**1. Introduction**

The world is currently facing global environmental challenges related to energy crisis, depletion of fossil fuel, and global warming [1,2]. To overcome these challenges, numerous research works have been carried out on high efficiency energy conversion technologies, as well as on the effective utilization of energy resources. One of the major sources of energy waste is automobile vehicles, where more than 50% of fuel energy is exhausted as heat [1]. Hence, the recovery and conversion of automobile exhaust heat using suitable conversion technology is a prerequisite in reducing fuel consumption as well as excessive demand of fossil fuel and natural resources. A thermoelectric generator is the most promising and convenient technology to convert waste heat into electrical energy [3].

Thermoelectric generators convert energy without degrading the environment. In addition, no mechanical moving parts are involved in the operation, and it is hence free of noise and vibration [4]. However, low conversion efficiency restricts their globalization in various application fields [5]. This low conversion efficiency is due to two major challenges: the lower Figure of Merit (*ZT*) of the

thermoelectric materials and various equipment-level challenges. Several new materials with higher *ZT* have been developed or the *ZT* of existing materials improved to overcome the low conversion efficiency. However, research works on thermoelectric equipment level challenges are comparatively less, especially those that seek to improve the performance of thermoelectric generators.

The thermo-electrical and thermo-mechanical behavior of four thermoelectric leg geometries wer compared by Erturun et al. for the non-segmented and segmented configurations [1]. For effective utilization of heat sources or sinks that are circular in shape, Shen et al. presented the concept and superiority of an annular segmented thermoelectric generator over an annular non-segmented thermoelectric generator [2]. Jin predicted the thermal stresses and deformations induced in a multilayer thin film thermoelectric module by developing the laminate and strength of material models [3]. Ma et al. showed the superiority of the performance of a multifunctional gradient thermoelectric module over a single functional gradient thermoelectric module [4]. Patil et al. investigated the performance characteristics of symmetrical segmented as well as asymmetrical segmented thermoelectric generators for various geometrical parameters and operating conditions [5]. Erturun et al. reported the influence of rectangular and cylindrical leg geometries, leg height, leg width/diameter, and leg spacing on power output, conversion efficiency, and thermal stresses. The predicted power, efficiency, and stress values using statistical models show accuracies of 8.85%, 1.22%, and 6.56%, respectively, with the corresponding simulated values using finite element analysis [6]. Wu et al. numerically investigated the effects of thermoelectric leg spacing, thickness of conducting plate, thickness of soldering layers, and thickness of ceramic plate on the power, efficiency, and thermal stresses of a thermoelectric generator under various heat flux conditions [7]. Erturun et al. proved that, compared to cylindrical legs, cylindrical coaxial legs show 7% higher efficiency and 10% reduced stresses, whereas rotated cylindrical legs showed 1.2% lower stresses and 0.3% higher efficiency [8]. Jia et al. concluded that a linear-shaped thermoelectric generator had superior performance compared to conventional π shaped thermoelectric generators [9]. Yilbas et al. conducted thermal stress analysis on a thermoelectric module with horizontally arranged rectangular as well as tapered leg geometries. The tapered leg geometry showed a lower value of thermal stress compared to the rectangular leg geometry [10]. Merbati et al. proved that thermoelectric legs with the cold side area twice the area of the hot side show lower values of thermal stress with small improvement in thermoelectric performance [11]. Ali et al. investigated the power generation and conversion efficiency of pin-shaped thermoelectric legs for various temperature ratios as well as external load resistance ratios [12]. Erturun et al. compared six different thermoelectric leg geometries and concluded that the cylindrical legs show less thermal stress and deformation with enhancement in thermoelectric power generation [13]. Yilbas et al. showed improvements in the first and second law efficiencies of the thermoelectric generator with double tapered leg geometry under various conditions of external load resistance and temperature ratio [14]. Gao et al. conducted thermal stress analysis on a single stage thermoelectric generator constructed with cubical leg geometry and Bi2Te<sup>3</sup> 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 *ZT* [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]. *Symmetry* **2020**, *12*, x FOR PEER REVIEW 3 of 41

The comparison of single stage, segmented, and two-stage arrangements of thermoelectric modules with various combinations of leg geometries and materials based on their electrical and structural performances has not been explored yet. Therefore, the objective of this study was to numerically compare single stage, two-stage, and single stage segmented arrangements with three different leg geometries, namely, square prism legs, cylindrical legs, and trapezoidal legs, constructed with SiGe, Bi2Te3, and a combination of SiGe+Bi2Te<sup>3</sup> 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. The comparison of single stage, segmented, and two-stage arrangements of thermoelectric modules with various combinations of leg geometries and materials based on their electrical and structural performances has not been explored yet. Therefore, the objective of this study was to numerically compare single stage, two-stage, and single stage segmented arrangements with three different leg geometries, namely, square prism legs, cylindrical legs, and trapezoidal legs, constructed with SiGe, Bi2Te3, and a combination of SiGe+Bi2Te3 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. Methodology**

The single stage, two-stage, and single stage segmented arrangements with square prism legs, cylindrical legs, and trapezoidal legs, as well as SiGe, Bi2Te3, and combination of SiGe+Bi2Te<sup>3</sup> 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. The single stage, two-stage, and single stage segmented arrangements with square prism legs, cylindrical legs, and trapezoidal legs, as well as SiGe, Bi2Te3, and combination of SiGe+Bi2Te3 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 2.1. Geometry and Material Specifications*

