**3. Results and Discussion**

The results and discussion of this study includes the validation of numerically predicted values of maximum power, maximum efficiency, and maximum stress with data published by Ma et al. [4] and Al-Merbati et al. [11], as well as the values calculated using the theoretical approach. The comparison of different configurations of the thermoelectric module based on maximum power, maximum efficiency, and maximum stress is considered and the effect of soldering layers on the performance of the thermoelectric module is discussed.

#### *3.1. Validation Symmetry* **2020**, *12*, x FOR PEER REVIEW 14 of 41

The maximum power and maximum stress predicted in this study using ANSYS codes were validated with the corresponding data published by Ma et al. [4] and Al-Merbati et al. [11]. The validation of codes for maximum power and stress are presented and the comparison between the numerical and theoretical values of the various performance parameters are discussed. validation of codes for maximum power and stress are presented and the comparison between the numerical and theoretical values of the various performance parameters are discussed. 3.1.1. Code Validation

#### 3.1.1. Code Validation The maximum power values evaluated in the present study using the thermal electric solver

The maximum power values evaluated in the present study using the thermal electric solver were found to be ±2% of the corresponding maximum power values reported by Ma et al. [4]. Similarly, the maximum stress and centerline stress values evaluated numerically in the present study using the static structure solver were validated at ±5% with the corresponding maximum stress and centerline stress values reported by Al-Merbati et al. [11]. The maximum power and maximum stress evaluated using ANSYS codes in the present study were found to be in the error range of ±5% with the corresponding data of Ma et al. [4] and Al-Merbati et al. [11]. Hence, the application of numerical approach using the validated ANSYS codes is justified for the further analysis. Figure 4a,b show the maximum power and stress validation with the corresponding previous studies of Ma et al. [4] and Al-Merbati et al. [11]. were found to be ±2% of the corresponding maximum power values reported by Ma et al. [4]. Similarly, the maximum stress and centerline stress values evaluated numerically in the present study using the static structure solver were validated at ±5% with the corresponding maximum stress and centerline stress values reported by Al-Merbati et al. [11]. The maximum power and maximum stress evaluated using ANSYS codes in the present study were found to be in the error range of ±5% with the corresponding data of Ma et al. [4] and Al-Merbati et al. [11]. Hence, the application of numerical approach using the validated ANSYS codes is justified for the further analysis. Figure 4a,b show the maximum power and stress validation with the corresponding previous studies of Ma et al. [4] and Al-Merbati et al. [11].

(**a**) Maximum power

**Figure 4.** *Cont*.

range.

**Figure 4.** Validation with previous study for (**a**) maximum power and (**b**) stress. **Figure 4.** Validation with previous study for (**a**) maximum power and (**b**) stress.

#### 3.1.2. Validation of Numerical Results with Theoretical Results 3.1.2. Validation of Numerical Results with Theoretical Results

The theoretical maximum power for the thermoelectric module with leg geometries, materials, and arrangements were calculated using correlations with Table 3. The maximum power for various configurations of the thermoelectric module were calculated using Equations (8) and (13). Similarly, the maximum efficiency for various configurations of the thermoelectric module was calculated theoretically using Equations (9) and (15). The simulated values of maximum power as well as maximum efficiency were calculated using Equations (22) and (23), respectively, which were compared with the corresponding theoretical values of maximum power and maximum efficiency. Figure 5a,b show the comparison between the numerical and theoretical results of maximum power and maximum efficiency, respectively, for the various configurations of the thermoelectric module. The numerical values of maximum power as well as maximum efficiency were validated with the corresponding theoretical values at ± 5% with the entire temperature difference range for all configurations of the thermoelectric module. The theoretical maximum power for the thermoelectric module with leg geometries, materials, and arrangements were calculated using correlations with Table 3. The maximum power for various configurations of the thermoelectric module were calculated using Equations (8) and (13). Similarly, the maximum efficiency for various configurations of the thermoelectric module was calculated theoretically using Equations (9) and (15). The simulated values of maximum power as well as maximum efficiency were calculated using Equations (22) and (23), respectively, which were compared with the corresponding theoretical values of maximum power and maximum efficiency. Figure 5a,b show the comparison between the numerical and theoretical results of maximum power and maximum efficiency, respectively, for the various configurations of the thermoelectric module. The numerical values of maximum power as well as maximum efficiency were validated with the corresponding theoretical values at ±5% with the entire temperature difference range for all configurations of the thermoelectric module.

