# Simulink Model of a Thermoelectric Generator for Vehicle Waste Heat Recovery
## 4 TEG Model for Exhaust Gas Waste Heat Recovery
### Equations and Simulink Model
The reduced Reynolds number has an experimental range of \(0.2\leq\mathrm{Re}^{*}\leq 200\)[25] and is calculated as:  
Figure 6:2  
The proposed equation for the Nusselt number is:  
Figure 6:3  
If the flow is fully turbulent and Re * exceeds its range limits, then the Nusselt number can be calculated using the Gnielinski relation [26]:  
Figure 6:4  
where \(\mathrm{Re}_{\mathrm{Hx,channel}}\) is the channel Reynolds number and \(\mathrm{f_{G}}\) is the friction factor in turbulent flow in smooth tubes determined from the first Petukhov equation [27]:  
Figure 6:5  
Figure 6:6  
Knowing the Nusselt number, the gas-fin heat transfer coefficient can then be calculated with the following expression:  
Figure 6:7  
To calculate the efficiency of the heat exchanger, as well as the thermal resistance, the efficiency of a single fin must be determined [28], along with the thermal resistances of all other intermediate layers. A single fin efficiency is calculated based on the \(\mathrm{m}_{\mathrm{fin}}\) parameter and the fin height, which, in this case is the same as the heat exchanger height:  
Figure 6:8  
Figure 6:9  
Figure 7:0  
Figure 9b shows the thermal resistance network used to calculate the hot (\(\mathrm{T_{h}}\) [K]) and cold side (\(\mathrm{T_{c}}\) [K]) junction temperatures along with the heat exchanger efficiency. To keep the model simple, and since it represents under 2% of the total heat transferred from the hot side [29], radiation losses are neglected. The resistances in Figure 9 are defined for all layers presented in Figure 8 based on their thickness (\(\mathrm{t_{layer-name}}\)), thermal conductivity (\(\mathrm{k_{layer-name}}\)), and surface area (\(\mathrm{A_{layer-name}}\)). The equivalent gas-fin heat transfer resistance and the thermal resistances of the base, grease, and ceramic are calculated according to Equations (52)-(55):  
Figure 7:1  
Figure 7:2  
Figure 7:3  
Figure 7:4  
The equivalent thermal resistance of the TEM is calculated based on the hot and cold junction temperatures and the heat flow to the cold side:  
Figure 7:5  
Figure 7:6 (**a**) Thermoelectric generator (TEG) temperatures of interest; (**b**) thermal resistance network for a CV; (**c**) simplified thermal resistance network for calculating \(\mathrm{T_{c}}\); and (**d**) simplified thermal resistance network for calculating \(\mathrm{T_{h}}\). Similarly, an equivalent load resistance is defined based on the hot junction temperature and the temperature on the cold side (\(\mathrm{T_{\infty_{\mathrm{c}}}}=1/2*(\mathrm{T_{\infty_{\mathrm{c}},\mathrm{in}}}+ \mathrm{T_{\infty_{\mathrm{c}},\mathrm{out}}})\) [K], where \(\mathrm{T_{\infty_{\mathrm{c}},\mathrm{in}}}\) and \(\mathrm{T_{\infty_{\mathrm{c}},\mathrm{out}}}\) are the temperature of the gas at the cold side heat exchanger inlet and outlet respectively):  
Figure 7:7  
Based on Equations (52)-(55), an equivalent thermal resistance of the hot side can be determined:  
Figure 7:8  
Similarly, the equivalent thermal resistance of the cold side is:  
Figure 7:9  
Often, an aluminum block is used to act as a thermal spreader that is placed between the HX and the TEM as in \[\mathrm{A_{ht,TC}}=\mathrm{A_{P}}+\mathrm{A_{n}}. \tag{34}\]0 In this case, the thermal resistance of the base is replaced by the thermal resistance of the aluminum block:  
\[\mathrm{A_{ht,TC}}=\mathrm{A_{P}}+\mathrm{A_{n}}. \tag{34}\]1  
It is important to note that, in such cases, the HX is usually also made of aluminum which allows for a simplification. Namely, the addition of the HX base thickness to the aluminum block thickness. They can, however, be introduced as separate blocks. In the present study, the former solution has been adopted.