Ravi Anant
Kishore
,
Prashant
Kumar
and
Shashank
Priya
*

Center for Energy Harvesting Materials and Systems (CEHMS), Virginia Tech., Blacksburg, VA 24061, USA. E-mail: spriya@vt.edu

Received
8th September 2017
, Accepted 4th October 2017

First published on 1st November 2017

Low-grade waste heat recovery is a promising source of renewable energy; however, there are practical challenges in the recovery process. Thermoelectric generators (TEGs) are a viable solution, but their efficiency remains low, thereby limiting their implementation. The performance of TEGs can be enhanced by optimizing the module configuration; however, optimizing all the parameters experimentally using traditional experimental techniques will require several trials, and therefore, it is cumbersome and expensive. Here, we demonstrate the Taguchi method for optimizing TEG modules and demonstrate that full optimization can be achieved in just 25 experiments. The optimization has been achieved in four stages. In the first stage, a numerical model of thermoelectricity is developed and used in the second stage to optimize key geometric parameters of TEGs through the Taguchi method. In the third stage, the Taguchi method is used to optimize the geometric dimensions of the heat sink. Lastly, in the fourth stage, the effect of operating conditions on the performance of TEGs is investigated. The results reveal that the Taguchi method is capable of predicting the near-optimal configuration of TEGs.

The performance of TE materials is measured in terms of the non-dimensional figure-of-merit, ZT, defined as^{5}

ZT = α^{2}σT/κ | (1) |

Theoretically, an efficient TE material should have a high Seebeck coefficient, high electrical conductivity, and low thermal conductivity. However, for most materials, thermal and electrical conductivities are directly related and these properties increase or decrease in the same manner. Also, the Seebeck coefficient and electrical conductivity vary reciprocally, causing difficulties in achieving any drastic improvement in the figure of merit ZT.^{5,6} Improvement in material performance has to be complemented with improvement in module design to achieve higher device level ZT. The thermal-to-electrical energy conversion efficiency of a TEG is defined as

(2) |

For a TEG operating between fixed hot-side and cold-side temperatures, T_{h} and T_{c}, there exists an optimal external load, where the output power is maximum. The maximum energy conversion efficiency of an ideal TEG operating under the optimal conditions is given in terms of ZT as:^{7}

(3) |

Some of the conventional TE materials include bismuth telluride (Bi_{2}Te_{3}), lead telluride (PbTe), silicon germanium (SiGe), and cobalt triantimonide (CoSb_{3}) based skutterudites. The selection of a suitable material depends upon the hot-side temperature of the source. Most of the conventional TE materials possess average ZT less than unity in low temperature regime. In applications below 200 °C, such as that for low-grade waste heat recovery, Bi_{2}Te_{3}-based alloys are the optimum TE materials. Bi_{2}Te_{3} alloyed with Sb and Se has slightly higher average ZT than the unmodified material, but the average value is still close to unity in the low temperature regime. Some techniques such as nanostructuring,^{8–13} doping with Cu or Ag atoms,^{14,15} and adjusting the atomic ratio of Sb/Bi atoms^{16} have been found to improve the ZT of Bi_{2}Te_{3}-based TE materials. Hao et al.^{17} have demonstrated that by suppressing the intrinsic excitation in Bi_{2}Te_{3}-based TE materials through addition of a small amount of electron acceptors such as Cd, Cu, and Ag lowers the lattice thermal conductivity and increases the ZT. The study further demonstrated that compound Bi_{0.5}Sb_{1.5−x}Cu_{x}Te_{3} (x = 0.005) has a maximum ZT of 1.4 and an average ZT of 1.2 between 100 and 300 °C. The TEG module fabricated using this material was found to exhibit an energy conversion efficiency up to 6.0% with a hot-side temperature of 512 K and a cold-side temperature of 295 K, which was about 30% higher than the non-optimized BiTe-based modules.

