*Performance*


#### **Appendix A. Voltage–Electrical Current and Electrical Power–Electrical Current Characteristics:** *p***and** *n***-Type Materials**

The voltage–electrical current characteristics (green curve) and the electrical power–electrical current characteristics (red curve) of a thermoelectric material with either *p*-type or *n*-type conduction are given in Figure A1.

**Figure A1.** Voltage Δ*ϕ* – electrical current *Iq* characteristics (green curves) and electrical power *P*el – electrical current characteristics *Iq* (red curves) for materials with: (**a**) Seebeck coefficient *α* being positive, which refers to *p*-type conduction and (**b**) Seebeck coefficient *α* being negative, which refers to *n*-type conduction. Here, Δ*T* = *T*hot−*T*cold *T*hot is the temperature difference along a thermoelectric material of length *L* and cross-sectional area *A*. These quantities, together with the (isothermal) electrical conductivity *σ* and the Seebeck coefficient, determine the electrical current *I*SC under electrical short-circuited conditions. The voltage Δ*ϕ*OC under electrical short-circuited conditions is determined by the Seebeck coefficient and the temperature difference. When the electrical power *P*el is negative (electrical power output), the material is in generator mode (thermal-to-electrical power conversion). When the electrical power *P*el is positive (electrical power input), the material is in entropy pump mode (electrical-to-thermal power conversion).

#### **Appendix B. Thermal-to-Electrical Power Conversion: Calculations and Established Models**

*Appendix B.1. Maximum Electrical Power Point (MEPP): Material in Generator Mode*

The MEPP is found by looking for the vanishing first derivative of the electrical power.

$$\begin{aligned} \mathbf{0} &= \frac{\partial P\_{\text{el}}}{\partial \mathbf{i}} &= -\frac{A}{L} \cdot \boldsymbol{\sigma} \cdot \mathbf{a}^2 \cdot (\Delta T)^2 \cdot \frac{\partial}{\partial \boldsymbol{i}} \left( \boldsymbol{i} - \boldsymbol{i}^2 \right) \\\\ &= -\frac{A}{L} \cdot \boldsymbol{\sigma} \cdot \mathbf{a}^2 \cdot (\Delta T)^2 \cdot (\mathbf{1} - \mathbf{2} \cdot \boldsymbol{i}) \end{aligned} \tag{A1}$$

The derivative vanishes if the term in the brackets vanishes, and the normalized current at the MEPP is as follows.

$$i\_{\rm MEPP} = \frac{1}{2} \tag{A2}$$

At the MEPP, the maximum electrical power is obtained as follows.

$$\begin{array}{rcl} P\_{\text{el,max}} & = P\_{\text{el}} \left( i\_{\text{MEPP}} \right) & = -\frac{A}{L} \cdot \sigma \cdot a^2 \cdot \left( \Delta T \right)^2 \cdot \left( i\_{\text{MEPP}} - i\_{\text{MEPP}}^2 \right) \\\\ & = -\frac{A}{L} \cdot \sigma \cdot a^2 \cdot \left( \Delta T \right)^2 \cdot \left( \frac{1}{2} - \left( \frac{1}{2} \right)^2 \right) \\\\ & = -\frac{A}{L} \cdot \sigma \cdot a^2 \cdot \left( \Delta T \right)^2 \cdot \left( \frac{1}{2} - \frac{1}{4} \right) \\\\ & = -\frac{1}{4} \cdot \frac{A}{L} \cdot \sigma \cdot a^2 \cdot \left( \Delta T \right)^2 \end{array} \tag{A3}$$

The 2nd-law power conversion efficiency at the MEPP is then obtained as follows.

$$\begin{array}{rcl} \eta\_{\text{II,gen,MEPP}} &=& \eta\_{\text{II,gen}} \left( i\_{\text{MEPP}} \right) & =& \frac{i\_{\text{MEPP}} - i\_{\text{MEPP}}}{i\_{\text{MEPP}} + \frac{1}{\pi T}} \\\\ &=& \frac{\frac{1}{2} - \left(\frac{1}{2}\right)^2}{\frac{1}{2} + \frac{1}{\pi T}} \\\\ &=& \frac{\frac{1}{2} - \frac{1}{\pi}}{\frac{1}{2} + \frac{1}{\pi T}} \\\\ &=& \frac{1}{4} \cdot \frac{1}{\frac{1}{2} + \frac{1}{\pi T}} \\\\ &=& \frac{zT}{4} \cdot \frac{1}{\frac{\pi T}{2} + 1} \\\\ &=& \frac{1}{2} \cdot \frac{zT}{\pi T + 2} \end{array} \tag{A4}$$

*Appendix B.2. Maximum Conversion Efficiency Point (MCEP): Material in Generator Mode*

The 2nd-law power conversion efficiency for a thermoelectric material operated in generator mode is obtained as follows.

