# Thermodynamics of Thermoelectric Phenomena and Applications

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## Abstract

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## 1. Introduction

#### 1.1. Historical Notes

#### 1.2. The Thermoelectric Engine

## 2. Forces & Fluxes

#### 2.1. Irreversible Thermodynamics and the Onsager–de Groot–Callen Model

#### 2.2. Forces and Fluxes

#### 2.3. Energy Flux and Heat Flux

#### 2.4. Thermoelectric Coefficients

#### 2.4.1. Decoupled Processes

#### 2.4.2. Coupled Processes

#### 2.5. The Entropy per Carrier

#### 2.6. Kinetic Coefficients and Transport Parameters

${L}_{11}$ | ${L}_{12}={L}_{21}$ | ${L}_{22}$ |

$\frac{T}{{e}^{2}}{\sigma}_{T}$ | $\frac{{T}^{2}}{{e}^{2}}{\sigma}_{T}{S}_{N}$ | $\frac{{T}^{3}}{{e}^{2}}{\sigma}_{T}{S}_{N}^{2}+{T}^{2}{\kappa}_{J}$ |

#### 2.7. The Dimensionless Figure of Merit z T

## 3. Heat & Entropy

#### 3.1. Volumetric Heat Production

#### 3.2. Entropy Production Density

## 4. Thermoelectric Generator, Cooler, and Heater

- Thermoelectric heater: TEH.
- Thermoelectric cooler: TEC.
- Thermoelectric generator: TEG.

- ${\alpha}_{n,p}\phantom{\rule{0.166667em}{0ex}}({T}_{\mathrm{h}}-{T}_{\mathrm{c}}):$ Seebeck voltage,
- ${T}_{\mathrm{h}}-{T}_{\mathrm{c}}:$ temperature difference,

- ${\alpha}_{n,p}\phantom{\rule{0.166667em}{0ex}}{T}_{\text{out}}I:$ Peltier heat flow.
- ${K}_{n,p}\phantom{\rule{0.166667em}{0ex}}\Delta T:$ conductive heat flow [113].

#### 4.1. Thermoelectric Generator

- Incoming thermal power:$${Q}_{\text{in}}=\alpha \phantom{\rule{0.166667em}{0ex}}{T}_{\mathrm{h}}\phantom{\rule{0.166667em}{0ex}}I-\frac{1}{2}\phantom{\rule{0.166667em}{0ex}}{R}_{\text{in}}\phantom{\rule{0.166667em}{0ex}}{I}^{2}+K\phantom{\rule{0.166667em}{0ex}}\left(\right)open="("\; close=")">{T}_{\mathrm{h}}-{T}_{\mathrm{c}}$$
- Outgoing thermal power:$${Q}_{\text{out}}=\alpha {T}_{\mathrm{c}}I+\frac{1}{2}\phantom{\rule{0.166667em}{0ex}}{R}_{\text{in}}\phantom{\rule{0.166667em}{0ex}}{I}^{2}+K\phantom{\rule{0.166667em}{0ex}}\left(\right)open="("\; close=")">{T}_{\mathrm{h}}-{T}_{\mathrm{c}}$$
- Electrical power produced:$${P}_{\text{pro}}={Q}_{\text{in}}-{Q}_{\text{out}}=\alpha \phantom{\rule{0.166667em}{0ex}}I\phantom{\rule{0.166667em}{0ex}}\left(\right)open="("\; close=")">{T}_{\mathrm{h}}-{T}_{\mathrm{c}}$$
- Open voltage:$${V}_{0}=\alpha \phantom{\rule{0.166667em}{0ex}}\left(\right)open="("\; close=")">{T}_{\mathrm{h}}-{T}_{\mathrm{c}}$$