The maximum stress values were calculated theoretically using Equations (18) to (20). The calculated theoretical values of the maximum stress were compared with the corresponding predicted values. The comparison of theoretical and numerical results of maximum stress for the single stage arrangement of the thermoelectric module with three leg geometries and two materials is shown in Figure 5c. The equations show that the theoretical values of stress are not dependent on leg geometry; hence, the stress values are same for all leg geometries. However, different materials have different theoretical values of maximum stress. For the same material, the numerical stress values for cylindrical legs show closer agreement, whereas square prism legs as well as trapezoidal legs with two-area configurations show almost equal error with the corresponding theoretical stress values. For the same material, the numerically predicted maximum stress for all the leg geometries was found at ±7% with the corresponding theoretical stress with the entire temperature difference The maximum stress values were calculated theoretically using Equations (18)–(20). The calculated theoretical values of the maximum stress were compared with the corresponding predicted values. The comparison of theoretical and numerical results of maximum stress for the single stage arrangement of the thermoelectric module with three leg geometries and two materials is shown in Figure 5c. The equations show that the theoretical values of stress are not dependent on leg geometry; hence, the stress values are same for all leg geometries. However, different materials have different theoretical values of maximum stress. For the same material, the numerical stress values for cylindrical legs show closer agreement, whereas square prism legs as well as trapezoidal legs with two-area configurations show almost equal error with the corresponding theoretical stress values. For the same material, the numerically predicted maximum stress for all the leg geometries was found at ±7% with the corresponding theoretical stress with the entire temperature difference range.


**Table 3.** Maximum power correlations derived for various leg geometries, materials, and arrangements. **Table 3.** Maximum power correlations derived for various leg geometries, materials, and

*Symmetry* **2020**, *12*, x FOR PEER REVIEW 16 of 41

SL means soldering layer.

(**a**) Maximum power of all the leg geometries

**Figure 5.** *Cont*.

*3.2. Optimum Temperature* 

*Symmetry* **2020**, *12*, x FOR PEER REVIEW 17 of 41

(**b**) Maximum efficiency of all the leg geometries

(**c**) Maximum stress for single stage arrangement

**Figure 5.** Comparison of numerical and theoretical results for (**a**) maximum power, (**b**) maximum efficiency, and (**c**) maximum stress. **Figure 5.** Comparison of numerical and theoretical results for (**a**) maximum power, (**b**) maximum efficiency, and (**c**) maximum stress.

#### *3.2. Optimum Temperature* melting point temperature of SiGe or Bi2Te3 material. If the two-stage and segmented arrangements

