http://pubs.acs.org/journal/acsodf Article Geometry Optimization for Miniaturized Thermoelectric Generators Gustavo G. Dalkiranis,* João H. C. Bocchi, Osvaldo N. Oliveira, Jr., and Gregório C. Faria Cite This: ACS Omega 2023, 8, 9364−9370 Read Online ACCESS Metrics & More Article Recommendations * sı Supporting Information ABSTRACT: Thermoelectric materials capable of converting heat into electrical energy are used in sustainable electric generators, whose efficiency has been normally increased with incorporation of new materials with high figure of merit (ZT) values. Because the performance of these thermoelectric generators (TEGs) also depends on device geometry, in this study we employ the finite element method to determine optimized geometries for highly efficient miniaturized TEGs. We investigated devices with similar fill factors but with different thermoelectric leg geometries (filled and hollow). Our results show that devices with legs of hollow geometry are more efficient than those with filled geometry for the same length and cross-sectional area of thermoelectric legs. This behavior was observed for thermoelectric leg lengths smaller than 0.1 mm, where the leg shape causes a significant difference in temperature distribution along the device. It was found that for reaching highly efficient miniaturized TEGs, one has to consider the leg geometry in addition to the thermal conductivity. 1. INTRODUCTION The search for technologies to allow humankind to harvest energy from renewable sources is crucial to reduce global warming and partially solve the energy crisis.1,2 While photovoltaic and wind power technologies are already maturing and available in the market, the same does not apply to technologies for harvesting heat waste. These technologies may positively impact the energetic matrix3 since roughly 60% of the energy obtained is wasted as heat in industrial or domestic processes.4 This energy harvesting Figure 1. Thermoelectric generator with two thermoelectric legs. The may also be obtained with microgenerators to power wearable displacement of the charge carriers is caused by the temperature gradient. devices, smart fabrics, and small medical gadgets, such as pacemakers.5−10 In this context, thermoelectric generators (TEGs) which transform thermal energy (heat) into electrical terms in the thermal and electrical conductivities cancel each energy can be used to recover part of the energy lost in heat form.11−13 These are devices with two (or more) thermo- other out, eq 1 can be written from .17 Because the electric legs, one p-type and the other n-type, thermally Seebeck coefficient and electrical and thermal conductivities connected in parallel and electrically in series, named as can depend on the temperature, this equation is not always thermocouples.14−16 Figure 1 shows a simplified scheme of a valid, especially with high temperature differences.18 Through- thermoelectric generator (TEG). out this article, however, we will keep the temperature gradient The capacity of a thermoelectric generator to convert heat not higher than 4 K, which is within the reliability of eq 1. into electrical energy is given by the figure of merit (ZT) Since a thermoelectric generator is basically a heat engine defined using eq 117 converting heat into electrical energy to perform work, its Received: December 12, 2022 (1) Accepted: February 17, 2023 where Sdevice is the Seebeck coefficient, Tavg is the average Published: March 1, 2023 temperature between the temperature of the hot (TH) and cold (TC) sides, Rdevice is the internal electrical resistance, and Gdevice is the thermal conductance of the device. Since the geometric © 2023 The Authors. Published by American Chemical Society https://doi.org/10.1021/acsomega.2c07916 9364 ACS Omega 2023, 8, 9364−9370 ACS Omega http://pubs.acs.org/journal/acsodf Article Figure 2. (a) Scheme of TEG geometries used in the simulations. A top view of the thermoelectric leg geometries in the cases: filled (b) and hollow (c). efficiency is limited by the Carnot cycle efficiency, corrected by geometry to the thermoelectric efficiency. Additional strategies a reducing thermoelectric factor. It can be calculated using eq are used to increase the temperature gradient without 214−16,19,20 negatively affecting other