*2.2. Mesh Convergence*

Mesh dependency of maximum power, maximum efficiency, and maximum stress for the single stage thermoelectric module constructed with square prism legs and SiGe material is shown in Figure 2a–c, respectively. Maximum power and maximum efficiency showed negligible variations within ±0.00008% and ±0.00002%, respectively, for the selected mesh numbers; however, maximum stress showed a variation within ±0.7% above mesh number of 115,401. Hence, the mesh with 115,401 nodes was selected for the single stage thermoelectric module with square prism legs and SiGe material to evaluate maximum power, maximum efficiency, as well as maximum stress. Similarly, mesh dependency was conducted for the thermoelectric module with combinations of leg geometries, materials, and arrangements. The number of nodes for the same arrangement of thermoelectric module with materials and leg geometries are the same, but the number of nodes for varying arrangements are different. Therefore, the number of nodes for various combinations of leg geometries and materials after mesh dependency was selected as 115,401 for the single stage arrangement, 540,706 for the two-stage arrangement, and 242,138 for the single stage segmented arrangement. Maximum power, maximum efficiency, and maximum stress were converged as ±1% above 115,401 (number of nodes) for the single stage arrangement, 540,706 for the two-stage arrangement, and 242,138 for the single stage segmented arrangement of the thermoelectric module with leg geometries and materials [28]. Meshing for the single stage, two-stage, and segmented arrangements of the thermoelectric module with square prism leg geometry is shown in Figure 3a–c, respectively.

(**a**) Maximum power

(**b**) Maximum efficiency

**Figure 2.** *Cont*.

stress.

*Symmetry* **2020**, *12*, x FOR PEER REVIEW 8 of 41

(**c**) Maximum stress **Figure 2.** Mesh independency test for (**a**) maximum power, (**b**) maximum efficiency, and (**c**) maximum

**Figure 2.** Mesh independency test for (**a**) maximum power, (**b**) maximum efficiency, and (**c**) maximum **Figure 2.** Mesh independency test for (**a**) maximum power, (**b**) maximum efficiency, and (**c**) maximum stress. stress.

(**a**) Single stage arrangement

(**a**) Single stage arrangement **Figure 3.** *Cont*.

*Symmetry* **2020**, *12*, x FOR PEER REVIEW 9 of 41

(**b**) Two-stage arrangement

(**c**) Single stage segmented

**Figure 3.** Meshing for (**a**) single stage, (**b**) two-stage, and (**c**) single stage segmented arrangements with square prism leg geometries. **Figure 3.** Meshing for (**a**) single stage, (**b**) two-stage, and (**c**) single stage segmented arrangements with square prism leg geometries.

#### *2.3. Boundary Conditions 2.3. Boundary Conditions*

In order to predict temperature distribution, power, conversion efficiency, and thermal stress, a thermoelectric module with various configurations was investigated under four boundary conditions, temperatures of hot and cold junctions, and voltage conditions at high and low potential sides. These boundary conditions were applied to the four different faces, as shown in Figure 1a–c. Temperature of the cold junction was kept constant at 20 °C and voltage at the low potential side was fixed at 0 V. Temperature of the hot junction and voltage at the high potential were not constant for the various configurations of the thermoelectric module. The boundary conditions for the hot junction temperature and the high potential voltage of leg geometries, materials, and arrangements were shown in Table 2. The hot side temperature was varied with a step size of 50 °C and the high potential voltage with a step size of 0.001 V. The maximum temperatures for the hot junction of various configurations were decided based on the melting point temperature of the corresponding In order to predict temperature distribution, power, conversion efficiency, and thermal stress, a thermoelectric module with various configurations was investigated under four boundary conditions, temperatures of hot and cold junctions, and voltage conditions at high and low potential sides. These boundary conditions were applied to the four different faces, as shown in Figure 1a–c. Temperature of the cold junction was kept constant at 20 ◦C and voltage at the low potential side was fixed at 0 V. Temperature of the hot junction and voltage at the high potential were not constant for the various configurations of the thermoelectric module. The boundary conditions for the hot junction temperature and the high potential voltage of leg geometries, materials, and arrangements were shown in Table 2. The hot side temperature was varied with a step size of 50 ◦C and the high potential voltage with a step size of 0.001 V. The maximum temperatures for the hot junction of various configurations were decided based on the melting point temperature of the corresponding material

material of construction. The high potential voltages varied from 0 V to voltage value at which current

of construction. The high potential voltages varied from 0 V to voltage value at which current and power became zero. During variations in high potential voltages, there existed an optimum voltage, which showed maximum power and maximum efficiency. Optimum voltage values were different with different configurations of the thermoelectric module because the variation ranges of the voltage were different for different configurations. In addition, the optimum voltage varied with the temperature difference for each configuration of the thermoelectric module. Maximum power and maximum efficiency were simulated at the optimum voltage for each configuration of the thermoelectric module at different temperature difference conditions. Apart from these four boundary conditions, the convection boundary condition with heat transfer coefficient of 1 × 10−<sup>6</sup> W/mm<sup>2</sup> ◦C was applied to all remaining faces of each configuration of the thermoelectric module. The heat transfer from the surfaces with the convection boundary condition to its surroundings was negligible.


