# 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 layering of the TEG, which will be used for calculating the amount of heat transferred to the hot side and from the cold side, is presented in Figure 8. The TEG is composed of two heat exchangers (hot and cold side), a thermal grease used to eliminate surface  
Figure 6: Hot side heat exchanger. Figure 7: Side view of module section. imperfections in the interface area, and two ceramic plates acting as electric insulators. The electrical conductor height is relatively small and, since it has a respectively high thermal conductivity and low electrical resistivity, it was neglected in the present model. As a first step, all relevant areas for the heat transfer are calculated starting from the area available for heat transfer in one TC (\(\mathrm{A_{ht,TC}}\) [m\({}^{2}\)]):  
\[\mathrm{A_{ht,TC}}=\mathrm{A_{P}}+\mathrm{A_{n}}. \tag{34}\]  
For a TEM with \(\mathrm{n_{TC}}\) thermocouples, the area available for heat transfer (\(\mathrm{A_{ht,TEM}}\) [m\({}^{2}\)]) will be:  
\[\mathrm{A_{ht,TEM}}=\mathrm{n_{TC}}\big{(}\mathrm{A_{P}}+\mathrm{A_{n}}\big{)}. \tag{35}\]  
Thus, for a TEG with \(\mathrm{n_{TEM}}\) [-] TEMs, the total area of the HX base in a control volume (\(\mathrm{A_{ht,CV,base-TEM}}\) [m\({}^{2}\)]) available for heat transfer will be:  
\[\mathrm{A_{ht,CV,base-TEM}}=\mathrm{n_{TEM}}\mathrm{n_{TC}}\big{(}\mathrm{A_{ P}}+\mathrm{A_{n}}\big{)}. \tag{36}\]  
The HX base area for one control volume (\(\mathrm{A_{ht,CV,base}}\) [m\({}^{2}\)]) is defined as:  
\[\mathrm{A_{ht,CV,base}}=\mathrm{W_{HK}}\cdot\mathrm{L_{CV}} \tag{37}\]  
where \(\mathrm{L_{CV}}\) is the length of the CV. The part of the HX base area that is not covered by TEMs is considered to be insulated. To calculate the amount of heat extracted from the hot gases, the total area (\(\mathrm{A_{ht,CV,total}}\)) available for heat exchange with the hot gases in a control volume must be determined:  
\[\mathrm{A_{ht,CV,total}}=\mathrm{n_{fin}}\Big{[}2(\mathrm{t_{HK,fin}}+\mathrm{ L_{CV}})\mathrm{H_{HK}}+\mathrm{z_{HK,fin-spacing}}\mathrm{L_{CV}}\Big{]}. \tag{38}\]  
Note that the equations presented are for a HX installed on the hot side, but the equations can also be used for an HX on the cold side. To show the fraction of the HX base surface that is used for heat transfer to the TCs, a HX surface usage coefficient is proposed:  
\[\mathrm{f_{Hx,Area}}=\frac{\mathrm{A_{ht,CV,base-TEM}}}{\mathrm{A_{ht,CV,base }}}. \tag{39}\]  
When only the flow velocity (\(\mathrm{V_{gas}}\) [m/s]) is available, the crossflow area of the HX is required to compute the mass flow:  
\[\mathrm{A_{Hx,cross}}=\mathrm{n_{fin}}\mathrm{z_{HK,fin-spacing}}\mathrm{H_{ HK}} \tag{40}\]  
consequently, the gas mass flow rate will be  
Figure 6:0 Integration of thermoelectric module in a thermoelectric generator. Figure 6:1  
If, however, the mass flow is available, then Equation (41) can be used to compute the gas flow velocity. The next step is to determine the forced convection heat transfer coefficient between the gases and the HX surface. This is respectively difficult since the flow ranges from developing to fully developed. On this topic, [PERSON] et al. [24] proposed and validated a model that uses a composite solution based on the limiting cases of flow between isothermal parallel plates. In the transitional flow region, an average Nusselt number is calculated as a function of the heat exchanger geometry and fluid flow velocity. To use this model, two conditions must be met: Air must be contained within the channels (this is achieved since a closed HX is considered) and the fin spacing must be considerably smaller than the fin height (high aspect ratio).