2. Theoretical and experimental analysis of symmetrical and asymmetrical legs

According to previous theoretical studies, the investigation of the geometric structure of thermoelectric legs is essential, as their geometry affects the performance of devices. It has been reported that asymmetrical shape of thermoelectric legs can lead to a decrease in the thermal and electrical conductance, which in turn improve the Seebeck voltage due to an increase in the temperature difference [41]. In the present work, the temperature differences of the proposed asymmetrical legs are calculated by using the Fourier law. Figure 1a shows the proposed asymmetrical thermoelectric leg. For the sake of simplicity in the analysis, it is considered that such leg is constituted of four simpler geometrical legs connected thermally in parallel. By analyzing this simpler leg as shown in Figure 1b, it is possible to determine the temperature profile of the asymmetrical leg. Based on the Fourier law, the heat conduction along the simpler leg is defined by

$$\frac{dT}{dx} + \frac{\dot{Q}}{k} \left[ \frac{L}{\left(m\chi + b\right)^2} \right] = 0 \tag{1}$$

T xð Þ¼ <sup>Q</sup>\_ 4k

where a is the half-length of the bigger cross-section end.

L m mx ð Þ þ b 

Similarly, for a rectangular thermoelectric leg, the temperature profile is given by

2x<sup>0</sup> � x a2 

The thermoelectric modules experimentally analyzed in this chapter have rectangular thermoelectric legs based on Bi2Te3 with a typical dimension of around 1.7 � 1.7 � 2.1 mm. Based on this dimensions and using Eqs. (2) and (3), Figure 3a shows the temperature profiles of an asymmetrical thermoelectric leg with a slant angle of 10� and a symmetrical thermoelectric leg under different heat fluxes, respectively. The upper solid lines (color online) represent the temperature profiles of the asymmetrical legs, whilst the lower dashed lines (color online) correspond to the temperature profiles of the symmetrical thermoelectric legs. As expected, as the heat flux decreases (Q<sup>1</sup> > Q<sup>2</sup> > Q3), the temperature difference across the leg also decreases. Besides, a larger temperature difference is obtained across the asymmetrical leg as compared with the symmetrical one for a given heat flux. Therefore, by lowering the overall thermal conductance of the device via asymmetrical legs, the temperature gradient in the legs is increased, thus Seebeck voltage across terminals must be significantly increased. Besides, for simplicity it is worth to mention that the thermal conductivity k in Eqs. (2) and (3) during the modeling is taken as a constant, that is temperature independent; however, despite such simplification, uncertainty between both results is not so significant as it is seen in Figure 2.

Figure 3b shows the temperature profiles of the asymmetrical thermoelectric leg as a function of the position for different slant angles of the pyramid for a given heat flux. As expected, as the slant angle increases, the temperature difference across the leg increases; however, such increase is limited to a critical slant angle θ<sup>c</sup> (see inset), at which a maximum temperature difference across the leg can be achieved without affect the leg length; such limit angle is given by

tan ð Þ¼ θ<sup>c</sup>

Wherefrom, the limit angle for a thermoelectric leg with the length of 2.1 mm is around 22�, for this reason in Figure 2b as the slant angle achieves this value, the temperature rise increase drastically. It is worth to mention that in such modeling, convective effects were not taken into consideration because experimentally the device was tested under vacuum in order to avoid heat losses. Clearly, asymmetrical legs could help in two ways, by lowering the overall thermal conductance of the device so as to increase the temperature gradient in the legs, and by harnessing the Thomson effect, that depends on the temperature gradient in the legs and the temperature variation of Seebeck coefficient of the material in the operating temperature range, which is generally neglected in conventional rectangular thermoelectric legs. Thomson coefficient is given by τð Þ¼ T TdS=dT, when a temperature gradient is imposed on a thermoelectric material S varies from point to point along of the length of the thermoelectric element.

a

T xð Þ¼ QL\_ 4k

Hence, the main conclusions and results of the modeling are still valid.

� Q\_ 4k

> � QL\_ 4k

L m mx ð Þ <sup>0</sup> þ b 

Thermoelectric Devices: Influence of the Legs Geometry and Parasitic Contact Resistances on ZT

x0 a2  þ T<sup>0</sup> (2)

105

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þ T<sup>0</sup> (3)

<sup>L</sup> (4)

Where, <sup>Q</sup>\_ is the heat flux through the simpler leg, <sup>k</sup> is the thermal conductivity, <sup>m</sup> <sup>¼</sup> tan ð Þ <sup>θ</sup> represents the slope of the pyramid, b is the half-length of the smaller cross section end, and L is the length of the sample. By solving Eq. (1) using the boundary conditions T xð Þ¼ T<sup>0</sup> at x ¼ x0, and considering the fact that four simpler legs in parallel to form a complete asymmetrical thermoelectric leg, the temperature profile of the thermoelectric leg of pyramidal shape is given by

Figure 1. Schematic diagram of (a) a pyramidal-shaped thermoelectric leg and (b) simplified asymmetrical leg for modeling of the pyramidal-shaped thermoelectric legs.

