# Power Generation, Efficiency and Thermal Stress of Thermoelectric Module with Leg Geometry, Material, Segmentation and Two-Stage Arrangement

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## Abstract

**:**

_{2}Te

_{3}material was suggested as the best combination with maximum power of 0.73 W, maximum efficiency of 13.2%, and maximum thermal stress of 0.694 GPa.

## 1. Introduction

_{2}Te

_{3}material in order to optimize its mechanical performance [15]. Two-stage and three-stage thermoelectric generators were designed by Zhang et al. [16] and Kanimba et al. [17] in order to achieve high conversion efficiency. Wilbrecht et al. presented a two-stage thermoelectric module with 44.2% higher efficiency and 22.8% lower power output compared to the single stage thermoelectric module for diesel electric locomotives [18]. For a cryogenic nitrogen-based waste cold heat recovery system, Weng et al. presented four two-layer cascaded thermoelectric generators with power generation of 0.93 W [19]. Chen et al. concluded that multistage thermoelectric generators show better performance than single stage thermoelectric generators [20]. Kaibe et al. developed a cascaded thermoelectric generator with conversion efficiency of 12.1% [21]. Thermoelectric generators arranged thermally as well as electrically in three different series/parallel connections were compared by Almeida et al. based on the equivalent $Z\overline{T}$ [22]. Ibrahim et al. optimized the shape and length size of thermoelectric legs to enhance the power output and conversion efficiency of a thermoelectric generator [23]. Under steady state temperature conditions, Turenne et al. showed that thermal stresses in larger thermoelectric generators consisting of a number of thermoelectric modules were reduced due to the provision of soldering layers [24]. Kunal et al. showed the applicability of thermoelectric modules in waste heat recovery for lower power generation [25].

_{2}Te

_{3}, and a combination of SiGe+Bi

_{2}Te

_{3}materials in ANSYS 19.1 commercial software under various boundary conditions of temperature difference and voltage load. The comparison was made for maximum power, maximum efficiency, and maximum thermal stress. Additionally, in order to achieve higher power, higher efficiency, and lower thermal stress, an optimum configuration was suggested with leg geometry, material, segmentation, and two-stage arrangement.

## 2. Methodology

_{2}Te

_{3}, and combination of SiGe+Bi

_{2}Te

_{3}materials were modelled in order to investigate the behavior of power generation, efficiency, and stress of the thermoelectric module. Modelled configurations of the thermoelectric module were analyzed numerically using the thermal-electric and static structure solvers of ANSYS 19.1 under various temperature difference and voltage load conditions.

#### 2.1. Geometry and Material Specifications

_{legs, coldside}> A

_{legs, hotside}and A

_{legs, hotside}> A

_{legs, coldside}were also modelled. The larger side area had dimensions of 1.2 mm × 1.0 mm and the smaller side area had dimensions of 0.8 mm × 1.0 mm. The trapezoidal legs were also modelled with same cross-section area, height, spacing, and volume as that of the square prism legs and cylindrical legs. However, the cross-section areas at the base of the square prism and cylindrical legs were identical to the cross-section area at the center of the trapezoidal legs. The spacing in square prism and cylindrical legs was constant for leg height, which varied in the trapezoidal legs with 0.8 mm spacing at the larger side area, continuously increasing toward the smaller side area. Each pair of p- and n- type thermoelectric legs was sandwiched between three copper conducting plates—two at the bottom and one at the top. The bottom two plates had the same dimensions: 1.4 mm × 1.4 mm area and 0.25 mm thickness. In order to construct various leg geometries with the same cross-section area, spacing, and volume, the area of the top conducting plate was kept different in each case. The top conducting plates with square prism legs, cylindrical legs, and trapezoidal legs with A

_{legs, hotside}> A

_{legs, coldside}and A

_{legs, coldside}> A

_{legs, hotside}had areas of 2.8 mm × 1.0 mm [4], 3.06 mm × 1.0 mm, 3.2 mm × 1.0 mm and 2.8 mm × 1.0 mm, respectively. However, the thickness of the top conducting plate was 0.25 mm in each case. The thermoelectric legs in the single stage configuration were either made of SiGe or Bi

_{2}Te

_{3}material in the two-stage configurations; the thermoelectric legs of both the stages were made of SiGe or Bi

_{2}Te

_{3}or a combination of SiGe+Bi

_{2}Te

_{3}materials. For the segmented configuration, the thermoelectric legs were made of a combination of SiGe+Bi