TEGs broadly consist of three main components: p-type and n-type TE legs, conductive electrodes that connect the TE legs electrically in series and thermally in parallel, and ceramic plates on two sides to electrically insulate the TE module. In addition, TEGs may also have metallic fins as the heat sink on the cold-side to dissipate the heat to the surroundings or to the cooling fluid. The performance of TEGs, not only depends on the ZT of the TE material but also on the operating conditions and configuration of the TE module. The operating conditions include the heat source and heat sink temperatures, the modes of heat transfer, and the environmental conditions such as ambient temperature, pressure, wind speed, and humidity. In addition, TEGs also have an optimal electric load where the device has the best performance. The performance of TEGs can be enhanced through three possible ways. First, by improving ZT of the thermoelectric materials. Second, by optimizing the geometric parameters of TEG modules, such as the number of p–n legs, leg length and cross-sectional area, and third, by optimizing the modes of heat transfer, which includes design improvement in the heat exchanger and heat sink. TE devices have been extensively examined analytically, numerically, and experimentally in the past to optimize both geometric parameters and operating conditions. Niu et al.,^{18} Hsu et al.,^{19} and Gou et al.^{20} have presented experimental studies on low-temperature TEGs for waste heat recovery. These studies have provided promising results indicating the potential for harvesting energy from low-temperature waste heat sources, especially from industrial waste. However, evaluating the influence of all the operating conditions and geometric parameters requires numerous experimentation; therefore, experimental studies on TE devices are often substantiated by analytical and numerical modelling, which can either be a simplified one-dimensional model^{21,22} or a complex three-dimensional model.^{23–26} A significant amount of research related to the optimization of p–n leg geometry, i.e. the length,^{27} the cross-sectional area,^{28} and the number of thermocouples,^{29} can be found in the literature. Some of the prior studies have focused on finding the length to the cross-sectional area ratio to determine the optimum shape parameter,^{30} aspect ratio (leg length/leg area),^{31} and slenderness ratio ((A_{p}/L_{p})/(A_{n}/L_{n}))^{32} of the p–n thermocouple. While high efficiency was found to occur when the length of the couple was large, output power was obtained at some intermediate value.^{33} For micro-thermoelectric generators, output power was found to decline with the cross-sectional area of the thermocouples, whereas efficiency was found to exhibit an opposite trend.^{34} In addition, since n-type telluride-based materials are usually weaker than their p-type counterparts, a slenderness ratio less than one was suggested for the peak performance of the TEG modules.^{35} There have been attempts^{36–38} on coupling the thermoelectric equations with the heat transfer equations of the heat sink in order to simultaneously optimize the TEG and heat sink geometries. The results have shown that the performance of TEGs is highly dependent on leg dimensions along with heat sink geometry and the method of heat transfer from the heat source/sink to the TE modules. In order to reduce the complexity in the TE module design and minimize the design parameters, various modelling techniques have been proposed. Fraisse^{39} attempted to study the leg geometry using thermoelectric element modelling based on an electrical analogy. Cheng and Lin^{40} proposed genetic algorithms (GAs) for geometric optimization of thermoelectric coolers in a confined volume. Another design method based on dimensional analysis for optimizing thermoelectric devices was developed by Lee^{41} and Yamanashi.^{42} Huang^{43} proposed a simplified conjugate-gradient method for geometry optimization of thermoelectric coolers. These previous studies have provided several insights into modelling of the TEGs and proposed pathways for understanding the role of controlling parameters. In practice, designers need a modelling methodology that can provide a near-optimum TEG configuration in the least number of trials. This not only saves time and resources but also improves the robustness as a diverse set of materials can be investigated in a rapid manner under varying operating conditions. Here we address this need through implementation of the Taguchi method. We believe that this methodology will provide a toolset for the design of modules to a broader community attempting to utilize thermoelectric materials.

