# Thermoelectric Generator-Powered Long-Range Wireless Real-Time Steam Leak Detection in Steam Traps
## 4 Steam Leak Detection
### TEG Design  
#### 4.1.1 Thermal Model  
This device has the mission of generating all the power required for the IIoT and all three sensors. The overall thermoelectrical converter performance depends on the TEG cell efficiency in the conversion of thermal into electricity. The working principle of the Peltier cell involves the physical effects named Seebeck, Peltier, Thomson, and Joule. * The Seebeck effect is responsible for the electricity generation in the cell. It takes place whenever two semiconductor materials \(A\) and \(B\) are joined together at their ends by a conductive material and there is a temperature difference between the two joints. This leads to the conversion of heat into electricity. The Seebeck coefficients account for electromotive force generation (Equation (1)). \[\frac{\partial E_{T}}{\partial T}=\alpha_{a}-\alpha_{b}\] (1) where \(\frac{\partial E_{T}}{\partial T}\) is the derivative of the electromotive force with respect to temperature. It represents the voltage gradient generated per unit of temperature change in the system. It is measured in volts per kelvin. \(\alpha_{a}\)\(\alpha_{b}\) is the Seebeck coefficient (or thermoelectric power) of material a or b, expressed in V/KV. It describes how efficiently material a or b converts a temperature difference into a voltage. * The Peltier effect is the cooling or heating effect that occurs when an electric current passes through a junction of two materials, as opposed to the Seebeck effect. The heat flux absorbed or rejected by this effect \(\dot{Q}_{Peltier}\) is given by Equation (2). \[\dot{Q}_{Peltier}=\pm IT(\alpha_{a}-\alpha_{b})\] (2) where \(I\) represents the current and \(T\) the temperature in the junction. Figure 5: The heat-powered IIoT device is composed of three modules; rendered (**a**) and real (**b**). * The Joule effect accounts for the heating up of a material when current passes through a conductor material. The heat flux \(\dot{Q}_{Joule}\) that is produced by this effect is represented by Equation (3), where \(R_{0}\) is the internal electrical resistance that accounts for this effect. \[\dot{Q}_{Joule}=R_{0}I\] (3)
* The Thomson effect takes place when current passes through a single material that is subject to a temperature gradient along its length. This can lead to both absorption and generation of heat depending on the value of the Thomson coefficients. The heat flux (Equation (4)) is dependent on the Thomson parameter \(\sigma\). \[\dot{Q}_{Thomson}=-\sigma I\Delta T\] (4)  
The conversion efficiency of a Peltier cell is given as a function of the parameter Z (units 1/K), or the parameter ZT (no units), where T represents the average temperature in the cell. The Z factor relates the energy supplied by the cell to the load with respect to the heat transfer that the cell has. Equations (5) and (6) give the maximum efficiency in terms of the figure of merit ZT used in thermoelectric devices. ZT depends on the temperature T, and the thermoelectric parameters \(\alpha\) (Seebeck coefficient), \(\sigma\) (electrical conductivity) and k (thermal conductivity). The term \(\mathrm{h_{c}}=1-\mathrm{T_{H}}/\mathrm{T_{C}}\) is the Carnot Efficiency. In [13], these equations are applied to the model. \[ZT=\frac{\sigma\alpha^{2}T}{\kappa} \tag{5}\]  
\[\eta_{max}=\frac{T_{H}-T_{\mathrm{C}}}{T_{H}}\ \frac{\sqrt{1-Z\overline{T}}-1}{ \sqrt{1+Z\overline{T}}-\frac{T_{\mathrm{C}}}{T_{H}}} \tag{6}\]  
Thanks to this model, it is possible to design the most efficient heat harvester and radiator to achieve the maximum \(\Delta\)T. In detail, the aluminum pipeline (Figure 6a) adapter transfers the heat harvested from the steam pipeline to the TEG hot surface (THot). Thanks to the thermal resistance of the Peltier cell and the radiator coupled to the cold surface (TCold), it is possible to achieve the corresponding \(\Delta\)T required for energy production. This \"heat exchanger\" (Figure 6b) extracts all the heat that passes through the Peltier cell and dissipates it through its cooling elements.