$$|\eta\_{\rm II,gcn} = |\ \frac{P\_{\rm el}}{P\_{\rm th}}| = \frac{\boldsymbol{\alpha} \cdot \boldsymbol{I\_{q}} \cdot \boldsymbol{\Delta T} - \frac{\boldsymbol{I\_{q}^{2}}}{\frac{\boldsymbol{I}\_{\rm q}}{\boldsymbol{L}} \cdot \boldsymbol{\sigma}}}{\boldsymbol{\alpha} \cdot \boldsymbol{I\_{q}} \cdot \boldsymbol{\Delta T} + \frac{\boldsymbol{A}}{\boldsymbol{L}} \cdot \boldsymbol{\Lambda\_{\rm OC}} \cdot \left(\boldsymbol{\Delta T}\right)^{2}} = \frac{\frac{\boldsymbol{I\_{q}}}{\boldsymbol{I\_{q}\rm S\boldsymbol{C}}} - \left(\frac{\boldsymbol{I\_{q}}}{\boldsymbol{I\_{q}\rm S\boldsymbol{C}}}\right)^{2}}{\frac{\boldsymbol{I\_{q}}}{\boldsymbol{I\_{q}\rm S\boldsymbol{C}}} + \frac{\boldsymbol{\Lambda\_{\rm OC}}}{\boldsymbol{\sigma} \cdot \boldsymbol{a^{2}}}}\tag{A5}$$

Substituting in Equation (A5) the dimensionless normalized electrical current *i* = |*<sup>I</sup>*q| |*<sup>I</sup>*q,SC| and the figure-of-merit *zT* = *<sup>σ</sup>*·*α*<sup>2</sup> ΛOC , the 2n<sup>d</sup> law power conversion efficiency can be written as follows.

$$
\eta\_{\rm II,gen} = \frac{i - i^2}{i + \frac{1}{zT}} \tag{A6}
$$

The maximum power conversion efficiency point (MCEP) can then be found by the first derivative to vanish. *∂η*II,gen

$$\begin{array}{l} 0 & = \quad ^{\frac{-1 + i \log n}{\log n}} \\\\ & = \frac{\partial}{\partial t} \left( \frac{i - i^2}{i + \frac{1}{\pi T}} \right) \\\\ & = \frac{(1 - 2 \cdot i) \cdot \left( i + \frac{1}{\pi T} \right) - \left( i - i^2 \right) \cdot 1}{\left( i + \frac{1}{\pi T} \right)^2} \\\\ & = \frac{i + \frac{1}{\pi T} - 2 \cdot i^2 - 2 \cdot \frac{1}{\pi T} \cdot i - i + i^2}{\left( i + \frac{1}{\pi T} \right)^2} \\\\ & = \frac{-i^2 - 2 \cdot \frac{1}{\pi} \cdot i + \frac{1}{\pi T}}{\left( i + \frac{1}{\pi T} \right)^2} \end{array} \tag{A7}$$

The derivative vanishes when the numerator vanishes.

$$i^2 + 2 \cdot \frac{1}{zT} \cdot i - \frac{1}{zT} = 0\tag{A8}$$

This quadratic equation has two solutions, from which only one gives a positive-definite normalized current *i* at the maximum conversion efficiency point (MCEP).

$$\begin{array}{l} \mathsf{MCEP}\_{\mathsf{gen}} &= \frac{\sqrt{1+zT}-1}{zT} \\\\ &= \frac{\left(\sqrt{1+zT}-1\right)\cdot\left(\sqrt{1+zT}+1\right)}{zT\cdot\left(\sqrt{1+zT}+1\right)} \\\\ &= \frac{1+zT+\sqrt{1+zT}-\sqrt{1+zT}-1}{zT\cdot\left(\sqrt{1+zT}+1\right)} \\\\ &= \frac{zT}{zT\cdot\left(\sqrt{1+zT}+1\right)} \\\\ &= \frac{1}{\sqrt{1+zT}+1} \end{array} \tag{A9}$$

At the MCEP, the maximum power conversion efficiency of the thermoelectric material in generator mode is then obtained as follows.

*η*II,gen,max = *η*II,gen *<sup>i</sup>*MCEP,gen = *<sup>i</sup>*MCEP,gen−*<sup>i</sup>*2MCEP,gen *<sup>i</sup>*MCEP,gen<sup>+</sup> 1*zT* = *<sup>i</sup>*MCEP,gen · <sup>1</sup>−*i*MCEP,gen *<sup>i</sup>*MCEP,gen<sup>+</sup> 1*zT* = *<sup>i</sup>*MCEP,gen · 1 *<sup>i</sup>*MCEP,gen −1 1+ 1 *<sup>i</sup>*MCEP,gen·*zT* = √ 1 1+*zT*+1 · 1 √ 1 1+*zT*+1 −1 1+ 1 √ 1 1+*zT*+1 ·*zT* = √ 1 1+*zT*+1 · √<sup>1</sup>+*zT*+1−<sup>1</sup> 1+ √<sup>1</sup>+*zT*+<sup>1</sup> *zT* = √ 1 1+*zT*+1 · *zT*·√<sup>1</sup>+*zT zT*+√<sup>1</sup>+*zT*+<sup>1</sup> = √ 1 1+*zT*+1 · *zT*·√<sup>1</sup>+*zT* <sup>1</sup>+*zT*+√<sup>1</sup>+*zT* = √ 1 1+*zT*+1 · √ *zT* 1+*zT*+1 = *zT* (√<sup>1</sup>+*zT*+<sup>1</sup>)<sup>2</sup> = 1+*zT*−1 (<sup>1</sup>+√<sup>1</sup>+*zT*)<sup>2</sup> = (√<sup>1</sup>+*zT*+<sup>1</sup>)·(√<sup>1</sup>+*zT*−<sup>1</sup>) (<sup>1</sup>+√<sup>1</sup>+*zT*)<sup>2</sup> = √√<sup>1</sup>+*zT*−<sup>1</sup> 1+*zT*+1 (A10)

By combining Equation (A9) and Equation (A10), the dependence of the maximum second-law efficiency on the electrical current can be shown to be linear.