*3.3. Maximum Power* 

The single stage as well as two-stage arrangements with either SiGe or Bi2Te<sup>3</sup> material were operated at maximum temperature differences of 980 ◦C and 480 ◦C, respectively, lower than the melting point temperature of SiGe or Bi2Te<sup>3</sup> material. If the two-stage and segmented arrangements of the thermoelectric module with SiGe+Bi2Te<sup>3</sup> material are operated at the maximum temperature difference of 980 ◦C, the operating temperature of SiGe material is below its melting point temperature, but the operating temperature of Bi2Te<sup>3</sup> material is higher than its melting point temperature limit. Therefore, optimum temperature was introduced for the two-stage and segmented arrangements of the thermoelectric module with SiGe+Bi2Te<sup>3</sup> material in order to operate them below the melting point temperature limit without failure. The optimum temperature is the hot junction temperature of the second stage for the two-stage arrangement of the thermoelectric module with SiGe+Bi2Te<sup>3</sup> material and the interface temperature between two segments for the segmented arrangement of the thermoelectric module with SiGe+Bi2Te<sup>3</sup> material [4,17]. The maximum value of the optimum temperature is the melting point temperature of Bi2Te<sup>3</sup> material, which is around 585 ◦C. Figure <sup>6</sup> shows the optimum temperature values of the two-stage and segmented arrangements of the thermoelectric module with SiGe+Bi2Te<sup>3</sup> material at various temperature difference conditions. The optimum temperature increases linearly with the temperature difference, as shown in the figure. As presented in the figure, in order to maintain optimum temperature of 585 ◦C or below, the two-stage arrangement with square prism legs are operated at the maximum temperature difference of 880 ◦C or lower, the two-stage arrangement with cylindrical and trapezoidal legs are operated at the maximum temperature difference of 830 ◦C or lower, and the segmented arrangement with square prism and cylindrical legs are operated at the maximum temperature difference of 730 ◦C or lower. of the thermoelectric module with SiGe+Bi2Te3 material are operated at the maximum temperature difference of 980 °C, the operating temperature of SiGe material is below its melting point temperature, but the operating temperature of Bi2Te3 material is higher than its melting point temperature limit. Therefore, optimum temperature was introduced for the two-stage and segmented arrangements of the thermoelectric module with SiGe+Bi2Te3 material in order to operate them below the melting point temperature limit without failure. The optimum temperature is the hot junction temperature of the second stage for the two-stage arrangement of the thermoelectric module with SiGe+Bi2Te3 material and the interface temperature between two segments for the segmented arrangement of the thermoelectric module with SiGe+Bi2Te3 material [4,17]. The maximum value of the optimum temperature is the melting point temperature of Bi2Te3 material, which is around 585 °C. Figure 6 shows the optimum temperature values of the two-stage and segmented arrangements of the thermoelectric module with SiGe+Bi2Te3 material at various temperature difference conditions. The optimum temperature increases linearly with the temperature difference, as shown in the figure. As presented in the figure, in order to maintain optimum temperature of 585 °C or below, the twostage arrangement with square prism legs are operated at the maximum temperature difference of 880 °C or lower, the two-stage arrangement with cylindrical and trapezoidal legs are operated at the maximum temperature difference of 830 °C or lower, and the segmented arrangement with square prism and cylindrical legs are operated at the maximum temperature difference of 730 °C or lower. For the single stage arrangement of the thermoelectric module with SiGe material, trapezoidal legs with Alegs, hotside> Alegs, coldside showed higher average stress compared to trapezoidal legs with Alegs, coldside> Alegs, hotside [11], which is discussed in detail in Sections 3.5.1. Therefore, out of the two

*Symmetry* **2020**, *12*, x FOR PEER REVIEW 18 of 41

For the single stage arrangement of the thermoelectric module with SiGe material, trapezoidal legs with Alegs, hotside > Alegs, coldside showed higher average stress compared to trapezoidal legs with Alegs, coldside > Alegs, hotside [11], which is discussed in detail in Section 3.5.1. Therefore, out of the two configurations of trapezoidal legs, only trapezoidal legs with Alegs, coldside > Alegs, hotside were considered for maximum power, maximum efficiency, and maximum stress analyses for single stage arrangement with Bi2Te<sup>3</sup> material and two-stage arrangement with SiGe, Bi2Te3, and SiGe+Bi2Te<sup>3</sup> materials. configurations of trapezoidal legs, only trapezoidal legs with Alegs, coldside> Alegs, hotside were considered for maximum power, maximum efficiency, and maximum stress analyses for single stage arrangement with Bi2Te3 material and two-stage arrangement with SiGe, Bi2Te3, and SiGe+Bi2Te3 materials.

**Figure 6.** Optimum temperature variation with respect to temperature difference.

**Figure 6.** Optimum temperature variation with respect to temperature difference.

#### *3.3. Maximum Power* arrangements of the thermoelectric module with various leg geometries and materials are presented

The comparison of maximum power for the single stage, two-stage, and single stage segmented arrangements of the thermoelectric module with various leg geometries and materials are presented in this section. in this section. 3.3.1. Single Stage Arrangement The comparison of maximum power with temperature difference for the single stage

The comparison of maximum power for the single stage, two-stage, and single stage segmented

#### 3.3.1. Single Stage Arrangement arrangement of the thermoelectric module with three leg geometries and two semiconductor

