Finite-Element Simulations of a Thermoelectric Generator and Their Experimental Validation

Thermoelectric Measurement Setup

To estimate the thermoelectric characteristics of the TEG, it was placed between a heat source (heated copper plate) and a heat sink (water-cooled copper plate). The device was fixed with a clamping force of 0.6 kN, as indicated by a dynamometer. A schematic illustration of the measurement setup is shown in Figure 1. To provide good thermal surface contact, thermal grease with a specific thermal energy conductivity of λ>1.46W(mK)−1 in the temperature range of ca. 300–550 K was used as the interface material. The temperature was monitored using thermocouples. The voltage U=Δφ was measured as the drop of the electric potential on the external load Rload. Due to the low internal resistance Rmodule of the thermoelectric generator (see Table 4), the electric current I was estimated indirectly:

The electric output power Pel was estimated according to:

Figure 1:

Schematic illustration of the measurement setup for determining the thermoelectric characteristics of the TEG with thermal grease as the interface material, after Hejtmanek et al. (2014).

Estimation of the Geometric and Thermoelectric Properties of the Thermoelectric Generator

A commercially available thermoelectric generator was purchased from Conrad Electronics SE (item number 193593–62, model number 1–7105). To construct the TEG in the model system, the geometry of the device, including the thermoelectric legs, the electric copper connectors and the alumina cover plates, were determined using the graphical tool ImageJ, which was applied to photomicrographs; see Schneider, Rasband, and Eliceiri (2012). Figure 2 presents photographs of the TEG and of the device modeled based on the estimated geometry for each individual component.

Figure 2:

Photographs of the TEG for estimating the geometrical parameters and display of the modeled device; (a) inner top view of the TEG, (b) side view of the TEG, (c) modeled device with n- and p-type thermoelectric material and electric copper connectors, (d) modeled device with Al2O3 cover plates.

The TEG includes 122−2=142 legs. The space between each pair of legs is 1 mm. Table 1 lists the geometric properties of the TEG without considering the tin solder. The geometric properties were estimated by analyzing the photographs shown in Figure 2. The relevant parameters are the length L of each material parallel to the entropy flux and its cross-sectional area A. The filling factor of the module, i.e., the ratio of thermoelectric active to inactive areas, is equal to 32%. For the copper connectors and alumina cover plates, the values for the specific resistivity ρ and specific thermal energy (heat) conductivity λ were taken from the literature; see Lide (2008). For the thermoelectric Bi2−xSbxTe3 materials, the specific thermal energy (heat) conductivity λ was also taken from the literature; see Kim et al. (2012). The temperature-averaged values for the specific entropy conductivity Λ in Table 1 were calculated for a medial temperature of 350 K according to:

The specific electrical resistivity ρ=1σ of the thermoelectric material is the reciprocal of the specific electrical conductivity σ, and it was estimated under the temperature drops listed in Table 4. The systematic error, which results from the deviation from isothermal conditions (i.e. ∇T=0) according to requirements of eq. [3], however, is assumed to be small.

The Seebeck coefficient α was experimentally determined under electric open-circuited conditions (i.e. j⃗q=0), for which eq. [3] provides:

As expressed by eq. [7], for balanced gradients of potentials referring to a sample of length L, the gradients ∇ can be substituted by the drops Δ of potentials along this length.

Microstructure Analysis

A basic unit was cut from the thermoelectric generator and polished using diamond-lapping films (Allied High Tech Multiprep) for field-emission scanning electron microscopy (FE-SEM) investigations using a JEOL JSM-6700F, which was equipped with an Oxford Instruments INCA 300 energy-dispersive X-ray spectrometer (EDXS) for elemental analysis. The phase composition of the bulk n- and p-type Bi2−xSbxTe3-legs was analyzed by X-ray diffraction (XRD) using a Bruker D8 Advance with Cu-Kα radiation. The legs were isolated from the TEG and prepared for XRD measurements by pressing the ductile material onto the sample holder to increase the detectable area of the samples. This preparation approach caused an orientation in the crystal structure, which was considered in the analysis of the diffraction data. The XRD measurement results were analyzed using the Rietveld method as implemented in Topas 4.2. The reference data for the structure analysis were taken from the Inorganic Crystal Structure Database (ICSD).

