3.5.1. Single Stage Arrangement

The maximum stress variation with temperature difference for the single stage thermoelectric module with various leg geometries and various materials is shown in Figure 9a. For the same leg geometry, the Bi2Te<sup>3</sup> material showed higher values of maximum stress compared to the SiGe material till a temperature difference of 480 ◦C; above that, the SiGe material showed increase in maximum stress with the highest value at a temperature difference of 980 ◦C due to the increase in temperatures. The coefficient of the thermal expansion of the Bi2Te<sup>3</sup> material was higher than that of the SiGe material. Therefore, the thermal stress induced in the Bi2Te<sup>3</sup> material was higher than that induced in the SiGe material at the same temperature difference condition. Further, for the same material, the cylindrical legs showed fewer maximum stress than the other two leg geometries [1,10] and square prism legs, trapezoidal legs with Alegs, coldside > Alegs, hotside, and Alegs, hotside > Alegs, coldside showed almost equal values of maximum stress due to the same geometrical structure with sharp corner edges. Square prism and trapezoidal legs have sharp edges, which are absent in cylindrical legs; therefore, the latter show lower thermal stress. Square prism legs, trapezoidal legs with Alegs, coldside > Alegs, hotside and trapezoidal legs with Alegs, hotside > Alegs, coldside with the SiGe material showed average stress of approximately 39 MPa, 38 MPa, and 41 MPa, respectively. The intensity of stress was high near the hot junction plate [1,10,11] and in the case of trapezoidal legs with Alegs, coldside > Alegs, hotside, the area of legs exposed to the hot side plate was less compared to the area of legs exposed to the hot side plate by the square prism legs and trapezoidal legs with Alegs, hotside > Alegs, coldside. Therefore, the average stress induces in the trapezoidal legs with Alegs, coldside > Alegs, hotside was lower than that of the square prism and trapezoidal legs with Alegs, hotside > Alegs, coldside [11]. Based on the area of legs exposed to the hot junction plate and the average stress values, the square prism legs and trapezoidal legs with Alegs, coldside > Alegs, hotside was preferred over the trapezoidal legs with <sup>A</sup>legs, hotside <sup>&</sup>gt; <sup>A</sup>legs, coldside. For the SiGe material and at a temperature difference of 980 ◦C, the square prism legs and trapezoidal legs with Alegs, coldside > Alegs, hotside and Alegs, hotside > Alegs, coldside showed maximum stress of 0.96 GPa and the cylindrical legs showed maximum stress of 0.91 GPa. Similarly, for the Bi2Te<sup>3</sup> material and at a temperature difference of 480 ◦C, the square prism legs and trapezoidal legs with Alegs, coldside > Alegs, hotside showed a maximum stress value of 0.61 GPa, whereas the cylindrical legs showed maximum stress of 0.58 GPa. The maximum stress of the single stage arrangement of the thermoelectric module with all leg geometries and materials increased linearly with the temperature difference.

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(**a**) Single stage arrangement

(**b**) Two-stage arrangement

**Figure 9.** *Cont*.

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(**c**) Single stage segmented arrangement

**Figure 9.** Maximum stress for (**a**) single stage arrangement (**b**) two-stage arrangement, and (**c**) single stage segmented arrangement. **Figure 9.** Maximum stress for (**a**) single stage arrangement (**b**) two-stage arrangement, and (**c**) single stage segmented arrangement.

#### 3.5.2. Two-stage Arrangement 3.5.2. Two-Stage Arrangement

The maximum stress shows linear variation with the temperature difference for the two-stage thermoelectric module with various leg geometries and materials, as shown in Figure 9b. In the case of SiGe, Bi2Te3 and SiGe+Bi2Te3 materials, the square prism and trapezoidal legs showed almost same maximum stress variation with temperature difference, which were higher than the corresponding maximum stress values for the cylindrical legs [1,10]. The cylindrical legs have a smooth geometrical structure; hence, they presented lower maximum stress compared to the other two leg geometries. For the same leg geometry, the SiGe+Bi2Te3 material showed higher maximum stress, followed by the Bi2Te3 and SiGe materials. The Bi2Te3 material showed higher thermal stress than the SiGe material due to its higher coefficient of thermal expansion. The SiGe+Bi2Te3 material showed higher thermal stress than the Bi2Te3 material because when two different materials with different thermal properties are connected at higher temperature conditions, it results in higher stress generation [1]. For the SiGe material and at temperature difference of 980 °C, the square prism and trapezoidal legs showed maximum stress of 1.62 GPa and the cylindrical legs showed maximum stress of 1.38 GPa. Similarly, for the Bi2Te3 material and at a temperature difference of 480 °C, the square prism and trapezoidal legs showed maximum stress of 0.82 GPa and the cylindrical legs showed maximum stress of 0.7 GPa. For the SiGe+Bi2Te3 material, the square prism legs showed maximum stress of 1.91 GPa at a temperature difference of 880 °C and the cylindrical as well as trapezoidal legs showed maximum stress of 1.56 GPa and 1.81 GPa, respectively, at a temperature difference of 830 °C. The maximum stress shows linear variation with the temperature difference for the two-stage thermoelectric module with various leg geometries and materials, as shown in Figure 9b. In the case of SiGe, Bi2Te<sup>3</sup> and SiGe+Bi2Te<sup>3</sup> materials, the square prism and trapezoidal legs showed almost same maximum stress variation with temperature difference, which were higher than the corresponding maximum stress values for the cylindrical legs [1,10]. The cylindrical legs have a smooth geometrical structure; hence, they presented lower maximum stress compared to the other two leg geometries. For the same leg geometry, the SiGe+Bi2Te<sup>3</sup> material showed higher maximum stress, followed by the Bi2Te<sup>3</sup> and SiGe materials. The Bi2Te<sup>3</sup> material showed higher thermal stress than the SiGe material due to its higher coefficient of thermal expansion. The SiGe+Bi2Te<sup>3</sup> material showed higher thermal stress than the Bi2Te<sup>3</sup> material because when two different materials with different thermal properties are connected at higher temperature conditions, it results in higher stress generation [1]. For the SiGe material and at temperature difference of 980 ◦C, the square prism and trapezoidal legs showed maximum stress of 1.62 GPa and the cylindrical legs showed maximum stress of 1.38 GPa. Similarly, for the Bi2Te<sup>3</sup> material and at a temperature difference of 480 ◦C, the square prism and trapezoidal legs showed maximum stress of 0.82 GPa and the cylindrical legs showed maximum stress of 0.7 GPa. For the SiGe+Bi2Te<sup>3</sup> material, the square prism legs showed maximum stress of 1.91 GPa at a temperature difference of 880 ◦C and the cylindrical as well as trapezoidal legs showed maximum stress of 1.56 GPa and 1.81 GPa, respectively, at a temperature difference of 830 ◦C.

