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Geometrical optimization of a thermoelectric device:
numerical simulations
S. Ferreira-Teixeira1, A. M. Pereira1, a)
1IFIMUP-IN, Department of Physics and Astronomy, Faculty of Sciences of University
of Porto, Portugal
a) Corresponding author: ampereira@fc.up.pt
Keywords: thermoelectric, energy harvesting, heat recovering, numerical simulations,
optimization
Abstract
In the present work, through the use of the COMSOL Multiphysics software,
thermocouples (TCs), constituted by a p and a n type leg, and thermoelectric (TE) devices
with different geometries are numerically simulated aiming for an optimized geometry.
The semiconductor material used in the simulation is bismuth telluride (Bi2Te3) and
copper (Cu) is used to achieve the electrical contact between the TE pillars. Two
geometries of the thermocouple legs are studied, cubic and cylindrical geometries,
revealing that both present identical performances under the same conditions. From these
simulations, the optimal ratios achieved between the various geometrical parameters of
the TC’s are analysed. It was verified that it exists an optimal ratio between the height
and the width of the TE legs with the value 5×10-3. Moreover, it is shown that increasing
the cross-section area of the legs enhances the power produced by a TC. It was also
observed that the length of the copper contacts should be smaller than 0.05 times the
width of the TE legs for the best performance to be achieved. It was also established that
there is an optimal ratio between the copper contacts height and the TE legs height, being,
approximately, 40. For a TE device, an increase in the number of TC’s is favourable
towards a better performance.
1. Introduction
The paradigm of world energy consumption is a concern that must be addressed in the
near future. There is a demand for the use of new sources of energy, namely generation
of sustainable energy using efficient methods. This paradigm leads to the concept of
energy harvesting or energy scavenging which is gaining more importance in the last
years [1, 2]. Several technologies of energy recovering with promising outputs already
exist. These use mechanical movement, pressure or temperature [2]. Taking advantage of
mechanical movements or pressure, Triboelectric and Piezoelectric generators are
extensively studied in parallel to Magnetic Induction based devices [1, 3]. Triboelectric
generators were invented as new way of harvesting energy using the coupling of the
triboelectric and electrostatic effects. Recently, nanostructures have been used, which
increases the efficiency up to 100%, making Triboelectric generators the technology with
the most prominent growth [4]. Other technologies that use movements or pressure (i.e.
Piezoelectric and Magnetic Induction), in spite of, nowadays also being at the nanoscale,
did not achieve outputs like Triboelectrics [1-3]. On the other end, Photovoltaic panels
and Biomass are technologies already present in the market.
All these technologies are being developed or effectively applied, using energy that
otherwise would go to waste. However, heat is still the form of energy that is widely
wasted. The thermoelectric effect is the unique phenomena that can directly convert waste
heat into electrical energy and vice-versa. It is based on the Seebeck effect which is the
generation of electricity due to a temperature difference. This effect is the basis of TE
generators [5, 6]. Although there is an enormous amount of waste heat all around us
(home heating, cars, industries…), these generators are not yet widely used because their
efficiency is not sufficient to attend our world’s demands. Therefore, there is a need of
enhancing the performance of TE generators [5-7].
A typical TE device contains a certain number of thermocouples (TCs) that are made of
p-type and n-type semiconductor legs (the TE material) connected electrically in series
and thermally in parallel. A temperature difference is imposed between the top and
bottom of the legs and an electrical voltage is generated [5]. Until now, the creation of
complex TE materials with enhanced figures of merit or the improvement of the
fabricated TE devices are the main efforts for the advance of TE generators. For example,
Ref. [6] is a review of the working principle of a TE generator and coolers, where an
overview of the current research into TE materials is also given. These materials can be
characterised into three categories such as semiconductors, ceramics (metallic oxides)
and polymers (carbon nanotubes in polymer-matrixes). On Ref. [7], the material’s
transport properties that influence the efficiency of a TE material are explained and
several complex TE materials being developed nowadays are discussed. These materials
are, for example, alloys of Bi, Te and Sb as well as skutterudite alloys like CoSb3.
