# Discrepancy between Constant Properties Model and Temperature-Dependent Material Properties for Performance Estimation of Thermoelectric Generators

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## Abstract

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## 1. Introduction

## 2. Methods, Results and Discussion

#### 2.1. Role of the T Dependence of Material Properties in Performance Estimation

_{2}Te

_{3}and PbTe have significantly different ${\kappa}_{\mathrm{h}}$ and ${\kappa}_{\mathrm{c}}$ and Figure A2a, showing that the weight of Joule and Thomson heat to ${Q}_{\mathrm{in}}$ is comparably large for these materials.

_{2}(Si,Sn) (referred to as n-Mg

_{2}X), Figure 2b. All profiles are calculated for the optimum current for maximum efficiency of the real material. Here, in addition to the 2TD materials, 1TD materials were also involved. $\alpha \left(T\right)$ and $\kappa \left(T\right)$ play a dominating role in the shaping of the temperature profile, which is reflected by the closeness of the $\alpha \left(T\right)\ne const.$, $\kappa \left(T\right)\ne const.$ case to the real material.

_{2}X from Figure 1a (red dots) can be discussed in terms of the hot side slopes of the corresponding temperature profiles (red lines) in Figure 2b when comparing between cases with the same $\kappa \left(T\right)$. The downward ${\frac{dT}{dx}}_{\mathrm{h}}$ for the 2TD material with $\alpha \left(T\right)=const.$(red solid line) indicates an increase in the inflowing Fourier due to missing Thomson heat, compared to the actual case (dark green line). Simultaneously, but only partly compensated in the ${Q}_{\mathrm{in}}$ balance by missing Thomson heat, less Peltier heat is absorbed at the hot side and therefore the efficiency is overestimated (Figure 1a left side, red dot). The 2TD $\kappa \left(T\right)=const.$(red dotted line) deforms the T profile considerably but hardly increases the heat input (Equation (6)) compared to the real material, as the SpAv of $\kappa \left(T\right)$ maintains an unchanged thermal resistance of the TE leg. We can conclude that replacing the $T$ dependence of $\alpha \left(T\right)$ and $\kappa \left(T\right)$ by adequate constants will, although significantly changing the $T$ profile, influence the inflowing heat and thus efficiency to a much lower extent due to compensating effects. The RD of CPM efficiency in effect arises mainly from a redistribution of internal Joule and Thomson heat due to considerable deformation of the T profile by neglecting the T dependence of $\kappa \left(T\right)$ and $\alpha \left(T\right)$ and local redistribution of reversible heat generation as a consequence of neglect of the T dependence of the convective entropy flux.

#### 2.2. Peltier–Thomson Heat Balance and the Resulting Uncertainty in CPM Efficiency

_{2}X, a particular example is given in Appendix A.3.2 (Figure A2c) where, with $\alpha \left(T\right)$ weakly changing between ${T}_{\mathrm{c}}$ and ${T}_{\mathrm{h}}$ but peaking inside, this compensation can also be almost perfect, or, as for SnSe (Figure 4, Figure 5b and Figure 6), overcompensation may even occur.

_{2}Te

_{3}, which has an exceptionally higher ${\kappa}_{\mathrm{h}}$ compared to the cold side (Figure A1a in Appendix A.2) together with high Joule release (Figure 4b) and almost compensation of the Peltier–Thomson balance. Thus, the Joule contribution dominates, leading to an underestimation of the efficiency. Additionally, SnSe behaves somewhat differently from the general trend, with a falling $\alpha \left(T\right)$ curve (Appendix A.2 Figure A1b) and the over-resistivity at the cold side (Appendix A.2 Figure A1c). Moreover, ${\kappa}_{\mathrm{h}}$ is much lower than ${\kappa}_{\mathrm{c}}$. As an effect, Joule heat is preferentially led to the cold side; consequently, hot side Joule heat is greatly overestimated in the CPM (Figure 4b), but as the relative contribution of Joule heat to ${Q}_{\mathrm{in}}$ is small (Figure A2a), the resulting trend towards the overestimation of performance in the CPM remains moderate. On the other hand, as seen from Appendix A.3.2 Figure A1b, Thomson heat is absorbed in the leg as $\alpha \left(T\right)$ for SnSe is a falling curve and is mainly bound to the hot side. As seen from Figure 4b, for SnSe, the hot side Peltier–Thomson heat will, unlike for most of the other materials, be overestimated by the CPM. However, the resulting underestimation of efficiency in the CPM will be overcompensated by the counteracting Joule heat distribution.