Among the three arrangements of the thermoelectric module, the single stage was baseline and the two-stage and single stage segmented arrangements were considered as modified cases of the single stage arrangement. Figure 1 shows the schematic diagrams of the single stage, two-stage, and single stage segmented arrangements of a thermoelectric module. A thermoelectric module with single stage arrangement comprised of one pair of p- and n-type thermoelectric legs and three conducting plates. The two-stage arrangement comprised of two single stage arrangements, positioned in the form of steps with an intermediate plate between them. The two pairs of thermoelectric legs in the two-stage arrangement were either made up of the same or different materials [17,18], whereas the single stage segmented arrangement was obtained by dividing the thermoelectric legs of the single stage arrangement into two equal halves. The upper half was named Segment 1 and the lower half Segment 2. Both segments were made of different materials [1,4]. Among the three arrangements of the thermoelectric module, the single stage was baseline and the two-stage and single stage segmented arrangements were considered as modified cases of the single stage arrangement. Figure 1 shows the schematic diagrams of the single stage, two-stage, and single stage segmented arrangements of a thermoelectric module. A thermoelectric module with single stage arrangement comprised of one pair of p- and n-type thermoelectric legs and three conducting plates. The two-stage arrangement comprised of two single stage arrangements, positioned in the form of steps with an intermediate plate between them. The two pairs of thermoelectric legs in the two-stage arrangement were either made up of the same or different materials [17,18], whereas the single stage segmented arrangement was obtained by dividing the thermoelectric legs of the single stage arrangement into two equal halves. The upper half was named Segment 1 and the lower half Segment 2. Both segments were made of different materials [1,4].

(**a**) Schematic of single stage arrangement (**b**) Schematic of two-stage arrangement

**Figure 1.** *Cont*.

*Symmetry* **2020**, *12*, x FOR PEER REVIEW 4 of 41

(**c**) Schematic of single stage segmented arrangement

(**j**) two-stage trapezoidal Alegs, hotside> Alegs, coldside (**k**) single stage segmented square (**l**) single stage segmented

cylindrical

(**m**) single stage square with soldering layer (**n**) two-stage square with soldering layer (**o**) two-stage square—second stage with soldering layer

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

The single stage and two-stage arrangements of the thermoelectric module were modelled with square prism legs, cylindrical legs, and trapezoidal legs; whereas the single stage segmented arrangement of the thermoelectric module was modelled with square prism and cylindrical leg geometries. The square prism legs were modelled with a 1.0 mm × 1.0 mm cross-section area (base area) and 0.96 mm height [4]. The spacing between the legs was kept constant at 0.8 mm [4]. By having the same cross-section area (base area), height, spacing, and volume as the square prism legs, cylindrical legs with 1.13 mm diameter were modelled. Trapezoidal legs with two different leg configurations, Alegs, coldside> Alegs, hotside and Alegs, hotside> Alegs, 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 Alegs, hotside> Alegs, coldside and Alegs, coldside> Alegs, 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 Bi2Te3 material in the two-stage configurations; the thermoelectric legs of both the stages were made of SiGe or Bi2Te3 or a combination of SiGe+Bi2Te3 materials. For the segmented configuration, the thermoelectric legs were made of a combination of SiGe+Bi2Te3 materials. In the two-stage arrangement with combination of SiGe+Bi2Te3, the upper stage was made of SiGe material and the bottom stage of Bi2Te3 material. Similarly, in the segmented legs, the upper half was made of SiGe material and the lower half of Bi2Te3 material. For a combination of SiGe+Bi2Te3 materials, the SiGe material was used near the hot side and the Bi2Te3 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 Bi2Te3 material as well as the two-stage arrangements with square prism legs and SiGe+Bi2Te3 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 The single stage and two-stage arrangements of the thermoelectric module were modelled with square prism legs, cylindrical legs, and trapezoidal legs; whereas the single stage segmented arrangement of the thermoelectric module was modelled with square prism and cylindrical leg geometries. The square prism legs were modelled with a 1.0 mm × 1.0 mm cross-section area (base area) and 0.96 mm height [4]. The spacing between the legs was kept constant at 0.8 mm [4]. By having the same cross-section area (base area), height, spacing, and volume as the square prism legs, cylindrical legs with 1.13 mm diameter were modelled. Trapezoidal legs with two different leg configurations, Alegs, coldside > Alegs, hotside and Alegs, hotside > Alegs, 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 Alegs, hotside > Alegs, coldside and Alegs, coldside > Alegs, 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 Bi2Te<sup>3</sup> material in the two-stage configurations; the thermoelectric legs of both the stages were made of SiGe or Bi2Te<sup>3</sup> or a combination of SiGe+Bi2Te<sup>3</sup> materials. For the segmented configuration, the thermoelectric legs were made of a combination of SiGe+Bi2Te<sup>3</sup> materials. In the two-stage arrangement with combination of SiGe+Bi2Te3, the upper stage was made of SiGe material and the bottom stage of Bi2Te<sup>3</sup> material. Similarly, in the segmented legs, the upper half was made of SiGe material and the lower half of Bi2Te<sup>3</sup> material. For a combination of SiGe+Bi2Te<sup>3</sup> materials, the SiGe material was used near the hot side and the Bi2Te<sup>3</sup> 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 Bi2Te<sup>3</sup> material as well as the two-stage arrangements with square prism legs and SiGe+Bi2Te<sup>3</sup> 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

cold plates with an area of 1.0 mm × 1.0 mm and thickness of 0.08 mm [4]. Hence, when soldering

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.


**Table 1.** Material specifications used in the numerical analysis [1,4,26,27].