Another way to optimize TE devices is to perform numerical simulations. So far, these
simulations focus either on optimizing a certain material property or on studying a type
of device under a set of experimental conditions, to draw a comparison between the
theoretical behaviour and the experimental results. One example of these type of TE
generators development is the work of references [8, 9]. As a first part of this work, they
used commercially available TE generators to realise experimentally a heat exchanger
based on cold and hot fluid flows. On the second part of their work, they simulated using
finite element methods their TE generator based heat exchanger, in order to have a
comparison between the experimental results they obtained and the theoretical expected
behaviour. References [10, 11] are also two examples of simulations of TE generators
being used to have a greater insight into a certain application of the generators or for a
comparison with experimental results. The COMSOL Multiphysics software has been
widely used in simulations of the TE effect. The previously referred work [9] was
performed using COMSOL Multiphysics, as well as Ref. [12], where the numerical
simulations where conducted to have a comparison with the experimental results of a TE
generator. However, simulations performed using this software focus on studying only a
certain novel device. Studies of the geometry of TE devices using simulations are still
ongoing. On Ref. [13], the authors simulate two types of devices with distinct TC’s and
study how their efficiency depends on the width of the legs and number of these TC’s.
However, their main focus was also the simulation of the heat source and a comparison
and validation of the numerical results with the experimental studies. There is still a lack
of systematic geometrical studies of a TE device, with not only a focus on the geometry
of the TE material but also on the electrical contacts geometry. Thus, the aim of this work
is to numerically simulate the influence of the geometric parameters of the TE material
and of the electrical contacts towards the optimization of the TE device. The optimization
was made in three phases: i) the study of the geometry of the TE legs of a TC; ii) the
temperature dependence of the power output, iii) the study of the geometry of the
electrical contacts and iv) the study of the optimal number of TCs on a TE device. These
studies were made by numerical simulations performed using the COMSOL Multiphysics
software.
2. Numerical considerations
The 3D geometries studied and their materials are represented in Figure 1. For the
numerical simulation, it was considered that both geometries are constituted by two legs
of a material with physical properties similar to bismuth telluride (Bi2Te3) with their
difference being the signal of the Seebeck coefficient, allowing one leg to be considered
as a p-type material and the other n-type. Moreover, to insure the electrical contact
between the legs, three layers of copper (Cu) were also included. Insulating layers as in
classical TE generators were not included due to the performed studies being the ideal
behaviour, without heat losses. The geometrical parameters of the legs were height (H)
and width (L) for the square geometry whereas for the cylindrical geometry it was
considered radii (R). In both geometries, the copper contacts have a height (h) and length
(D) when connecting the TE legs, with the bottom ones having the same base as the legs
and the one at the top achieving the electrical contact between the two TE legs. The TE
device is constituted by a certain number of TCs as the ones described, with a p-type leg
connected to two n-type legs by layers of copper, as represented in Figure 2.
Figure 1 Schematic geometries of the thermocouples considered: cubic and cylindrical, and boundary
conditions.
Figure 2 Schematic geometry of the thermoelectric device considered with cubic thermocouples and
boundary conditions.
The COMSOL Multiphysics 4.4 was the software used to perform the simulations. The
TE module inside the heat conduction module was the basis for all the simulations. This
module uses finite element methods (FEM) to solve the equations that describe the TE
effect: the heat conduction equation in solids and the continuity of electric charge
equation with the constitutive TE equations [14-16]. FEM is a numerical method in which
the system is divided into smaller systems (finite elements) where the simpler equations
that govern each of the finite elements are assembled into a large system of equations,
which are then solved using variational methods to minimise the associated error
function. The studies carried out are stationary and without a magnetic field. Considering
this, the system of coupled TE equations that the module solves is:
(1)
where T, V, S, k and σ are the absolute temperature, the electrical potential, the Seebeck
coefficient, the thermal conductivity and the electrical conductivity, respectively [14, 17].