device parameters. Glatz et al. studied the thermal resistance between the thermal and electrical contacts, suggesting that it could be optimized to enhance the temperature gradient between the thermoelectric (2) legs.29 The group of Professor LeBlanc explored several leg shapes and their influence on the power output of TEGs. where TH and TC are the temperatures of the hot and cold Although some of the shapes are difficult to realize sides, respectively. Therefore, for a fixed TH and TC, the higher experimentally, their work confirmed the importance of leg the ZT, the higher the efficiency is. It is worth mentioning that geometry on optimized TEGs.30,31 A thorough review on this the COMSOL’s thermoelectric module considers temperature- topic brought a compilation of results, in which device dependent material parameters and the Thomson effect. geometries are correlated with the thermoelectric efficiency Therefore, the heat flux parameter calculated in our modeling and power output.32 protocol considers the Thomson effect, validating the use of eq Here, we focused on understanding the influence of the 2 to calculate the overall device efficiency. Several strategies thermoelectric leg geometry on miniaturized TEG parameters. have been reported to increase the efficiency (ηTE), ranging Simulations with the thermoelectric module of COMSOL from synthesizing new materials to producing hybrid software were carried out for two types of thermoelectric leg devices.21,22 The most common route, however, is still the geometries, as shown in Figure 2, i.e., filled and hollow. They search for new materials and composites with high ZT had similar cross-sectional areas, and a fixed temperature values.23−28 Nonetheless, it is known that the overall device gradient was applied to the devices. The device efficiency was geometry can play an important role on the properties of calculated varying the length and cross-sectional area of the electrical and thermal systems.29−31 Of great interest here is thermoelectric legs. Our results indicate that for a leg length the optimization of the geometry of the thermoelectric legs.32 smaller than 0.1 mm, its shape affects the TEG efficiency by For instance, when reducing the length of the thermoelectric changing the temperature profile along the device. Further- legs to about 55% of their original size, there was a 10% more, we present guidelines for constructing optimized increase in the conversion efficiency and a 48% increase in the miniaturized TEGs, considering that the temperature distribu- output efficiency.33 It is clear that longer thermoelectric legs tion along the device depends not only on the thermal increase the overall thermal resistance, thus decreasing thermal conductance but also on the leg geometry. conductance. A reduction in thermal conductance is normally desirable to ensure that the temperature difference between the leg extremes is preserved and optimized. For automobile waste 2. MODELING heat recovery, Kumar et al. observed that the efficiency of The influence of leg geometry on the TEG efficiency was thermoelectric legs depends on their length, with optimization studied with two geometries for the thermoelectric legs: (a) of the length and filling fraction yielding a maximum output filled (Figure 2b) and (b) hollow (Figure 2c). The cross- power even reducing the amount of material used.34 On the sectional area (fill factor) and length (L) were varied for both other hand, the longer the leg, the higher the internal geometries to allow for a comparison. The simulations were resistance is, negatively impacting the efficiency. This en performed using a device with two thermoelectric legs of p- passant analysis suggests that there will be an optimal value for and n-type Bi2Te3, with electrical and thermal contacts of the length of the devices, revealing the importance of device copper and Al2O3, respectively. The electrical and thermal 9365 https://doi.org/10.1021/acsomega.2c07916 ACS Omega 2023, 8, 9364−9370 ACS Omega http://pubs.acs.org/journal/acsodf Article Figure 3. (a) Efficiency of TEGs with filled and hollow thermoelectric leg geometries as a function of the length of the thermoelectric leg. (b) Relative increase percentage in the TEG efficiency for the hollow compared with the filled case. Figure 