_{2}Te

_{3}materials. In the two-stage arrangement with combination of SiGe+Bi

_{2}Te

_{3}, the upper stage was made of SiGe material and the bottom stage of Bi

_{2}Te

_{3}material. Similarly, in the segmented legs, the upper half was made of SiGe material and the lower half of Bi

_{2}Te

_{3}material. For a combination of SiGe+Bi

_{2}Te

_{3}materials, the SiGe material was used near the hot side and the Bi

_{2}Te

_{3}material near the cold side due to their melting point temperatures. The configurations of the thermoelectric module were modelled and compared without considering soldering layers. However, in order to show the effect of soldering layers on the performance characteristics of the thermoelectric module, the single stage and two-stage arrangements with square prism legs and Bi

_{2}Te

_{3}material as well as the two-stage arrangements with square prism legs and SiGe+Bi

_{2}Te

_{3}material were modelled separately with the soldering layers. Soldering layers were provided between the thermoelectric legs and the hot plate as well as the thermoelectric legs and the cold plates with an area of 1.0 mm × 1.0 mm and thickness of 0.08 mm [4]. Hence, when soldering layers were provided in those cases, the height of the legs reduced to 0.8 mm. The soldering layers were made up of 63Sn-37Pb material. The two-stage configuration carried an intermediate plate between both the stages of the thermoelectric modules. An intermediate plate in each case of the two-stage arrangement was made up of ceramic material having an area of 3.6 mm × 1.0 mm and thickness of 0.25 mm. Figure 1 shows the various configurations of the thermoelectric module with different combinations of arrangements and leg geometries. Table 1 shows the material specifications used in the numerical analysis.

#### 2.2. Mesh Convergence

#### 2.3. Boundary Conditions

^{−6}W/mm

^{2}°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.

#### 2.4. Governing Equations

#### 2.5. Data Reduction

#### 2.5.1. Physical Assumptions

#### 2.5.2. Theoretical Analysis

_{,}the power equation was obtained. To obtain maximum power, constraint of $\frac{\partial P}{\partial {R}_{L}}=0$ was applied on the power equation, which gave the maximum power condition of $R={R}_{L}$. When $R={R}_{L}$ is used in the power equation, Equation (8) is derived, which calculates the theoretical maximum power of the thermoelectric module [4,33]:

_{A}is the ratio of the top side area of the trapezoidal leg (${A}_{T}$) to the bottom side area of the trapezoidal leg (${A}_{B}$) from Equation (12). ${A}_{0}$ is the equivalent uniform cross-section area of the trapezoidal leg:

#### 2.5.3. Numerical Analysis

## 3. Results and Discussion

#### 3.1. Validation

#### 3.1.1. Code Validation

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

#### 3.2. Optimum Temperature

_{2}Te

_{3}material were operated at maximum temperature differences of 980 °C and 480 °C, respectively, lower than the melting point temperature of SiGe or Bi

_{2}Te

_{3}material. If the two-stage and segmented arrangements of the thermoelectric module with SiGe+Bi

_{2}Te

_{3}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 Bi

_{2}Te

_{3}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+Bi

_{2}Te

_{3}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+Bi

_{2}Te

_{3}material and the interface temperature between two segments for the segmented arrangement of the thermoelectric module with SiGe+Bi

_{2}Te

_{3}material [4,17]. The maximum value of the optimum temperature is the melting point temperature of Bi

_{2}Te

_{3}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+Bi

_{2}Te

_{3}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.

_{legs, hotside}> A

_{legs, coldside}showed higher average stress compared to trapezoidal legs with A

_{legs, coldside}> A

_{legs, 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 A

_{legs, coldside}> A

_{legs, hotside}were considered for maximum power, maximum efficiency, and maximum stress analyses for single stage arrangement with Bi

_{2}Te

_{3}material and two-stage arrangement with SiGe, Bi

_{2}Te

_{3}, and SiGe+Bi

_{2}Te

_{3}materials.

#### 3.3. Maximum Power

#### 3.3.1. Single Stage Arrangement

_{2}Te

_{3}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 Bi

_{2}Te

_{3}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 Bi

_{2}Te

_{3}material showed maximum power of 0.31 W at a temperature difference of 480 °C.

#### 3.3.2. Two-Stage Arrangement

_{2}Te

_{3}material showed significant enhancement in maximum power, compared to both stages either made of SiGe or Bi

_{2}Te

_{3}alone [17,18]. In the case of SiGe, Bi

_{2}Te

_{3}, and SiGe+Bi

_{2}Te

_{3}material, the square prism and trapezoidal legs with A

_{legs, coldside}> A

_{legs, 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+Bi

_{2}Te

_{3}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 Bi

_{2}Te

_{3}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.