$$
\begin{aligned}
\eta\_{\text{II,gen,max}} \left( \dot{\iota}\_{\text{MCEP,gen}} \right) &= \frac{\sqrt{1+zT}-1}{\sqrt{1+zT}+1} \\ &= \frac{\sqrt{1+zT}+1-2}{\sqrt{1+zT}+1} \\ &= \frac{\frac{1}{\text{MCE,gen}}-2}{\frac{1}{\text{MCE,gen}}}
\end{aligned}
\tag{A11}
$$

$$=1-2 \cdot i\_{\mathsf{MCEP}\_{\mathsf{f}}\mathsf{gen}}$$

The electrical power at the MCEP is as follows.

$$\begin{aligned} \text{P}\_{\text{elMCEP}} &= \text{P}\_{\text{el}} \left( \text{\textdegree\text{MCEP}} \text{gen} \right) &= -\frac{A}{L} \cdot \sigma \cdot \mathbf{a}^2 \cdot \left( \Delta T \right)^2 \cdot \left( \mathbf{\hat{M}}\_{\text{MCEP}} \text{gen} - \mathbf{\hat{M}}\_{\text{MCEP}} \text{gen} \right) \\ &= -\frac{A}{L} \cdot \sigma \cdot \mathbf{a}^2 \cdot \left( \Delta T \right)^2 \cdot \left( \frac{1}{\sqrt{1 + \tau T} + 1} - \frac{1}{\left( \sqrt{1 + \tau T} + 1 \right)^2} \right) \\ &= -\frac{A}{L} \cdot \sigma \cdot \mathbf{a}^2 \cdot \left( \Delta T \right)^2 \cdot \left( \frac{\sqrt{1 + \tau T} + 1}{\left( \sqrt{1 + \tau T} + 1 \right)^2} - \frac{1}{\left( \sqrt{1 + \tau T} + 1 \right)^2} \right) \\ &= -\frac{A}{L} \cdot \sigma \cdot \mathbf{a}^2 \cdot \left( \Delta T \right)^2 \cdot \frac{\sqrt{1 + \tau T}}{\left( \sqrt{1 + \tau T} + 1 \right)^2} \\ &= -\frac{1}{4} \cdot \frac{A}{L} \cdot \sigma \cdot \mathbf{a}^2 \cdot \left( \Delta T \right)^2 \cdot \frac{4 \cdot \sqrt{1 + \tau T}}{\left( \sqrt{1 + \tau T} + 1 \right)^2} \\ &= P\_{\text{el,max}} \cdot \frac{4 \cdot \sqrt{1 + \tau T}}{\left( \sqrt{1 + \tau T} + 1 \right)^2} \end{aligned} \tag{A12}$$

#### *Appendix B.3. Comparison to Power Conversion Efficiency after Fuchs: Thermogenerator Device*

By accepting temperature and entropy as primitive quantities, Fuchs [33] has created aggregate dynamical models of a Peltier device. Suggesting the Peltier device to function analogously to a battery, he has derived linear voltage-electrical current characteristics and identified the only two dissipative processes, which are the diffusion of electric charge and the diffusion of entropy. For the case of the device being operated as a thermogenerator, Fuchs [33] has derived its 2nd-law efficiency by the ratio of useful to available power and expressed the efficiency with respect to the internal resistance of the device *R*TEG and an external load resistance *R*ext.

$$\eta\_{\rm II,TEG} = \frac{R\_{\rm ext}}{R\_{\rm TEG} + (R\_{\rm TEG} + R\_{\rm ext}) \cdot \frac{1}{zT}} \cdot \frac{R\_{\rm TEG}}{R\_{\rm TEG} + R\_{\rm ext}} \tag{A13}$$

For a given figure-of-merit *zT*, the 2nd-law efficiency of the device has its maximum at.

$$R\_{\rm ext} = \sqrt{1 + zT} \cdot R\_{\rm TEG} \tag{A14}$$

Thus, the maximum 2nd-law power conversion efficiency is as follows.

$$\begin{array}{l} \text{III}\_{\text{TEC,max}} & = \frac{\sqrt{1+z^2}\text{Rig}}{\text{TEC}+\sqrt{1+z^2}\text{T}+\text{TEC}+\frac{\text{Rig}}{\text{T}}} \cdot \frac{\text{Rig}}{\text{Rig}+\sqrt{1+z^2}\text{T}+\text{TEC}} \\\\ & = \frac{\sqrt{1+z^2}\text{Rig}}{1+\left(1+\sqrt{1+z^2}\right)\frac{z}{\text{T}}} \cdot \frac{\text{T}}{1+\sqrt{1+z^2}} \\\\ & = \frac{\sqrt{1+z^2}}{zT+\left(1+\sqrt{1+z^2}\right)} \cdot \frac{zT}{1+\sqrt{1+z^2}} \\\\ & = \frac{\sqrt{1+z^2}}{1+z^2T+\sqrt{1+z^2}} \cdot \frac{zT}{1+\sqrt{1+z^2}} \\\\ & = \frac{zT}{\left(1+z\sqrt{1+z^2}\right)} \\\\ & = \frac{1+zT-1}{\left(1+\sqrt{1+z^2}\right)^2} \\\\ & = \frac{\left(\sqrt{1+z^2}\right)\cdot\left(\sqrt{1+z^2}-1\right)}{\left(1+\sqrt{1+z^2}\right)^2} \\\\ & = \frac{\sqrt{1+z^2}-1}{\left(1+\sqrt{1+z^2}\right)^2} \end{array} \tag{A15}$$

Of note, Fuchs has neglected the Joule "heat", which would only have a small impact when the device is operated in generator mode. Note that Equation (A15) is equivalent to what has been obtained in this work for a thermoelectric material apart from a device (cf. Equation (A10)).