4. (a) TGE efficiency for hollow and filled thermoelectric legs as a function of the cross-sectional area. (b) Relative increase percentage in the TEG efficiency for the hollow compared with the filled case. contact resistances between the thermoelectric legs and the resistance was determined using Ohm’s law, . Due to electrical contacts were considered in the model according to the experimental values.35,36 All the physical parameters at 300 the low electric current applied, the internal electrical K are summarized in Table S1 in the Supporting Information. resistance was attained at 300 K, and an assumption was We have used the temperature-dependent physical parameters made of a uniform temperature along the thermoelectric legs. from the COMSOL database. Figure 2a presents the overall In subsidiary simulations, a temperature difference in the scheme of the simulated TEG geometries. The calculation of thermoelectric legs was considered, but there was no the figure of merit (ZT) requires the Seebeck coefficient (S), significant influence on the results (see Table S2 in the thermal conductance (G), and internal resistance (R) of the Supporting Information). The internal resistances were device, which were obtained using the thermoelectric module calculated assuming a temperature gradient (Tvarying) and of COMSOL software. The Seebeck coefficient and thermal without the temperature gradient (Tfixed). The difference in conductance were obtained simultaneously, within the same resistance between the two conditions is smaller than 1.4%, calculation run. A temperature difference (ΔT = 302 − 298 = justifying the use of a fixed internal resistance. Therefore, we 4 K) was applied to the borders of the device (between the hot assumed no change in temperature to calculate the internal and cold sides) to calculate the Seebeck voltage (VS). The resistance, reducing the computational cost. In addition, the Seebeck coefficient was calculated by dividing the generated steady-state analysis in all the simulations considered the heat Seebeck voltage by the temperature difference applied to the conduction through the thermoelectric leg, as well as the electric potential generated by the leg due to the Seebeck device . Throughout the simulation process, the effect. Fourier’s law determines the heat conduction mentioned device was considered as thermally isolated and under vacuum, above through the equation transferring heat only through thermal contact. To determine the thermal conductance (G), the cross-sectional heat flux (ϕq) (4) through the copper contact to the thermoelectric legs was probed (in W/m2). Thermal conductance was calculated using where q̅ is the conductive heat flux vector, k is the thermal eq 3 conductivity of the material, and ∇T is the temperature gradient. Using the Seebeck relationship, the electric potential at each point can be calculated, thus (3) (5) The TEG internal electrical resistance was obtained with a where E̅ is the vector of the electric field intensity, S is the 10 μA direct current feeding the device, and the electric Seebeck coefficient, and T is the temperature. For further potential in each electrical contact (Cu) was probed. This details, we refer readers to the work by the group of Prof. 9366 https://doi.org/10.1021/acsomega.2c07916 ACS Omega 2023, 8, 9364−9370 ACS Omega http://pubs.acs.org/journal/acsodf Article Figure 5. (a) Relative percentage increase for G, R, and S as a function of the cross-sectional area of the thermoelectric leg. (b) Temperature difference between the thermoelectric legs and Seebeck voltage generated by the devices as a function of the cross-sectional area of the filled and hollow geometries. (c) Temperature profile along the TEG with filled and hollow thermoelectric leg geometries. (d) Thermal conductance and temperature difference applied to thermoelectric legs as a function of the hollow thermoelectric leg position. LeBlanc,30 where a comprehensive summary of the main resistance to increase, thus optimizing the temperature equations and modeling procedures related to COMSOL’s difference between the thermoelectric legs. For large areas, thermoelectric module is presented. the efficiency drops considerably, tending asymptotically to the same value, regardless of the geometry. The