Thermoelectric Devices: Influence of the Legs Geometry and Parasitic Contact Resistances on ZT http://dx.doi.org/10.5772/intechopen.75790 105

$$T(\mathbf{x}) = \frac{\dot{Q}}{4k} \left[ \frac{L}{m(m\mathbf{x} + b)} \right] - \frac{\dot{Q}}{4k} \left[ \frac{L}{m(m\mathbf{x}\_0 + b)} \right] + T\_0 \tag{2}$$

Similarly, for a rectangular thermoelectric leg, the temperature profile is given by

$$T(\mathbf{x}) = \frac{\dot{Q}L}{4k} \left[ \frac{2\mathbf{x}\_0 - \mathbf{x}}{a^2} \right] - \frac{\dot{Q}L}{4k} \left[ \frac{\mathbf{x}\_0}{a^2} \right] + T\_0 \tag{3}$$

where a is the half-length of the bigger cross-section end.

2. Theoretical and experimental analysis of symmetrical and

dT dx <sup>þ</sup> <sup>Q</sup>\_ k

According to previous theoretical studies, the investigation of the geometric structure of thermoelectric legs is essential, as their geometry affects the performance of devices. It has been reported that asymmetrical shape of thermoelectric legs can lead to a decrease in the thermal and electrical conductance, which in turn improve the Seebeck voltage due to an increase in the temperature difference [41]. In the present work, the temperature differences of the proposed asymmetrical legs are calculated by using the Fourier law. Figure 1a shows the proposed asymmetrical thermoelectric leg. For the sake of simplicity in the analysis, it is considered that such leg is constituted of four simpler geometrical legs connected thermally in parallel. By analyzing this simpler leg as shown in Figure 1b, it is possible to determine the temperature profile of the asymmetrical leg. Based on the Fourier law, the heat conduction

> L ð Þ mx <sup>þ</sup> <sup>b</sup> <sup>2</sup> " #

Where, <sup>Q</sup>\_ is the heat flux through the simpler leg, <sup>k</sup> is the thermal conductivity, <sup>m</sup> <sup>¼</sup> tan ð Þ <sup>θ</sup> represents the slope of the pyramid, b is the half-length of the smaller cross section end, and L is the length of the sample. By solving Eq. (1) using the boundary conditions T xð Þ¼ T<sup>0</sup> at x ¼ x0, and considering the fact that four simpler legs in parallel to form a complete asymmetrical thermoelectric leg, the temperature profile of the thermoelectric leg of pyramidal shape is

Figure 1. Schematic diagram of (a) a pyramidal-shaped thermoelectric leg and (b) simplified asymmetrical leg for

¼ 0 (1)

asymmetrical legs

104 Bringing Thermoelectricity into Reality

along the simpler leg is defined by

modeling of the pyramidal-shaped thermoelectric legs.

given by

The thermoelectric modules experimentally analyzed in this chapter have rectangular thermoelectric legs based on Bi2Te3 with a typical dimension of around 1.7 � 1.7 � 2.1 mm. Based on this dimensions and using Eqs. (2) and (3), Figure 3a shows the temperature profiles of an asymmetrical thermoelectric leg with a slant angle of 10� and a symmetrical thermoelectric leg under different heat fluxes, respectively. The upper solid lines (color online) represent the temperature profiles of the asymmetrical legs, whilst the lower dashed lines (color online) correspond to the temperature profiles of the symmetrical thermoelectric legs. As expected, as the heat flux decreases (Q<sup>1</sup> > Q<sup>2</sup> > Q3), the temperature difference across the leg also decreases. Besides, a larger temperature difference is obtained across the asymmetrical leg as compared with the symmetrical one for a given heat flux. Therefore, by lowering the overall thermal conductance of the device via asymmetrical legs, the temperature gradient in the legs is increased, thus Seebeck voltage across terminals must be significantly increased. Besides, for simplicity it is worth to mention that the thermal conductivity k in Eqs. (2) and (3) during the modeling is taken as a constant, that is temperature independent; however, despite such simplification, uncertainty between both results is not so significant as it is seen in Figure 2. Hence, the main conclusions and results of the modeling are still valid.

Figure 3b shows the temperature profiles of the asymmetrical thermoelectric leg as a function of the position for different slant angles of the pyramid for a given heat flux. As expected, as the slant angle increases, the temperature difference across the leg increases; however, such increase is limited to a critical slant angle θ<sup>c</sup> (see inset), at which a maximum temperature difference across the leg can be achieved without affect the leg length; such limit angle is given by

$$\tan\left(\theta\_{\ell}\right) = \frac{a}{L} \tag{4}$$

Wherefrom, the limit angle for a thermoelectric leg with the length of 2.1 mm is around 22�, for this reason in Figure 2b as the slant angle achieves this value, the temperature rise increase drastically. It is worth to mention that in such modeling, convective effects were not taken into consideration because experimentally the device was tested under vacuum in order to avoid heat losses. Clearly, asymmetrical legs could help in two ways, by lowering the overall thermal conductance of the device so as to increase the temperature gradient in the legs, and by harnessing the Thomson effect, that depends on the temperature gradient in the legs and the temperature variation of Seebeck coefficient of the material in the operating temperature range, which is generally neglected in conventional rectangular thermoelectric legs. Thomson coefficient is given by τð Þ¼ T TdS=dT, when a temperature gradient is imposed on a thermoelectric material S varies from point to point along of the length of the thermoelectric element.