#### *Appendix B.4. Comparison to Power Conversion Efficiency after Altenkirch: Thermogenerator Device*

Altenkirch [55] has estimated the power conversion efficiency for a thermogenerator (called thermopile at that time), which has been assumed to be made of two legs of dissimilar materials. For a small temperature difference along the device, which will cause only a small thermally-induced electrical current and allows neglect the Joule heating as well as the Thomson effect, he has derived his Equation (4) for the 1st-law power conversion efficiency. Altenkirch [55] has factorized the 1s<sup>t</sup> law power conversion efficiency into the Carnot efficiency and what we call here the 2nd-law power conversion efficiency *η*II. The latter has been of the following form.

$$\begin{array}{l} \eta \text{ II,TEG} &=& \frac{zT \cdot \frac{R\_{\text{TEG}}}{R\_{\text{TEG}}}}{zT \cdot \left(1 + \frac{R\_{\text{TEG}}}{R\_{\text{TEG}}}\right) + \left(1 + \frac{R\_{\text{out}}}{R\_{\text{TEG}}}\right)^2} \\\\ &=& \frac{zT \cdot R\_{\text{out}} \cdot R\_{\text{TEG}}}{zT \cdot \left(R\_{\text{TEG}}^2 + R\_{\text{TEG}} \cdot R\_{\text{ext}}\right) + \left(R\_{\text{TEG}} + R\_{\text{out}}\right)^2} \\\\ &=& \frac{R\_{\text{out}} \cdot R\_{\text{TEG}}}{R\_{\text{TEG}} \cdot \left(R\_{\text{TEG}} + R\_{\text{out}}\right) + \left(R\_{\text{TEG}} + R\_{\text{out}}\right)^2 \cdot \frac{1}{zT}} \\\\ &=& \frac{R\_{\text{out}}}{R\_{\text{TEG}} + \left(R\_{\text{TEG}} + R\_{\text{out}}\right) \cdot \frac{1}{zT}} \cdot \frac{R\_{\text{TEG}}}{R\_{\text{TEG}} + R\_{\text{out}}} \end{array} \tag{A16}$$

Here, Altenkirchs's nomenclature has been substituted by *R*ext *R*TEG = *x* and *zT* = 10<sup>7</sup> · *η*. In his treatment, the factor 10<sup>7</sup> appeared due to the use of the calorie as the energy units, and "*η*" was called the effective thermopower of the device, which however contained the Seebeck coefficient multiplied with the square root of the ratio of specific thermal and specific electrical conductivities of the thermoelectric materials involved. Equation (A16) is equivalent to the result observed by Fuchs (cf. Equation (A13)).

Subsequently, Altenkirch derived the efficiency to be maximized for the following.

$$\mathbf{x} = \frac{R\_{\text{ext}}}{R\_{\text{TEG}}} = \sqrt{1 + zT} \tag{A17}$$

Note that Equation (A17) is equivalent to the result obtained by Fuchs (cf. Equation (A14)).

For the thermoelectric generator (TEG), Altenkirch derived the maximum 2nd-law power conversion efficiency *η*II,TEG,max to be (see Altenkirch [55], Equation (5)) as follows.

$$
\eta\_{\text{II,TEG,max}} = \frac{\sqrt{1+zT}-1}{\sqrt{1+zT}+1} \tag{A18}
$$

Note that Equation (A18) is equivalent to the result obtained by Fuchs (cf. Equation (A15)).

Even though Altenkirch did not use the term figure-of-merit (compare Altenkirch [55], Figure 3), he plotted the maximum 2n<sup>d</sup> law power conversion efficiency *η*II,TEG,max as a function of *x* = *R*ext *R*TEG for different values of his "*η*", which despite a dimensionless factor has been identified with *zT*. In the plot, he indicated the shift of the MCEP with varied figure-of-merit.

Altenkirch extended his approach by considering the impact of the Thomson effect on the power conversion efficiency. Moreover, he added remarks on the rate of thermal power exchange of the device with a hot reservoir and cold reservoir and its impact on the effective temperature difference along the device.

#### *Appendix B.5. Comparison to Power Conversion Efficiency after Ioffe: Thermogenerator Device*

Ioffe [56] has considered a thermocouple in which legs of materials 1 and 2 of equal length are joined by a metallic bridge. The Seebeck coefficient of the device has been estimated from those of the two legs: *α* =| *α*1 | + | *α*2 |. From equal length and the individual values of the electrical resistivities (*ρ*1, *ρ*2), "heat" conductivities (*λ*OC,1, *λ*OC,2) and cross-sectional area, he has calculated the total electrical resistance *R*TEG and thermal conductance of the device *K*TEG (see Ioffe [56], p. 36). To calculate the efficiency of thermal-to-electrical power conversion of the device, he has neglected the Thomson "heat". Furthermore, he made an assumption regarding the Joule "heat" (see Ioffe [56], p. 38): "Of the total Joule 'heat' *Iq*2 · *R*TEG generated in the thermoelement, half passes to the hot junction, returning the power 12 · *Iq*2 · *R*TEG and the rest is transferred to the cold junction." As a result, the temperatures of the hot *T*hot and cold junction *T*cold appear in the maximum second-law efficiency.

