# TEG Design for Waste Heat Recovery at an Aviation Jet Engine Nozzle
## 3 Results  
Due to the thermal resistance of the convective boundary layers at the core and bypass flow, only a fraction of the gas temperature difference can be used at the TE material, which limits the maximum power output and efficiency of the thermocouples. In presence of external thermal resistances the temperature difference at the TE material is a function of the leg length. For small leg lengths, the reduced thermal resistance of the TE leg yields a high incident heat flow but low temperature difference across the legs. Consequently, the power output and the efficiency also become low. Though high leg lengths provide an increase of the temperature difference, the increasing thermal resistance of the thermocouple lowers the incident heat flow, which yields a reduction of the power output above a certain leg length, likewise. Hence, as explained above, maximum output power is achieved for an optimal intermediate leg length based on matching leg and parasitic thermal resistances [30]. For the set HTC at both boundary layers (\(\alpha_{\text{H}}=100\) WK\({}^{-1}\)m\({}^{-2}\)/\(\alpha_{\text{C}}=64\) WK\({}^{-1}\)m\({}^{-2}\)), the optimal leg length for maximised power output equals \(l_{\text{opt}}=21.2\) mm at \(F=25\%\) (Figure 6). Each thermocouple contributes 12.18 mW to the total power output of the TEM working at a temperature difference of 169 K across the TE legs. When considering the resulting heat flux, the efficiency equals \(\eta=8.2\%\), which is a result of the optimized HPVVM with its fitted temperature characteristic of transport properties with regard to the application temperature range. With the assumptions made on geometries and chosen material properties, the uni-couple exhibits the following operational characteristics at the  
Figure 5: Example for an occupation of the nozzle surface by TE modules (TEM) (Courtesy of [PERSON], TU Braunschweig). The shown design implies the application of vertical bars for improved heat exchange at the bypass flow, which is not considered in the simulation. Inset: Overall vision of TEM alignment on an aviation nozzle. nozzle (Figure 6). Resulting values of the mass, power output, and the power density of the TE module and the entire TEG are summarized in Table 4.  
Increasing \(\alpha_{\mathrm{H}}\) from 100 \(\mathrm{WK^{-1}m^{-2}}\) to 500 \(\mathrm{WK^{-1}m^{-2}}\) at the hot side yields a reduction of the optimal leg length for thermal resistance match, and in turn leads to an increase of the incident heat flux (Figure 7a). The lower mass translates into a higher gravimetric power density, which is likewise achievable by an assumed increase of the _PF_ (Figure 7b). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline _F = 25\%_ & _m [g]_ & _Pmax [W]_ & _P\({}_{\mathrm{G}}\) [Wkg\({}^{-1}\)]_ & _P\({}_{\mathrm{A\_TEM}}\) [Wcm\({}^{-2}\)]_ & _P\({}_{\mathrm{A\_TE}}\) [Wcm\({}^{-2}\)]_ \\ \hline TEM & 58.4 & 3.8 & & \\ TEG & 25812.8 & 1679.67 & 65.06 & 0.152 & 0.61 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Mass \(m\), maximum power output \(P_{\mathrm{max}}\), gravimetric power density \(P_{\mathrm{G}}\), and power densities per area for a HPVM-based TEM with \(F\) = 25% at \(\alpha_{\mathrm{H}}\) = 100 \(\mathrm{WK^{-1}m^{-2}}\) and \(\alpha_{\mathrm{C}}\) = 64 \(\mathrm{WK^{-1}m^{-2}}\). _P\({}_{\mathrm{A\_TEM}}\)_ relates the areal power output to the total footprint of the TEM, while _P\({}_{\mathrm{A\_TE}}\)_ is the ratio between power output and the active surface of the TEM occupied by TE legs only. Values for mass and power output are scaled up for the entire thermoelectric generators (TEG) at the nozzle, which consists of 442 TEM. Figure 6: Simulation results of a thermocouple with \(F\) = 25% and for convective heat transfer coefficients of \(\alpha_{\mathrm{H}}\) = 100 \(\mathrm{WK^{-1}m^{-2}}/\alpha_{\mathrm{C}}\) = 64 \(\mathrm{WK^{-1}m^{-2}}\). 3 rd order polynomial fits of power output (**a**), temperature difference (**b**), efficiency (**c**), and hot side heat flux (**d**) as functions of the leg length.