# TEG Design for Waste Heat Recovery at an Aviation Jet Engine Nozzle
## 4 Discussion  
Using the HPVVM with \(F\) = 25% and optimized leg length, a minimum \(\alpha_{\rm H}\) \(>\) 350 WK\({}^{-1}\)m\({}^{-2}\) is required to fulfil the \"APD + CFD\"-limit of 100 Wkg\({}^{-1}\). The requirement of the \"APD\"-limit (173 Wkg\({}^{-1}\)) and the limit for a range invariant TEG installation (520 Wkg\({}^{-1}\)) are not achievable at all (Figure 7b). Neither, increasing the \(PF\) of the HPVVM by 30% at \(\alpha_{\rm H}\) = 100 WK\({}^{-1}\)m\({}^{-2}\) does fulfil the criteria for a beneficial TEG installation. Generally, the increase of \(\alpha_{\rm H}\) reduces the thermal resistance within the heat transmission path, allowing for a reduced leg length and mass saving for the configuration. Varying \(\alpha_{\rm H}\) from 100 WK\({}^{-1}\)m\({}^{-2}\) to 500 WK\({}^{-1}\)m\({}^{-2}\) yields a reduction of the optimal leg length by 30%, which translates into lowered thermocouple mass of 14.7%. Due to the matching of thermal resistances, the effective temperature difference at the TE legs remains almost constant, while the heat flux increases significantly for smaller leg lengths. This leads to an increase of gravimetric power density. Due to an unchanged \(\alpha_{\rm C}\), the mean temperature rises along the thermocouple with increasing \(\alpha_{\rm H}\). Since the material properties of the HPVM are fixed with regard to their initially set temperature characteristics, a higher mean temperature yields a continuously increasing bipolar contribution due to the approach to the intrinsic temperature domain of the supposed TE properties. In connection with non-vanishing temperature drops at the boundary, nozzle, and insulation layers, this results in a flattening of power density curves at higher \(\alpha_{\rm H}\), irrespective of the lower optimal leg lengths. Since the minimal gravimetric power requirement of the aircraft system of 100 Wkg\({}^{-1}\) is only satisfied at \(F\) = 25% by an improved convective heat exchange with a concurrent significant increase of materials performance, it is apparently not expedient to apply TE modules with higher filling factors for the application at the aircraft nozzle. This might take into account that progress towards maximal \(ZT\)-values\(\sim\)2 would represent a significant improvement for future TE materials with technological relevance and high stability from today's point of view. Against this background and in conjunction with reasonable assumptions on the performance for the convective heat exchange at the nozzle, a beneficial TEG application is attainable only for smaller filling factors. However, the minimum applicable filling factor \(F_{\rm min}\) is limited for several reasons. First, the spreading resistance of components for heat conduction within the heat transmission path can lower the effective temperature difference at the TE materials, since heat has increasingly to be provided laterally through substrates or insulation layers to the thermocouple legs at low \(F\). For instance, the United States National Aeronautics and Space Administration (NASA) demonstrated the feasibility of \(F\) = 3.4% within their GPHS-RTG [42] and even lower values for \(F_{\rm min}\) = 1% have been suggested as a practical limit, once materials with high thermal conductivity are used to connect the thermocouples to external heat exchangers [43]. Secondly, \(F_{\rm min}\) is limited due to the reduction of necessitated leg lengths for matching of thermal resistances. A decreasing leg length puts higher requirements on the electrical contact resistance between bridges and the TE legs in order to maintain a high power output of the TEM at low \(F\). At typical values for the electrical conductivity of TE materials (\(\sim\)10\({}^{3}\) Scm\({}^{-1}\)), contact resistances between 10\({}^{-6}\)-10\({}^{-5}\)\(\Omega\)cm\({}^{2}\) allow for a reduction of the leg length down to 0.1 mm, which was already successfully demonstrated in commercial TE devices for cooling [44] and generator applications [45].