In this study, we utilize thermal and electrical properties of a high performance Bi_{0.5}Sb_{1.5−x}Cu_{x}Te_{3} (x = 0.005) alloy reported by Hao et al.^{17} and numerically simulate the performance of TEGs built using this composition under different geometric and operating conditions. The Taguchi method of optimization is a statistical tool that predicts the optimal performance with far less number of trials than the conventional optimization techniques, where only one factor is normally varied at a given instance. The Taguchi optimization method allows us to vary multiple factors at a given instance in a controlled manner, thereby reducing the total number of trials required. This method is widely used in diverse fields of research and engineering, such as manufacturing processes,^{44–47} thermal analysis,^{48–50} and chemical and biochemical studies.^{51–53} Chen et al.^{54} used the Taguchi method to optimize the dimensions, length, width, and height, of the heat sink along with the hot-side temperature and resistive load for TEGs. An n-type polycrystalline silicon layer for a micro thermoelectric generator (TEG) was designed using the Taguchi method by Kim et al.^{55} Kishore et al.^{56} have applied the Taguchi method to optimize the cooling capacity and coefficient of performance of thermoelectric coolers.

Here, we demonstrate the optimization of TEGs in four subsequent stages. In the first stage, we validate the numerical model using published experimental data and then investigate the effect of contact resistances on the output power and efficiency. In the second stage, the geometric parameters of TEGs, namely the cross-sectional area and height of p–n legs, along with the resistive load are optimized using Taguchi method. In the third stage, the Taguchi method is used to optimize the geometric parameters of the heat sink. In the fourth and last stage, we study the effect of operating conditions, more specifically the total heat transfer coefficient and the ambient temperature, on the performance of TEGs.

(4) |

(5) |

The output power P and efficiency η are given as:^{21}

(6) |

(7) |

The one-dimensional analytical equations presented above are valid under a small temperature gradient and constant thermoelectric properties. In order to account for temperature-dependent TE properties at a larger temperature difference, the coupled thermoelectric equations need to be solved. The coupled thermal–electrical governing equations in the steady-state are given as^{62}

(8) |

(9) |

(10) |

As part of the mesh independency test, we varied the element count by refining the mesh size at three stages. The mesh independency test is essential to ensure that the numerical results are independent of the grid size. Fig. 1(b) depicts the medium-size mesh structure with an element count of 12129. The temperature dependent material properties of the TE material Bi_{0.5}Sb_{1.5−x}Cu_{x}Te_{3} (x = 0.005) from the literature^{17} have been used for all the calculations. For the mesh independency test, the hot-side and cold-side temperatures were fixed at 513 K and 295 K, respectively. Fig. 1(c) and (d) depict the numerical results for output power and efficiency versus load resistance obtained at three different element counts. The maximum change in results between the first case (element count # 7760) and the second case (element count # 12129) was found to be around 1.7% for output power and 2.0% for efficiency. The maximum change in results between the second case (element count # 12129) and the third case (element count # 33960) was found to be less than 0.1%. Therefore, we have used a medium-size meshing strategy in the rest of paper.

(11) |

(12) |

(13) |

(14) |

Fig. 2(a) and (b) show the key geometric parameters and the boundary conditions for the TEG module considered in the first step of the optimization study. The overall area of the TEG module was fixed to be 30 mm × 30 mm. The height of the module varies with the change in the height of p–n legs. The thickness of the copper electrode was fixed at 0.1 mm and the ceramic substrates had 0.8 mm thickness. The gap distance among p–n legs is fixed at 0.15 mm. A constant electrical contact resistance of 1.0 × 10^{−9} Ω m^{2} and thermal contact resistance of 2.2 × 10^{−4} m^{2} K W^{−1} are considered. The length, L, and width, W, of p–n legs are considered equal and varied together; whereas, the height, H, of the p–n legs is varied independently to optimize the leg geometry. The number of p–n legs also changes with change in cross-sectional area and can be calculated using the model. In addition to the geometric parameters, the resistive load also has a strong effect on the performance of TEGs; therefore, it needs to be varied. The hot-side temperature, T_{h} and the cold-side temperature, T_{c} are fixed at 513 K and 295 K, respectively, in this case.