- Efficiency:$$\eta =\frac{{P}_{\text{pro}}}{{Q}_{in\phantom{\rule{4.pt}{0ex}}}}=\frac{\Delta T}{{T}_{\mathrm{h}}}\frac{M}{M+1+\frac{{\left(\right)}^{M}}{2}Z{T}_{\mathrm{h}}}$$As we can notice the efficiency is the product of the reversible Carnot efficiency ${\eta}_{\mathrm{C}}=\frac{\Delta T}{{T}_{\mathrm{h}}}=\frac{{T}_{\mathrm{h}}-{T}_{\mathrm{c}}}{{T}_{\mathrm{h}}}$ with the irreversible factor $\raisebox{1ex}{$M$}\!\left/ \!\raisebox{-1ex}{$\left(\right)$}\right.-\frac{1}{2}\frac{\Delta T}{{T}_{\mathrm{h}}}.$
- Maximal efficiency:From the derivation $\frac{\partial \eta}{\partial M}=0$, we get after a few algebra steps$${\eta}_{max}={\eta}_{c}\phantom{\rule{0.166667em}{0ex}}\frac{{M}_{\eta}-1}{{M}_{\eta}+\frac{{T}_{\mathrm{c}}}{{T}_{\mathrm{h}}}}$$
- Maximal electrical power output:$$\frac{\partial {P}_{\text{pro}}}{\partial I}=0=\alpha \left(\right)open="("\; close=")">{T}_{\mathrm{h}}-{T}_{\mathrm{c}}$$$${I}_{P}^{max}=\frac{\alpha \left(\right)open="("\; close=")">{T}_{\mathrm{h}}-{T}_{\mathrm{c}}}{}2\phantom{\rule{0.166667em}{0ex}}{R}_{\text{in}}$$The last equation tells us that the maximal power output is obtained when the electrical load resistance ${R}_{\text{load}}$ is equal to the internal resistance of the TEG. Then the maximal output power is obtained for ${M}_{P}=M=1$. The maximal power expression reduces to$${P}_{\text{pro}}^{max}=\frac{{\alpha}^{2}\phantom{\rule{0.166667em}{0ex}}\Delta {T}^{2}}{4\phantom{\rule{0.166667em}{0ex}}{R}_{\text{in}}}$$One can notice that the condition for maximal efficiency (${M}_{\eta}=\sqrt{1+Z\phantom{\rule{0.166667em}{0ex}}{T}_{\mathrm{m}}}$) and maximal output power (${M}_{P}=1$) are different. This means that given a fabricated TE device where the geometric lengths and areas are fixed, more power will be produced if additional heat is supplied and higher current drawn than in the maximum efficiency configuration. However, when designing an optimal device for a particular application, the optimum design will have the geometry and current for maximum efficiency because this will provide more power with the same designed heat input.