The comparison of maximum power with temperature difference for the single stage arrangement of the thermoelectric module with three leg geometries and two semiconductor materials is shown in Figure 7a. For the same material, leg geometry had no significant effect on maximum power because of the same internal resistance as well as the same optimum voltage values [1,6,10]. At the same temperature difference, the optimum voltage of the thermoelectric module was the same and leg geometry had the same volume and base area with equal internal resistance for all the legs. However, materials with high *ZT* showed higher maximum power than those with lower *ZT* [4,17]. Hence, for the same leg geometry, the Bi2Te<sup>3</sup> material showed higher maximum power than the SiGe material at the same temperature difference. This behavior was observed up to a temperature difference of 480 ◦C as the Bi2Te<sup>3</sup> material could not be operated at a higher temperature difference because operation above that temperature difference increased the hot side temperature above its melting point. Hence, beyond the temperature difference of 480 ◦C, the SiGe material showed an increase in maximum power with the highest value at the temperature difference of 980 ◦C. For all the materials and leg geometries with the single stage arrangement of the thermoelectric module, the maximum power increased with the temperature difference. All the leg geometries with the SiGe material showed maximum power of 0.65 W at a temperature difference of 980 ◦C, while all the leg geometries with the Bi2Te<sup>3</sup> material showed maximum power of 0.31 W at a temperature difference of 480 ◦C. materials is shown in Figure 7a. For the same material, leg geometry had no significant effect on maximum power because of the same internal resistance as well as the same optimum voltage values [1,6,10]. At the same temperature difference, the optimum voltage of the thermoelectric module was the same and leg geometry had the same volume and base area with equal internal resistance for all the legs. However, materials with high ത showed higher maximum power than those with lower ത [4,17]. Hence, for the same leg geometry, the Bi2Te3 material showed higher maximum power than the SiGe material at the same temperature difference. This behavior was observed up to a temperature difference of 480 °C as the Bi2Te3 material could not be operated at a higher temperature difference because operation above that temperature difference increased the hot side temperature above its melting point. Hence, beyond the temperature difference of 480 °C, the SiGe material showed an increase in maximum power with the highest value at the temperature difference of 980 °C. For all the materials and leg geometries with the single stage arrangement of the thermoelectric module, the maximum power increased with the temperature difference. All the leg geometries with the SiGe material showed maximum power of 0.65 W at a temperature difference of 980 °C, while all the leg geometries with the Bi2Te3 material showed maximum power of 0.31 W at a temperature difference of 480 °C.

(**a**) Single stage arrangement

**Figure 7.** *Cont*.

(**b**) Two-stage arrangement

(**c**) Single stage segmented arrangement

**Figure 7.** Maximum power for (**a**) single stage arrangement, (**b**) two-stage arrangement, and (**c**) single stage segmented arrangement. **Figure 7.** Maximum power for (**a**) single stage arrangement, (**b**) two-stage arrangement, and (**c**) single stage segmented arrangement.

#### 3.3.2. Two-stage Arrangement 3.3.2. Two-Stage Arrangement

The variation of maximum power with the temperature difference for the two-stage thermoelectric module with various leg geometries and materials is shown in Figure 7b. In the twostage arrangement of the thermoelectric module with the same leg geometry, a combination of materials in the first stage made up of SiGe material and the second stage made up of Bi2Te3 material showed significant enhancement in maximum power, compared to both stages either made of SiGe The variation of maximum power with the temperature difference for the two-stage thermoelectric module with various leg geometries and materials is shown in Figure 7b. In the two-stage arrangement of the thermoelectric module with the same leg geometry, a combination of materials in the first stage made up of SiGe material and the second stage made up of Bi2Te<sup>3</sup> material showed significant enhancement in maximum power, compared to both stages either made of SiGe or Bi2Te<sup>3</sup> alone [17,18].

In the case of SiGe, Bi2Te3, and SiGe+Bi2Te<sup>3</sup> material, the square prism and trapezoidal legs with Alegs, coldside > Alegs, hotside showed similar maximum power with the temperature difference but the cylindrical legs showed enhancement in maximum power at the corresponding same temperature difference, although the degree of increase was less [1,6,10]. In the case of the SiGe+Bi2Te<sup>3</sup> material, maximum power of 0.46 W was obtained for square prism legs at a temperature difference of 880 ◦C, whereas the cylindrical legs showed 0.43 W and trapezoidal legs showed 0.41 W maximum power at a temperature difference of 830 ◦C. For the SiGe material and temperature difference of 980 ◦C, the square prism and trapezoidal legs showed maximum power of 0.25 W and the cylindrical legs showed maximum power of 0.27 W. For the Bi2Te<sup>3</sup> material and temperature difference of 480 ◦C, the square prism legs and trapezoidal legs showed maximum power of 0.12 W and the cylindrical legs showed maximum power of 0.13 W. The maximum power increased with the temperature difference in all the cases, as shown in Figure 7b.