Fig. 2(c) and (d) show the key geometric parameters and the boundary conditions of the heat sink considered in the second step of the optimization study. The geometric dimensions of p–n legs are fixed based on the results obtained from the first step. Other parameters such as the thickness of the copper electrode and ceramics substrates and contact resistances are the same as mentioned in the first step. The heat sink is assumed to be made up of aluminum and the thermal contact resistance between the ceramic substrate and the aluminum base is assumed to be 2.2 × 10^{−4} m^{2} K W^{−1}.^{72} The gap, b, between the fins is fixed to be 0.5 mm, whereas fin thickness, a, and the height, c, are varied along with the external resistive load. The number of fins also changes with change in its cross-sectional area, which can be calculated since the overall size of the TEG module is fixed. The hot-side temperature is fixed at 513 K and a constant total heat transfer coefficient of 20 W m^{−2} K^{−1} is considered on the cold-side. The ambient temperature is fixed at 295 K. Table S2 in the ESI document† shows the different steps followed during the optimization process, the overall objective, and the range of parameters considered.

Control factors | Levels | |||||
---|---|---|---|---|---|---|

(1) | (2) | (3) | (4) | (5) | ||

(A) | Cross-sectional area (mm^{2}) |
1.0 × 1.0 | 1.25 × 1.25 | 1.50 × 1.50 | 1.75 × 1.75 | 2.0 × 2.0 |

(B) | Height (mm) | 1.0 | 1.25 | 1.50 | 1.75 | 2.0 |

(C) | External resistance (Ω) | 2.0 | 4.0 | 6.0 | 8.0 | 10.0 |

Trial | Control factors | P (W) | S/N_{P} (dB) |
η (%) | S/N_{η} (dB) |
||
---|---|---|---|---|---|---|---|