_{2}X, Mg

_{2}Si and PbTe, which have larger Thomson contributions (Figure A2a), leading to larger discrepancies of the CPM efficiency estimate.

#### 2.3. Refining the CPM Efficiency Estimate

_{2}Te

_{3}) where Thomson heat is released to the cold side but absorbed from the hot side. A rule to treat all of the cases likewise is needed. Figure 5a,b and Figure A2c,d accordingly show scenarios where $\alpha \left(T\right)$ contains almost linear intervals along with strongly bowed ones, where $\alpha \left(T\right)$ is monotonous or contains a maximum, where ${\alpha}_{\mathrm{h}}$ and $\overline{\alpha}$ are far from each other or close together or where $\alpha \left(T\right)$ crosses the $\overline{\alpha}$ horizontal once or twice. The position of the extrema (maxima or minima) of $\Delta {T}_{\mathrm{Thomson}}\left(x\right)$ is marked in each diagram by a brown line. Accordingly, the area corresponding to the uncompensated heat might be more complex than is shown in Figure 3, e.g., see Figure 5a. The area to the left of the $\alpha \left(T\right)$ curve to the $\alpha $-axis from this point up to the hot side ${\alpha}_{\mathrm{h}}$ (marked by a red border) represents ${\dot{Q}}_{\tau ,\mathrm{h}}$. The fact that the respective area also contains negatively counted parts when $\alpha \left(T\right)$ goes through a maximum is also taken into account. Accordingly, the upper slim boat-shaped area in Figure 5a counts as negative; symbolically, it is mirrored in the green area.

## 3. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Note from Section 2.1

#### Appendix A.2. Material Data and Boundary Conditions

**Figure A1.**Temperature-dependent thermoelectric material properties of representative material classes: (

**a**) thermal conductivity, (

**b**) Seebeck coefficient, (

**c**) electrical resistivity and (

**d**) figure of merit. Since SnSe has much higher resistivity, the scale for it is given on the right y-axis. All the raw experimental data taken from the literature [20,21,22,23,24] were fitted with appropriate polynomials (usually 3rd or 4th order). For SnSe, a 9th order polynomial fit was used owing to the complex T dependence and hence shows an unusually high $zT$

_{max}. However, this does not affect the physics discussed and hence these fitted data were used throughout the manuscript.

**Table A1.**Temperature range of analysis for all materials of Figure A1.

Material | Temperature Range of Analysis |
---|---|

p-Mg_{2}(Si,Sn) | 723 K to 383 K |

n-Mg_{2}(Si,Sn) | 723 K to 383 K |

HMS | 833 K to 298 K |

Mg_{2}Si | 833 K to 298 K |

p-Bi_{2}Te_{3} | 553 K to 301 K |

SnSe | 973 K to 373 K |

PbTe | 850 K to 320 K |

#### Appendix A.3. Additional Information

#### Appendix A.3.1. Finding Individual Contributions to the Total $T$ Profile

_{2}X, Figure 2b.