#### 3.3.3. Single Stage Segmented Arrangement

_{2}Te

_{3}material. Maximum power increased with the temperature difference for both the combinations. The leg geometry had no effect on the maximum power [1,6,10], as shown in Figure 7c, due to same internal resistance and same optimum voltage condition. The square prism as well as the cylindrical legs showed maximum power of 0.73 W at a temperature difference of 730 °C.

#### 3.4. Maximum Efficiency

#### 3.4.1. Single Stage Arrangement

_{2}Te

_{3}material showed superior increase in maximum efficiency compared to the SiGe material over the entire temperature difference range due to high $Z\overline{T}$ [4]. Although the Bi

_{2}Te

_{3}material was operated up to a temperature difference of 480 °C due to its low melting point temperature, it still showed the highest value of maximum efficiency at a temperature difference of 480 °C, compared to the maximum efficiency value for the SiGe material at a temperature difference of 980 °C. At temperature difference of 480 °C, the Bi

_{2}Te

_{3}material had a higher maximum power, thus showing higher maximum efficiency than SiGe. From a temperature difference of 480 to 980 °C, the SiGe material showed increase in maximum power and the heat absorbed also increased due to increase in temperature difference, which resulted in lower maximum efficiency for the SiGe material than that of the Bi

_{2}Te

_{3}material. Maximum efficiency of 12.2% was observed at a temperature difference of 480 °C for all the leg geometries with the Bi

_{2}Te

_{3}material. However, maximum efficiency of 5.1% was obtained for all the leg geometries with the SiGe material at a temperature difference of 980 °C.

#### 3.4.2. Two-Stage Arrangement

_{2}Te

_{3}material showed the highest values of maximum efficiency with temperature difference, compared to the SiGe+Bi

_{2}Te

_{3}and SiGe materials but up to a temperature difference of 480 °C. However, from temperature difference of 480 °C, the SiGe+Bi

_{2}Te

_{3}material showed increase in maximum efficiency up to the temperature difference corresponding to the optimum temperature condition. The SiGe material showed the lowest values of maximum efficiency until the temperature difference of 980 °C. Although Bi

_{2}Te

_{3}as well as SiGe+Bi

_{2}Te

_{3}materials are operated till a particular temperature difference because of the limitation of their melting point temperatures, they still showed comparatively higher values of maximum efficiencies than the SiGe material. For the same material, leg geometry had very less effect on maximum efficiency because of the same optimum voltage load and same internal resistance [1,6,10]. In the case of SiGe, Bi

_{2}Te

_{3}and SiGe+Bi

_{2}Te

_{3}materials, square prism, cylindrical, and trapezoidal legs showed almost the same maximum efficiency variation with temperature difference. At a temperature difference of 980 °C and for the SiGe material, the square prism legs and cylindrical legs showed maximum efficiency of 4.6% and the cylindrical legs showed maximum efficiency of 4.7%. For the Bi

_{2}Te

_{3}material and temperature difference of 480 °C, the square prism legs and cylindrical legs showed maximum efficiency of 10.8% and the cylindrical legs showed maximum efficiency of 11.1%. In the case of the SiGe+Bi

_{2}Te

_{3}material, the square prism legs showed maximum efficiency of 15% at a temperature difference of 880 °C and the cylindrical as well as trapezoidal legs showed maximum efficiencies of 14.6% and 14.4%, respectively, at a temperature difference of 830 °C. The maximum efficiency of the two-stage arrangement of the thermoelectric module with leg geometries and materials increased over the entire temperature difference range.

#### 3.4.3. Single Stage Segmented Arrangement

_{2}Te

_{3}material increased with the temperature difference. The maximum efficiency variation with the temperature difference is shown in Figure 8c. Like maximum power, the leg geometries had no effect on the maximum efficiency of the single stage segmented arrangement of the thermoelectric module with two leg geometries and combination of the SiGe+Bi

_{2}Te

_{3}material [1,6,10]. At a temperature difference of 730 °C, the square prism and cylindrical legs showed maximum efficiency of 13.2%.

