{4.pt}{0ex}}\text{system,}\phantom{\rule{4.pt}{0ex}}\text{heterogeneous}$ |

${Z}_{eq-series}$ | $\text{Equivalent}\phantom{\rule{4.pt}{0ex}}\text{figure}\phantom{\rule{4.pt}{0ex}}\text{of}\phantom{\rule{4.pt}{0ex}}\text{merit}\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{the}\phantom{\rule{4.pt}{0ex}}\text{series}\phantom{\rule{4.pt}{0ex}}\text{system}\phantom{\rule{4.pt}{0ex}}$ |

${K}_{conv}$ | $\text{Thermal}\phantom{\rule{4.pt}{0ex}}\text{conductance}\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{the}\phantom{\rule{4.pt}{0ex}}\text{heat}\phantom{\rule{4.pt}{0ex}}\text{conveyed}\phantom{\rule{4.pt}{0ex}}\text{by}\phantom{\rule{4.pt}{0ex}}\text{the}\phantom{\rule{4.pt}{0ex}}\text{electrical}\phantom{\rule{4.pt}{0ex}}\text{current}$ |

${K}_{eq}$ | $\text{Thermal}\phantom{\rule{4.pt}{0ex}}\text{conductance}\phantom{\rule{4.pt}{0ex}}\text{of}\phantom{\rule{4.pt}{0ex}}\text{the}\phantom{\rule{4.pt}{0ex}}\text{TEG}$ |

${I}_{eq}$ | $\text{Equivalent}\phantom{\rule{4.pt}{0ex}}\text{electrical}\phantom{\rule{4.pt}{0ex}}\text{current}$ |

${I}_{Qeq}$ | $\text{Equivalent}\phantom{\rule{4.pt}{0ex}}\text{heat}\phantom{\rule{4.pt}{0ex}}\text{flux}\phantom{\rule{4.pt}{0ex}}\text{inside}\phantom{\rule{4.pt}{0ex}}\text{the}\phantom{\rule{4.pt}{0ex}}\text{TEG}$ |

${\alpha}_{eq}$ | $\text{Equivalent}\phantom{\rule{4.pt}{0ex}}\text{Seebeck}\phantom{\rule{4.pt}{0ex}}\text{coefficient}$ |

$\Delta V$ | Voltage |

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**MDPI and ACS Style**

Vargas-Almeida, A.; Olivares-Robles, M.A.; Lavielle, F.M.
Performance of a Composite Thermoelectric Generator with Different Arrangements of SiGe, BiTe and PbTe under Different Configurations. *Entropy* **2015**, *17*, 7387-7405.
https://doi.org/10.3390/e17117387

**AMA Style**

Vargas-Almeida A, Olivares-Robles MA, Lavielle FM.
Performance of a Composite Thermoelectric Generator with Different Arrangements of SiGe, BiTe and PbTe under Different Configurations. *Entropy*. 2015; 17(11):7387-7405.
https://doi.org/10.3390/e17117387

**Chicago/Turabian Style**

Vargas-Almeida, Alexander, Miguel Angel Olivares-Robles, and Federico Méndez Lavielle.
2015. "Performance of a Composite Thermoelectric Generator with Different Arrangements of SiGe, BiTe and PbTe under Different Configurations" *Entropy* 17, no. 11: 7387-7405.
https://doi.org/10.3390/e17117387