$$\eta\_{\text{II,TEG,max}} = \frac{\sqrt{1+zT}-1}{\sqrt{1+zT} + \frac{T\_{\text{heat}}}{T\_{\text{cold}}}} \tag{A19}$$

The aforementioned argument, which was probably inspired by Altenkirch's [55] article, is based on misunderstanding the dissipation, which in the author's opinion is thermal energy to leave the system together with produced entropy. The entropy, and thus the thermal energy, will not have driving force to flow to higher temperature. Anyway, following the argument, the thermal input power is diminished by half of the dissipated Joule "heat". In this work, it has been outlined that the effect of Joule "heat" would be a diminished thermal power supply due to a changed temperature profile (cf. Section 4.2 in the the main text).

Neglecting the Joule "heat", Ioffe has derived the following equation (see Ioffe [56], p. 40).

$$\eta\_{\text{II,TEG,max}} = \frac{\sqrt{1+z^{\overline{T}}}-1}{\sqrt{1+z^{\overline{T}}}+1} \tag{A20}$$

Note that this is equivalent to what has been obtained in this work for a thermoelectric material apart from a device.

In the factor *z*, which Ioffe deduced (see Ioffe [56], p. 39), the cross-sectional areas *A*1 and *A*2 and length *L* cancel out, so it depends only on the thermoelectric properties of both materials but not their dimensions.

$$z = \frac{a^2}{K\_{\rm TEG} \cdot R\_{\rm TEG}} = \frac{a^2}{\left(\sqrt{\lambda\_{\rm OC,1} \cdot \rho\_1} + \sqrt{\lambda\_{\rm OC,2} \cdot \rho\_2}\right)^2} = \frac{a^2}{\left(\sqrt{\frac{\lambda\_{\rm OC,1}}{\sigma\_1}} + \sqrt{\frac{\lambda\_{\rm OC,2}}{\sigma\_2}}\right)^2} \tag{A21}$$

In the case that the electrical conductivities (*σ* = *σn* = *<sup>σ</sup>p*) and "heat" conductivities (*λ*OC = *λ*OC,1 = *λ*OC,2) are equal in both legs of the device, respectively, Equation (A21) becomes the following.

$$z = \frac{\sigma \cdot a^2}{\lambda\_{\text{OC}}} \tag{A22}$$

Ioffe used Equation (A22) when discussing a thermoelectric cooler (see Ioffe [56], p. 100) but derived an equivalent expression – using the thermal conductance instead of the thermal conductivity – when discussing the thermogenerator (see Ioffe [56], p. 38ff.). Anyway, in Equations (A19) and (A20) for the maximum power conversion efficiency, there appears not the factor *z* but this factor multiplied with the average temperature *T*.

$$z\mathcal{T} = z \cdot \mathcal{T} = z \cdot \frac{T\_{\text{hot}} + T\_{\text{cold}}}{2} \tag{A23}$$

Because of Ioffe's Equations (A21)–(A23), the figure-of-merit of a thermoelectric material is currently termed *zT* or *zT*.

#### **Appendix C. Electrical-to-Thermal Power Conversion: Calculations and Established Models**

#### *Appendix C.1. Power Conversion Efficiency*

When the thermoelectric material is used in a cooler, the coefficient of performance *COP* is the ratio of the thermal power removed from the cold side *T*cold · *IS* related to the electrical power *P*el.

$$\begin{array}{rcl} \text{COP}\_{\text{cooker}} & = \left| \begin{array}{c} \frac{T\_{\text{coalg}} \cdot L\_{\text{Si}}}{P\_{\text{td}}} \; | \; | \; | \; | \; | \; | \; \frac{P\_{\text{td}}}{P\_{\text{td}}} \; | \; | \; | \; | \; \frac{P\_{\text{td}}}{P\_{\text{td}}} \; | \; | \; | \; | \; | \; | \; | \; | \end{array} \right. \\\\ & &= \frac{T\_{\text{coalg}}}{\Lambda T} \cdot \frac{\frac{\dot{\Delta}}{\dot{\Delta}} \cdot \left( \Lambda\_{\text{OC}} + r n^{2} \cdot i \right) \cdot (\Lambda T)^{2}}{\frac{1}{\dot{\Delta}} \cdot r \cdot a^{2} \cdot (\Lambda T)^{2} \cdot (i - i^{2})} \\\\ &= \frac{T\_{\text{coalg}}}{\Lambda T} \cdot \frac{\frac{\Lambda\_{\text{OC}}}{\dot{\Delta}} + i}{\frac{r}{\dot{\Delta}} - i^{2}} \\\\ &= \frac{T\_{\text{coalg}}}{\Lambda T} \cdot \frac{i + \frac{1}{r^{2}}}{-r^{2} + i} \\\\ &= \frac{T\_{\text{coalg}}}{\Lambda T} \cdot \eta \text{u}\_{\text{c}} \end{array} \tag{A24}$$

When the thermoelectric material is used in a heater, the coefficient of performance *COP* is the ratio of the thermal power released to the hot side *T*hot · *IS* related to the electrical power *P*el.