(A) | (B) | (C) | |||||

1 | 1 | 1 | 1 | 1.668 | 4.445 | 1.695 | −35.418 |

2 | 1 | 2 | 2 | 2.480 | 7.889 | 2.799 | −31.060 |

3 | 1 | 3 | 3 | 2.900 | 9.248 | 3.599 | −28.876 |

4 | 1 | 4 | 4 | 3.108 | 9.849 | 4.209 | −27.516 |

5 | 1 | 5 | 5 | 3.199 | 10.100 | 4.694 | −26.57 |

6 | 2 | 1 | 2 | 3.602 | 11.132 | 4.316 | −27.298 |

7 | 2 | 2 | 3 | 3.798 | 11.591 | 5.119 | −25.816 |

8 | 2 | 3 | 4 | 3.793 | 11.579 | 5.673 | −24.923 |

9 | 2 | 4 | 5 | 3.709 | 11.385 | 6.096 | −24.299 |

10 | 2 | 5 | 1 | 2.037 | 6.180 | 3.126 | −30.099 |

11 | 3 | 1 | 3 | 3.926 | 11.879 | 4.402 | −27.128 |

12 | 3 | 2 | 4 | 3.849 | 11.706 | 4.857 | −26.272 |

13 | 3 | 3 | 5 | 3.710 | 11.388 | 5.203 | −25.674 |

14 | 3 | 4 | 1 | 3.511 | 10.908 | 4.498 | −26.939 |

15 | 3 | 5 | 2 | 4.090 | 12.235 | 5.936 | −24.531 |

16 | 4 | 1 | 4 | 1.831 | 5.2557 | 2.718 | −31.316 |

17 | 4 | 2 | 5 | 1.790 | 5.0551 | 2.999 | −30.461 |

18 | 4 | 3 | 1 | 3.550 | 11.006 | 5.568 | −25.086 |

19 | 4 | 4 | 2 | 3.207 | 10.121 | 5.828 | −24.690 |

20 | 4 | 5 | 3 | 2.810 | 8.975 | 5.732 | −24.834 |

21 | 5 | 1 | 5 | 0.941 | −0.523 | 1.513 | −36.402 |

22 | 5 | 2 | 1 | 2.968 | 9.450 | 4.765 | −26.440 |

23 | 5 | 3 | 2 | 2.299 | 7.229 | 4.292 | −27.346 |

24 | 5 | 4 | 3 | 1.896 | 5.558 | 3.985 | −27.991 |

25 | 5 | 5 | 4 | 1.632 | 4.257 | 3.796 | −28.414 |

Tables 3 and 4, respectively, show the mean response for raw data and S/N data for output power and efficiency. The mean response refers to the mean value of the output response and is calculated for every level of all control factors. For example, the mean response of the raw data for the output power for factor A at level 1 implies the mean of all the power data for factor A at level 1 in column 5 of Table 2. Likewise, the mean response of S/N data for output power for factor B at level 3 implies the average of all S/N_{P} data for parameter B at level 3 in column 6 of Table 2. Similarly, we can calculate the means at all levels of different control factors, which are presented in Tables 3 and 4. The data are plotted in Fig. S1(a) and (b) shown in the ESI.† The level that has the highest mean of S/N ratios implies the optimal level of a control factor and the combination of all optimal levels constitutes the optimal setting. It can be noted from Table 4 that the mean response for S/N_{P} is highest when factor A is at level 3, factor B is at level 3, and factor C is at level 2. Therefore, combination A_{3}B_{3}C_{2} (cross-sectional area: 1.5 × 1.5 mm^{2}, height: 1.5 mm, and resistive load: 4.0 Ω) is the optimal control factor setting for the highest output power. Similarly, the combination A_{3}B_{4}C_{3} (cross-sectional area: 1.5 × 1.5 mm^{2}, height: 1.75 mm, and resistive load: 6.0 Ω) is the optimal setting for the highest efficiency. The optimal number of p–n legs can also be calculated since the overall area of the TEG module is fixed to be 30 × 30 mm^{2}. The optimal number of p–n leg pairs is found to be 81 for both cases, though the height of p–n legs is different for optimal output power and optimal efficiency.

Level 1 | Level 2 | Level 3 | Level 4 | Level 5 | ||
---|---|---|---|---|---|---|

Means of raw data for power, (W) | A | 2.671 | 3.388 | 3.817 | 2.638 | 1.947 |

B | 2.394 | 2.977 | 3.25 | 3.086 | 2.754 | |

C | 2.747 | 3.136 | 3.066 | 2.843 | 2.67 | |

Means of raw data for efficiency, (%) | A | 3.399 | 4.866 | 4.979 | 4.569 | 3.670 |

B | 2.929 | 4.108 | 4.867 | 4.923 | 4.657 | |

C | 3.930 | 4.634 | 4.567 | 4.251 | 4.101 |

Table 5 shows the ANOVA table for output power data. The ANOVA table highlights the percentage contribution by each of the three factors on the output power. It can be noted that the percentage contribution from the cross-sectional area, height, and resistive load is 54.22%, 11.33%, and 4.18%, respectively. Therefore, the cross-sectional area of p–n legs is the most prominent factor that affects the output power, followed by the height of p–n legs and resistive load. The ANOVA for efficiency is shown in Table 6. It can be seen that percentage contribution by the cross-sectional area, height, and resistive load on efficiency is 26.28%, 35.10%, and 4.61%, respectively. Clearly, the height of p–n legs is the most prominent factor in this case, followed by the cross-sectional area and then the resistive load. It is also important to note that the contribution from the error term is quite large (30.27% and 33.98%, respectively), indicating a significant effect from few external factors that were not included in the study. Some process factors are complex and difficult to be controlled experimentally and are thus are not taken into account. In the case of TEGs, temperature-dependent material properties have a significant impact on the performance of the device. Although the hot-side and the cold-side temperatures are fixed, the temperature distribution inside the TEG module varies with change in geometric parameters and resistive load, resulting in variation in the performance of the TE material and consequently in the performance of the device. Therefore, the error term containing the uncontrollable factors like temperature-dependent material properties has appeared to have a significant impact on the performance of TEGs.