#### 3.5.3. Single Stage Segmented Arrangement 3.5.3. Single Stage Segmented Arrangement

Maximum stress varies linearly with temperature difference for the single stage segmented arrangement of the thermoelectric module with both the leg geometries and the SiGe+Bi2Te3 material, as shown in Figure 9c. The cylindrical leg geometry showed the lowest values of maximum stress, Maximum stress varies linearly with temperature difference for the single stage segmented arrangement of the thermoelectric module with both the leg geometries and the SiGe+Bi2Te<sup>3</sup> material, as shown in Figure 9c. The cylindrical leg geometry showed the lowest values of maximum stress, compared to the square prism legs [1,10] over the entire temperature difference range due to no

sharp edges and a smooth geometrical structure. At a temperature difference of 730 ◦C and for the SiGe+Bi2Te<sup>3</sup> material, the square prism legs showed maximum stress of 0.72 GPa and the cylindrical legs showed maximum stress of 0.69 GPa. edges and a smooth geometrical structure. At a temperature difference of 730 °C and for the SiGe+Bi2Te3 material, the square prism legs showed maximum stress of 0.72 GPa and the cylindrical legs showed maximum stress of 0.69 GPa.

### 3.5.4. Stress Variation along Thermoelectric Leg Height 3.5.4. Stress Variation along Thermoelectric Leg Height

Stress variation along the selected locations of the thermoelectric module with various combinations of leg geometries, materials, and arrangements are discussed here. The selected centerline locations in the vertical direction for various arrangements of the thermoelectric module with square prism legs are shown in Figure 10a–c. In Figure 10a–c, the dotted line of locations on the thermoelectric legs shows the direction from the bottom of the thermoelectric legs to the top. Similar locations are selected in the cylindrical and trapezoidal leg geometries. The variation of stress for the selected locations were considered at the maximum operating temperature difference for each combination. Stress variation along the selected locations of the thermoelectric module with various combinations of leg geometries, materials, and arrangements are discussed here. The selected centerline locations in the vertical direction for various arrangements of the thermoelectric module with square prism legs are shown in Figure 10a–c. In Figure 10a–c, the dotted line of locations on the thermoelectric legs shows the direction from the bottom of the thermoelectric legs to the top. Similar locations are selected in the cylindrical and trapezoidal leg geometries. The variation of stress for the selected locations were considered at the maximum operating temperature difference for each combination.

(**a**) Single stage square legs

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(**b**) Two-stage square legs

(**c**) Single stage segmented stage square legs

(**d**) p-type semiconductors in single stage arrangement

(**e**) n-type semiconductors in single stage arrangement

(**f**) p-type semiconductors in two-stage arrangement

(**g**) n-type semiconductors in two-stage arrangement

(**h**) p-type semiconductors in single stage segmented arrangement

(**i**) n-type semiconductors in single stage segmented arrangement

**Figure 10.** Selected centerline locations in the vertical direction for (**a**) single stage square legs (**b**), two-stage square legs (**c**), single stage segmented stage square legs and variation of the stress along the selected centerline locations of (**d**) p type semiconductors in single stage arrangement, (**e**) n-type semiconductors in single stage arrangement, (**f**) p type semiconductors in two-stage arrangement, (**g**) n-type semiconductors in two-stage arrangement, (**h**) p type semiconductors in single stage segmented arrangement, and (**i**) n-type semiconductors in single stage segmented arrangement. **Figure 10.** Selected centerline locations in the vertical direction for (**a**) single stage square legs (**b**), two-stage square legs (**c**), single stage segmented stage square legs and variation of the stress along the selected centerline locations of (**d**) p type semiconductors in single stage arrangement, (**e**) n-type semiconductors in single stage arrangement, (**f**) p type semiconductors in two-stage arrangement, (**g**) n-type semiconductors in two-stage arrangement, (**h**) p type semiconductors in single stage segmented arrangement, and (**i**) n-type semiconductors in single stage segmented arrangement.
