# Simulink Model of a Thermoelectric Generator for Vehicle Waste Heat Recovery
## 2 Thermoelectric Model Equations  
A thermoelectric couple (TC) consists of p-type and n-type semiconductor legs that are connected thermally in parallel and electrically in series (Figure 1). When heat is applied to produce electric current, the TC operates as a generator. If, however, current is applied at the terminals, then the TC operates as a heat pump. For this effort, the focus is on thermoelectric generation and the use of the model as a heat pump is left to future work. If the two junctions of the TC are subjected to different temperatures (\(\Delta\)T = T\({}_{\rm h}-\) T\({}_{\rm c}\) [K], where T\({}_{\rm h}\) [K]--hot side temperature, T\({}_{\rm c}\)--cold side temperature), a potential difference proportional to the temperature variation, will appear (V\({}_{\rm OC,TC}\) [V]--open circuit voltage V\({}_{\rm Load,TC}\) [V]--potential difference across a load connected to the TC):  
\[\rm V_{OC,TC}=\alpha_{TC}\cdot(T_{\rm h}-T_{\rm c}) \tag{1}\]  
where \(\alpha_{TC}\) [V/K] is the relative Seebeck coefficient:  
\[\rm\alpha_{TC}=\alpha_{p}-\alpha_{n}. \tag{2}\]  
If the TC is connected to an external load with a resistance R\({}_{\rm Load,TC}\) [\(\Omega\)], a current flow will appear (I [A]). Consequently, the potential difference V\({}_{\rm Load,TC}\) can be expressed as:  
\[\rm V_{Load,TC}=I\cdot R_{Load,TC}. \tag{3}\]  
In addition, there is an internal electrical resistance of the TC, R\({}_{\rm TC}\) [\(\Omega\)], which is defined based on the characteristics of the two legs: Electrical resistivity (\(\rm\rho_{P}\) [\(\Omega\)m]--electrical resistivity of the p-type leg, \(\rm\rho_{n}\) [\(\Omega\)m]--electrical resistivity of the n-type leg), length (L\({}_{\rm p}\) [m]--length of the p-type leg, L\({}_{\rm n}\) [m]--length of the n-type leg), and cross-sectional area (A\({}_{\rm p}\) [m\({}^{2}\)]--cross-section area of the p-type leg, A\({}_{\rm n}\) [m\({}^{2}\)]--cross-section area of the n-type leg). \[\rm R_{TC}=\rm\rho_{p}\frac{L_{p}}{A_{p}}+\rm\rho_{n}\frac{L_{n}}{A_{n}} \tag{4}\]  
Knowing that the relation between V\({}_{\rm Load,TC}\) and V\({}_{\rm OC,TC}\) is  
\[\rm V_{Load,TC}=\rm V_{OC,TC}-IR_{TC} \tag{5}\]  
which results in a derivation of the current. Figure 1: Thermoelectric couple operating as a generator. \[\mathrm{I}=\frac{\mathrm{\alpha_{TC}(T_{h}-T_{c})}}{\mathrm{R_{TC}+R_{Load,TC}}}. \tag{6}\]  
This flow of current leads to three additional effects: Peltier heating or cooling, Joule heating, and the Thomson effect. The latter is relatively small and is usually neglected to simplify the analysis, as is the case in this study. If current flows through the junction, heat must be continuously added or rejected to maintain a constant temperature. Furthermore, this heat flow is proportional to the current flow and changes direction if the current is reversed. This is called the Peltier effect and is defined by the Peltier coefficient \(\mathrm{\pi_{TC,junction}}\)[J/A]:  
\[\mathrm{\pi_{TC,junction}}=\frac{\mathrm{\dot{Q}_{junction}}}{\mathrm{I}} \tag{7}\]  
where \(\mathrm{\dot{Q}_{junction}}\) [W] is the rate of heat absorption/rejection at the hot/cold junction. Using Kelvin relationships, the Peltier coefficient can be expressed based on the Seebeck coefficient, which is respectively easier to measure:  
\[\mathrm{\pi_{TC,junction}}=\mathrm{\alpha_{TC}T_{junction}} \tag{8}\]  
where \(\mathrm{T_{junction}}\) [K] is the temperature of the hot side and cold side junction. Thus, the rate of heat absorption/rejection at this junction can be expressed as:  
\[\mathrm{\dot{Q}_{junction}}=\mathrm{\alpha_{TC}T_{junction}}. \tag{9}\]  
In addition to Joule heating, there is another irreversible phenomenon that accompanies the reversible thermoelectric effect, namely the conduction of heat. In a TC, heat conduction causes a transfer of heat from the hot side to the cold side as a means of achieving thermal equilibrium.