*Fluxes*




#### **Appendix A. Input Data for FEM-Simulation**

**Table A1.** Thermoelectric parameters of module 1 (*p*-type Bi0.5Sb1.5Te3 [37], *n*-type Bi2Te3-xSbx [38]) used for FEM simulations. For all material parameters a linear behavior within the applied temperature range has been assumed and the average value has been used for the calculation of the *<sup>A</sup>*n/*A*p ratio.


**Table A2.** Thermoelectric parameters of module 2 (*p*-type FeNb0.88Hf0.12Sb [39], *n*-type Hf0.6Zr0.4NiSn0.995Sb0.005 [40]) used for FEM simulations. For all material parameters a linear behavior within the applied temperature range has been assumed and the average value has been used for the calculation of the *<sup>A</sup>*n/*A*p ratio.


**Table A3.** Thermoelectric parameters of module 3 (*p*-type Ca3Co4O9 [41], *n*-type In1.95Sn0.05O3 [41]) used for FEM simulations. For all material parameters a linear behavior within the applied temperature range has been assumed and the average value has been used for the calculation of the *<sup>A</sup>*n/*A*p ratio.


#### **Appendix B. An/Ap Optimization for Matching Short-Circuit Current**

Here, the optimized *<sup>A</sup>*n/*A*p ratio for matching *<sup>I</sup>*q,SC is derived. The idea of this optimization is as follows: the working points of the respective thermoelectric materials overlap, if the flux of charge in both materials is the same. Then, the working points of the materials overlap, so

$$I\_{\mathbf{q}, \text{MEPP}, \mathbf{n}} = I\_{\mathbf{q}, \text{MEPP}, \mathbf{p}} \tag{A1}$$

By including

$$I\_{\rm q,MEPP,n} = \frac{|a\_{\rm n}|(\Delta T)^2}{2R\_{\rm n}}\tag{A2}$$

and

$$I\_{\rm q,MEPP,p} = \frac{\kappa\_{\rm P} (\Delta T)^2}{2R\_{\rm P}} \tag{A3}$$

with the electrical resistance of the materials

$$R\_{\mathbf{n}} = \rho\_{\mathbf{n}} \cdot \frac{l\_{\mathbf{n}}}{A\_{\mathbf{n}}} \tag{A4}$$

and

$$R\_{\mathbb{P}} = \rho\_{\mathbb{P}} \cdot \frac{l\_{\mathbb{P}}}{A\_{\mathbb{P}}} \tag{A5}$$

the following relation is received:

$$\frac{|\alpha\_{\rm n}| \cdot (\Delta T)^2}{2\rho\_{\rm n}\frac{l\_{\rm D}}{A\_{\rm R}}} = \frac{\alpha\_{\rm P} \cdot (\Delta T)^2}{2\rho\_{\rm P}\frac{l\_{\rm P}}{A\_{\rm P}}}\tag{A6}$$

After rearrangemen<sup>t</sup> and with the assumed same length of the thermolegs *l*n = *l*p the result for a *<sup>A</sup>*n/*A*p ratio for matching *<sup>I</sup>*q,SC is:

$$\frac{A\_{\rm n}}{A\_{\rm P}} = \frac{a\_{\rm P}}{|a\_{\rm n}|} \cdot \frac{\rho\_{\rm n}}{\rho\_{\rm P}} \tag{A7}$$

#### **Appendix C. An/Ap Optimization for Maximum Power**

The maximum power output of a TE module is a function of the electrical current *I*q of the module at the MEPP *<sup>I</sup>*q,MEPP,TEG and the voltage *U* of the module at the MEPP *U*MEPP,TEG, which are calculated according to Equations (A8) and (A9):

$$I\_{\rm q,MEPP,TEG} = \frac{(\alpha\_{\rm P} - \alpha\_{\rm n}) \cdot \Delta T}{2R} \tag{A8}$$

$$\mathcal{U}\_{\text{MEPP,TEG}} = \frac{(\mathfrak{a}\_{\text{P}} - \mathfrak{a}\_{\text{n}}) \cdot \Delta T}{2} \tag{A9}$$

From this, the maximum electrical power output of a module at the MEPP can be derived as

$$P\_{\rm el,max,TEG} = \frac{(\mathfrak{a}\_{\rm P} - \mathfrak{a}\_{\rm n})^2 \cdot (\Delta T)^2}{4R\_{\rm TEG}}\tag{A10}$$

> with the internal electrical resistance of the module *R*TEG

$$R\_{\rm TEG} = \rho\_{\rm P} \frac{l\_{\rm P}}{A\_{\rm P}} + \rho\_{\rm n} \frac{l\_{\rm n}}{A\_{\rm n}} \tag{A11}$$