#### 3.5. Maximum Stress

#### 3.5.1. Single Stage Arrangement

_{2}Te

_{3}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 Bi

_{2}Te

_{3}material was higher than that of the SiGe material. Therefore, the thermal stress induced in the Bi

_{2}Te

_{3}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 A

_{legs, coldside}> A

_{legs, hotside}, and A

_{legs, hotside}> A

_{legs, 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 A

_{legs, coldside}> A

_{legs, hotside}and trapezoidal legs with A

_{legs, hotside}> A

_{legs, 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 A

_{legs, coldside}> A

_{legs, 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 A

_{legs, hotside}> A

_{legs, coldside.}Therefore, the average stress induces in the trapezoidal legs with A

_{legs, coldside}> A

_{legs, hotside}was lower than that of the square prism and trapezoidal legs with A

_{legs, hotside}> A

_{legs, 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 A

_{legs, coldside}> A

_{legs, hotside}was preferred over the trapezoidal legs with A

_{legs, hotside}> A

_{legs, coldside}. For the SiGe material and at a temperature difference of 980 °C, the square prism legs and trapezoidal legs with A

_{legs, coldside}> A

_{legs, hotside}and A

_{legs, hotside}> A

_{legs, coldside}showed maximum stress of 0.96 GPa and the cylindrical legs showed maximum stress of 0.91 GPa. Similarly, for the Bi

_{2}Te

_{3}material and at a temperature difference of 480 °C, the square prism legs and trapezoidal legs with A

_{legs, coldside}> A

_{legs, 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.

#### 3.5.2. Two-Stage Arrangement

_{2}Te

_{3}and SiGe+Bi

_{2}Te

_{3}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+Bi

_{2}Te

_{3}material showed higher maximum stress, followed by the Bi

_{2}Te

_{3}and SiGe materials. The Bi

_{2}Te

_{3}material showed higher thermal stress than the SiGe material due to its higher coefficient of thermal expansion. The SiGe+Bi

_{2}Te

_{3}material showed higher thermal stress than the Bi

_{2}Te

_{3}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 Bi

_{2}Te

_{3}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+Bi

_{2}Te

_{3}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

_{2}Te

_{3}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+Bi

_{2}Te

_{3}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

#### Single Stage Arrangement

_{2}Te

_{3}material, the cylindrical legs presented lower stress than the other leg geometries. The higher temperature difference showed higher thermal stress. Therefore, the Bi

_{2}Te

_{3}material showed lower stress than the SiGe material for all three leg geometries because stress variation along the selected locations was presented at a temperature difference of 480 °C for the Bi

_{2}Te

_{3}material and 980 °C for the SiGe material.

#### Two-Stage Arrangement

_{2}Te

_{3}and SiGe materials showed higher values of stress than the Bi

_{2}Te

_{3}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+Bi

_{2}Te

_{3}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.

#### Single Stage Segmented Arrangement

_{2}Te

_{3}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+Bi

_{2}Te

_{3}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.

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

_{2}Te

_{3}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+Bi

_{2}Te

_{3}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+Bi

_{2}Te

_{3}material is suggested as the optimum configuration of the thermoelectric module.

_{2}Te

_{3}materials is 2100 s.

#### 3.7. Effect of Soldering Layers

_{2}Te

_{3}material showed higher thermal stress compared to the SiGe material. Hence, soldering layers were only provided on the thermoelectric legs constructed with the Bi

_{2}Te

_{3}material [4]. In order to investigate the effect of the soldering layers, the single stage arrangement of the thermoelectric module with the square prism legs and the Bi

_{2}Te

_{3}material as well as the two-stage arrangement of the thermoelectric module with the square prism legs and the Bi

_{2}Te

_{3}material are compared with and without soldering layers. In addition, the two-stage arrangement with square prism legs with the SiGe+Bi

_{2}Te

_{3}material is also compared with and without soldering layers. In the case of the SiGe+Bi

_{2}Te

_{3}material with the soldering layers, the second stage of the Bi

_{2}Te

_{3}material is provided with soldering layers. Addition of the soldering layers reduces the thermal stress compared to the corresponding same configuration without soldering layers. The soldering layers experience most of the deformation compared to the other parts of the thermoelectric module and sometimes undergo plastic deformation. Hence, excessive thermal stress on the thermoelectric legs are reduced because the larger effect of the thermal stress is absorbed by the soldering layers. The thickness of the soldering layers is very small; hence, the resistance offered is a small contribution to the total effective resistance of the thermoelectric module. Therefore, the soldering layers have no significant effect on maximum power or maximum efficiency. The melting point temperature of the soldering layers is 185 °C; hence, various configurations of the thermoelectric module with soldering layers are operated at a maximum hot junction temperature below 185 °C. Therefore, a comparison of configurations of the thermoelectric module provided with and without soldering layers was done for temperature difference of 150 °C. Maximum stress for the various configurations with the soldering layers were considered only for the thermoelectric legs, not the entire thermoelectric module, as is shown in Table 6.