#### 4.2. Thermoelectric Cooler

- Incoming thermal power:$${Q}_{\mathrm{c}}=\alpha \phantom{\rule{0.166667em}{0ex}}{T}_{\mathrm{c}}\phantom{\rule{0.166667em}{0ex}}I-\frac{1}{2}\phantom{\rule{0.166667em}{0ex}}{R}_{\text{in}}\phantom{\rule{0.166667em}{0ex}}{I}^{2}-K\phantom{\rule{0.166667em}{0ex}}({T}_{\mathrm{h}}-{T}_{\mathrm{c}})$$
- Outgoing thermal power:$${Q}_{\text{rej}}=\alpha \phantom{\rule{0.166667em}{0ex}}{T}_{\mathrm{h}}\phantom{\rule{0.166667em}{0ex}}I+\frac{1}{2}\phantom{\rule{0.166667em}{0ex}}{R}_{\text{in}}\phantom{\rule{0.166667em}{0ex}}{I}^{2}-K\phantom{\rule{0.166667em}{0ex}}({T}_{\mathrm{h}}-{T}_{\mathrm{c}})$$
- Consumed electrical power:$${P}_{\text{rec}}={Q}_{\text{rej}}-{Q}_{\mathrm{c}}=\alpha \phantom{\rule{0.166667em}{0ex}}({T}_{\mathrm{h}}-{T}_{\mathrm{c}})\phantom{\rule{0.166667em}{0ex}}I+{R}_{\text{in}}\phantom{\rule{0.166667em}{0ex}}{I}^{2}$$
- Maximal cooling power:$${I}_{{Q}_{\mathrm{c}}^{\text{max}}}=\frac{\alpha {T}_{\mathrm{c}}}{{R}_{\text{in}}}$$$${Q}_{\mathrm{c}}^{\text{max}}=\frac{1}{2}\phantom{\rule{0.166667em}{0ex}}\frac{{\alpha}^{2}{T}_{\mathrm{c}}^{2}}{{R}_{\text{in}}}-K\phantom{\rule{0.166667em}{0ex}}({T}_{\mathrm{h}}-{T}_{\mathrm{c}})=K\phantom{\rule{0.166667em}{0ex}}\left(\right)open="("\; close=")">\frac{1}{2}\phantom{\rule{0.166667em}{0ex}}{T}_{\mathrm{c}}^{2}Z-({T}_{\mathrm{h}}-{T}_{\mathrm{c}})$$One can notice that the maximal cooling power is directly driven by the figure of merit of the material and of the device, respectively.
- Coefficient of performance ${\phi}_{\mathrm{c}}$ [120]:$${\phi}_{\mathrm{c}}=\frac{{Q}_{\mathrm{c}}}{{P}_{\text{rec}}}=\frac{\alpha \phantom{\rule{0.166667em}{0ex}}{T}_{\mathrm{c}}\phantom{\rule{0.166667em}{0ex}}I-\frac{1}{2}\phantom{\rule{0.166667em}{0ex}}{R}_{\text{in}}\phantom{\rule{0.166667em}{0ex}}{I}^{2}-K\phantom{\rule{0.166667em}{0ex}}({T}_{\mathrm{h}}-{T}_{\mathrm{c}})}{\alpha \phantom{\rule{0.166667em}{0ex}}({T}_{\mathrm{h}}-{T}_{\mathrm{c}})\phantom{\rule{0.166667em}{0ex}}I+{R}_{\text{in}}\phantom{\rule{0.166667em}{0ex}}{I}^{2}}$$
- Maximal ${\phi}_{\mathrm{c}}:$$$\frac{\partial {\phi}_{\mathrm{c}}}{\partial {V}_{0}}=0\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\text{with}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{V}_{0}=\alpha ({T}_{\mathrm{h}}-{T}_{\mathrm{c}})\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\text{gives}$$$${\phi}_{\mathrm{c}}^{max}=\frac{{T}_{\mathrm{c}}}{({T}_{\mathrm{h}}-{T}_{\mathrm{c}})}\frac{\sqrt{1+Z{T}_{\mathrm{m}}}-\frac{{T}_{\mathrm{h}}}{{T}_{\mathrm{c}}}}{\sqrt{1+Z{T}_{\mathrm{m}}}+1}$$${V}_{0}$ is open voltage, a specific value of $\mathcal{V}$. The Carnot factor ${\phi}_{\text{Car}}=\frac{{T}_{\mathrm{c}}}{({T}_{\mathrm{h}}-{T}_{\mathrm{c}})}$ is the reversible term of ${\phi}_{\mathrm{c}}^{max}$. The second term contains the irreversible contributions, $\left(\right)open="("\; close=")">\sqrt{1+Z{T}_{\mathrm{m}}}-\frac{{T}_{\mathrm{h}}}{{T}_{\mathrm{c}}}$It should be noticed that Equation (56) is similar to the expression obtained for the TEG configuration, see Equation (49). Both formulae are well-known and often written as [27,119,121,122]$$\begin{array}{cc}\hfill {\eta}_{\text{max}}& =\frac{{T}_{\mathrm{h}}-{T}_{\mathrm{c}}}{{T}_{\mathrm{h}}}\frac{\sqrt{1+Z{T}_{\mathrm{m}}}-1}{\sqrt{1+Z{T}_{\mathrm{m}}}+{T}_{\mathrm{c}}/{T}_{\mathrm{h}}}\hfill \end{array}$$$$\begin{array}{cc}\hfill \frac{1}{{\phi}_{\mathrm{c}}^{\text{max}}}& =\frac{{T}_{\mathrm{h}}-{T}_{\mathrm{c}}}{{T}_{\mathrm{h}}}\frac{\sqrt{1+Z{T}_{\mathrm{m}}}+1}{\sqrt{1+Z{T}_{\mathrm{m}}}-{T}_{\mathrm{h}}/{T}_{\mathrm{c}}}\hfill \end{array}$$
- Maximum coolingThe maximum temperature difference is achieved for ${Q}_{\mathrm{c}}^{\text{max}}=0$ and hence for ${\phi}_{\mathrm{c}}=0$. In this case we get from Equation (54)$$\Delta {T}_{max}={({T}_{\mathrm{h}}-{T}_{\mathrm{c}})}_{max}=\frac{1}{2}\phantom{\rule{0.166667em}{0ex}}Z\phantom{\rule{0.166667em}{0ex}}{T}_{\mathrm{c}}^{2}$$