Considering, that the effecive area *A* is a sum of the cross-sectional areas *A*n and *<sup>A</sup>*p, Equation (A11) can be differentiated and has to be equal 0 for its maximum. So

$$-(a\_{\rm p} - a\_{\rm n})^2 (\Delta T)^2 \cdot \frac{\frac{\rho\_{\rm P} l\_{\rm P}}{(A + A\_{\rm p})^2} - \frac{\rho\_{\rm R} l\_{\rm R}}{A\_{\rm p}^2}}{(\frac{\rho\_{\rm P} l\_{\rm P}}{A\_{\rm p}} + \frac{\rho\_{\rm R} l\_{\rm R}}{A - A\_{\rm P}})^2} = 0$$

This Equation (A12) is zero, if the numerator of the fraction is zero, so

$$\frac{\rho\_{\mathbb{P}} l\_{\mathbb{P}}}{(A + A\_{\mathbb{P}})^2} - \frac{\rho\_{\mathbb{n}} l\_{\mathbb{n}}}{A\_{\mathbb{P}}^2} = 0 \tag{A13}$$

After rearrangement, the optimum *<sup>A</sup>*n/*A*p ratio for maximum power output is received as

$$\frac{A\_{\rm n}}{A\_{\rm P}} = \sqrt{\frac{\rho\_{\rm n}}{\rho\_{\rm P}}} \tag{A14}$$

The final Equation (A14) derived corresponds to the reported ratio for maximum power output of Xing et al. [36] .

#### **Appendix D. Efficiency of the Module**

The maximum first-law efficiency *η*I,TEG,max of a module is the product of the Carnot efficiency *η*Carnot and the second-law efficiency *η*II,TEG,max [9,11] and can be determined as

$$
\eta\_{\rm I,TEG,max} = \eta\_{\rm Carnot} \cdot \eta\_{\rm II,TEG,max} = \frac{T\_{\rm hot} - T\_{\rm cold}}{T\_{\rm hot}} \cdot \frac{\sqrt{1 + Z\overline{T}} - 1}{\sqrt{1 + Z\overline{T}} + 1} \tag{A15}
$$

Here, *T* ¯ is the average temperature and *Z* is a function of the materials thermoelectric parameters:

$$Z = \frac{a^2}{R \cdot K} \tag{A16}$$

with

$$
\boldsymbol{\alpha} = (\boldsymbol{\alpha}\_{\mathbb{P}} - \boldsymbol{\alpha}\_{\mathbb{N}}) \tag{A17}
$$

$$R = \frac{1}{\sigma\_{\text{P}}} \cdot \frac{l\_{\text{P}}}{A\_{\text{P}}} + \frac{1}{\sigma\_{\text{n}}} \cdot \frac{l\_{\text{n}}}{A\_{\text{n}}} \tag{A18}$$

and

$$K = \lambda\_{\rm P} \cdot \frac{A\_{\rm P}}{l\_{\rm P}} + \lambda\_{\rm n} \cdot \frac{A\_{\rm n}}{l\_{\rm n}} \tag{A19}$$

So, the maximum first-law efficiency *η*I,TEG,max of a module can be determined as a function of the materials thermoelectric parameter and the respective cross-sectional areas *A*n and *<sup>A</sup>*p [30].

#### **Appendix E. Simulated Module Fluxes**

**Figure A1.** Analyzed path *x* along a central line through the respective thermoleg on the example of module 1.

**Figure A2.** Distribution of (**a**) temperature *T*(*x*), (**b**,**<sup>c</sup>**) flux density of thermal energy *j*E,th(*x*), (**d**) voltage *U*(*x*) and (**<sup>e</sup>**,**f**) flux density of electrical charge *j*q(*x*) in module 1.

**Figure A3.** Distribution of (**a**) temperature *T*(*x*), (**b**,**<sup>c</sup>**) flux density of thermal energy *j*E,th(*x*), (**d**) voltage *U*(*x*) and (**<sup>e</sup>**,**f**) flux density of electrical charge *j*q(*x*) in module 2.

**Figure A4.** Distribution of (**a**) temperature *T*(*x*), (**b**,**<sup>c</sup>**) flux density of thermal energy *j*E,th(*x*), (**d**) voltage *U*(*x*) and (**<sup>e</sup>**,**f**) flux density of electrical charge *j*q(*x*) in module 3 with *zT*-optimized geometry.

**Figure A5.** Distribution of (**a**) temperature *T*(*x*), (**b**,**<sup>c</sup>**) flux density of thermal energy *j*E,th(*x*), (**d**) voltage *U*(*x*) and (**<sup>e</sup>**,**f**) flux density of electrical charge *j*q(*x*) in module 3 with geometry for matching *<sup>I</sup>*q,SC.

**Figure A6.** Distribution of (**a**) temperature *T*(*x*), (**b**,**<sup>c</sup>**) flux density of thermal energy *j*E,th(*x*), (**d**) voltage *U*(*x*) and (**<sup>e</sup>**,**f**) flux density of electrical charge *j*q(*x*) in module 3 with power-optimized geometry.