$$\begin{array}{rcl} \text{COP}\_{\text{hoater}} & = \begin{vmatrix} \frac{T\_{\text{hoet}} \cdot I\_{\text{S}}}{P\_{\text{el}}} \end{vmatrix} & = \begin{vmatrix} \frac{T\_{\text{hoet}} \cdot I\_{\text{S}}}{P\_{\text{lb}}} \end{vmatrix} \cdot \begin{vmatrix} \frac{P\_{\text{lb}}}{P\_{\text{el}}} \end{vmatrix} & = \begin{array}{rcl} \frac{T\_{\text{hoet}}}{\Delta T} \cdot \left| \begin{array}{c} \frac{P\_{\text{lb}}}{P\_{\text{el}}} \end{array} \right| \end{array} \\\\ & = \begin{array}{rcl} \frac{T\_{\text{hoet}}}{\Delta T} \cdot \frac{i + \frac{1}{\Delta T}}{-i^{2} + i} & \\\\ & = \frac{T\_{\text{hoet}}}{\Delta T} \cdot \eta \text{II}\_{\text{eep}} \\\\ & = \frac{\eta \text{II}\_{\text{eep}}}{\eta \text{C}} \end{array} \end{array} \tag{A25}$$

The second-law efficiency for the thermoelectric material in entropy pump mode *η*II,ep is as follows.

$$\begin{array}{lcl}\eta\_{\text{II},\text{eq}} & = \begin{array}{c} \frac{P\_{\text{th}}}{P\_{\text{el}}} \; | \; \; \end{array} & = \begin{array}{c} \frac{\Phi \cdot \left(\Lambda\_{\text{CC}} + \tau a^2 \cdot i\right) \cdot \left(\Lambda T\right)^2}{\frac{\Phi}{i} \cdot \sigma \cdot a^2 \cdot \left(\Lambda T\right)^2 \cdot \left(i - i^2\right)}\\ & = \begin{array}{c} \frac{\Lambda\_{\text{CC}}}{i - i^2} + i\\ \frac{\sigma a^2}{i - i^2}\\ = \frac{i + \frac{1}{\tau T}}{-i^2 + i} \end{array} \end{array} \tag{A26}$$

The electrical power *P*el used in Equations (A24)–(A26) is available by the fall of electric charge along the electrical potential difference Δ*ϕ*. It drives the pumping of entropy from the material's cold side to its hot side. The thermal power *P*th = Δ*T* · *IS* = *T*hot · *IS* − *T*cold · *IS* is needed for lifting entropy along the temperature difference Δ*T*. Some illustration is given in Figure A2.

**Figure A2.** When the thermoelectric material is operated in entropy pump mode, electrical power *P*el, which is available by the fall of electric charge along Δ*ϕ*, drives the pumping of entropy from the cold side to hot side. The thermal power *P*th = Δ*T* · *IS* = *T*hot · *IS* − *T*cold · *IS* for lifting entropy along the temperature difference Δ*T* adds to the thermal power removed from the cold side *T*cold · *IS* to give the thermal power released to the hot side *T*hot · *IS*. Different width of arrows refers to different magnitudes of thermal power at the opposite sides of the material, which is due to thermoelectric power conversion.

#### *Appendix C.2. Maximum Conversion Efficiency Point (MCEP): Material in Entropy Pump Mode*

The maximum power conversion efficiency point (MCEP) follows when the first derivative of the 2nd-law power conversion efficiency, as given by Equation (A26), vanishes.

$$\begin{aligned} 0 &= \frac{\partial \eta\_{\text{eff}}}{\partial i} \\ &= \frac{\partial}{\partial i} \left( \frac{i + \frac{1}{\tau I}}{-i^2 + i} \right) \\\\ &= \frac{\mathbf{i} \cdot \left( -i^2 + i \right) - \left( i + \frac{1}{\tau I} \right) \cdot (-2 \cdot i + 1)}{\left( -i^2 + i \right)^2} \\\\ &= \frac{-i^2 + i + 2 \cdot i^2 + \frac{2}{\tau} \cdot i - i - \frac{1}{\tau I}}{\left( -i^2 + i \right)^2} \\\\ &= \frac{i^2 + \frac{2}{\tau} \cdot i - \frac{1}{\tau I}}{\left( -i^2 + i \right)^2} \end{aligned} \tag{A27}$$

The derivative vanishes when the numerator vanishes.

$$i\dot{I}^2 + \frac{2}{zT} \cdot i - \frac{1}{zT} = 0\tag{A28}$$

The quadratic Equation (A28) has two solutions.

$$\begin{array}{rcl} \dot{m}\_{1,2} &= -\frac{1}{\overline{z}T} \pm \sqrt{\left(\frac{1}{\overline{z}T}\right)^2 + \frac{1}{\overline{z}T}} \\\\ &= -\frac{1}{\overline{z}T} \pm \frac{1}{\overline{z}T} \cdot \sqrt{1 + \overline{z}T} \end{array} \tag{A29}$$

From the two solutions shown in Equation (A29) only one fulfils the requirement *i* ≤ − 1*zT* for the material's maximum conversion efficiency point (MCEP) in entropy pump mode. Thus, the normalized electrical current at the maximum conversion efficiency point (MCEP) is obtained as follows.

 $\rho\_{\text{MCE,cp}} = -\frac{1}{zT} - \frac{1}{zT} \cdot \sqrt{1 + zT}$ 

$$= -\frac{1 + \sqrt{1 + zT}}{zT}$$

$$= -\frac{\sqrt{1 + zT} + 1}{zT} \qquad \text{!} \tag{A30}$$

$$= -\frac{\sqrt{1 + zT} + 1}{1 + zT - 1}$$

$$= -\frac{\sqrt{1 + zT} + 1}{\left(\sqrt{1 + zT} + 1\right)\left(\sqrt{1 + zT} - 1\right)}$$

$$= -\frac{1}{\sqrt{1 + zT} - 1}$$

The maximum 2nd-law power conversion efficiency for a thermoelectric material operated in entropy pump mode is then as follows.