## 4. Conclusions

- (a)
- For all configurations of the thermoelectric module, the numerically predicted values of maximum power and maximum efficiency were validated at ±5% error and the numerically predicted values of maximum stress were validated at ±7% error with their theoretical values, respectively.
- (b)
- For the same arrangement and the same material, leg geometries with the same volume and same base area have a negligible effect on maximum power and maximum efficiency. However, the cylindrical legs showed lower values of maximum stress than that of the square prism and trapezoidal legs.
- (c)
- For the same arrangement and leg geometry, the SiGe+Bi
_{2}Te_{3}material for the thermoelectric module showed higher maximum power and maximum efficiency than the other individual materials. The thermal stress of the thermoelectric module with the SiGe+Bi_{2}Te_{3}material was higher than those of the thermoelectric module with the SiGe and Bi_{2}Te_{3}materials. In addition, the intensity of stress in the thermoelectric module with the SiGe+Bi_{2}Te_{3}material could be reduced by using soldering layers without affecting maximum power and maximum efficiency. - (d)
- The segmented arrangement of the thermoelectric module showed a higher maximum power and maximum efficiency and lower maximum stress in all combinations of the thermoelectric module. Therefore, the segmented arrangement of the thermoelectric module with cylindrical leg geometry and a combination of SiGe+Bi
_{2}Te_{3}materials was selected as the optimum configuration for the thermoelectric module with maximum power of 0.73 W, maximum efficiency of 13.2%, and maximum thermal stress of 0.69 GPa.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

A | Leg cross-sectional area (m^{2}) |

E | Yong modulus (GPa) |

$\overrightarrow{E}$ | Electric field intensity (N/C or V/m) |

${H}_{a}$ | Heat absorbed (W) |

${I}_{theoretical,opt}$ | Theoretical optimum current (A) |

${I}_{numerical,opt}$ | Numerical optimum current (A) |

$\overrightarrow{J}$ | Electric current intensity (A/ m^{2}) |

K | Curvature (m^{−1}) |

${k}_{1}$ | Thermal conductivity of segment 1 (W/m °C) |

${k}_{2}$ | Thermal conductivity of segment 2 (W/m °C) |

L | Leg length (m) |

${P}_{theoretical,max}$ | Theoretical maximum power (W) |

${P}_{numerical,max}$ | Numerical maximum power (W) |

p | Peltier coefficient (V) |

R | Internal resistance ($\mathsf{\Omega})$ |

${R}_{equivalent}$ | Equivalent resistance ($\mathsf{\Omega})$ |

R_{L} | External load resistance ($\mathsf{\Omega})$ |

${T}_{H}$ | Hot junction temperature (°C) |

${T}_{C}$ | Cold junction temperature (°C) |

$\overline{T}$ | Average temperature (°C) |

$\Delta T$ | Temperature difference (°C) |

$\nabla T$ | Temperature gradient |

${V}_{theoretical,opt}$ | Theoretical optimum voltage (V) |

${V}_{numerical,opt}$ | Numerical optimum voltage (V) |

$Z\overline{T}$ | Figure of Merit |

${Z}_{c}$ | Coordinate (m) |

${Z}_{equivalent}\overline{T}$ | Equivalent Figure of Merit |

$\alpha $ | Seebeck coefficient (V/K) |

${\u0273}_{theoretical,max}$ | Theoretical maximum efficiency |

${\u0273}_{numerical,max}$ | Numerical maximum efficiency |

${\rho}_{p}$ | Electrical resistivity of p-type (Ω m) |

${\rho}_{n}$ | Electrical resistivity of n-type (Ω m) |

${\sigma}_{p}$ | Electrical conductivity of p-type (Ω^{−1} m^{−1}) |

${\sigma}_{n}$ | Electrical conductivity of n-type (Ω^{−1} m^{−1}) |

${\alpha}_{equivalent}$ | Equivalent Seebeck coefficient (V/K) |

${\alpha}_{1}$ | Seebeck coefficient of segment 1 (V/K) |

${\alpha}_{2}$ | Seebeck coefficient of segment 2 (V/K) |

${\rho}_{1}$ | Electrical resistivity of segment 1 (Ω m) |

${\rho}_{2}$ | Electrical resistivity of segment 2 (Ω m) |

${\sigma}_{v}$ | von-Mises stress (GPa) |

${\sigma}_{T}and{\sigma}_{ij}$ | Thermal stress (i, j = 1,2,3) (GPa) |

${\sigma}_{xx}$ | Normal stress in longitudinal direction (GPa) |

${\alpha}_{T}$ | Coefficient of thermal expansion (K^{−1}) |

$\epsilon $ | Strain |

${\epsilon}_{n}$ | Normal strain |

$v$ | Poisson’s ratio |

$\nabla \varnothing $ | Electrical potential gradient |

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**Figure 1.**Various configurations of the thermoelectric module with different combinations of arrangements and leg geometries.

