# Simulink Model of a Thermoelectric Generator for Vehicle Waste Heat Recovery
## 4 TEG Model for Exhaust Gas Waste Heat Recovery
### Equations and Simulink Model
For the case presented in Figure 10, the heat exchanger efficiency is defined as:  
\[\mathrm{A_{ht,TC}}=\mathrm{A_{P}}+\mathrm{A_{n}}. \tag{34}\]2  
To compute the junction temperatures, one more thermal resistance is needed, namely the equivalent thermal resistance of the out (output and heat rejection) side:  
\[\mathrm{A_{ht,TC}}=\mathrm{A_{P}}+\mathrm{A_{n}}. \tag{34}\]3 Integration of thermoelectric module in a thermoelectric generator. \[\mathrm{A_{ht,TC}}=\mathrm{A_{P}}+\mathrm{A_{n}}. \tag{34}\]4  
Based on these resistances, the hot and cold side junction temperatures can be expressed with respect to T\({}_{\infty,h}\) (T\({}_{\infty,h}\) = 1/2 * (T\({}_{\infty,h,in}\) + T\({}_{\infty,h,out}\)) [K], where T\({}_{\infty,h,in}\) and T\({}_{\infty,h,out}\) are the temperature of the gas at the hot side heat exchanger inlet and outlet respectively and T\({}_{\infty,c}\):  
\[\mathrm{A_{ht,TC}}=\mathrm{A_{P}}+\mathrm{A_{n}}. \tag{34}\]5  
\[\mathrm{A_{ht,TC}}=\mathrm{A_{P}}+\mathrm{A_{n}}. \tag{34}\]6  
The amount of heat extracted from the hot gases can be determined from the CV inlet and outlet enthalpy represented by the heat capacity times the difference in temperature:  
\[\mathrm{A_{ht,TC}}=\mathrm{A_{P}}+\mathrm{A_{n}}. \tag{34}\]7  
However, in this effort, the outlet temperature is an output value. Neglecting the heat losses through radiation \(Q_{gas,Hx}=Q_{h}\), therefore, the outlet temperature is:  
\[\mathrm{A_{ht,TC}}=\mathrm{A_{P}}+\mathrm{A_{n}}. \tag{34}\]8  
When installing a heat exchanger on the exhaust path of an ICE, there is also the issue of the increased backpressure that negatively influences engine performance and emissions. Consequently, the pressure drop must be estimated. Here, there are three effects that need to be considered: Viscous drag effects, flow expansion, and flow contraction. For a CV, the pressure loss due to this viscous drag effect is:  
\[\mathrm{A_{ht,TC}}=\mathrm{A_{P}}+\mathrm{A_{n}}. \tag{34}\]9  
where f is the friction factor defined using the [PERSON] equation [30]:  
\[\mathrm{A_{ht,TEM}}=\mathrm{n_{TC}}\big{(}\mathrm{A_{P}}+\mathrm{A_{n}}\big{)}. \tag{35}\]0  
The [PERSON] equation was chosen because it covers both developing and fully developed internal flows. The Reynolds number is calculated using Equation (45), while A and B are:  
\[\mathrm{A_{ht,TEM}}=\mathrm{n_{TC}}\big{(}\mathrm{A_{P}}+\mathrm{A_{n}}\big{)}. \tag{35}\]1  
\[\mathrm{A_{ht,TEM}}=\mathrm{n_{TC}}\big{(}\mathrm{A_{P}}+\mathrm{A_{n}}\big{)}. \tag{35}\]2  
To calculate A, two additional parameters are required, namely the surface roughness (\(\epsilon_{surface}\)), which can be taken from literature and the channel hydraulic diameter (D\({}_{hydr,channel}\)), which is calculated from the channel cross-sectional area and perimeter:  
\[\mathrm{A_{ht,TEM}}=\mathrm{n_{TC}}\big{(}\mathrm{A_{P}}+\mathrm{A_{n}}\big{)}. \tag{35}\]3  
For small gas density changes along the length of the heat exchanger, in case of n\({}_{CV}\) control volumes (CV\({}_{length}\) = L\({}_{HK}\)/n\({}_{CV}\)), Equation (67) can be simplified to:\[\mathrm{A_{ht,TEM}}=\mathrm{n_{TC}}\big{(}\mathrm{A_{P}}+\mathrm{A_{n}}\big{)}. \tag{35}\]4  
Changes in the pipe cross-section area along the flow direction also lead to pressure losses by causing the flow to either expand or contract. To estimate these losses, the Borda-Carnot correlations are used [31]. These correlations require knowledge about cross sectional areas, as well as the gas density and velocity. For a TEG installed on the exhaust pipe of an ICE (Figure 11), these cross-sectional areas are: Exhaust pipe upstream of the HX (A\({}_{\mathrm{inlet-pipe,cross}}\) [m\({}^{2}\)]), HX inlet (A\({}_{\mathrm{HX,inlet,cross}}\) [m\({}^{2}\)]) and outlet (A\({}_{\mathrm{HX,outlet,cross}}\) [m\({}^{2}\)]), as well as the exhaust pipe downstream of the HX (A\({}_{\mathrm{outlet-pipe,cross}}\) [m\({}^{2}\)]).