## 4 Model Verification and Analysis  
In order to assess the disparity in output power between the TEG and the FTEG, the study initially examines the test parameters specified by the manufacturer of the TGM1-127-1.0-0.8 thermoelectric chip. At a hot side temperature (Th) of 200 \({}^{\circ}\)C and a cold side temperature (Tc) of 30 \({}^{\circ}\)C, the FTEG chip generates a maximum power output of 5.1 W [40]. Furthermore, to compare the simulated values of the FTEG with the test values provided by the manufacturer, a replica of the TGM1-127-1.0-0.8 thermoelectric chip module was created and tested under identical temperature conditions. This test yielded an open-circuit voltage of 13.42 V and an output power of 5.81 W. Although there is a 0.12% discrepancy between the actual data and simulation results of the TGMI-127-1.0-0.8 thermoelectric chip, this study will base subsequent validations on the simulation results to highlight the achievements in developing the TTEG model. The simulation experiment for the TEG reveals that, by altering the shape of the thermoelectric element (leg) within the TTEG, increasing the contact area with the heat source, and reducing the length of the thermoelectric element, the TTEG produces an open-circuit voltage of 0.32 V and an output power of 9.44 W (as depicted in Figure 7). The inner and outer ceramic layers are maintained at constant temperatures of 200 \({}^{\circ}\)C and 30 \({}^{\circ}\)C, respectively. This signifies a 62.5% surge in output power about the TGM1-127-1.0-0.8 thermoelectric chip module. In order to verify the accuracy of these simulation findings, the temperature on the cold side of the TTEG was maintained at a constant 30 \({}^{\circ}\)C.  
[PERSON] and colleagues [43] improved the thermoelectric performance of FTEG modules by using heat pipes for enhanced cooling, resulting in a 15.49% increase in maximum power density. However, ATEGs and TTEGs still outperform FTEGs in efficiency [44]. Our study's TTEG prototype, the first to apply full-ring thermoelectric elements in fluid waste heat recovery, is still in the experimental phase due to manufacturing limitations. Optimizing the thermoelectric module under the same volume conditions of thermoelectric materials, TTEG showed a 62.5% increase in maximum output power. This suggests that a complete ring structure enhances thermal contact and conversion efficiency, and that improvements in the heat exchanger design can yield greater thermoelectric performance gains than local system enhancements.  
In comparison, the temperature on the hot side was systematically adjusted between 200 \({}^{\circ}\)C and 350 \({}^{\circ}\)C to monitor the corresponding alterations in the thermoelectric module's output power. The output power generated by annular thermoelectric elements, as computed by [PERSON] et al. [41], was compared with the output power values derived from this simulation experiment. According to Figure 8, The estimated maximum power values derived from the thermoelectric material properties in the literature are slightly higher than the calculated values obtained through simulation experiments in this study. This comparison result is consistent with the findings of [PERSON] and others [27]. There was a variance of around 4% between the two, which suggests a substantial similarity and confirms the accuracy of the simulation results. This error margin also aligns with the findings of [PERSON] et al. [40], who tested the performance of the TEC1-12706 thermoelectric chip through experiments involving varying load resistances and compared it to the research results of [PERSON] and others [45].  
Figure 7: This diagram depicts the temperature distribution across a Tubular Thermoelectric Module (TTEG). It illustrates the spatial variation in temperature within the module, highlighting areas of heat concentration and dispersion which are vital for assessing the module’s thermal efficiency and performance characteristics.