*η*II,ep,max = *η*II,ep,max *<sup>i</sup>*MCEP,ep = *<sup>i</sup>*MCEP,ep<sup>+</sup> 1*zT* −*<sup>i</sup>*2MCEP,ep+*i*MCEP,ep = − √<sup>1</sup>+*zT*+<sup>1</sup> *zT* + 1*zT* − √<sup>1</sup>+*zT*+<sup>1</sup> *zT* 2− √<sup>1</sup>+*zT*+<sup>1</sup> *zT* = *zT zT* · −√<sup>1</sup>+*zT*−1+<sup>1</sup> − 1 *zT* ·(√<sup>1</sup>+*zT*+<sup>1</sup>)<sup>2</sup>−(√<sup>1</sup>+*zT*+<sup>1</sup>) = 1 −(√<sup>1</sup>+*zT*+<sup>1</sup>) · −√<sup>1</sup>+*zT* 1 *zT* ·(√<sup>1</sup>+*zT*+<sup>1</sup>)+<sup>1</sup> = √ *zT* 1+*zT*+1 · √<sup>1</sup>+*zT* (√<sup>1</sup>+*zT*+<sup>1</sup>)+*zT* = √ *zT* 1+*zT*+1 · √<sup>1</sup>+*zT* <sup>1</sup>+*zT*+√<sup>1</sup>+*zT* = √ *zT* 1+*zT*+1 · √ 1 1+*zT*+1 = √ 1+*zT*−1 1+*zT*+1 · √ 1 1+*zT*+1 = (√<sup>1</sup>+*zT*+<sup>1</sup>)·(√<sup>1</sup>+*zT*−<sup>1</sup>) √<sup>1</sup>+*zT*+<sup>1</sup> · √ 1 1+*zT*+1 = √√<sup>1</sup>+*zT*−<sup>1</sup> 1+*zT*+1 (A31)

By combining Equations (A30) and (A31), the dependence of the maximum second-law efficiency on the electrical current can be shown to be hyperbolic.

$$\begin{array}{l} \eta\_{\text{Il.eq,max}} \left( i\_{\text{MCE,cp}} \right) &= \frac{\sqrt{1+z}T - 1}{\sqrt{1+z}T + 1} \\\\ &= \frac{\sqrt{1+z}T - 1}{\sqrt{1+z}T - 1 + 2} \\\\ &= \frac{\frac{-1}{\text{MCE,cp}}}{\frac{-1}{\text{MCE,cp}} + 2} \\\\ &= \frac{1}{1 - 2 \cdot \text{MCE,cp}} \end{array} \tag{A32}$$

The normalized thermal power (cf. Appendix C.3) at the MCEP is obtained by combining Equations (A35) and (A30).

$$\begin{aligned} p\_{\text{th,MCP}} &= p\_{\text{th}} \left( \dot{y}\_{\text{MCP,dp}} \right) &= 4 \cdot |\frac{1}{zT} - \frac{\sqrt{1 + zT} - 1}{\sqrt{1 + zT} - 1}| \\ &= \frac{4}{zT} \cdot |1 - \frac{zT}{\sqrt{1 + zT} - 1}| \\ &= \frac{4}{zT} \cdot |1 - \frac{1 + zT - 1}{\sqrt{1 + zT} - 1}| \\ &= \frac{4}{zT} \cdot |1 - \frac{\left(\sqrt{1 + zT} + 1\right) \cdot \left(\sqrt{1 + zT} - 1\right)}{\sqrt{1 + zT} - 1}| \\ &= \frac{4}{zT} \cdot |1 - \left(\sqrt{1 + zT} + 1\right)| \\ &= \frac{4}{zT} \cdot |1 - \sqrt{1 + zT} - 1| \\ &= \frac{4}{zT} \cdot |-\sqrt{1 + zT}| \\ &= 4 \cdot |-\frac{\sqrt{1 + zT}}{zT}| \\ &= 4 \cdot \frac{\sqrt{1 + zT}}{zT} \end{aligned} \tag{A3.31}$$

The absolute thermal power at the MCEP in entropy pump mode, which is related to the MEPP in generator mode, is thus the following:

$$\begin{aligned} \mid P\_{\text{th,MCEP}} \mid \quad &= p\_{\text{th,MCEP}} \cdot P\_{\text{el,max}}\\ &= 4 \cdot \frac{\sqrt{1+zT}}{zT} \cdot P\_{\text{el,max}} \end{aligned} \tag{A34}$$

*Appendix C.3. Normalized Thermal Power*

> The normalized thermal power *p*th is obtained as follows.

$$\begin{array}{rcl} p\_{\text{th}} &=& \frac{|P\_{\text{th}}|}{P\_{\text{el,max}}} \\\\ &=& \frac{|\frac{d}{d} \cdot \left(\Lambda\_{\text{OC}} + \sigma a^2 \cdot i\right) \cdot \left(\Lambda T\right)^2|}{\frac{1}{4} \cdot \frac{d}{L} \cdot \sigma \cdot a^2 \cdot \left(\Lambda T\right)^2} \\\\ &=& 4 \cdot \left|\begin{array}{c} \Lambda \underline{\text{OC}} \\ \sigma a^2 \end{array} + i \right| \\\\ &=& 4 \cdot \left|\begin{array}{c} \frac{1}{\tau T} \ + i \end{array} \right| \end{array} \tag{A35}$$

