Two-Stage Arrangement

Figure 10f,g show the stress variation along the selected center line locations in the two-stage arrangement with various thermoelectric leg geometries and various materials. The variation of stress in various leg geometries with different p-type semiconductor materials is shown in Figure 10f. Figure 10g shows the stress variation in different leg geometries with various n-type semiconductor materials. In the two-stage arrangement of the thermoelectric module, the variation of stress was the same for the same p-type and n-type semiconductor material with a small variation in their values due to same thermal properties [1]. In the second stage of the two-stage arrangement, the variation trend of stress was similar as of the single stage arrangement case, the minimum value of stress was at the bottom of the leg near the cold junction, and increased towards the hot side of the legs. The intensity of stress for the second stage of the two-stage arrangement with various materials and leg geometries was high near the intersection of the thermoelectric legs of the second stage and the intermediate plate. A height of 0.00096 m to 0.001210 m points to the intermediate plate, and stress variation in the intermediate plate is not considered here, as this section deals only with stress variation in the thermoelectric legs. The variation trend of stress in the first stage of the two-stage arrangement is different compared to the second stage of the two-stage arrangement. In the first stage of the two-stage arrangement, higher values of stress occurred at two locations—the first was at the interconnection of the intermediate plate with the thermoelectric legs of the first stage, and the second one was at the interconnection of the thermoelectric legs of the first stage and the hot side plate. When materials with different properties are connected with each other at a higher temperature, it results in higher stress [1]. However, the values of stress in the thermoelectric legs near the hot side plate were higher than that near the intermediate plate because of a higher temperature of the hot side plate than that of the intermediate plate. As shown in Figure 10f,g, the minimum values of stress for the first stage of the two-stage arrangement occurred at the middle of the thermoelectric legs—at a location between 0.00154 m to 0.00176 m. For the same leg geometry, the SiGe+Bi2Te<sup>3</sup> and SiGe materials showed higher values of stress than the Bi2Te<sup>3</sup> material because the stress variation of the latter for the selected locations was at a temperature difference of 480 ◦C, which is lower than the temperature difference at which stress variation were presented for the SiGe+Bi2Te<sup>3</sup> and SiGe materials. In the case of the

second stage of the two-stage arrangement, the square prism and trapezoidal legs showed higher stress than the cylindrical legs for the same material, while in the case of the first stage of the two-stage arrangement, the cylindrical legs showed higher stress than the square prism and trapezoidal legs for the same material. Thus, for the selected centerline locations on the thermoelectric legs, the intensity of stress was high in the cylindrical legs for the first stage of the two-stage arrangement, but it was low in the cylindrical legs for the second stage of the two-stage arrangement. As the stress effect is considered for the whole leg geometry of the two-stage arrangement, the cylindrical legs were found to have lower stress than the other two leg geometries. *Symmetry* **2020**, *12*, x FOR PEER REVIEW 33 of 41

trapezoidal legs showed higher stress than the cylindrical legs for the same material, while in the case

square prism and trapezoidal legs for the same material. Thus, for the selected centerline locations

#### Single Stage Segmented Arrangement of the first stage of the two-stage arrangement, the cylindrical legs showed higher stress than the

this study.