[equations 42]. Nevertheless, the nonlinear component in the temperature dependence of the intrinsic S has never been into consideration in the conventional thermal rate equations; thus when S varies nonlinearly with temperature, a portion of the nonlinear component of τ(T) remains in the thermal rate [equations 42]. Hence, when a thermoelectric element is subjected to a temperature gradient an effective Seebeck coefficient S1 will appear. In this sense, previous experimental and analytical studies have demonstrated that an enhanced effective Seebeck coefficient can be obtained by combining the Thomson effect with the intrinsic Seebeck coefficient of a thermoelectric element because of the nonlinear changes in the intrinsic Seebeck coefficient with the temperature [43]. Hence, asymmetrical thermoelectric elements should present an enhancement in the effective Seebeck coefficient as compared to symmetrical ones because of the higher temperature gradient generated along the thermoelectric element as it is

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Although in this chapter the influences of the temperature dependence of the thermal conductivity and Seebeck coefficient are not considered in the model given by Eqs. (2) and (3) for the sake of simplicity; however, the main conclusions arising from the model are still valid. In addition, a simulation study has been carried out using a finite element simulation on 3D geometries for a thermoelectric device consisting of nine pairs of pyramidal legs and rectangular legs respectively using the dedicated software "Thermoelectric module of COMSOL Multiphysics in Seebeck mode". Figure 4a, b show the simulation results of the temperature profiles and terminal voltages in the asymmetrical device respectively due to the presence of a given heat flux. For comparison, Figure 4c, d show the temperature profile and the terminal voltage in the symmetrical device respectively. Clearly, asymmetrical legs based thermoelectric device presents a higher open circuit terminal voltage than its rectangular counterpart as a consequence of the larger temperature difference generated in the legs because of their asym-

The modules were made using p-type and n-Type Bi2Te3 thermoelectric materials available from Thermal Electronics Corp with ZT~1. For clarity, the full process so as to fabricate the thermoelectric legs is shown in Figure 5. The raw thermoelectric material in the form of a rod is covered by a layer of wax (wax-70) so as to hold the material in the cutting base during the cutting process, see Figure 5a. Then, it is cut into slices of 2.1 mm in thickness, which is the length of the thermoelectric leg as shown in Figure 5b. The wax is cleaned up using a warm solution of water and aquaclean-900 at a concentration of 10 mg/ml after slices cutting process as shown in Figure 5c. Next, a layer of Ni ranging from 0.5 and 1 μm is coated on both surfaces of the slide by electroplating. The Ni layer works as a diffusion barrier between the solder (Sn/Pb 60/40) and the thermoelectric legs, see Figure 5d. Prior to cutting, the Nielectroplated slide is again fixed on a graphite plate using wax (wax-70) with the aim of keeping the legs during cutting, as you can see in Figure 5e. Subsequently, thermoelectric legs with the regular geometry of 1.7 1.7 2.1 mm are obtained using a circular saw cutting machine (Accutom-100), see Figure 5f. Finally, the wax is removed from legs as

In order to obtain the asymmetrical legs, a similar process as described above is employed. However, in this case, a tilted base is used to hold the graphite plate during cuttings. Hence, so as to obtain the asymmetrical geometry (truncated square pyramid), it was necessary to do a

shown in Figure 3a.

metry.

previously indicated in Figure 5c.

Figure 2. Comparison of temperature profiles taking κ as a function of the temperature as well as a constant, for symmetrical and asymmetrical thermoelectric legs.

Figure 3. Temperature profiles as a function of the position for asymmetrical and rectangular thermoelectric legs at (a) different heat fluxes and at (b) different slant angle of the pyramid at room temperature.

The linear component in the temperature dependence of the intrinsic S can be taken into account so as to appear in the conventional thermal rate equations, but in fact it cancels out with the linear component of the Thomson coefficient and never appears in the thermal rate [equations 42]. Nevertheless, the nonlinear component in the temperature dependence of the intrinsic S has never been into consideration in the conventional thermal rate equations; thus when S varies nonlinearly with temperature, a portion of the nonlinear component of τ(T) remains in the thermal rate [equations 42]. Hence, when a thermoelectric element is subjected to a temperature gradient an effective Seebeck coefficient S1 will appear. In this sense, previous experimental and analytical studies have demonstrated that an enhanced effective Seebeck coefficient can be obtained by combining the Thomson effect with the intrinsic Seebeck coefficient of a thermoelectric element because of the nonlinear changes in the intrinsic Seebeck coefficient with the temperature [43]. Hence, asymmetrical thermoelectric elements should present an enhancement in the effective Seebeck coefficient as compared to symmetrical ones because of the higher temperature gradient generated along the thermoelectric element as it is shown in Figure 3a.