Finite-Element Simulations

Using a script-based input process referring to the Analysis System (ANSYS), changes in thermoelectric properties as a result of changing material parameters, such as the composition and geometry of the considered materials, and the number of legs can be rematched very easily. The used ANSYS version is 15.0 academic. The script-based input tool uses the ANSYS Parametric Design Language (APDL). The simulation process can be separated into three basic steps: preprocessing, solving and postprocessing. During the preprocessing step, the geometric properties are created, the material’s properties are set, the model is meshed and additional conditions are defined. Meshing determines the number of elements of the model and their nodes. The accuracy is given by the network’s density. After the preprocessing is complete, solving of the grid points is executed. During the postprocessing step, the results are exported as plots, tables or figures. In this work, as mentioned in Section “Estimation of the Geometric and Thermoelectric Properties of the Thermoelectric Generator”, a commercially available TEG was measured to estimate the average data for the thermoelectric properties and to evaluate a model system created using the FEM simulation tool. The modeled device consists of 72048 elements. Each thermoelectric leg is build up by 36 elements (total number of elements for the thermoelectric legs is 5,112) while each electric copper connector contains 220 elements (total number of elements for the electric copper connectors is 31,460). Each alumina cover plate consists of 17,138 (total number of elements for the alumina cover plates is 35,476). The non-linear solution converged after equilibrium iteration 2.

Microstructure of Materials

To estimate the elemental composition of the thermoelectric materials using the EDXS method in the FE-SEM, a two-leg fragment was cut from the TEG and polished. The SEM micrograph in Figure 3(a) shows the thermoelectric n- and p-type Bi2−xSbxTe3, the electric copper connectors and the alumina cover plates. The geometrical parameters are the same as those obtained from the photomicrograph of Figure 2(a), (b). Figure 3(b)–(f) shows the EDXS elemental distribution in the device by bright contrast.

Figure 3:

Side view of basic unit of the TEG exhibiting p- and n-type thermoelectric legs; (a) secondary electron micrograph, (b) Bi-M, (c) Sb-L, (d) Te-L, (e) Al-K, (f) Cu-K. Rectangularly marked areas in (a) refer to EDX spectra shown in Figure 4.

Figure 4 presents the EDX spectra of the n- and p-type material (areas of analysis are marked in Figure 3(a)). The EDXS analysis does not detect any amount of antimony inside the n-type thermoelectric semiconductor. However, Kouhkarenko et al. (2001) reported that doping with a certain amount of antimony, approximately 25 at.% in the bismuth telluride structure, leads to an improvement in the Seebeck coefficient α of the n-type thermoelectric material. The thermoelectric properties of Bi2−xSbxTe3 are also influenced by the ratio of (Bi,Sb) and Te; see Fleurial et al. (1988). The ideal ratio for a stoichiometric composition (Bi,Sb)2Te3 is (Bi,Sb)Te=23=0.667. Within the accuracy of quantitative EDXS, both materials (n-type and p-type) match this value, which supports the assumption that the materials have been made according to the strategy described by Kouhkarenko et al. (2001).

Figure 4:

EDX spectra of n- and p-type thermoelectric legs according to the rectangular areas marked in Figure 3(a).

The composition of the analyzed materials of the TEG is as expected. As indicated by the XRD pattern in Figure 5, the structure is preferentially orientated with the c-axis perpendicular to the sample holder, which is confirmed by the appearance of lattice reflections from the (0 0 l) planes with l = 9, 12, 15 (main reflection), 18, and 21. Furthermore, reflections from the (1 0 l) planes with l = 10, 13, 25, 28 are present, which, however, indicates only a slight tilt away from the c-axis. For refinement of the crystal structure, the preferred orientation resulting from the preparation (pressing of ductile material onto the sample holder) has to be considered. Because of the considerable amount of the Sb in the p-type material, the lattice parameters for this composition are smaller than those for the n-type material. The radii for structure-building elements can be extracted from Shannon (1976). With a medial coordination number of 6, it is 76 pm for Sb3+, 103 pm for Bi3+, and 221 pm for Te2−. For the n-type material, which contains only Bi on the cationic site, the diffraction data were fitted to the reference structure of Bi2Te3. For the p-type material, which contains a large amount of Sb on the cationic site, the diffraction data were fitted to the reference structure of Sb2Te3. The lattice parameters that were calculated by fitting the diffraction data to the aforementioned reference structures are listed in Table 2.

Figure 5:

Scanned (black curve) and refined (red curve) X-ray diffraction data with difference curve (light gray); (a) n-type Bi2Te3, (b) p-type Sb2−xBixTe3.

The results for the atomic positions considering the goodness of the Rietveld fit are listed in Table 3. The coordinates x = 0 and y = 0 are fixed. The goodness of fit (GOF) is expressed by the R-weighted pattern Rwp and R-expected Rexp. For the fit criteria used in Topas 4.2 see Young (1993).