Source of variation | Degree of freedom (DOF) | Sum of squares (SS) | Variance (V) | F-Value (F) | Percentage contribution |
---|---|---|---|---|---|

(A) Cross-sectional area | 4 | 10.538 | 2.6345 | 5.37 | 54.22% |

(B) Height | 4 | 2.2026 | 0.5507 | 1.12 | 11.33% |

(C) Load | 4 | 0.8125 | 0.2031 | 0.41 | 4.18% |

Error | 12 | 5.8825 | 0.4902 | 30.27% | |

Total | 24 | 19.436 | 100% |

Source of variation | Degree of freedom (DOF) | Sum of squares (SS) | Variance (V) | F-Value (F) | Percentage contribution |
---|---|---|---|---|---|

(A) Cross-sectional area | 4 | 0.001031 | 0.000258 | 2.32 | 26.28% |

(B) Height | 4 | 0.001377 | 0.000344 | 3.1 | 35.10% |

(C) Load | 4 | 0.000181 | 0.000045 | 0.41 | 4.61% |

Error | 12 | 0.001333 | 0.000111 | 33.98% | |

Total | 24 | 0.003923 | 100% |

The optimal output power at optimal setting A_{3}B_{3}C_{2} can be predicted by eqn (8), which can be modified as

(15) |

It is important to note that both power and efficiency are maximum at an intermediate value of leg cross-sectional area: 1.5 × 1.5 mm^{2}. The performance of a TE device normally increases with an increase in leg cross-sectional area; however, since the overall area of the TEG module is fixed at 30 × 30 mm^{2}, increasing the cross-sectional area beyond 1.5 × 1.5 mm^{2} drastically reduces the number of p–n legs, thereby causing both power and efficiency to decrease. Our finding regarding the optimal leg height (1.5 mm for power and 1.75 mm for efficiency) is in agreement with the results shown by other researchers. Rowe^{33} reported that in order to obtain high efficiency, the TEG module should be designed with long thermocouples. However, if a large power per unit area is needed the thermocouple length should be optimized at a relatively shorter length. Finally, considering the optimal value of load resistance, previous studies have reported that TEGs perform the best when external load resistance equals the effective internal resistance;^{18} therefore, an intermediate value of load resistance (4.0 Ω for power and 6.0 Ω for efficiency) has appeared as the optimal resistive load. In addition, since the optimal control factors for output power and efficiency are different, it is apparent that the optimum design of a TEG module is likely to be a compromise between obtaining high efficiency or large output power. The difference in output power at the two optimal settings, A_{3}B_{3}C_{2} and A_{3}B_{4}C_{3}, is only 0.1 W; therefore, we have considered the second setting A_{3}B_{4}C_{3} as the optimal setting and have fixed the cross-sectional area of 1.50 × 1.50 mm^{2} and height 1.75 mm for the remainder of the paper. These are nearly the same geometric parameters of the p–n legs as that used in ref. 17.

Control factors | Levels | |||||
---|---|---|---|---|---|---|

(1) | (2) | (3) | (4) | (5) | ||

(A) | Fin thickness (mm) | 0.5 | 0.75 | 1.0 | 1.25 | 1.5 |

(B) | Fin height (mm) | 10 | 15 | 20 | 25 | 30 |

(C) | External resistance (Ω) | 2.0 | 5.0 | 8.0 | 11.0 | 15.0 |

Trial | Control factors | P (W) | S/N_{P} (dB) |
η (%) | S/N_{η} (dB) |
||
---|---|---|---|---|---|---|---|