increase in the 3. RESULTS AND DISCUSSION performance of TGEs with the hollow thermoelectric leg in Figure 4b shows a gain of more than 98%, for areas smaller We simulated the device efficiency for several lengths of the than 0.0591 mm2. thermoelectric legs with a fixed cross-sectional area of 0.19 The increase in efficiency is more prominent in the hollow mm2. Figure 3a summarizes the results, where hollow devices geometry at smaller cross-sectional areas (A < 0.1 mm2), which tend to have higher output efficiencies. This is better is illustrated in Figure 5a for the relative percentage between summarized in Figure 3b, which shows the increase in the device efficiency in the hollow geometry by calculating the the geometries for all thermoelectric parameters, i.e., relative output efficiency . The max- , where p stands for thermal conductance, imum ηr is ca. 40% for lengths smaller than 500 nm, which G; the internal resistance is R; and the Seebeck coefficient is S. means that a simple geometry optimization already produces The Seebeck coefficient (S) increases up to ca. 90% for devices outstanding results. It is worth mentioning that similar results with thermoelectric leg area between 0.01 and 0.04 mm2, were obtained when simulating the same structure/geometry contributing to the increase in efficiency with the hollow with distinct materials, e.g., with PbTe2 replacing Bi2Te3. geometry. Furthermore, the internal resistance decreases for Efficiencies were calculated from the simulated results of hollow legs�reduction of approximately 20% for cross- thermal conductance, internal electrical resistance, and sectional areas smaller than 0.012 mm2, influencing on the Seebeck coefficient (see Figure S1 in Supporting Information efficiency of TEGs. In contrast, the thermal conductance has a for details). maximum increase of ∼90% for hollow geometries for an area We then kept the length of the thermoelectric legs fixed at of the thermoelectric leg between 0.01 and 0.04 mm2. Figure 300 nm and simulated the thermal conductance (G), internal S2 in the Supporting Information compares the absolute values resistance (R), Seebeck coefficient (S), and the figure of merit of G, R, and S for both geometries and several cross-sectional (ZT), varying the cross-sectional area of the thermoelectric areas. legs. Figure 4 shows that the device efficiency increases with Studying the effects from the three parameters (G, R, and S) decreasing cross-sectional area, as expected from the on the device efficiency is useful, but it does not allow us to literature.37 Indeed, a reduction in area causes the thermal understand the origin of the differences caused by geometry. 9367 https://doi.org/10.1021/acsomega.2c07916 ACS Omega 2023, 8, 9364−9370 ACS Omega http://pubs.acs.org/journal/acsodf Article Figure 6. Thermal conductance (a) and temperature difference applied to the legs for TEGs (b) with filled and hollow thermoelectric legs with a cross-sectional area of 0.0591 mm2, for several lengths of the thermoelectric legs. Since the Seebeck coefficient depends on the Seebeck voltage The influence from the temperature distribution along the , we decided to verify the importance of the TEGs caused by the thermoelectric leg geometry occurs when temperature difference in the thermoelectric legs. Figure 5b the influence of the thermoelectric leg length on the thermal shows that the temperature differences (ΔTLeg) in the hollow resistance is no longer maximum. Also, the temperature devices are higher than those with filled geometry, and the difference tends to be a plateau for thermoelectric legs longer same applies to the Seebeck voltage (VS). This begs the than 0.1 mm (dominance of thermal conductance). question of why the temperature difference between the It is worth mentioning that a similar trend was found when thermoelectric legs is larger for the hollow geometry if their analyzing the simulated open-circuit voltage and output power thermal conductance is also higher. One should expect that a curves for different leg geometries (see Figure S5 in the higher thermal conductance facilitates thermal equilibrium and Supporting Information). generates smaller temperature differences. This apparent contradiction was resolved by investigating the temperature 4. CONCLUSIONS profile