**Figure 2.**Mesh independency test for (

**a**) maximum power, (

**b**) maximum efficiency, and (

**c**) maximum stress.

**Figure 3.**Meshing for (

**a**) single stage, (

**b**) two-stage, and (

**c**) single stage segmented arrangements with square prism leg geometries.

**Figure 5.**Comparison of numerical and theoretical results for (

**a**) maximum power, (

**b**) maximum efficiency, and (

**c**) maximum stress.

**Figure 7.**Maximum power for (

**a**) single stage arrangement, (

**b**) two-stage arrangement, and (

**c**) single stage segmented arrangement.

**Figure 8.**Maximum efficiency 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.

**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 11.**Maximum thermal stress contours in (

**a**) single stage square prism legs, (

**b**) single stage cylindrical legs, (

**c**) single stage trapezoidal legs A

_{legs, hotside}> A

_{legs, coldside}, (

**d**) single stage trapezoidal legs A

_{legs, coldside}> A

_{legs, hotside}, (

**e**) two-stage square prism legs, (

**f**) two-stage cylindrical legs, (

**g**) two-stage trapezoidal legs A

_{legs, coldside}> A

_{legs, hotside}, (

**h**) single stage segmented square prism legs, and (

**i**) single stage segmented cylindrical legs.

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

Material | SiGe | Bi_{2}Te_{3} | Copper | 63Sn-37Pb | Al_{2}O_{3} |
---|---|---|---|---|---|

T_{M} (°C) | 1177 | 585 | 1083 | 183 | 2072 |

α (μV/K) | 115 (p-type) −115 (n-type) | 227 (p-type) −227 (n-type) | - | - | - |

Z (K^{−1}) | 2.38 × 10^{−4} | 1.53 × 10^{−3} | - | - | - |

ρ (Ω m) | 1 × 10^{−5} | 1.95 × 10^{−5} | 1.84 × 10^{−8} | 1.45 × 10^{−7} | 1 × 10^{12} |

k (W/m K) | 5.56 | 1.73 | 394.5 | 50 | 17.2–37.2 |

E (GPa) | 155 | 49.7 | 100–119 | 12–36 | 366–380 |

v | 0.3 | 0.28 | 0.31 | 0.40 | 0.26 |

α_{T} (10^{−6}/K) | 4.2 | 16.8 | 16.7–17.6 | 24 | 4.89–6.68 |

Material | Arrangement | Thermoelectric Legs | Boundary Conditions |
---|---|---|---|

Temperature Conditions | |||

SiGe | Single stage, two-stage | Square prism, cylindrical, trapezoidal | 50 °C–1000 °C |

Bi_{2}Te_{3} (without soldering layer) | Single stage, two-stage | Square prism, cylindrical, trapezoidal | 50 °C–500 °C |

Bi_{2}Te_{3} (with soldering layer) | Single stage, two-stage | Square prism | 50 °C–150 °C |

SiGe+Bi_{2}Te_{3} (without soldering layer) | Two-stage | Square prism, cylindrical, trapezoidal | 50 °C–1000 °C |

SiGe+Bi_{2}Te_{3} (without soldering layer) | Segmented | Square prism, cylindrical | 50 °C–1000 °C |

SiGe+Bi_{2}Te_{3} (with soldering layer) | Two-stage | Square prism | 50 °C–150 °C |

Voltage Conditions | |||

All configurations | Single stage, two-stage, segmented | Square prism, cylindrical, trapezoidal | 0 V to voltage value at which current and power become zero |

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

Arrangements | Material | Leg Geometry | Maximum Power (W) |
---|---|---|---|

Single stage | SiGe | Square Cylindrical | $6.89\times {10}^{-7}\times {(\Delta T)}^{2}$ |

Single stage | SiGe | Trapezoidal (A_{leg,hotside}>A_{leg,coldside}) | $6.80\times {10}^{-7}\times {(\Delta T)}^{2}$ |

Single stage | SiGe | Trapezoidal (A_{leg,coldside}>A_{leg,hotside}) | $6.80\times {10}^{-7}\times {(\Delta T)}^{2}$ |

Single stage | Bi_{2}Te_{3} | Square Cylindrical | $1.38\times {10}^{-6}\times {(\Delta T)}^{2}$ |