Although in this chapter the influences of the temperature dependence of the thermal conductivity and Seebeck coefficient are not considered in the model given by Eqs. (2) and (3) for the sake of simplicity; however, the main conclusions arising from the model are still valid. In addition, a simulation study has been carried out using a finite element simulation on 3D geometries for a thermoelectric device consisting of nine pairs of pyramidal legs and rectangular legs respectively using the dedicated software "Thermoelectric module of COMSOL Multiphysics in Seebeck mode". Figure 4a, b show the simulation results of the temperature profiles and terminal voltages in the asymmetrical device respectively due to the presence of a given heat flux. For comparison, Figure 4c, d show the temperature profile and the terminal voltage in the symmetrical device respectively. Clearly, asymmetrical legs based thermoelectric device presents a higher open circuit terminal voltage than its rectangular counterpart as a consequence of the larger temperature difference generated in the legs because of their asymmetry.

The modules were made using p-type and n-Type Bi2Te3 thermoelectric materials available from Thermal Electronics Corp with ZT~1. For clarity, the full process so as to fabricate the thermoelectric legs is shown in Figure 5. The raw thermoelectric material in the form of a rod is covered by a layer of wax (wax-70) so as to hold the material in the cutting base during the cutting process, see Figure 5a. Then, it is cut into slices of 2.1 mm in thickness, which is the length of the thermoelectric leg as shown in Figure 5b. The wax is cleaned up using a warm solution of water and aquaclean-900 at a concentration of 10 mg/ml after slices cutting process as shown in Figure 5c. Next, a layer of Ni ranging from 0.5 and 1 μm is coated on both surfaces of the slide by electroplating. The Ni layer works as a diffusion barrier between the solder (Sn/Pb 60/40) and the thermoelectric legs, see Figure 5d. Prior to cutting, the Nielectroplated slide is again fixed on a graphite plate using wax (wax-70) with the aim of keeping the legs during cutting, as you can see in Figure 5e. Subsequently, thermoelectric legs with the regular geometry of 1.7 1.7 2.1 mm are obtained using a circular saw cutting machine (Accutom-100), see Figure 5f. Finally, the wax is removed from legs as previously indicated in Figure 5c.

In order to obtain the asymmetrical legs, a similar process as described above is employed. However, in this case, a tilted base is used to hold the graphite plate during cuttings. Hence, so as to obtain the asymmetrical geometry (truncated square pyramid), it was necessary to do a

The linear component in the temperature dependence of the intrinsic S can be taken into account so as to appear in the conventional thermal rate equations, but in fact it cancels out with the linear component of the Thomson coefficient and never appears in the thermal rate

Figure 3. Temperature profiles as a function of the position for asymmetrical and rectangular thermoelectric legs at (a)

different heat fluxes and at (b) different slant angle of the pyramid at room temperature.

Figure 2. Comparison of temperature profiles taking κ as a function of the temperature as well as a constant, for

symmetrical and asymmetrical thermoelectric legs.

106 Bringing Thermoelectricity into Reality

Figure 4. COMSOL simulation results: (a) temperature profile and (b) open circuit voltage for the asymmetrical thermoelectric module, (c) temperature profile and (d) open circuit voltage for the symmetrical thermoelectric module, respectively at room temperature.

installed between heating and cooling plates. Next, a dc voltage with regular increments is applied to the 130 Ω resistive heater of the heating plate by using the BK Precision 9184 dc power supply. This action induces a constant heat flow Q which flows through the test specimens in the stationary temperature state, as well as provides several temperature biases ΔT across the thermoelectric module surfaces. It is worth to note that the side of the applied heat source does not matter in symmetrical modules; however, in the asymmetrical module, the heat source must be applied necessarily on the smaller area side because it has the smallest thermal conductance along the pyramid. In Figure 7c, it is plotted ΔT vs Q wherefrom the slope it can be calculated the thermal resistance of the devices; evidently, the asymmetrical legs

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Parallel, the open circuit thermal voltage is measured along with the temperature rise across the device, such voltage is detected by using the Metrohm Autolab B. V. system; then, it is also plotted V vs Q as shown in Figure 7d. Again, the asymmetrical legs module presents a higher open circuit thermal voltage than its symmetrical counterpart; clearly, such enhancement must be related to the thermoelectric performance of the asymmetrical thermoelectric module. It is worth to mention that by continuing applying heat fluxes the temperature rise across the device will continue increasing, and therefore the open circuit output voltage also increases. However, if this heat flux becomes excessive, it could damage the device because of the

On the other hand, when a thermoelectric device is connected to any load it is desirable that such device be able to transfer the greatest amount of power to the load. In this sense, applying

module presents a higher thermal resistance than the symmetrical legs module.

melting of the weld joining the thermoelectric legs.

