## 5 Results and Discussion
### Optimization of Thermocouple Numbers  
As the length of the TEG increases, there are corresponding changes in the temperature distribution at both the cold and hot ends, the internal resistance, and the module's output power. This study brings engine exhaust at a temperature of 550 \({}^{\circ}\)C and a rate of 0.032 kg/s into the hot end of the TTEG. Simultaneously, ethylene glycol coolant, at a temperature of 90 \({}^{\circ}\)C and a rate of 0.237 kg/s, is introduced at the cold end. This study examines how the output power of the TTEG changes when the number of thermocouples is varied, while keeping the cold and hot end inlet temperatures and mass flow rates constant. The goal is to identify the ideal number of thermocouples resulting in the highest output power. Figures 13 and 14 illustrate the temperature changes at the cold and hot ends of a TTEG with 125 thermocouple pairs. Figure 13 demonstrates that the temperature at the cold end of the TTEG is variable rather than constant. The module's temperature rises progressively as the number of thermocouples increases due to heat absorption from the hot engine exhaust through the thermocouples, with the coolant arriving from the right. In contrast, Figure 14 illustrates a decline in temperature near the high-temperature end of the TEG. In this scenario, the engine exhaust enters from the right side and, as the module extends, the thermocouples convert more heat, resulting in a progressive decline in temperature. Increasing the number of thermocouples from 125 to 750 pairs results in a rise in the output temperature of the coolant from 93.71 \({}^{\circ}\)C to 103.89 \({}^{\circ}\)C. Similarly, the exhaust gas's temperature at the TTEG's exit reduces from 477.62 \({}^{\circ}\)C for 125 pairs of thermocouples to 293.02 \({}^{\circ}\)C for 750 pairs. These data emphasize the intricate relationship between the quantity of thermocouples, temperature distribution, and heat transport in a TTEG. Comprehending these interactions is essential for maximizing the efficiency and functionality of TTEGs, especially in cases where they are used to recover waste heat from high-temperature sources such as engine exhausts. The study offers valuable insights into optimizing the design of TTEGs by altering the number of thermocouples to obtain the appropriate equilibrium between heat absorption and conversion efficiency. Figure 12: (**a**–**h**) showcase the correlation between the external load and the thermal efficiency of a Tubular Thermoelectric Module (TTEG) under various temperature conditions. According to Ohm's Law, \(V=I\times R\), where \(V\) is voltage, \(I\) is current, and \(R\) is resistance. Ohm's Law states that voltage (\(V\)) is equal to the product of current (\(I\)) and resistance (\(R\)), expressed as \(V=I\times R\). The internal resistance of a thermoelectric module grows linearly as the number of thermocouples within the module increases. While adding more thermocouples can expand the surface area that absorbs heat and raises the open-circuit voltage [41], it is worth noting that the temperature differential between the cold and hot ends of the TTEG is inconsistent, as evidenced by Figures 13 and 14. Hence, the TREG's voltage does not correlate directly with the module's length. Nevertheless, the current tends to diminish when the internal resistance increases in direct proportion. The power output formula, \(P=I^{2}\times R_{L}\) (where \(R_{L}=R_{I}\), the load resistance equals the internal resistance), states that, if the increase in internal resistance (caused by an increased number of thermocouples) is smaller than the square of the decrease in current, the maximum output power of the thermoelectric module will decrease. The optimal length for achieving the maximum power output is the point at which the rise in internal resistance matches the square of the reduction in current for the TTEG.