#### Appendix A.3.2. Contributions to ${\dot{\mathrm{Q}}}_{\mathrm{in}}$

_{2}X and Bi

**Te**

_{2}_{3}are given in Figure A2c,d, respectively. Due to the bowed shape of the $\alpha \left(T\right)$ graph and relatively close values of ${\alpha}_{\mathrm{h}}$ to ${\alpha}_{\mathrm{c}}$ for p-Mg

_{2}X, the difference between ${\dot{Q}}_{\pi ,\mathrm{h}}$ and ${\dot{Q}}_{\pi ,\mathrm{h}}^{\mathrm{CPM}}$ is almost negligible, but ${\dot{Q}}_{\tau ,\mathrm{h}}$ amounts to more than twice the amount of ${\dot{Q}}_{\pi ,\mathrm{h}}-{\dot{Q}}_{\pi ,\mathrm{h}}^{\mathrm{CPM}}$. Nevertheless, this did not affect the efficiency deviation ${\mathsf{\delta}\mathsf{\eta}}_{\mathrm{max}}$ too much, as ${\dot{Q}}_{\tau ,\mathrm{h}}$ is quite small in absolute terms. In the case of Bi

_{2}Te

_{3}, ${\dot{Q}}_{\pi ,\mathrm{h}}^{\mathrm{CPM}}$ is even higher than ${\dot{Q}}_{\pi ,\mathrm{h}}$ again due to the curved shape of $\alpha \left(T\right)$, affecting the position of ${\alpha}_{\mathrm{TAv}}$. However, ${\dot{Q}}_{\tau ,\mathrm{h}}$ almost completely compensates for this Peltier heat difference, keeping the influence on the efficiency deviation negligible.

**Figure A2.**(

**a**) Ratio of individual heat contributions to ${\dot{Q}}_{\mathrm{in}}$(Equation (A1)) calculated from the corresponding partial temperature profiles (for comparison, all quantities are counted as positive when flowing into the element) (left y-axis), and distribution factors (right y-axis) for Thomson and Joule heat. (

**b**) Thomson $T$ profiles for all example materials (

**c**) $\dot{S}\left(T\right)$ diagram for p-Mg2X showing the area between $I\alpha \left({T}_{\mathrm{h}}\right)$ and $I{\alpha}_{\mathrm{TAv}}$ (corresponding to the Peltier heat difference between the CPM and real case), which is very small due to the shape of $\alpha \left(T\right)$. The position of the first peak in the Thomson partial $T$ profile is marked as a brown vertical line. (

**d**) $\dot{S}\left(T\right)$ diagram for Bi

_{2}Te

_{3},where ${\alpha}_{\mathrm{TAv}}>\alpha \left({T}_{\mathrm{h}}\right)$. Hence, ${\dot{Q}}_{\pi ,\mathrm{h}}^{\mathrm{CPM}}$ is higher than ${\dot{Q}}_{\pi ,\mathrm{h}}$.

#### Appendix A.4. Thomson Heat Distribution and Entropy

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**Figure 1.**(

**a**) Comparison of the relative deviation of the calculated maximum efficiency of 2TD (two temperature-dependent property) model materials (one of the thermoelectric properties kept constant,) to their real counterpart for the example materials, (

**b**) $T$ profile bending caused by Joule heat for example materials. Distinct asymmetry is observed particularly for PbTe and SnSe, correlated to maximum offset values in the middle part of Figure (

**a**).

**Figure 2.**(

**a**) Bending of $T$ profiles for the real materials at $j=0$ (dotted lines) and $j={j}_{\mathrm{opt}}$ (solid lines), normalized to $\Delta T$, (

**b**) $T$ profile bending for the 1TD and 2TD model materials in comparison to the full $T$ -dependent case and the constant properties case, along with the individual contributions to the fully $T$-dependent profile for an n-Mg

_{2}(Si,Sn) TE leg with ${T}_{\mathrm{h}}$ = 723 K and ${T}_{\mathrm{c}}$ = 383 K.

**Figure 3.**Schematic representation of reversible heat exchange in a TE leg for a linear $\alpha \left(T\right)$ curve (black line) in a plot of a convective 1D entropy flow with a constant current $I$. According to the relation ${\dot{Q}}_{\pi}=I\alpha T$, areas in the $\dot{S}\left(T\right)$ diagram represent certain amounts of (flowing or exchanged) Peltier (including Thomson) heat. The dark blue and light blue rectangles—in- and outflowing Peltier heat; trapezium above the $\dot{S}\left(T\right)$ curve—Thomson heat (marked with slant lines); trapezium below the $\dot{S}\left(T\right)$ curve (marked in checked lines) —gross electrical power generated (${V}_{o}I)$; red trapezium—Thomson heat flowing to the hot side; orange rectangle—hot side Peltier heat (CPM). The green triangle indicates part of the difference in the amount of absorbed Peltier heat at the hot side between the actual and the CPM cases that is not compensated in the real material by backflowing Thomson heat ${\dot{Q}}_{\tau ,\mathrm{h}}$.