#### 4.3. Thermoelectric Heater

- Incoming thermal power:$${Q}_{\text{w}}=\alpha \phantom{\rule{0.166667em}{0ex}}{T}_{\mathrm{h}}\phantom{\rule{0.166667em}{0ex}}I+\frac{1}{2}\phantom{\rule{0.166667em}{0ex}}{R}_{\text{in}}\phantom{\rule{0.166667em}{0ex}}{I}^{2}-K\phantom{\rule{0.166667em}{0ex}}({T}_{\mathrm{h}}-{T}_{\mathrm{c}})$$
- Outgoing thermal power:$${Q}_{\text{rej}}=\alpha \phantom{\rule{0.166667em}{0ex}}{T}_{\mathrm{c}}\phantom{\rule{0.166667em}{0ex}}I-\frac{1}{2}\phantom{\rule{0.166667em}{0ex}}{R}_{\text{in}}\phantom{\rule{0.166667em}{0ex}}{I}^{2}-K\phantom{\rule{0.166667em}{0ex}}({T}_{\mathrm{h}}-{T}_{\mathrm{c}})$$
- Consumed electrical power:$${P}_{\text{rec}}={Q}_{\text{w}}-{Q}_{\text{rej}}=\alpha \phantom{\rule{0.166667em}{0ex}}({T}_{\mathrm{h}}-{T}_{\mathrm{c}})\phantom{\rule{0.166667em}{0ex}}I+{R}_{\text{in}}\phantom{\rule{0.166667em}{0ex}}{I}^{2}$$
- Coefficient of performance:$${\phi}_{\text{w}}=\frac{{Q}_{\text{w}}}{{P}_{\text{rec}}}=\frac{\alpha \phantom{\rule{0.166667em}{0ex}}{T}_{\mathrm{h}}\phantom{\rule{0.166667em}{0ex}}I+\frac{1}{2}\phantom{\rule{0.166667em}{0ex}}{R}_{\text{in}}\phantom{\rule{0.166667em}{0ex}}{I}^{2}-K\phantom{\rule{0.166667em}{0ex}}({T}_{\mathrm{h}}-{T}_{\mathrm{c}})}{\alpha ({T}_{\mathrm{h}}-{T}_{\mathrm{c}})\phantom{\rule{0.166667em}{0ex}}I+{R}_{\text{in}}\phantom{\rule{0.166667em}{0ex}}{I}^{2}}$$
- Maximal ${\phi}_{\mathrm{w}}:$$${\phi}_{\text{w}}^{max}=\frac{{T}_{\mathrm{h}}}{\left(\right)}$$The Carnot factor for ϕ is here ${\phi}_{\text{Car}}=\frac{{T}_{\mathrm{h}}}{\left(\right)}$, whereas the irreversible contribution is given by the second term.