For the TE module to solve the system of equations, two different computational meshes
were applied to the simulated volume. It was used a quadrangular mesh for the cubic TC:
to the Bi2Te3 legs was applied a mesh that had a refinement equal to their height divided
in at least four parts, with at least 400 nodes. To the copper connections the COMSOL’S
swept mesh was applied, with the same refinement of the TE legs. For the cylindrical TC
a triangular mesh (the COMSOL’s free tetrahedral mesh) was used. To the simulated TE
devices, it was applied a triangular mesh. The software’s meshes (free tetrahedral) had a
fine to finer refinement of the COMSOL software, depending on the magnitude of the
geometrical parameters. The convergence criteria of the simulations was the
automatically provided by COMSOL, having been verified that these criteria were
enough to obtain precise results.
The materials used are in the material’s library of the COMSOL Multiphysics software.
The properties of bismuth telluride (Seebeck coefficient, electrical and thermal
conductivities) vary with temperature. These properties are defined in the software only
within the range [200,400] K of temperature. Studies of the temperature were
consequently within this range. All copper properties were constant and exist in the
library of the software, except for the Seebeck coefficient. It was used Scopper = 6.5×10-6
V/K [17], which does not vary in a significate way in the range of temperatures studied.
In all simulations performed, a temperature difference between the bottom and top of the
TC was applied. The bottom was at a constant temperature TC = 0 °C in all the simulations,
and the top was at a temperature TH that was constant in each simulation but varied in
studies of temperature. All the other surfaces were thermally and electrically isolated, by
assuming that the simulated systems were placed in vacuum. The copper layer of the first
leg of the TC was grounded (V = 0 V) in all the simulations, as it is seen in Figure 1.
The simulations of the TE device were made with cubic TCs, having the optimal
geometrical parameters found in the simulations performed in this work prior to the TE
device simulations. As before, a temperature difference was applied between the bottom
and top of the device, with the bottom at TC = 0 °C and the top at TH = 47 °C leading to
a temperature difference ΔT = 47 °C, and all other surfaces were thermally isolated. The
copper layer of the first leg was grounded, as seen in Figure 2. The number of TCs varied
between 1 and 5.
The voltage difference between the two legs of the TC and the first and last leg of the TE
devices was thus obtained at the probe points presented in Figure 1 and 2. The temperature
was also obtained at the indicated probe points, which are situated in the geometrical
centre of the leg’s cross section. The maximum power output is given by equation (2)
[18]. The ideal case of device performance is assumed. This case corresponds to
performance studies of the device using a loading resistance of the same value of the
device resistance R. The device resistance is given by the sum of the resistance of the
Bi2Te3 legs and of the copper contacts [18]. It can be approximated by equation (3), where
n is the number of TC’s in a TE device (for the TC studies, n = 1) and σ is the electrical
conductivity of the considered materials (Bi2Te3 and Cu).
(2)
(3)
3. Numerical results
From the numerical simulations performed, it was discovered that the considered cubic
and cylindrical geometries, with the same leg length and area, had precisely the same
results under the same conditions. This shows that the shape of the leg does not play a
role on the TE device performance. Thus, the results throughout this manuscript were
obtained using the cubic TC.
