# Simulink Model of a Thermoelectric Generator for Vehicle Waste Heat Recovery
## 2 Thermoelectric Model Equations
Considering the thermal conductivities of the two legs (\(\mathrm{k_{P}}\)[W/mK]--thermal conductivity of the p-type leg, \(\mathrm{k_{n}}\)[W/mK]--thermal conductivity of the n-type leg), one can define the thermal conductance (\(\mathrm{K_{TC}}\)[W/K]) of the TC:  
\[\rm\alpha_{TC}=\alpha_{p}-\alpha_{n}. \tag{2}\]0  
Joule heating, \(\mathrm{\dot{Q}_{oule,h/c}}\)[W], is the process where electrical energy is converted into thermal energy as current flows through a resistance. For a TC, Joule heating decreases the rate of heat absorption at the hot junction and increases the rate of heat transfer to the cold junction, in equal shares (i.e., half of the heat generated by the Joule effect passes to the hot side and the other half to the cold side) [16]:  
\[\rm\alpha_{TC}=\alpha_{p}-\alpha_{n}. \tag{2}\]1  
Neglecting the Thomson effect, by applying the first law of thermodynamics at the hot junction of a TC, the rate of heat absorption, \(\mathrm{\dot{Q}_{h,TC}}\)[W], is:  
\[\rm\alpha_{TC}=\alpha_{p}-\alpha_{n}. \tag{2}\]2  
whereas the rate of heat rejection, \(\mathrm{\dot{Q}_{c,TC}}\)[W], at the cold side is:  
\[\rm\alpha_{TC}=\alpha_{p}-\alpha_{n}. \tag{2}\]3  
Overall, the power delivered to the load, \(\mathrm{P_{el,TC}}\)[W], can be calculated in two ways:  
\[\rm\alpha_{TC}=\alpha_{p}-\alpha_{n}. \tag{2}\]4  
or as:\[\rm\alpha_{TC}=\alpha_{p}-\alpha_{n}. \tag{2}\]5  
The thermoelectric performance of the material is assessed using the figure of merit [1/K]  
\[\rm\alpha_{TC}=\alpha_{p}-\alpha_{n}. \tag{2}\]6  
which, for a TC becomes  
\[\rm\alpha_{TC}=\alpha_{p}-\alpha_{n}. \tag{2}\]7  
This figure of merit, like the thermoelectric parameters, is temperature dependent. However, often, average values are assumed with errors found to be below 10% of the true value [17]. The conversion (or thermal) efficiency of the TC is the ratio of the power output to the heat input at the hot side. To simplify the final expression, the following notations are employed for the load resistance ratio (m [-]) and Carnot efficiency (\(\eta_{\mathrm{C}}\) [-]), respectively,  
\[\rm\alpha_{TC}=\alpha_{p}-\alpha_{n}. \tag{2}\]8  
\[\rm\alpha_{TC}=\alpha_{p}-\alpha_{n}. \tag{2}\]9  
Thus, the conversion efficiency is:  
\[\rm V_{Load,TC}=I\cdot R_{Load,TC}. \tag{3}\]0  
Expressions of maximum parameters (current, voltage, power, and efficiency) are also introduced here due to their importance in determining TEG parameters, which will be shown later. The maximum voltage occurs at open circuit when I = 0:  
\[\rm V_{Load,TC}=I\cdot R_{Load,TC}. \tag{3}\]1  
The maximum current, however, occurs at short circuit, where \(\mathrm{R_{Load,TC}}=0\); thus leading to the following expression:  
\[\rm V_{Load,TC}=I\cdot R_{Load,TC}. \tag{3}\]2  
To obtain the maximum power, it can be shown that the load resistance must equal the internal resistance (thus: m = 1), which leads to the following expression:  
\[\rm V_{Load,TC}=I\cdot R_{Load,TC}. \tag{3}\]3  
The output of a single TC is generally small; e.g., well below 1 [W]. Therefore, many TCs are connected to form a thermoelectric module (TEM) (Figure 2); thus, increasing the power output. At this point it is worth recalling that a TEM can operate both as a generator (often designated with TEG in literature) and/or as a heat pump (usually TEC--thermoelectric cooler). In the present study, the TEM operates as a generator, but the notation of TEM is used to avoid confusion with the acronym TEG that involves the overall system including heat exchangers (shown later). The output of a TEM can be expressed based on the number of thermocouples (\(\eta_{\rm TC}\) [-]) and the parameters of the TC, except for electrical current that is independent of the number of thermocouples.