(A) | (B) | (C) | |||||

1 | 1 | 1 | 1 | 0.7160 | −2.9024 | 1.810 | −34.844 |

2 | 1 | 2 | 2 | 1.4498 | 3.2262 | 3.319 | −29.581 |

3 | 1 | 3 | 3 | 1.7774 | 4.9954 | 3.895 | −28.189 |

4 | 1 | 4 | 4 | 1.8620 | 5.3998 | 4.000 | −27.959 |

5 | 1 | 5 | 5 | 1.7867 | 5.0409 | 3.825 | −28.348 |

6 | 2 | 1 | 2 | 0.8940 | −0.9732 | 2.548 | −31.877 |

7 | 2 | 2 | 3 | 1.3082 | 2.3333 | 3.299 | −29.632 |

8 | 2 | 3 | 4 | 1.5160 | 3.6141 | 3.594 | −28.890 |

9 | 2 | 4 | 5 | 1.5473 | 3.7916 | 3.561 | −28.969 |

10 | 2 | 5 | 1 | 1.3495 | 2.6033 | 2.594 | −31.720 |

11 | 3 | 1 | 3 | 0.8214 | −1.7088 | 2.570 | −31.800 |

12 | 3 | 2 | 4 | 1.1209 | 0.9914 | 3.060 | −30.286 |

13 | 3 | 3 | 5 | 1.2558 | 1.9782 | 3.197 | −29.904 |

14 | 3 | 4 | 1 | 1.0909 | 0.7560 | 2.295 | −32.783 |

15 | 3 | 5 | 2 | 1.74793 | 4.8505 | 3.683 | −28.677 |

16 | 4 | 1 | 4 | 0.70524 | −3.0331 | 2.391 | −32.428 |

17 | 4 | 2 | 5 | 0.9275 | −0.6538 | 2.728 | −31.283 |

18 | 4 | 3 | 1 | 0.82569 | −1.6644 | 1.963 | −34.143 |

19 | 4 | 4 | 2 | 1.42423 | 3.0716 | 3.286 | −29.666 |

20 | 4 | 5 | 3 | 1.6497 | 4.3479 | 3.742 | −28.538 |

21 | 5 | 1 | 5 | 0.60036 | −4.4317 | 2.169 | −33.276 |

22 | 5 | 2 | 1 | 0.57639 | −4.7857 | 1.603 | −35.903 |

23 | 5 | 3 | 2 | 1.11481 | 0.94403 | 2.875 | −30.828 |

24 | 5 | 4 | 3 | 1.37312 | 2.75417 | 3.386 | −29.406 |

25 | 5 | 5 | 4 | 1.48065 | 3.40903 | 3.549 | −28.998 |

Table S5 in the ESI document† shows the ANOVA table for the output power data. The percentage contribution from the fin thickness, fin height, and resistive load on output power can be seen to be 19.60%, 60.06%, and 19.40%, respectively. Therefore, the fin height is the most prominent factor that affects the output power followed by the fin thickness and the resistive load. The ANOVA for efficiency is shown in Table S6 in the ESI document.† The percentage contribution from the fin thickness, fin height, and resistive load on efficiency is 11.40%, 37.39%, and 50.52%, respectively. Therefore, the height of the fins and the resistive load are the two most prominent factors, followed by the fin thickness. The contribution by the error term is very small (0.94% for output power and 0.60% for efficiency), indicating the insignificant effect from the external factors that were not included in this study.

Using eqn (8), the predicted value of optimal output power was found to be 2.03 W. In addition, using eqn (9), the confidence interval (CI) at the 85% confidence level for the predicted value of the output power was calculated to be ±0.103. It implies that at the confidence level of 85%, the optimal value of output power lies in the range of 1.93 W to 2.14 W. The confirmation run performed at the optimal setting A_{1}B_{5}C_{3} shows the output power of 2.1 W, which is well within the predicted range. Also, the difference between the actual and the predicted values of power is about 3.2%. Similarly, the optimal efficiency at the optimal setting A_{1}B_{5}C_{3} can also be predicted. The predicted value of the optimal efficiency η_{opt} was found to be 4.23%. The confirmation simulation at the optimal setting of A_{1}B_{5}C_{3} shows the efficiency of 4.25%, which is 0.6% higher than the predicted value.