Single stage | Bi_{2}Te_{3} | Trapezoidal | $1.36\times {10}^{-6}\times {(\Delta T)}^{2}$ |

Single stage (SL) | Bi_{2}Te_{3} | Square | $1.65\times {10}^{-6}\times {(\Delta T)}^{2}$ |

Two-stage—1st stage | SiGe | Square Cylindrical | $6.89\times {10}^{-7}\times {(\Delta T)}^{2}$ |

Two-stage—2nd stage | SiGe | Square Cylindrical | $6.89\times {10}^{-7}\times {(\Delta T)}^{2}$ |

Two-stage—1st stage | SiGe | Trapezoidal | $6.80\times {10}^{-7}\times {(\Delta T)}^{2}$ |

Two-stage—2nd stage | SiGe | Trapezoidal | $6.80\times {10}^{-7}\times {(\Delta T)}^{2}$ |

Two-stage—1st stage | Bi_{2}Te_{3} | Square Cylindrical | $1.38\times {10}^{-6}\times {(\Delta T)}^{2}$ |

Two-stage—2nd stage | Bi_{2}Te_{3} | Square Cylindrical | $1.38\times {10}^{-6}\times {(\Delta T)}^{2}$ |

Two-stage—1st stage | Bi_{2}Te_{3} | Trapezoidal | $1.36\times {10}^{-6}\times {(\Delta T)}^{2}$ |

Two-stage—2nd stage | Bi_{2}Te_{3} | Trapezoidal | $1.36\times {10}^{-6}\times {(\Delta T)}^{2}$ |

Two-stage—1st stage | SiGe | Square Cylindrical | $6.89\times {10}^{-7}\times {(\Delta T)}^{2}$ |

Two-stage—2nd stage | Bi_{2}Te_{3} | Square Cylindrical | $1.38\times {10}^{-6}\times {(\Delta T)}^{2}$ |

Two-stage—1st stage | SiGe | Trapezoidal | $6.80\times {10}^{-7}\times {(\Delta T)}^{2}$ |

Two-stage—2nd stage | Bi_{2}Te_{3} | Trapezoidal | $1.36\times {10}^{-6}\times {(\Delta T)}^{2}$ |

Two-stage—1st stage (SL) | Bi_{2}Te_{3} | Square | $1.65\times {10}^{-6}\times {(\Delta T)}^{2}$ |

Two-stage—2nd stage (SL) | Bi_{2}Te_{3} | Square | $1.65\times {10}^{-6}\times {(\Delta T)}^{2}$ |

Two-stage—1st stage | SiGe | Square | $6.89\times {10}^{-7}\times {(\Delta T)}^{2}$ |

Two-stage—2nd stage (SL) | Bi_{2}Te_{3} | Square | $1.65\times {10}^{-6}\times {(\Delta T)}^{2}$ |

Segmented | SiGe+Bi_{2}Te_{3} | Square Cylindrical | $1.41\times {10}^{-6}\times {(\Delta T)}^{2}$ |

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

Combination | Arrangements | Material | Leg Geometry | Maximum Temperature Difference (°C) | Maximum Power (W) | Maximum Efficiency (%) | Maximum Stress (GPa) |
---|---|---|---|---|---|---|---|

1 | Single stage | SiGe | Square | 980 | 0.6438 | 5.0973 | 0.962 |

2 | Single stage | SiGe | Cylindrical | 980 | 0.6472 | 5.1058 | 0.912 |

3 | Single stage | SiGe | Trapezoidal (A_{leg,hotside}> A_{leg,coldside}) | 980 | 0.6275 | 5.0910 | 0.968 |

4 | Single stage | SiGe | Trapezoidal (A_{leg,coldside}> A_{leg,hotside}) | 980 | 0.6280 | 5.0964 | 0.968 |

5 | Single stage | Bi_{2}Te_{3} | Square | 480 | 0.3132 | 12.156 | 0.607 |

6 | Single stage | Bi_{2}Te_{3} | Cylindrical | 480 | 0.3145 | 12.163 | 0.575 |

7 | Single stage | Bi_{2}Te_{3} | 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 | Bi_{2}Te_{3} | Square | 480 | 0.1192 | 10.811 | 0.822 |