along the device. Figure 5c shows that for a fixed We have performed a systematic study of the influence from temperature difference of 4 K, the geometric differences the thermoelectric leg geometry in thermoelectric generators between the filled and hollowed legs yield a distinct with two commonly used geometries. For a fixed temperature temperature distribution along the device. This increases the difference of 4 K and a fixed cross-sectional area of 0.19 mm2, a temperature gradient applied to the thermoelectric legs, even higher efficiency was observed for the thermoelectric generator for high temperature differences, e.g., 100 K (for details, see with the hollow geometry, with a relative gain of ca. 40% for Figure S3 in the Supporting Information). The results for the lengths smaller than 500 nm. The difference in efficiency was two geometries are the same, though, when the leg length is even higher (98%) in the hollow legs for the length of 300 nm larger than 0.1, as shown in Figure S4. To explore further the and for cross-sectional areas smaller than 0.06 mm2. In general, apparent contradiction between thermal conductance and more efficient devices can be obtained by increasing the temperature difference, these parameters were simulated for thermal resistance of a thermoelectric leg with an increase in its TEG with hollow thermoelectric legs (L = 300 nm and A = length and/or decrease in its cross-sectional area. Also, 0.0591 mm2), varying the position of the thermoelectric legs decreasing the temperature difference in the thermal and from the corner to the center of the electrical contact. This also electrical contacts yields efficient TEGs. Although the thermal changed the perimeter of the hollow thermoelectric leg. Figure conductance is lower in the case of filled legs, the temperature 5d indicates that the position and perimeter of hollow applied to thermoelectric leg is higher in the case of hollow thermoelectric legs affect thermal conductance and temper- legs. Since the temperature difference applied to TEGs is fixed, ature difference applied to the thermoelectric legs. Both the temperature distribution changes. This shows that for parameters reach a maximum when the thermoelectric leg is fabricating highly efficient miniaturized TGEs, one has to positioned around 0.1 mm from the corner of the electrical consider not only the thermal conductivity of the thermo- contact. Also, when the hollow thermoelectric legs become electric leg but also its geometry and temperature distribution similar to filled thermoelectric legs (x,y position at 0.375 mm), along the device. The leg geometry is especially relevant for the thermal conductance is the same for the two geometries lengths smaller than 0.1 mm. As for bulker devices, the thermal (filled and hollow legs). conductance seems to be the dominating factor over the leg The increase in the efficiency occurs when the lengths of the shape phenomena. thermoelectric legs are smaller than 0.1 mm, as indicated in Figure 3. Hence, it is also expected that the temperature difference applied to the thermoelectric leg and the device ■ * ASSOCIATED CONTENT sı Supporting Information thermal conductance are the same for thermoelectric legs The Supporting Information is available free of charge at longer than 0.1 mm since thermal conductance dominates, and https://pubs.acs.org/doi/10.1021/acsomega.2c07916. the influence of leg geometry becomes negligible. Figure 6a shows a higher thermal conductance for the hollow thermo- Physical properties of all materials; relative percentage electric legs with lengths up to 0.05 mm and area of 0.0591 internal resistance considering the temperature; relative mm2. For lengths smaller than 0.1 mm, the hollow geometry percentage increase for G, R, and S as a function of the provides higher temperature differences, as shown in Figure 6b. thermoelectric leg length; G, R, and S of the TEGs as a 9368 https://doi.org/10.1021/acsomega.2c07916 ACS Omega 2023, 8, 9364−9370 ACS Omega http://pubs.acs.org/journal/acsodf Article function of the thermoelectric leg cross-sectional area; (5) Ashraf, M.; Masoumi, N. A Thermal Energy Harvesting Power temperature profile along the TEG when a temperature Supply With an Internal Startup Circuit for Pacemakers. IEEE Trans. difference of 100 K is applied on the devices; Very Large Scale Integr. (VLSI) Syst. 2016, 24, 26−37. temperature profile along the TEG with a thermoelectric (6) Bahk, J.