Figure 5. Steps for the fabrication of the thermoelectric legs.

cutting in every face of the rectangular leg by rotating the graphite tilted base 90 during each cutting. The slope of the base depends on the desired slant angle in the thermoelectric legs as shown in Figure 6a, b, clearly, the graphite plate is tilted at an angle of 10. Besides, it has been attempted to fabricate the legs with a slant angle close to the critical angle of 22 so as to maximize the thermal gradient. However, it has been observed that angles higher than 10 produce legs with a small cross-section in the thin end. Such legs tend to be very fragile because of the mechanical properties of Bi2Te3, and they fracture during the assembling process due to the pressure and thermal treatments applied. Figure 6c, d show the side and top view of the asymmetrical legs used for the fabrication of the thermoelectric device.

Figure 7a, b show the images of the fabricated modules with symmetrical and asymmetrical legs, respectively, wherefrom differences in the geometry of the thermoelectric legs can be observed by comparing both images. At first instance, the performance of the devices has been evaluated by means of the hot-plate method. In such method, the thermoelectric modules are Thermoelectric Devices: Influence of the Legs Geometry and Parasitic Contact Resistances on ZT http://dx.doi.org/10.5772/intechopen.75790 109

Figure 5. Steps for the fabrication of the thermoelectric legs.

cutting in every face of the rectangular leg by rotating the graphite tilted base 90 during each cutting. The slope of the base depends on the desired slant angle in the thermoelectric legs as shown in Figure 6a, b, clearly, the graphite plate is tilted at an angle of 10. Besides, it has been attempted to fabricate the legs with a slant angle close to the critical angle of 22 so as to maximize the thermal gradient. However, it has been observed that angles higher than 10 produce legs with a small cross-section in the thin end. Such legs tend to be very fragile because of the mechanical properties of Bi2Te3, and they fracture during the assembling process due to the pressure and thermal treatments applied. Figure 6c, d show the side and

Figure 4. COMSOL simulation results: (a) temperature profile and (b) open circuit voltage for the asymmetrical thermoelectric module, (c) temperature profile and (d) open circuit voltage for the symmetrical thermoelectric module, respec-

tively at room temperature.

108 Bringing Thermoelectricity into Reality

top view of the asymmetrical legs used for the fabrication of the thermoelectric device.

Figure 7a, b show the images of the fabricated modules with symmetrical and asymmetrical legs, respectively, wherefrom differences in the geometry of the thermoelectric legs can be observed by comparing both images. At first instance, the performance of the devices has been evaluated by means of the hot-plate method. In such method, the thermoelectric modules are installed between heating and cooling plates. Next, a dc voltage with regular increments is applied to the 130 Ω resistive heater of the heating plate by using the BK Precision 9184 dc power supply. This action induces a constant heat flow Q which flows through the test specimens in the stationary temperature state, as well as provides several temperature biases ΔT across the thermoelectric module surfaces. It is worth to note that the side of the applied heat source does not matter in symmetrical modules; however, in the asymmetrical module, the heat source must be applied necessarily on the smaller area side because it has the smallest thermal conductance along the pyramid. In Figure 7c, it is plotted ΔT vs Q wherefrom the slope it can be calculated the thermal resistance of the devices; evidently, the asymmetrical legs module presents a higher thermal resistance than the symmetrical legs module.

Parallel, the open circuit thermal voltage is measured along with the temperature rise across the device, such voltage is detected by using the Metrohm Autolab B. V. system; then, it is also plotted V vs Q as shown in Figure 7d. Again, the asymmetrical legs module presents a higher open circuit thermal voltage than its symmetrical counterpart; clearly, such enhancement must be related to the thermoelectric performance of the asymmetrical thermoelectric module. It is worth to mention that by continuing applying heat fluxes the temperature rise across the device will continue increasing, and therefore the open circuit output voltage also increases. However, if this heat flux becomes excessive, it could damage the device because of the melting of the weld joining the thermoelectric legs.

On the other hand, when a thermoelectric device is connected to any load it is desirable that such device be able to transfer the greatest amount of power to the load. In this sense, applying

Figure 6. Schematic diagram of (a) the tilted base and (b) angled leg cutting suggested to obtain asymmetrical thermoelectric legs; micro-photographs of (c) side view and (d) top view of the thermoelectric legs.

the Theory of Maximum Power Transfer assumes a simple electrical circuit with a voltage source V which is the thermoelectric device; the source has an internal resistance Ri, a load resistance RL, a voltage drop VL, as shown in Figure 8.

The current is I = V/(Ri + RL) and the power delivered to the charge is

$$P = I^2 \mathcal{R}\_L = V^2 \mathcal{R}\_L / \left(\mathcal{R}\_L + \mathcal{R}\_i\right)^2 \tag{5}$$

Figure 7. Photographs of the fabricated module of nine pairs of legs: (a) with symmetrical legs and (b) with asymmetrical legs; (c) temperature rise across both modules as a function of input heat flux; and (d) open circuit voltage as a function of

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Figure 8. Basic electric circuit showing the internal resistance Ri of the thermoelectric device and the load resistance RL.

input heat flux.