For the single stage segmented arrangement of the thermoelectric module with two leg geometries and the SiGe+Bi2Te<sup>3</sup> material, the variation of stress along the selected center line locations in the p-type and n-type semiconductors is presented in Figure 10h,i. For the same leg geometry, the same behavior of the stress variation was observed for the p-type and n-type semiconductors [1] with the SiGe+Bi2Te<sup>3</sup> material due to same thermal properties. The maximum values of stress occur at the interconnection of both the materials (at the middle height) as well as at the interconnection of the thermoelectric legs and the hot side plate [1] because of the interconnections of different materials at both the locations. The hot side plate and thermoelectric legs with the different materials were interconnected at a higher temperature with a higher stress at the interconnection. In addition, two different materials with different thermal properties are interconnected at the middle height of the thermoelectric legs at a higher temperature with higher stress at the middle height of the legs. The maximum stress at the middle height and the hot side of the thermoelectric legs are almost equal. Minimum stress was observed at the cold side of the thermoelectric legs, as shown in Figure 10h,i, because the temperature in this location was low, resulting in lower stress. The square legs showed higher values of stress compared to the cylindrical legs due to its sharp edges. on the thermoelectric legs, the intensity of stress was high in the cylindrical legs for the first stage of the two-stage arrangement, but it was low in the cylindrical legs for the second stage of the two-stage arrangement. As the stress effect is considered for the whole leg geometry of the two-stage arrangement, the cylindrical legs were found to have lower stress than the other two leg geometries. Single Stage Segmented Arrangement For the single stage segmented arrangement of the thermoelectric module with two leg geometries and the SiGe+Bi2Te3 material, the variation of stress along the selected center line locations in the p-type and n-type semiconductors is presented in Figure 10h,i. For the same leg geometry, the same behavior of the stress variation was observed for the p-type and n-type semiconductors [1] with the SiGe+Bi2Te3 material due to same thermal properties. The maximum values of stress occur at the interconnection of both the materials (at the middle height) as well as at the interconnection of the thermoelectric legs and the hot side plate [1] because of the interconnections of different materials at both the locations. The hot side plate and thermoelectric legs with the different materials were interconnected at a higher temperature with a higher stress at the interconnection. In addition, two different materials with different thermal properties are interconnected at the middle height of the thermoelectric legs at a higher temperature with higher stress at the middle height of the legs. The maximum stress at the middle height and the hot side of the thermoelectric legs are almost equal.

Figure 11 shows the stress distribution contours of the thermoelectric legs of different configurations in the thermoelectric module. Figure 11 supports the graphical presentation and discussion of stress described for various configurations of the thermoelectric module in Section 3.5. Figure 11 also describes that the intensity of stress is high at the intersection of the hot side plate and the thermoelectric legs for all configurations of the thermoelectric module. In addition, for various leg geometries and materials, the intensity of stress was high at the interconnection and the intermediate plate of thermoelectric legs for the two-stage arrangement and at the middle height of thermoelectric legs for the single stage segmented arrangement. The strain generated in various configurations of the thermoelectric module, as presented in Equation (21), is a replica of the stress. Namely, the behavior of the strain is same as the stress for various configurations of the thermoelectric module. Therefore, detailed discussions on this strain are not considered further in this study. Minimum stress was observed at the cold side of the thermoelectric legs, as shown in Figure 10h,i, because the temperature in this location was low, resulting in lower stress. The square legs showed higher values of stress compared to the cylindrical legs due to its sharp edges. Figure 11 shows the stress distribution contours of the thermoelectric legs of different configurations in the thermoelectric module. Figure 11 supports the graphical presentation and discussion of stress described for various configurations of the thermoelectric module in Section 3.5. Figure 11 also describes that the intensity of stress is high at the intersection of the hot side plate and the thermoelectric legs for all configurations of the thermoelectric module. In addition, for various leg geometries and materials, the intensity of stress was high at the interconnection and the intermediate plate of thermoelectric legs for the two-stage arrangement and at the middle height of thermoelectric legs for the single stage segmented arrangement. The strain generated in various configurations of the thermoelectric module, as presented in Equation (21), is a replica of the stress. Namely, the behavior of the strain is same as the stress for various configurations of the thermoelectric module. Therefore, detailed discussions on this strain are not considered further in

(**a**) Single stage square prism legs (**b**) Single stage cylindrical legs

**Figure 11.** *Cont*.

*Symmetry* **2020**, *12*, x FOR PEER REVIEW 34 of 41

(**c**) Single stage trapezoidal legs Alegs, hotside> Alegs, coldside (**d**) Single stage trapezoidal legs Alegs, coldside> Alegs, hotside