## 5. The General Conductivity Matrix

#### 5.1. Derivation of the Conductivity Matrix

#### 5.2. The Peltier-Thomson Coefficient

- $\alpha T\nabla \phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\mathbf{J}$: equals zero due to particle conservation,
- $T\mathbf{J}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\nabla \alpha $ : “Peltier-Thomson” term,
- $\mathbf{J}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\alpha \nabla T=\mathbf{J}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\left(\right)open="("\; close=")">\mathbf{E}-\frac{1}{{\sigma}_{T}}\mathbf{J}$ electrical work production and dissipation,
- $\nabla \phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\left(\right)open="("\; close=")">-{\kappa}_{J}\nabla T$: change in thermal conduction due to heat produced or absorbed.

#### 5.3. The Peltier–Thomson Term

- Pure Peltier, isothermal junction between two materials:$$\mathbf{J}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\left(\right)open="("\; close=")">\nabla \Pi -\alpha \nabla T$$
- Thomson, homogeneous material under temperature gradient:$$\mathbf{J}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\left(\right)open="("\; close=")">\nabla \Pi -\alpha \nabla T\nabla T=\tau \phantom{\rule{0.166667em}{0ex}}\mathbf{J}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\nabla T$$$$\nabla \Pi =\frac{d\Pi}{dT}\nabla T\phantom{\rule{2.em}{0ex}}\mathrm{and}\phantom{\rule{2.em}{0ex}}\tau =\frac{d\Pi}{dT}-\alpha $$$$\nabla \phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathbf{J}}_{Q}=\tau \phantom{\rule{0.166667em}{0ex}}\mathbf{J}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\nabla T+\mathbf{J}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\mathbf{E}-\frac{{J}^{2}}{{\sigma}_{T}}-\nabla \phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\left(\right)open="("\; close=")">{\kappa}_{J}\nabla T$$$$\nabla \phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathbf{J}}_{Q}=\tau \phantom{\rule{0.166667em}{0ex}}\mathbf{J}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\nabla T+\mathbf{J}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\mathbf{E}-\frac{{J}^{2}}{{\sigma}_{T}}-{\kappa}_{J}{\nabla}^{2}T$$

#### 5.4. Local Energy Balance

## 6. Relative Current and Thermoelectric Potential

#### 6.1. Relative Current and Thermoelectric Potential

#### 6.2. Thermoelectric Potential and Local Reduced Efficiency for a Thermogenerator

#### 6.3. Compatibility Approach

## 7. Optimum Device Design and FGM

## 8. Conclusions

## Acknowledgements

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**Figure 2.**Dithermal thermodynamic engine: The cycling working fluid is marked as F and the thermal leak is marked as L. (

**a**) ideal model neglecting the engine walls; (

**b**): realistic model including the thermal leaks.

**Figure 5.**Reduced efficiency ${\eta}_{\mathrm{r}}=f\left(\frac{\Phi}{\alpha T}\right)$ for TEG (irreversible factor).

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**MDPI and ACS Style**

Goupil, C.; Seifert, W.; Zabrocki, K.; Müller, E.; Snyder, G.J.
Thermodynamics of Thermoelectric Phenomena and Applications. *Entropy* **2011**, *13*, 1481-1517.
https://doi.org/10.3390/e13081481

**AMA Style**

Goupil C, Seifert W, Zabrocki K, Müller E, Snyder GJ.
Thermodynamics of Thermoelectric Phenomena and Applications. *Entropy*. 2011; 13(8):1481-1517.
https://doi.org/10.3390/e13081481

**Chicago/Turabian Style**

Goupil, Christophe, Wolfgang Seifert, Knud Zabrocki, Eckhart Müller, and G. Jeffrey Snyder.
2011. "Thermodynamics of Thermoelectric Phenomena and Applications" *Entropy* 13, no. 8: 1481-1517.
https://doi.org/10.3390/e13081481