Then is considered a variation in power when the load resistance RL is changed, but internal resistance of thermoelectric device Ri as well as the voltage V are constants. When RL ¼ 0, the output power is clearly zero. Likewise, when RL is too big, the output power is zero as well. This suggests that an intermediate value of RL is maximum. When P is derived as a function of RL in Eq. (5) and then it is equal to zero, it is easily shown that the maximum power is when RL ¼ Ri.

In this sense, in order to evaluate the maximum power given by the modules the maximum output power between the asymmetrical and the symmetrical modules has been evaluated by Thermoelectric Devices: Influence of the Legs Geometry and Parasitic Contact Resistances on ZT http://dx.doi.org/10.5772/intechopen.75790 111

Figure 7. Photographs of the fabricated module of nine pairs of legs: (a) with symmetrical legs and (b) with asymmetrical legs; (c) temperature rise across both modules as a function of input heat flux; and (d) open circuit voltage as a function of input heat flux.

the Theory of Maximum Power Transfer assumes a simple electrical circuit with a voltage source V which is the thermoelectric device; the source has an internal resistance Ri, a load

Figure 6. Schematic diagram of (a) the tilted base and (b) angled leg cutting suggested to obtain asymmetrical thermo-

Then is considered a variation in power when the load resistance RL is changed, but internal resistance of thermoelectric device Ri as well as the voltage V are constants. When RL ¼ 0, the output power is clearly zero. Likewise, when RL is too big, the output power is zero as well. This suggests that an intermediate value of RL is maximum. When P is derived as a function of RL in Eq. (5) and then it is equal to zero, it is easily shown that the maximum power is when

In this sense, in order to evaluate the maximum power given by the modules the maximum output power between the asymmetrical and the symmetrical modules has been evaluated by

RL=ð Þ RL þ Ri

<sup>2</sup> (5)

resistance RL, a voltage drop VL, as shown in Figure 8.

110 Bringing Thermoelectricity into Reality

RL ¼ Ri.

The current is I = V/(Ri + RL) and the power delivered to the charge is

electric legs; micro-photographs of (c) side view and (d) top view of the thermoelectric legs.

P ¼ I 2 RL <sup>¼</sup> <sup>V</sup><sup>2</sup>

Figure 8. Basic electric circuit showing the internal resistance Ri of the thermoelectric device and the load resistance RL.

way of the maximum power transfer theorem. In this case, an identical heat flux of around 4 mW/mm<sup>2</sup> was supplied to both modules by applying an electrical current of 90 mA to the Ohmic heater; once modules reached the steady state different load resistances were connected into the module and the voltage across the resistance was recorded, by using those voltages and resistances the output power was estimated. Figure 9 shows the obtained results, it can be observed that asymmetrical module delivers more power than the symmetrical one once the load resistance equals the device internal resistance. In Fact, the asymmetrical thermoelectric module shows to have almost twofold the maximum delivered power as compared to conventional one with a constant square cross-section. Besides, by estimating the maximum available power per unit amount of material (mass of the legs) it has been obtained 433 μW/gram and 1.57 mW/gram for the symmetrical and the asymmetrical modules, respectively.

in Ref. [44] is in logarithmic scale, so it can be closely compared to ΔT = 20�C and ΔT = 30�C, for

Thermoelectric Devices: Influence of the Legs Geometry and Parasitic Contact Resistances on ZT

The thermoelectric figure of merit of the fabricated modules was also evaluated by using impedance spectroscopy technique [45]. In this method, the thermoelectric figure of merit is determined by measuring the adiabatic and isothermal responses of the module under electrical excitation. Under the adiabatic condition at steady state (i.e., ω = 0), the total impedance of

Where Rte is the thermoelectric resistance, and it refers to the resistance of the device which is the result of the temperature difference induced between the ends of the sample due to the Peltier effect, Riso is the isothermal resistance, and it is the resistance of the device Riso excluding thermal effects but including parasitic resistances such as contact and wire resistances, Rp. According to the Harman method, the thermoelectric figure of merit is given by the ratio between the thermoelectric resistance and the isothermal resistance of the system [46]; hence, by dividing Eq. (6) by Riso, ZT is calculated as a function of adiabatic resistance Rad, isothermal

> ZT <sup>¼</sup> Rad � Rp Riso � Rp

> > ZT <sup>¼</sup> Rte

Therefore, by using Eqs. (7) or (8), the effective thermoelectric figure of merit of a device can be accomplished. The adiabatic and isothermal resistances can be easily accessed via electrical impedance measurements [45]. Likewise, parasitic resistances Rc from module with symmetrical and asymmetrical legs are evaluated by applying Transmission Line Method (TLM) [47]

During the measurements, the samples were isolated and suspended to provide adiabatic conditions in a similar way as required in the Harman method [46]. Figure 10a, b show the experimental electrical impedance curves obtained for the symmetrical and asymmetrical leg modules, respectively. In both curves, the thermoelectric, adiabatic, and isothermal resistances are indicated in order to access to their respective values. Nevertheless, it is well known that a material has more than one contribution to its impedance response, which is often the case of thermoelectric materials where thermoelectric impedance, isothermal impedance, and contact impedance have distinct contributions. Hence, one can witness more than one semi-circle, often

Riso � Rp

As well as, in terms of the thermoelectric resistance ZT is given by

and so they can be removed from ZT as it seen in Eq. (7).

