# TEG Design for Waste Heat Recovery at an Aviation Jet Engine Nozzle
## 2 Materials and Methods
With regard to the system level requirements on the gravimetric power density, further assumptions on the mass density of the different components are made (Table 1). When considering light-weight silicide TE materials and contributions from the other components of the simulated configuration, the mass of the thermocouple is calculated, which equals, for instance, 187 mg for the design case with \(F=25\%\) at its optimal TE leg length of 21.2 mm (Table 1). Additional weight would be introduced by mounting hardware, cabling, power processing units, and possibly batteries, which was not considered in this study. For every studied parametric case, the thermal resistance of the thermocouple is matched with the sum of the effective heat resistances by convection plus the minor contributions of the TiAl and AlN layers by variation of the TE leg length in the range between 1 mm and 25 mm. A matching of thermal resistances is necessary for the maximization of electrical power output, since the product of the temperature difference across the TE materials and incident heat flow to the configuration is maximized under these conditions [30]. First, calculations are conducted for a filling factor of \(F=25\%\) using the HPVM (case 1 in Table 2). The variation of the power output with TE material performance is studied by increasing the power factor of the HPVM in three steps by 10%, each yielding a \(ZT_{\text{max}}\) of 1.56 and 1.95 for the p- and n-type material, respectively, at the last step (case 2). This is clearly a free \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Part** & **Material** & \(G\) **[mm]** & \(V\) **[mm\({}^{3}\)]** & \(\rho\) **[gcm\({}^{-3}\)]** & \(m\) **[mg]** \\ \hline Insulation & AlN & \(2\times 4\times 0.325\) & \(2\times 2.6=5.2\) & 3.2 [36] & 16.6 \\ TE & Mg\({}_{2}\)Si & \(1\times 1\times 21.2\) & \(2\times 21.2=42.4\) & 2.3 & 97.52 \\ Bridge & Cu & \(1\times 3\times 0.2\) & \(2\times 0.6=1.2\) & 8.92 [39] & 10.7 \\ Nozzle & TiAl & \(2\times 4\times 2\) & 16 & 3.9 [34] & 62.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material, geometry \(G\), volume \(V\), density \(\rho\), and mass \(m\) as properties of different components of the thermoelectric uni-couple configuration. The values are representative for a thermocouple design with a filling factor of \(F=25\%\) and an adjusted length of thermoelectric legs of 21.2 mm for matching of thermal resistance. Figure 4: Qualitative heat flow distribution on the hot side of the thermocouple for an exemplary set of boundary conditions and geometrical thermocouple design (**a**), areal interpolation of discrete node data on the normal component of the heat flux vector (**b**), and continuous area function of the heat flux on the hot side (**c**). parameter variation without any specific optimization strategies for materials performance behind. Anyway, the best known laboratory values for advanced TE materials to date already exceed these assumed \(ZT\) values [40; 41]. In order to optimize the gravimetric power density apart from an increase of the materials performance, further simulations are carried out on the base of the HPVM but with smaller filling factors \(F\) (case 3), which yields a reduction of the thermal convective resistances on both sides of the thermocouple. Reduction of \(F\) is accomplished within the model by increasing the area of TiAl and AlN surfaces per uni-couple on both sides of the air flows. Additionally, the effect of a presumed improvement of the convective heat transfer on the hot side is studied by choosing a higher HTC \(\alpha_{\mathrm{H}}\) as compared to the outcome of the CFD simulations within every simulated parameter configuration of the model. The increase of \(\alpha_{\mathrm{H}}\) reflects a hypothetic modification of the nozzle surface for an improved aerodynamic design, which was not optimized concerning its heat transfer in this study at all.