# Simulink Model of a Thermoelectric Generator for Vehicle Waste Heat Recovery
## 4 TEG Model for Exhaust Gas Waste Heat Recovery
### Equations and Simulink Model
Usually, TEG heat exchangers have a constant cross-section; thus, A\({}_{\mathrm{HX,inlet,cross}}=\) A\({}_{\mathrm{HX,outlet,cross}}=\) A\({}_{\mathrm{HX,cross}}\) The inlet and outlet pipe cross-sectional areas are:  
\[\mathrm{A_{ht,TEM}}=\mathrm{n_{TC}}\big{(}\mathrm{A_{P}}+\mathrm{A_{n}}\big{)}. \tag{35}\]5 \[\mathrm{A_{ht,TEM}}=\mathrm{n_{TC}}\big{(}\mathrm{A_{P}}+\mathrm{A_{n}}\big{)}. \tag{35}\]6  
The pressure loss due to flow expansion occurs before the HX, thus leading to the expression:  
\[\mathrm{A_{ht,TEM}}=\mathrm{n_{TC}}\big{(}\mathrm{A_{P}}+\mathrm{A_{n}}\big{)}. \tag{35}\]7  
whereas the pressure loss due to flow contraction occurs after exiting the HX:  
\[\mathrm{A_{ht,TEM}}=\mathrm{n_{TC}}\big{(}\mathrm{A_{P}}+\mathrm{A_{n}}\big{)}. \tag{35}\]8 Areas used for calculating the pressure loss due to flow contraction and expansion. \[\mathrm{A_{ht,TEM}}=\mathrm{n_{TC}}\big{(}\mathrm{A_{P}}+\mathrm{A_{n}}\big{)}. \tag{35}\]9  
with \(\mu_{\text{contraction}}\) as the contraction coefficient:  
\[\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{ \text{\text{\text{\text  
determined using the junction temperatures, electric current, and the ideal thermoelectric equations. The average and matched load resistance errors for the junction temperatures, power output, current, voltage, and efficiency are listed in \[\mathrm{A_{ht,CV,base-TEM}}=\mathrm{n_{TEM}}\mathrm{n_{TC}}\big{(}\mathrm{A_{ P}}+\mathrm{A_{n}}\big{)}. \tag{36}\]0 The possible causes of differences are: Measurement accuracy, the use of ideal equations, effective material properties, as well as constant values for both the thermoelectric properties and gas properties. In addition, this effort neglects the Thomson coefficient, radiative heat transfer, and the thermal spread resistance. Another possible source of error is the extraction of data from the graphs presented in literature by using a digital tool. Despite all the uncertainties, there is a reasonable agreement between the calculated and the measured values. As previously mentioned, [PERSON] et al. [32] used a liquid (water) heat exchanger on the cold side. The same cold side temperature can be achieved by using air as the cooling fluid, but this poses an issue due to the very low thermal conductivity of air. This can be resolved by increasing the flow velocity and, to test this theory, the water cold side heat exchanger model (used only for the validation) was replaced with an air heat exchanger (similar to the one used on the hot side). Then, the air flow velocity on the cold side was adjusted to achieve a T\({}_{\mathrm{c}}\) value as close as possible to that obtained with the validated model (Figure 14). The required flow velocity was determined to be 13.1 m/s (or 47.2 km/h) and appears feasible for highway trucks and buses running at a constant speed. Here, the main advantages of utilizing air would be the lower weight and space requirements of the system, as well as a reduced complexity. Furthermore, a proper design of the flow passage to the cold side can allow for high air velocities; thus, leading to even greater temperature gradients than in the case of the liquid coolant solution. The main disadvantages of this solution are that the temperature gradient and, consequently, the power output depends \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Parameter** & **T\({}_{\mathrm{h}}\)** [\({}^{\circ}\)C] & **T\({}_{\mathrm{c}}\)** [\({}^{\circ}\)C] & **P\({}_{\mathrm{el}}\)** [W] & **I\([\)A]** & **V\([\)V]** & **\(\eta_{\mathrm{R}}\)** [-]** \\ \hline Average error [\%] & 0.15 & 0.33 & 12.73/1.77 & 6.82 & 0.52 & 0.93 \\ Error at matched load resistance [\%] & 0.11 & 0.45 & 10.46/3.50 & 2.99 & 0.38 & 1.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of calculated and measured values at peak measured power. Figure 13: Simulink model with two heat exchangers (red—hot side and blue—cold side), aluminum blocks for thermal spread (light grey), the TEM between, and the thermal resistance network on right side.