3.1. Influence of the geometry of the thermoelectric material
Figure 3.a) presents the electrical potential as a function of the ratio H/L for three different
widths considering a temperature difference of ΔT = 47 °C. A full overlap of the curves
is observed for all widths studied (1, 10 and 100 μm). These results unveiled that the
electrical properties of the TC’s are only dependent on the geometrical ratios and not on
the values of the geometrical parameters. Concerning the values of the electrical potential,
they are in the range of 0 mV to -20 mV, approximately. This result is inconsistent with
the expected behaviour since considering the Seebeck relation for a constant temperature
difference, the generated voltage should be constant (ΔV=-S.ΔT). Thus, for two Bi2Te3
(S= 0.21 mV/K) legs in series under a temperature difference of ΔT = 47 °C, the expected
generated voltage is around -20 mV, which is the maximum value achieved in the
simulations i.e. for ratios only higher than 0.1.
In order to understand these results, the temperature distribution of the TC’s legs was
studied when a constant temperature difference of 47 °C was applied. In Figure 4, the
temperature at the three points of the TC’s Bi2Te3 along the leg (see Fig. 1) is plotted as
a function of the ratio H/L, considering the width L = 100 µm. It was verified that the
temperature has the same values for every cross-sectional plane of the legs perpendicular
to the direction of the gradient of temperature (z direction), due to the imposed boundary
conditions that consider the legs thermally and electrically isolated. Analysing the middle
probe point of the leg, the temperature is constant and independent of the ratio. However,
when considering the temperature at the bottom and top of the Bi2Te3 leg, as a function
of H/L, an increase of the temperature at the top and the same variation but with opposite
signal at the bottom is found. These results allow us to understand the behaviour obtained
in Figure 3, i.e. the achieved electrical potential behaviour arises from the variation of the
effective temperature difference, which increases with the increasing ratio until it reaches
the applied value. The effective temperature difference is equal to the corresponding
applied value for ratios H/L higher than 0.1, which is the same ratio found previously for
the electrical potential. This is also shown in Figure 3.b), which compares the electrical
potential results with the effective temperature difference at the Bi2Te3 legs as a function
of the ratio H/L. From these results, the effective Seebeck coefficient of the TC at this
temperature can also be obtained using the Seebeck relation. The calculated value was,
approximately, -0.42 mV/K which is in accordance with the expected value.
Figure 3 Electrical potential as a function of the ratio H/L for an applied temperature difference of 47°C: a) for three different widths:
L=100 µm, L=10 µm and L=1 µm; b) and effective temperature difference for L=100 μm.
Figure 4 Temperature throughout the Bi2Te3 leg’s as a function of the ratio H/L.
The power was calculated using equation (3) and is displayed in Figure 5 as a function of
the ratio H/L for the three different widths considered. All the widths have the same
behaviour, but the power decreases one order of magnitude with a decrease in the order
of magnitude of the width, namely, the maximum power is 26 mW, 2.6 mW and 0.26
mW for L=100 μm, 10 μm and 1 μm, respectively. The power as a function of the ratio
H/L has a peak at a maximum ratio of, approximately, 5x10-3. This is due to the
dependence of the power on the geometrical parameters. For smaller ratios, the electrical
potential is small, as seen in the previous results. This makes the power output lower,
explaining the first region before the maximum. After the maximum, the power output is
dominated by the resistance because the potential starts to have a constant value. Since
the resistance, for high H/L ratios, is dominated by the Bi2Te3 leg’s term, the power starts
to be proportional to the inverse of the ratio, explaining the behaviour obtained in Figure
5.
To verify that these results are valid for every temperature difference, this study was
performed under different applied temperature differences. The results presented in
Figure 6 were obtained. In Figure 6.a), the electric potential is depicted as a function of
the ratio H/L for four different applied temperature differences. It is observed that the
electric potential has the same behaviour for the different temperature differences
considered: approximately zero for small ratios increasing until achieving the highest
value possible. This behaviour occurred due to the temperature distribution throughout
the TC and it is thus independent of the applied temperature difference. Furthermore, it
is verified that the electric potential increases in value with increasing temperature. Figure
6.b) shows the power as a function of the ratio H/L for four different applied temperature
differences, where it is seen that the power increases with temperature. Moreover, it is
observed that the power has a peak for the four different temperature differences and that
their maximum is the same, being equal to 5x10-3. The conclusion is that the optimal ratio
H/L of a TC is independent of the temperature gradient.