Rad ¼ Rte þ Riso (6)

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� 1 (7)

Rp ¼ Rc þ Rw (9)

(8)

113

the asymmetrical module.

the module can be written as [45]

resistance Riso, and parasitic resistance Rp:

where

Evidently, these modules apparently present low output power, however by comparing these modules with several commercially available they have very competitive output power values [44]. For instance, by extrapolating the data shown in Ref. [44] to ΔT = 20C a TEG module based on Bi2Te3 model FERROTEG 9501/71/040B with 71 pairs, and 22 mm 22 mm generates a maximum output power around 1.5 mW. In our case, for modules with only nine pairs, we obtain 0.3 and 0.5 mW for symmetrical and asymmetrical modules, respectively. Nevertheless, by the projection of our modules to 71 pairs we would obtain 2.36 and 3.94 mW, respectively. Besides, if we compare our module against TEG-FERROTEG 9500/127/100B module based on Bi2Te3 with 127 pairs, and 40 40 mm under ΔT = 20C, which delivers an output power around 2.5 mW, we would obtain by a similar extrapolation 4.23 and 7.05 mW for symmetrical and asymmetrical modules, respectively, under ΔT = 20C. It is worth to mention that ΔT scale

Figure 9. Output power vs load resistance for symmetrical and asymmetrical modules.

in Ref. [44] is in logarithmic scale, so it can be closely compared to ΔT = 20�C and ΔT = 30�C, for the asymmetrical module.

The thermoelectric figure of merit of the fabricated modules was also evaluated by using impedance spectroscopy technique [45]. In this method, the thermoelectric figure of merit is determined by measuring the adiabatic and isothermal responses of the module under electrical excitation. Under the adiabatic condition at steady state (i.e., ω = 0), the total impedance of the module can be written as [45]

$$R\_{\text{ad}} = R\_{\text{tr}} + R\_{\text{iso}} \tag{6}$$

Where Rte is the thermoelectric resistance, and it refers to the resistance of the device which is the result of the temperature difference induced between the ends of the sample due to the Peltier effect, Riso is the isothermal resistance, and it is the resistance of the device Riso excluding thermal effects but including parasitic resistances such as contact and wire resistances, Rp.

According to the Harman method, the thermoelectric figure of merit is given by the ratio between the thermoelectric resistance and the isothermal resistance of the system [46]; hence, by dividing Eq. (6) by Riso, ZT is calculated as a function of adiabatic resistance Rad, isothermal resistance Riso, and parasitic resistance Rp:

$$ZT = \frac{R\_{ad} - R\_p}{R\_{iso} - R\_p} - 1\tag{7}$$

As well as, in terms of the thermoelectric resistance ZT is given by

$$ZT = \frac{R\_{tc}}{R\_{iso} - R\_p} \tag{8}$$

where

way of the maximum power transfer theorem. In this case, an identical heat flux of around 4 mW/mm<sup>2</sup> was supplied to both modules by applying an electrical current of 90 mA to the Ohmic heater; once modules reached the steady state different load resistances were connected into the module and the voltage across the resistance was recorded, by using those voltages and resistances the output power was estimated. Figure 9 shows the obtained results, it can be observed that asymmetrical module delivers more power than the symmetrical one once the load resistance equals the device internal resistance. In Fact, the asymmetrical thermoelectric module shows to have almost twofold the maximum delivered power as compared to conventional one with a constant square cross-section. Besides, by estimating the maximum available power per unit amount of material (mass of the legs) it has been obtained 433 μW/gram and

Evidently, these modules apparently present low output power, however by comparing these modules with several commercially available they have very competitive output power values [44]. For instance, by extrapolating the data shown in Ref. [44] to ΔT = 20C a TEG module based on Bi2Te3 model FERROTEG 9501/71/040B with 71 pairs, and 22 mm 22 mm generates a maximum output power around 1.5 mW. In our case, for modules with only nine pairs, we obtain 0.3 and 0.5 mW for symmetrical and asymmetrical modules, respectively. Nevertheless, by the projection of our modules to 71 pairs we would obtain 2.36 and 3.94 mW, respectively. Besides, if we compare our module against TEG-FERROTEG 9500/127/100B module based on Bi2Te3 with 127 pairs, and 40 40 mm under ΔT = 20C, which delivers an output power around 2.5 mW, we would obtain by a similar extrapolation 4.23 and 7.05 mW for symmetrical and asymmetrical modules, respectively, under ΔT = 20C. It is worth to mention that ΔT scale

1.57 mW/gram for the symmetrical and the asymmetrical modules, respectively.