**Figure 4.**Calculated relative deviation (RD) of (

**a**) the maximum efficiency, $\delta {\eta}_{\mathrm{max}}$, heat input, $\delta {\dot{Q}}_{\mathrm{in}}$, power at maximum efficiency, $\delta {P}_{{\eta}_{\mathrm{max}}}$, and optimum current, $\delta {I}_{\mathrm{opt},\eta}$; additionally, $\delta {\dot{Q}}_{\mathrm{in}}$ when neglecting $\delta {I}_{\mathrm{opt},\eta}$ (black stars), (

**b**) Joule heat, $\delta {\dot{\mathrm{Q}}}_{\mathrm{J}}^{\mathrm{h}}$, reversible heat, $\delta {\dot{\mathrm{Q}}}_{\mathsf{\pi}\mathsf{\tau}}^{\mathrm{h}}$, (see Equation (12)) and, for direct comparison, also $d{\dot{\mathrm{Q}}}_{\mathrm{J}}^{\mathrm{h}}/{\dot{\mathrm{Q}}}_{\mathsf{\pi}\mathsf{\tau}}^{\mathrm{h}}$.

**Figure 5.**Plot of the convective 1D entropy flow at constant current $I$ for (

**a**) PbTe and (

**b**) SnSe. Relevant areas are marked to determine the uncompensated Peltier–Thomson heat $d{\dot{Q}}_{\mathsf{\pi}\mathsf{\tau}}^{\mathrm{h}}$ (green area). Note that the $\frac{L}{2}$ temperature and the temperature ${T}_{\mathsf{\tau},\mathrm{ex}}$ according to the extremum of $\Delta {T}_{\mathrm{Thomson}}\left(x\right)$ may be located quite far apart (

**b**) whereas ${T}_{\mathsf{\tau},\mathrm{ex}}$ is very close to the crossing point of $\alpha \left(T\right)$ to $\overline{\alpha}$.

**Figure 6.**RD in maximum efficiency, $\delta {\eta}_{\mathrm{max}}^{\mathrm{corr}}$, corrected with respect to ${\mathrm{d}\dot{\mathrm{Q}}}_{\mathsf{\pi}\mathsf{\tau}}^{\mathrm{h},I=\mathrm{const}}$ (${T}_{\mathsf{\tau},\mathrm{ex}}$ according to the peak of the exact Thomson profile; blue), ${\mathrm{d}\dot{\mathrm{Q}}}_{\mathsf{\pi}\mathsf{\tau}}^{\mathrm{h}}$ (exact numerical calculation; red; compare also Equation (12)) and a first guess by the $\frac{L}{2}$ position.

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**MDPI and ACS Style**

Ponnusamy, P.; de Boor, J.; Müller, E.
Discrepancy between Constant Properties Model and Temperature-Dependent Material Properties for Performance Estimation of Thermoelectric Generators. *Entropy* **2020**, *22*, 1128.
https://doi.org/10.3390/e22101128

**AMA Style**

Ponnusamy P, de Boor J, Müller E.
Discrepancy between Constant Properties Model and Temperature-Dependent Material Properties for Performance Estimation of Thermoelectric Generators. *Entropy*. 2020; 22(10):1128.
https://doi.org/10.3390/e22101128

**Chicago/Turabian Style**

Ponnusamy, Prasanna, Johannes de Boor, and Eckhard Müller.
2020. "Discrepancy between Constant Properties Model and Temperature-Dependent Material Properties for Performance Estimation of Thermoelectric Generators" *Entropy* 22, no. 10: 1128.
https://doi.org/10.3390/e22101128