Figure 5 Power as a function of the ratio H/L for three different widths: a) L1 = 100 μm, b) L2 = 10 µm
and c) L3 = 1 µm.
Figure 6 a) Electric potential and b) power as a function of the ratio H/L for four different applied temperature differences.
The dependence on the width of the TE legs, L, was also studied for a constant H=100
µm, by analysing the ratio L/H. From the numerical results of this study, we discovered
that the electrical potential is constant and equal to -19.6 mV. This occurs due to the
gradient of temperature being constant throughout the TE material legs in this study (the
effective applied temperature is constant when varying L). Figure 7 shows the dependence
of the logarithm of the power on the logarithm of the ratio L/H with an inset of the power
variation with the ratio L/H. One recognizes the proportionality of the logarithms,
concluding that the power output is proportional to a power of the ratio L/H. From the
linear fit of the logarithms, it is established that the power is proportional to the square of
the ratio L/H. Although the voltage remains constant, the power, given by equation (2),
depends on the cross-section area of the legs, which is proportional to the square of the
width of the legs. A maximum power is thus obtained by increasing the cross-section area
of the legs, with 0.026W being achieved for a width of L=1 cm.
Figure 7 Logarithm of the power as a function of the logarithm of the ratio L/H with the inset of the
power as a funtion of the ratio L/H.
3.2. Influence of the applied temperature on the effective temperature and device
output
Having obtained the previous results, a deeper analysis of the applied temperature was
made. In Figure 8.a), the effective temperature difference through the thermoelectric
material is plotted against the applied temperature difference between the Cu contacts.
The effective temperature difference is the difference of the temperature between the
bottom and top probe points of temperature in Figure 1, for a TC with the obtained optimal
ratio H/L and a width of L = 100 µm. It is demonstrated that the effective temperature
difference is lower than the applied one, being proportional to it. By performing a linear
fit, it is achieved a constant of proportionality of, approximately, 0.86. In Figure 8.b) the
electric potential as a function of the effective temperature difference is shown, where a
linear behaviour is recognized as expected from the Seebeck effect [5, 6]. From this graph,
the Seebeck coefficient of the studied TC was calculated to be -0.440 mV/K, which is in
agreement with the value obtained in the study of the ratio H/L. Since the TE legs are
connected in series electrically, the expected value of the Seebeck coefficient of the TC
is approximately equal to two times the Seebeck coefficient of Bi2Te3 (-0.42 mV/K),
which is also in accordance with the value obtained from this study.
The power output as a function of the temperature difference was also obtained. In Figure
9, it is depicted the logarithm of the power as a function of the logarithm of the effective
temperature difference with an inset of the power as a function of the effective
temperature difference. It is discovered that the logarithms are proportional and from their
linear fit, it is concluded that the power is approximately proportional to the square of the
temperature. This arises because the power is proportional to the square of the potential
and thus to the applied temperature difference. Increasing the applied temperature
difference, increases the effective temperature difference, and it thus enhances the power
output of the TC, with 0.20 W being achieved for 100°C.
Figure 8 a) Effective temperature difference and b) electric potential as a function of the applied temperature difference.
Figure 9 Logarithm of the power as a function of the logarithm of the effective temperature difference
with an inset of the power as a function of the effective temperature difference.
3.3. Influence of the contact resistance
An important parameter in a TE is the contact resistance. Several simulations were
performed changing the dimensions of the Cu contacts, namely their height (h), their
length (D) and their area, in order to evaluate the contact influence.
The ratio h/H between the height of the copper layers and the height of the TE legs was
studied. Figure 10 shows the potential and the effective temperature difference of the
device as a function of the ratio for L = 100 μm and with the optimal H/L ratio. It is
determined that the effective temperature difference is equal to the applied temperature
difference for ratios smaller than 2.5. Moreover, the electric potential has the same
behaviour, reaching the expected value (approximately -20 mV) only for ratios smaller
than 2.5.