112 Bringing Thermoelectricity into Reality

Figure 9. Output power vs load resistance for symmetrical and asymmetrical modules.

$$R\_p = R\_c + R\_w \tag{9}$$

Therefore, by using Eqs. (7) or (8), the effective thermoelectric figure of merit of a device can be accomplished. The adiabatic and isothermal resistances can be easily accessed via electrical impedance measurements [45]. Likewise, parasitic resistances Rc from module with symmetrical and asymmetrical legs are evaluated by applying Transmission Line Method (TLM) [47] and so they can be removed from ZT as it seen in Eq. (7).

During the measurements, the samples were isolated and suspended to provide adiabatic conditions in a similar way as required in the Harman method [46]. Figure 10a, b show the experimental electrical impedance curves obtained for the symmetrical and asymmetrical leg modules, respectively. In both curves, the thermoelectric, adiabatic, and isothermal resistances are indicated in order to access to their respective values. Nevertheless, it is well known that a material has more than one contribution to its impedance response, which is often the case of thermoelectric materials where thermoelectric impedance, isothermal impedance, and contact impedance have distinct contributions. Hence, one can witness more than one semi-circle, often

overlapping each other which makes impossible to distinguish them. One of the ways to model such a behavior in a simple model can be using in three series–parallel RC elements circuit. In Figure 10a, b, the solid line corresponds to the obtained fitting results. For clarity, such resistance results, as well as the effective thermoelectric figure of merit of the symmetrical and asymmetrical modules are shown in Table 1. Evidently, the thermoelectric figure of merit of the asymmetrical module is almost two-fold the thermoelectric figure of merit of the symmetrical module, such result confirms the enhanced thermoelectric performance of the asymmetrical module as a consequence of the larger temperature rise generated in the legs because of their asymmetry. Hence, harnessing of the Thomson coefficient via asymmetrical legs could be an important strategy in order to accomplish thermoelectric devices with enhanced performance.

Moreover, according to the values shown in Table 1, evidently, the parasitic electrical resistances play an important role in the performance of the device. For instance, if we take into consideration only parasitic contact effects (i.e. parasitic electrical contact resistance between legs and ceramic plates) and neglect the effect of parasitic resistances generated by cable

Thermoelectric Devices: Influence of the Legs Geometry and Parasitic Contact Resistances on ZT

http://dx.doi.org/10.5772/intechopen.75790

115

Thermoelectric device engineering involves the formation of several intrinsic parasitic resistances that affect the thermoelectric module performance. In this sense, the TLM has been applied to discard the parasitic resistances and demonstrate that the increase on ZT of the device is mainly due to asymmetric effect in thermoelectric legs and consequently the non-

Figure 11a shows different lengths in the symmetrical and asymmetric thermoelectric legs as well as their respective electrical resistances as a function of length. The total measured

Where RW1 y RW2 are wiring resistances, RC1 y RC3 are contact resistances due to metal contacts, RC2 is associated with the metallic contact between the junction of the P-type and Ntype thermoelectric legs, and RP-TE and RN-TE define the internal resistance of the P-type and

Figure 11. (a) Diagram of the variation of the length in thermoelectric couples and analysis of the electrical resistance, and

RT ¼ RW<sup>1</sup> þ RC<sup>1</sup> þ Rp�TE þ RC<sup>2</sup> þ Rn�TE þ RC<sup>3</sup> þ RW<sup>2</sup> (10)

3. Impact of parasitic contact electrical resistances on ZT of the

wiring, a value of 0.43 and 0.73 on ZT is obtained.

thermoelectric device

linear Thomson effect that governs them.

resistance consists of several components:

N-type thermoelectric legs, respectively.

(b) analogous electrical circuit of thermally coupled pairs with welding.

On the other hand, it is worth to mention that the present experimental research is mainly focused on the development of devices for applications at room temperature (i.e. 300 K), in that case, it is not necessary to measure the temperature dependence of ZT. Besides, our devices are based on P and N-type Bi2Te3, it is well known that such materials present an optimal thermoelectric performance at around room temperature; hence, operation of such materials must be well below 100C, so an operation condition above this temperature will damage the device because by applying an excessive heat flux it could damage the device due to the melting of the weld joining the thermoelectric legs. In this sense, it is not possible to operate such device under a high-temperature rise away from room temperature would affect seriously their performance.

Figure 10. Experimental electrical impedance curves at room temperature for (a) symmetrical and (b) asymmetrical nine pairs of thermoelectric modules.


Table 1. Experimental parameters of the symmetrical and asymmetrical leg thermoelectric modules.

Moreover, according to the values shown in Table 1, evidently, the parasitic electrical resistances play an important role in the performance of the device. For instance, if we take into consideration only parasitic contact effects (i.e. parasitic electrical contact resistance between legs and ceramic plates) and neglect the effect of parasitic resistances generated by cable wiring, a value of 0.43 and 0.73 on ZT is obtained.