Thermoelectric Investigations

To estimate the thermoelectric properties of the TEG, the measurement data were analyzed and the average values for the specific electrical conductivity σ (T), the Seebeck coefficient α (T) from the thermovoltage U(T) and the temperature difference ΔT for the thermoelectric materials were calculated. The electronic and thermal quantities for copper and alumina were taken from Lide (2008) and are presented in Table 1. A summary of the electronic, thermal and thermoelectric parameters of the thermoelectric materials is given in Table 4. The specific resistivities ρleg of the legs were estimated from the absolute resistivity of the complete device Rmodule by considering the number of thermoelectric legs and their geometry. The Seebeck coefficient αleg was estimated from the open-circuit voltage UOC according to eq. [7]. Because the coefficient was averaged over all integrated legs, the same absolute values for n- and p-type legs are obtained, and the Seebeck coefficient becomes ±αleg. The maximum electric current ISC under electric short-circuited conditions (i.e. U=Δφ=0) and the maximum electric output power Pmax are listed. All quantities are related to the established temperature drops ΔT and the medial temperature of the device Tmedial.

To determine the thermoelectric parameters of the device, five temperature differences ΔT were established, which are indicated in Table 4 and Figures 6 and 7. Figure 6 shows U–I curves with good agreement between the FEM simulation (solid line) and measurement (data points) for all temperature conditions. Note, that the slope of the lines refers to the internal resistance Rmodule, as indicated in Table 4. The electric output power Pel was estimated in terms of different load resistivities Rload. Figure 7 shows good agreement between the experimental Pel−I curve (data points) and the FEM simulation (solid line).

Figure 6:

Comparison of measured (dots) and simulated (lines) decrease in voltage over electrical current in terms of load resistance RL and temperature difference ΔT.

Figure 7:

Comparison of measured (dots) power characteristics and simulated (lines) parameters in terms of load resistance RL and temperature difference ΔT.

The results of the thermoelectric behavior can be shown in a vectorial plot that refers to the density of transported quantities, the thermal energy flux density j⃗E,th and the electric flux density j⃗q. Figure 8 shows the results of the FEM simulation of the properties of the TEG’s components at the maximum electric power output for a chosen temperature difference of 58 K. The colors of the vectors refer to the local value of the density of the transported quantity, which is in indicated in the legend. The flux densities of thermal energy j⃗E,th and electric charge j⃗q depend on the material of the TEG’s components and on the local potential gradients according to the potential distributions of Figure 9. The Al2O3 cover plates are not displayed but the flux densities at their location are. From Figure 8(a), it is clear that the thermal power density j⃗E,th is distinctly smaller in the region of the Al2O3 cover plates (4W/m2) than in the copper connectors (2−4⋅104W/m2) or the thermoelectric Bi2−xSbxTe3 legs (4−5⋅104W/m2). Due to the extremely high electric resistivity of the Al2O3 cover plates (see Table 1), the electric current density j⃗q vanishes at their location. In the copper connectors, it amounts to j⃗q=1⋅10−17−2⋅105A/m2 with strong local variation, and it is highest in the thermoelectric legs (2⋅105A/m2). Note that fluxes of thermal energy and electric charge are co-aligned in the case of p-type legs and counter-aligned in the case of n-type legs.

Figure 8:

Flux densities of transported quantities with an established temperature difference of 58 K for conditions of maximum electric power output. (a) thermal energy flux density j⃗E,th, (b) electric flux density j⃗q.

Figure 9(a) shows the distribution of the entropy potential T, obtained from the FEM simulation and a cross-section of the TEG at maximum electric power output for a potential drop of ΔT=58K. Figure 9(b) shows the respective distribution of the electric potential U=Δφ along the electric serial connection of the assembled thermoelectric legs and copper connectors inside the module. When the potential at the electric input is at ground (i.e. φ=0), it continuously increases along the chain to be φ=571mV at the electric output. Note that this value is identical to the voltage at the electric power maximum (from Figure 7) in Figure 6 for the same temperature drop. The modeling of TEGs using the FEM method has the advantage of providing deep insight into the distribution of all relevant quantities throughout the entire module.

Figure 9:

Potential distributions in the TEG at the electric power maximum and referring to the temperature difference of 58 K as obtained from the finite-element simulation: (a) temperature distribution referring to the local entropy potential, (b) established electric potential.

The results from the finite-element simulation illustrate the relation between the entropy potential T and electric potential φ, which is given algebraically in eq. [3] by the respective gradients, and the obtained flux densities j⃗E,th (see eq. [1]) and j⃗q.