(**e**) Two-stage square prism legs (**f**) Two-stage cylindrical legs

(**g**) Two-stage trapezoidal legs Alegs, coldside> Alegs, hotside (**h**) Segmented square prism legs

**Figure 11.** *Cont*.

*Symmetry* **2020**, *12*, x FOR PEER REVIEW 35 of 41

(**i**) Segmented cylindrical legs

**Figure 11.** Maximum thermal stress contours in (**a**) single stage square prism legs, (**b**) single stage cylindrical legs, (**c**) single stage trapezoidal legs Alegs, hotside> Alegs, coldside, (**d**) single stage trapezoidal legs Alegs, coldside> Alegs, hotside, (**e**) two-stage square prism legs, (**f**) two-stage cylindrical legs, (**g**) two-stage trapezoidal legs Alegs, coldside> Alegs, hotside, (**h**) single stage segmented square prism legs, and (**i**) single stage segmented cylindrical legs. **Figure 11.** Maximum thermal stress contours in (**a**) single stage square prism legs, (**b**) single stage cylindrical legs, (**c**) single stage trapezoidal legs Alegs, hotside > Alegs, coldside, (**d**) single stage trapezoidal legs Alegs, coldside > Alegs, hotside, (**e**) two-stage square prism legs, (**f**) two-stage cylindrical legs, (**g**) two-stage trapezoidal legs Alegs, coldside > Alegs, hotside, (**h**) single stage segmented square prism legs, and (**i**) single stage segmented cylindrical legs.

#### A comparison of the thermoelectric module with combinations of leg geometry, material, *3.6. Selection of Optimum Configuration for the Thermoelectric Module*

*3.6. Selection of Optimum Configuration for the Thermoelectric Module* 

segmentation, and two-stage arrangement based on maximum power, maximum efficiency, and maximum stress at the corresponding maximum operating temperature difference is discussed in this section. Optimum configurations of the thermoelectric module with the combinations of leg geometry, material, segmentation, and two-stage arrangement were selected based on three performance parameters of maximum power, maximum efficiency, and maximum stress. Therefore, higher values of maximum power and maximum efficiency, and lower values of maximum stress of all combinations for the thermoelectric module were suggested as optimum configuration. A graphical presentation of the comparison of various configurations of the thermoelectric module based on maximum temperature difference, maximum efficiency, maximum power, and maximum stress is shown in Figure 12. In Figure 12 and Table 4, numbers 1 to 18 on the abscissa A comparison of the thermoelectric module with combinations of leg geometry, material, segmentation, and two-stage arrangement based on maximum power, maximum efficiency, and maximum stress at the corresponding maximum operating temperature difference is discussed in this section. Optimum configurations of the thermoelectric module with the combinations of leg geometry, material, segmentation, and two-stage arrangement were selected based on three performance parameters of maximum power, maximum efficiency, and maximum stress. Therefore, higher values of maximum power and maximum efficiency, and lower values of maximum stress of all combinations for the thermoelectric module were suggested as optimum configuration.