-H.; Fang, H.; Yazawa, K.; Shakouri, A. Flexible leg length of 0.2 mm; open-circuit voltage and output Thermoelectric Materials and Device Optimization for Wearable Energy Harvesting. J. Mater. Chem. C 2015, 3, 10362−10374. power curves for the TEG with filled and hollow (7) Li, M.; Chen, J.; Zhong, W.; Luo, M.; Wang, W.; Qing, X.; Lu, thermoelectric legs; and temperature profile along the Y.; Liu, Q.; Liu, K.; Wang, Y.; Wang, D. Large-Area, Wearable, Self- hollow TEG measured at two different points of the Powered Pressure−Temperature Sensor Based on 3D Thermoelectric hollow thermoelectric leg (PDF) Spacer Fabric. ACS Sens. 2020, 5, 2545−2554. ■ AUTHOR INFORMATION Corresponding Author (8) Wang, L.; Zhang, K. Textile-Based Thermoelectric Generators and Their Applications. Energy Environ. Mater. 2020, 3, 67−79. (9) Serrano-Claumarchirant, J. F.; Brotons-Alcázar, I.; Culebras, M.; Sanchis, M. J.; Cantarero, A.; Muñoz-Espí, R.; Gómez, C. M. Gustavo G. Dalkiranis − São Carlos Institute of Physics, Electrochemical Synthesis of an Organic Thermoelectric Power University of São Paulo, 13560-970 São Carlos, SP, Brazil; Generator. ACS Appl. Mater. Interfaces 2020, 12, 46348−46356. Catalan Institute of Nanoscience and Nanotechnology (10) Mazzotta, A.; Carlotti, M.; Mattoli, V. Conformable on-skin (ICN2), 08193 Barcelona, Spain; orcid.org/0000-0001- devices for thermo-electro-tactile stimulation: materials, design, and 9590-9249; Email: dalkiranis@gmail.com fabrication. Mater. Adv. 2021, 2, 1787−1820. (11) Yang, J. Potential Applications of Thermoelectric Waste Heat Authors Recovery in the Automotive Industry. In ICT 2005. 24th International João H. C. Bocchi − São Carlos Institute of Physics, University Conference on Thermoelectrics, 2005; pp 170−174. of São Paulo, 13560-970 São Carlos, SP, Brazil (12) Hendricks, T. J. Integrated Thermoelectric-Thermal System Osvaldo N. Oliveira, Jr. − São Carlos Institute of Physics, Resistance Optimization to Maximize Power Output in Thermo- University of São Paulo, 13560-970 São Carlos, SP, Brazil; electric Energy Recovery Systems- CORRIGENDUM. MRS Online orcid.org/0000-0002-5399-5860 Proc. Libr. 2014, 1642, 1. Gregório C. Faria − São Carlos Institute of Physics, University (13) Fitriani; Ovik, R.; Long, B. D.; Barma, M. C.; Riaz, M.; Sabri, of São Paulo, 13560-970 São Carlos, SP, Brazil; M. F. M.; Said, S. M.; Saidur, R. A Review on Nanostructures of High- orcid.org/0000-0001-6138-8473 Temperature Thermoelectric Materials for Waste Heat Recovery. Renewable Sustainable Energy Rev. 2016, 64, 635−659. Complete contact information is available at: (14) Rowe, D. M. Thermoelectrics Handbook: Macro to Nano; Taylor https://pubs.acs.org/10.1021/acsomega.2c07916 & Francis, 2005. (15) Kroon, R.; Mengistie, D. A.; Kiefer, D.; Hynynen, J.; Ryan, J. Author Contributions D.; Yu, L.; Müller, C. Thermoelectric plastics: from design to All authors have given approval to the final version of the synthesis, processing and structure-property relationships. Chem. Soc. manuscript. Rev. 2016, 45, 6147−6164. Funding (16) He, R.; Schierning, G.; Nielsch, K. Thermoelectric Devices: A This work was supported by INCT/INEO; Coordination of Review of Devices, Architectures, and Contact Optimization. Adv. Mater. Technol. 2018, 3, 1700256. Superior Level Staff Improvement (CAPES, Brazil) (grant (17) Ferrando-Villalba, P.; Pérez-Marín, A. P.; Abad, L.; Dalkiranis, number 88887.635729/2021-00); National Council for G. G.; Lopeandia, A. F.; Garcia, G.; Rodriguez-Viejo, J. Measuring Scientific and Technological Development (CNPq, Brazil) Device and Material ZT in a Thin-Film Si-Based Thermoelectric (grant numbers 406767/2018-1 and 3111184/2019-7); and Microgenerator. Nanomaterials 2019, 9, 653. Sao Paulo Research Foundation (FAPESP, Brazil) (grant (18) Kim, H. S.; Liu, W.; Chen, G.; Chu, C.-W.; Ren, Z. numbers 2018/22214-6, 2019/26375-7, and 2021/12458-8). Relationship between Thermoelectric Figure of Merit and Energy Notes Conversion Efficiency. Proc. Natl. Acad. Sci. 2015, 112, 8205−8210. The authors declare no competing financial interest. 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