The power as a function of the h/H ratio was also examined and it is depicted in Figure
11, for three different heights of the TE legs. It has a maximum at roughly h/H=40, for
all the studied heights. The maximum value is 0.03 for a TE leg height of 500nm, reducing
to 0.003 for a height of 50 nm. From this plot, the conclusion that there is an optimal ratio
between the heights of the copper contacts and the thermoelectric legs can be draw and
that this ratio is independent of the height of the TE legs.
Figure 10 Electrical potential and effective temperature difference as a function of the ratio h/H for an
applied temperature difference of 47°C.
To verify the consistency of this result, studies of this ratio for different applied
temperatures were performed. Figure 11.a) shows the potential as a function of the ratio
h/H. It has the same behaviour for all the considered temperatures. As previously, the
achieved maximum occurs only due to the geometric parameters and how they influence
the temperature distribution throughout the device and not on the actual values of the
applied temperature differences. The power output of the TC as a function of the ratio for
the considered temperatures is presented in Figure 12.b). It corroborates the existence of
a maximum for all the different applied temperature differences. This maximum is
independent of the temperature, being achieved at a ratio of approximately h/H=40, which
was the value obtained previously.
Figure 11 Power as a function of the ratio h/H for three different widths: a) H1 = 500 nm, b) H2 = 250 nm
and c) H3 = 50 nm.
Figure 12 a) Electric potential and b) power as a function of the ratio h/H for four different applied temperature differences.
The length of the copper contacts (the distance between the TE legs) was also
investigated. Using a TC with the other parameters optimal ratios, it was realised that the
electric potential is constant and independent of the distance between the legs, having
approximately the value of -15 mV. It was demonstrated that the potential has a
dependence on only the distribution of the temperature throughout the TE legs. Since a
change in the length of the copper contacts does not affect the temperature distribution,
the voltage remains constant. The power output of the TC as a function of the ratio
between the copper contact length and the width of the legs is presented in Figure 13, for
two different widths of the legs. While the potential is constant, it is observed that the
power output has a different behaviour that it is equal for the different considered widths.
Furthermore, from Figure 13, it is detected a plateau of maximum power for ratios D/L
smaller than 0.05. With this, it is concluded that to obtain the maximum power output,
the contact length D should be smaller than 0.05L. There is no influence on the power
output from the decrease of the contact length for values of the ratio smaller than 0.05,
since the power is constant and for example, equal to, approximately, 0.05W for a width
of L = 100 μm.
Figure 13 Power output as a function of the ratio D/L for two different TE leg’s widths.
The area of the copper contacts was also inspected. The copper contact width on the
bottom and top of the TE leg was reduced by a ratio r from the width L of the TE leg,
with the area given by (rL)2. The electric potential and power output were examined, for
a TC with the optimal values found in this work but a varying copper contact area. The
electric potential is constant and independent of the area of the copper contacts, having
the value -15 mV. However, the power has a different behaviour that is depicted in Figure
14 as a function of the ratio squared r2 for two different widths of the TE legs. It is detected
that the power output increases with the squared r for both the considered widths. This
occurs due to the decrease in the resistance, (equation (3)), specially of the copper
contacts. By increasing the area of the copper contacts, in this approximation, the charge
carriers have more area to cross the same height, which implies a decrease in the
resistance. The power thus increases with the area of the copper contacts. With this, a
higher copper contact area is needed to attain the highest power output, being
approximately 0.014W and 0.029 W for L = 50 μm and L = 100 μm, approximately.
Figure 14 Power output as a function of the ratio squared r2 for two different TE leg’s widths.