The optimal geometry of the heat sink predicted by the Taguchi method is in agreement with the findings of other researchers. Chen et al.^{54} obtained an optimal fin height of 28 mm from the range of 7 mm to 28 mm and an optimal fin thickness of 0.1 mm from the range 0.1 to 0.4 mm selected in their study. Clearly, the largest fin height and smallest fin thickness appeared as the optimal geometric dimensions of the heat sink. Likewise, Jang et al.^{73} reported that an increase of fin height leads to intensification of the heat transfer rate due to the development of new thermal boundary layers, thereby improving both the power and efficiency of TEGs. In addition, increasing the number of fins by decreasing the fin thickness increases the total fin surface area exposed for cooling, which eventually increases the heat transfer rate and improves the TEG performance.^{73,74} Theoretically, for a heat sink of the fixed base length and width, L_{HS}, fin height H_{f}, and fin thickness t_{f}, the effective heat transfer coefficient (h_{eff}) is expressed as^{54}

(16) |

Hence, for fixed values of other parameters, the effective heat transfer coefficient (h_{eff}) increases with an increase in fin height H_{f} and with a decrease in fin thickness t_{f}.

It is important to note that in this study, we have considered fixed electrical and thermal contact resistances as 1.0 × 10^{−9} Ω m^{2} and 2.2 × 10^{−4} m^{2} K W^{−1}, respectively, to reduce the number of invariants. In practice, however, it is very difficult to maintain a constant value of contact resistances due to their inherent nature. Contact resistance depends on numerous factors such as contact pressure, interfacial materials, surface deformations, surface roughness and quality of deposition. Therefore, contact quality between two interfaces depends not only on the material properties of the interfacial layer but also on the manufacturing techniques and operating conditions. Several studies^{75–78} in the past have attempted to enhance the performance of thermoelectric devices by reducing contact resistances using various evaluation and fabrication techniques. However, such attempts are limited and often concentrated to certain thermoelectric materials. The effect of contact resistances becomes more prominent in the case of a segmented thermoelectric generator. Finding the compatibility relationship among various thermoelectric materials, the diffusion barriers and the matching interface layers is a potential area of future research in the design of high efficiency thermoelectric generators.

• Under different levels of geometric parameters examined, it was found that p–n legs having a cross-sectional area of 1.5 × 1.5 mm^{2} and height of 1.5 mm provided the highest output power.

• The p–n legs having a cross-sectional area of 1.5 × 1.5 mm^{2} and height of 1.75 mm were found to provide the highest efficiency.

• Under different levels of fin dimensions examined, it was found that the performance of a TEG is highest when the fin thickness is 0.5 mm and fin height is 30 mm.

• The study also showed that a TEG designed under ideal laboratory conditions is expected to have much lower performance under actual operating conditions. For example, a TEG designed at a fixed hot-side temperature of 513 K and cold-side temperature of 295 K was found to have a maximum efficiency of about 6%. However, the efficiency of the TEG reduced to 4.25% when it was operated under an ambient condition of 295 K.

• The output power of the TEG was found to increase from 1.43 W to 2.83 W and efficiency from 3.46% to 4.92% on increasing the heat transfer coefficient from 10 to 60 W m^{−2} K^{−1} at a fixed ambient temperature of 295 K.

• The output power of the TEG was found to reduce from 2.56 W to 1.56 W and efficiency from 4.71% to 3.63% on increasing the ambient temperature from 273 K to 313 K at a fixed heat transfer coefficient of 20 W m^{−2} K^{−1}.

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## Footnote |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c7se00437k |

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