12 | Two-stage | Bi_{2}Te_{3} | Cylindrical | 480 | 0.1258 | 11.060 | 0.701 |

13 | Two-stage | Bi_{2}Te_{3} | Trapezoidal | 480 | 0.1169 | 10.833 | 0.821 |

14 | Two-stage | SiGe+Bi_{2}Te_{3} | Square | 880 | 0.4627 | 15.011 | 1.91 |

15 | Two-stage | SiGe+Bi_{2}Te_{3} | Cylindrical | 830 | 0.4302 | 14.551 | 1.56 |

16 | Two-stage | SiGe+Bi_{2}Te_{3} | Trapezoidal | 830 | 0.4053 | 14.365 | 1.81 |

17 | Segmented | SiGe+Bi_{2}Te_{3} | Square | 730 | 0.7293 | 13.172 | 0.716 |

18 | Segmented | SiGe+Bi_{2}Te_{3} | Cylindrical | 730 | 0.7325 | 13.165 | 0.694 |

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

Combination | Arrangements | Material | Leg Geometry | Computational Time (s)/Single Case |
---|---|---|---|---|

1 | Single stage | SiGe | Square | 1200 |

2 | Single stage | SiGe | Cylindrical | 1560 |

3 | Single stage | SiGe | Trapezoidal (A_{leg,hotside}> A_{leg,coldside}) | 1500 |

4 | Single stage | SiGe | Trapezoidal (A_{leg,coldside}> A_{leg,hotside}) | 1440 |

5 | Single stage | Bi_{2}Te_{3} | Square | 1260 |

6 | Single stage | Bi_{2}Te_{3} | Cylindrical | 1380 |

7 | Single stage | Bi_{2}Te_{3} | Trapezoidal | 1320 |

8 | Two-stage | SiGe | Square | 3300 |

9 | Two-stage | SiGe | Cylindrical | 6000 |

10 | Two-stage | SiGe | Trapezoidal | 4500 |

11 | Two-stage | Bi_{2}Te_{3} | Square | 3540 |

12 | Two-stage | Bi_{2}Te_{3} | Cylindrical | 5940 |

13 | Two-stage | Bi_{2}Te_{3} | Trapezoidal | 8940 |

14 | Two-stage | SiGe+Bi_{2}Te_{3} | Square | 3780 |

15 | Two-stage | SiGe+Bi_{2}Te_{3} | Cylindrical | 6240 |

16 | Two-stage | SiGe+Bi_{2}Te_{3} | Trapezoidal | 7140 |

17 | Segmented | SiGe+Bi_{2}Te_{3} | Square | 1920 |

18 | Segmented | SiGe+Bi_{2}Te_{3} | Cylindrical | 2100 |

Arrangements | Material | Maximum Power-Theoretical (W) | Maximum Power-Numerical (W) | Maximum Efficiency-Numerical (%) | Maximum Stress-Numerical (GPa) |
---|---|---|---|---|---|

Single stage | Bi_{2}Te_{3} | 0.0233 | 0.0230 | 3.7805 | 0.163 |

Single stage (SL) | Bi_{2}Te_{3} | 0.0279 | 0.0270 | 3.7486 | 0.022 |

Two-stage | Bi_{2}Te_{3} | 0.0091 | 0.0089 | 3.3790 | 0.212 |

Two-stage (SL) | Bi_{2}Te_{3} | 0.0104 | 0.0101 | 3.2696 | 0.024 |

Two-stage | SiGe+Bi_{2}Te_{3} | 0.0111 | 0.0109 | 2.7764 | 0.283 |

Two-stage (SL) | SiGe+Bi_{2}Te_{3} | 0.0120 | 0.0115 | 2.6337 | 0.151 |

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**MDPI and ACS Style**

Lee, M.-Y.; Seo, J.-H.; Lee, H.-S.; Garud, K.S.
Power Generation, Efficiency and Thermal Stress of Thermoelectric Module with Leg Geometry, Material, Segmentation and Two-Stage Arrangement. *Symmetry* **2020**, *12*, 786.
https://doi.org/10.3390/sym12050786

**AMA Style**

Lee M-Y, Seo J-H, Lee H-S, Garud KS.
Power Generation, Efficiency and Thermal Stress of Thermoelectric Module with Leg Geometry, Material, Segmentation and Two-Stage Arrangement. *Symmetry*. 2020; 12(5):786.
https://doi.org/10.3390/sym12050786

**Chicago/Turabian Style**

Lee, Moo-Yeon, Jae-Hyeong Seo, Ho-Seong Lee, and Kunal Sandip Garud.
2020. "Power Generation, Efficiency and Thermal Stress of Thermoelectric Module with Leg Geometry, Material, Segmentation and Two-Stage Arrangement" *Symmetry* 12, no. 5: 786.
https://doi.org/10.3390/sym12050786