3.4. Thermoelectric Device
With an optimized TC (L = 100 µm, H = 5×10-3L = 500 nm, h = 40H= 20 µm, D = 0.05L
= 5 µm, r = 1), TE devices were studied under an applied temperature of ΔT=47°C. The
number of TC’s, n, in a TE device was analysed and the results are presented in Figure
15. In this Figure, the power as a function of the number of TC’s, with an inset of the
voltage as a function of n is shown. One confirms that the potential is proportional to the
number of TCs of a device. Due to the electrical connection in series of the TCs, the
voltage difference between the first and last leg is the sum of the electrical potential that
each TC generates and it is thus proportional to the number of TCs. Since the power is
proportional to the square of the voltage, the power must also be proportional to the square
of the number of TC’s. This is corroborated by Figure 15, where the power fit is also
presented. From this, a higher power is obtained with the highest possible number of TC’s
in a TE device.
From these results, a comparison with Photovoltaic panels, one of the most used
renewable energy systems can be made. A Photovoltaic panel as a mean production of 10
W/cm2 [19]. The area of the studied TE device is given by equation (5). To obtain 10W,
at least 17 TC’s must be used, under an applied temperature of 47 ºC. A TE device with
17 TC’s has an area of 3.649×10-3 cm2, which is much smaller than the equivalent area
of a photovoltaic panel.
(5)
Figure 15 Power output as a function of the number of thermocouples for L = 100 µm with an inset of the
electric potential as a function of the number of thermocouples.
3.5. Experimental Validation
Using commercial thermoelectric generators (TEGs) devices (A: TEC1 -12706 40x40mm
127 TCs; B: TEC1 – 24105 50x50 mm 241 TCs), some experimental validations were
performed of the previously obtained results. Measurements of the geometrical
parameters of each individual TC of the used TE’s were performed, having obtained
H=1.2 mm, L=1.17mm, which corresponds to a H/L ratio approximately equal to 1. The
experimental setup and electrical circuits used to measure the power output of the TEGs
is presented in Figure 16. The temperature difference was applied by heating on side of
the TEGs with a hot plate and keeping the other at room temperature by using a heat
dissipater. The temperatures were measured within an error of 1ºC. Two circuits were
employed to measure the maximum power output of the TEGs. These are constituted by
a variable resistance [1, 106] Ω, a Load resistance Rload = 50±1 Ω and a Nanovoltmeter
(HP 34420A) with a precision of 0.001 mV. Figure 16.b) shows the circuit used to obtain
the current through the load resistance and Figure 16.c) presents the circuit used to obtain
the voltage output of the TEG. From these two measurements, the maximum power was
calculated.
Figure 16 Experimental Validation. a) Experimental Setup. The temperature difference was applied by
heating one side of the Thermoelectric Generators with a hot plate. Electric circuits used to measure the
maximum power output of the TEGs b) Electric current and c) Voltage output. d) Power output of the
thermoelectric generator A compared with the power output of a single thermocouple as a function of
temperature and normalised to the power output at 47 ºC.
The power output of each thermoelectric device was measured at room temperature when
a temperature difference of 50 ºC was applied. It was verified that device A had a power
output of 1.6mW whereas device B achieved a maximum power of 7.5mW. A model for
the power as a function of the number of thermocouples using numerical results was
obtained in the previous section, as presented in Figure 15. One verified that the power is
proportional to the square of the number of TCs and thus by calculating the ratio between
the experimentally obtained power values and the respective square of the number of TCs,
the law can be verified. It is calculated that the ratio between the powers is 0.21 and that
the ratio between the number of TCs n is 0.28. Considering how the measurement was
performed and the errors associated with it, namely room temperature fluctuations,
especially when a high temperature difference is applied, the experimental measured
values and numerical results are in great accordance with each other.