*Appendix C.4. Comparison to Power Conversion Efficiency after Altenkirch: Thermoelectric Cooler Device*

For a thermoelectric cooler made of two legs of dissimilar thermoelectric materials (called a thermopile) in steady-state condition, Altenkirch [57] has derived an expression for the minimum electrical power input related to a given cooling power (see Altenkirch [57], Equation (12)), which factorizes into a Carnot-type factor *T*hot−*T*cold *T*cold and the reciprocal of what he called the dissipation factor for the electro-thermal device. It must be emphasized that the Carnot-type factor introduced by Altenkirch is different from Carnot's efficiency because it relates the temperature difference *T*hot − *T*cold

to the temperature of the cold side *T*cold instead of the hot side *T*hot. This is due to the thermal energy current removed from the cold side being related to the electrical power input.

When Altenkirch's nomenclature is substituted by *zT* = 10<sup>7</sup> · *η*, his dissipation factor (see Altenkirch [57], Equation (13)) for the thermoelectric cooler (TEC), which is the device-related analogue of what we here call the maximum 2nd-law power conversion efficiency for a thermoelectric material operated in entropy pump mode *η*II,ep,max, becomes as follows.

$$\eta\_{\text{II,TEC,max}} = \frac{\sqrt{1+zT} - \frac{\tau\_{\text{flat}}}{T\_{\text{cold}}}}{\sqrt{1+zT}+1} \tag{A36}$$

Altenkirch [57] states that, for small temperature differences (i.e., *T*hot *T*cold ≈ 1), the maximum 2nd-law power conversion efficiency for thermoelectric cooler *η*II,TEC,max becomes the following.

$$
\eta\_{\text{II,TEC,max}} = \frac{\sqrt{1+zT}-1}{\sqrt{1+zT}+1} \tag{A37}
$$

Altenkirch's result of Equation (A37) for a device is identical to the maximum 2nd-law power conversion efficiency for a thermoelectric material operated in entropy pump mode *η*II,ep,max as obtained in this work (see Equation (A31)).

#### *Appendix C.5. Comparison to Power Conversion Efficiency after Ioffe: Thermoelectric Cooler Device*

For a thermoelectric cooler made of two legs of dissimilar thermoelectric materials, Ioffe [56] (see Ioffe [56], p. 99) has derived a maximum coefficient of performance *COP*, which he factorized into the inverse of a Carnot-type factor *T*cold *T*hot−*T*cold and what we here call the maximum 2nd-law efficiency *η*II,ep,max. After Ioffe [56], the device-related analogue of the latter has been as follows.

$$\eta\_{\text{II,TEC,max}} = \frac{\sqrt{1 + \frac{1}{2} \cdot z \cdot (T\_{\text{hot}} + T\_{\text{cold}})} - \frac{T\_{\text{hot}}}{T\_{\text{cold}}}}{\sqrt{1 + \frac{1}{2} \cdot z \cdot (T\_{\text{hot}} + T\_{\text{cold}}) + 1}} \tag{A.38}$$

In the case of small temperature difference (i.e., *T*cold *T*hot ≈ 1) and when identifying the average temperature *T* = 1 2 · (*<sup>T</sup>*hot + *<sup>T</sup>*cold), it becomes identical to the result of this work for a thermoelectric material (see Equation (A31)).

#### **References and Notes**



© 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

*Article*

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

#### **Prasanna Ponnusamy 1,\*, Johannes de Boor 1 and Eckhard Müller 1,2,\***


Received: 6 July 2020; Accepted: 30 September 2020; Published: 4 October 2020

**Abstract:** The e fficiency of a thermoelectric (TE) generator for the conversion of thermal energy into electrical energy can be easily but roughly estimated using a constant properties model (CPM) developed by Io ffe. However, material properties are, in general, temperature ( *T*)-dependent and the CPM yields meaningful estimates only if physically appropriate averages, i.e., spatial averages for thermal and electrical resistivities and the temperature average (TAv) for the Seebeck coe fficient (α), are used. Even though the use of αTAv compensates for the absence of Thomson heat in the CPM in the overall heat balance, we find that the CPM still overestimates performance (e.g., by up to 6% for PbTe) for many materials. The deviation originates from an asymmetric distribution of internally released Joule heat to either side of the TE leg and the distribution of internally released Thomson heat between the hot and cold side. The Thomson heat distribution di ffers from a complete compensation of the corresponding Peltier heat balance in the CPM. Both e ffects are estimated quantitatively here, showing that both may reach the same order of magnitude, but which one dominates varies from case to case, depending on the specific temperature characteristics of the thermoelectric properties. The role of the Thomson heat distribution is illustrated by a discussion of the transport entropy flow based on the α(*T*) plot. The changes in the lateral distribution of the internal heat lead to a di fference in the heat input, the optimum current and thus of the e fficiency of the CPM compared to the real case, while the estimate of generated power at maximum e fficiency remains less a ffected as it is bound to the deviation of the optimum current, which is mostly <1%. This deviation can be corrected to a large extent by estimating the lateral Thomson heat distribution and the asymmetry of the Joule heat distribution. A simple guiding rule for the former is found.

**Keywords:** TEG performance; device modeling; temperature profile; constant properties model; Fourier heat; Thomson heat; Joule heat