presents various combinations of the thermoelectric module and the thermoelectric module constructed with the segmented arrangement, cylindrical legs, and combination of the SiGe+Bi2Te3 material show the optimum values of maximum power, maximum efficiency, and maximum stress. The single stage segmented arrangement of the thermoelectric module with the cylindrical leg geometry and the SiGe+Bi2Te3 material showed a combination of higher maximum power, higher maximum efficiency, and lower maximum thermal stress [35]. Therefore, based on the overall effect of power, efficiency, and stress, a thermoelectric module constructed with the segmented arrangement, cylindrical legs, and combination of the SiGe+Bi2Te3 material is suggested as the optimum configuration of the thermoelectric module. The comparison of all 18 configurations of the thermoelectric module was carried out based on computational time/single case of simulation, which is presented in Table 5. From the table, it can be concluded that the two-stage arrangement has a higher computational time, followed by the segmented arrangement and single stage arrangement, respectively. The computational time for the selected optimum configuration of the thermoelectric module constructed with the segmented arrangement, cylindrical legs, and combination of SiGe+Bi2Te3 materials is 2100 s. A graphical presentation of the comparison of various configurations of the thermoelectric module based on maximum temperature difference, maximum efficiency, maximum power, and maximum stress is shown in Figure 12. In Figure 12 and Table 4, numbers 1 to 18 on the abscissa presents various combinations of the thermoelectric module and the thermoelectric module constructed with the segmented arrangement, cylindrical legs, and combination of the SiGe+Bi2Te<sup>3</sup> material show the optimum values of maximum power, maximum efficiency, and maximum stress. The single stage segmented arrangement of the thermoelectric module with the cylindrical leg geometry and the SiGe+Bi2Te<sup>3</sup> material showed a combination of higher maximum power, higher maximum efficiency, and lower maximum thermal stress [35]. Therefore, based on the overall effect of power, efficiency, and stress, a thermoelectric module constructed with the segmented arrangement, cylindrical legs, and combination of the SiGe+Bi2Te<sup>3</sup> material is suggested as the optimum configuration of the thermoelectric module.

The comparison of all 18 configurations of the thermoelectric module was carried out based on computational time/single case of simulation, which is presented in Table 5. From the table, it can be concluded that the two-stage arrangement has a higher computational time, followed by the segmented arrangement and single stage arrangement, respectively. The computational time for the selected optimum configuration of the thermoelectric module constructed with the segmented arrangement, cylindrical legs, and combination of SiGe+Bi2Te<sup>3</sup> materials is 2100 s.


**Table 4.** Comparison of various combinations of thermoelectric module based on maximum temperature difference, maximum power, maximum efficiency, and maximum stress.


**Combi**

**Table 4.** Comparison of various combinations of thermoelectric module based on maximum

1 Single stage SiGe Square 980 0.6438 5.0973 0.962 2 Single stage SiGe Cylindrical 980 0.6472 5.1058 0.912

5 Single stage Bi2Te3 Square 480 0.3132 12.156 0.607 6 Single stage Bi2Te3 Cylindrical 480 0.3145 12.163 0.575 7 Single stage Bi2Te3 Trapezoidal 480 0.3056 12.156 0.611 8 Two-stage SiGe Square 980 0.2590 4.5902 1.62 9 Two-stage SiGe Cylindrical 980 0.2710 4.6881 1.38 10 Two-stage SiGe Trapezoidal 980 0.2537 4.5982 1.62 11 Two-stage Bi2Te3 Square 480 0.1192 10.811 0.822 12 Two-stage Bi2Te3 Cylindrical 480 0.1258 11.060 0.701 13 Two-stage Bi2Te3 Trapezoidal 480 0.1169 10.833 0.821 14 Two-stage SiGe+Bi2Te3 Square 880 0.4627 15.011 1.91 15 Two-stage SiGe+Bi2Te3 Cylindrical 830 0.4302 14.551 1.56 16 Two-stage SiGe+Bi2Te3 Trapezoidal 830 0.4053 14.365 1.81

**Maximum Temperature Difference (°C)** 

**Maximum Power (W)**  **Maximum Efficiency (%)** 

980 0.6275 5.0910 0.968

980 0.6280 5.0964 0.968

**Maximum Stress (GPa)** 

temperature difference, maximum power, maximum efficiency, and maximum stress.

Trapezoidal (Aleg,hotside> Aleg,coldside)

Trapezoidal (Aleg,coldside> Aleg,hotside)

**nation Arrangements Material Leg Geometry** 

3 Single stage SiGe

4 Single stage SiGe

**Figure 12.** Comparison of various configurations of thermoelectric module based on maximum **Figure 12.** Comparison of various configurations of thermoelectric module based on maximum temperature difference, maximum efficiency, maximum power, and maximum stress.

**Table 5.** Comparison of various combinations of thermoelectric module based on computational time/single case of simulation.

temperature difference, maximum efficiency, maximum power, and maximum stress.