Following the studies of the maximum power for both devices, the power output of device
A as a function of temperature was studied, for temperature differences between 2 and 53
ºC. Figure 16.d) displays the measured power output normalised by the power output at
47ºC and the numerical results of Figure 9 also normalised by the power at 47ºC as a
function of applied temperature. One observes the great accordance between the
experimental and numerical results. The small deviations, as before, are due to the small
variations of the local room temperature of the cold side of the TEG at which the
experimental results were obtained.
4. Conclusion
In conclusion, this work allowed us to optimize a TE device. Our simulations
demonstrated that the Bi2Te3 legs of a TC can have a cubic or cylindrical geometry
without having a different performance. The obtained value of the ratio between the
height and the width of the TE material legs that optimizes the TC operation was 5×10-3.
Furthermore, our simulations showed that increasing the cross-section area of the legs
improves the power produced by a TC. The copper contact resistance was also studied
and it was verified that there is an optimal ratio between the contact height and the TE
leg height, being, approximately, 40. It was also observed that the maximum power output
of the TC occurs for ratios between the copper contact length and the width of the TE legs
smaller than 0.05. The area of the copper contacts should also be equal to the cross-
sectional area of the TE legs, for the TC to have a maximum power output. The power
produced by a TC is also proportional to the square of the temperature difference.
Moreover, the optimal operation of a TE device was achieved with the highest number of
TCs studied, indicating that an optimized device should have the maximum number of
TCs possible.
After the performed theoretical studies, an experimental verification of the obtained
results would complement the obtained results. However, in order to perform similar
analysis of the geometry of a TE device, various devices would have to be fabricated and
measured. The obtained numerical results predict the behaviour of devices with
geometrical parameters having a wide range of values, which vary orders of magnitude.
Nowadays, the fabrication processes used to fabricate TE devices allow for the fabrication
of devices with geometrical ratios only within a certain range. Thus, to fabricate all the
studied devices (from nanometric devices to microscopic) more than one fabrication
technique would have had to be used. This could lead to differences of performance
between devices that were fabricated by different methods, which would not allow for a
good comparative and quantitative study of their performance. With this, the fabrication
and production of the quantity of devices needed to verify the geometry theoretical studies
cannot be attained, and two commercial thermoelectric generators have been used to
verify experimentally the model used throughout the presented work.
5. Acknowledgements
The authors acknowledge funding from FEDER, ON2 and FCT through project
PTDC/CTM-NAN/5414/2014 and Associated Laboratory - IN.
A. Supplementary Information
Variable
Constant Parameters
H/L
L1=100 µm
D=L=100 µm
h=10 µm
ΔT=25, 47, 75, 100°C
r=1
L2=10 µm
D=L=100 µm
h=1 µm
ΔT=47°C
r=1
L3=1 µm
D=L=100 µm
h=0.1 µm
ΔT=47°C
r=1
L/H
H=100 µm
D=L=100 µm
h=10 µm
ΔT=47°C
r=1
ΔT
L=100 µm
H=5×10-3L=500 nm
h=10 µm
D=L=100 µm
r=1
h/H
L1=100 µm
H1=5×10-3L=500 nm
D=L=100 µm
ΔT=25, 47, 75, 100°C
r=1
L2=10 µm
H2=5×10-3L=50 nm
D=L=10 µm
ΔT=47°C
r=1
L3=50 µm
H3=5×10-3L=250 nm
D=L=50 µm
ΔT=47°C
r=1
D/L
L1=100 µm
H1=5×10-3L=500 nm
h1=40H1=20 µm
ΔT=47°C
r=1
L2=50 µm
H2=5×10-3L=250 nm
h2=40H2=10 µm
ΔT=47°C
r=1
r
L1=100 µm
H1=5×10-3L=500 nm
h1=40H1=20 µm
D1=0.05L1=5 µm
ΔT=47°C
L2=50 µm
H2=5×10-3L=250 nm
h2=40H2=10 µm
D1=0.05L2=2.5 µm
ΔT=47°C
Table 1 Studies of the thermocouple.
Variable
Constant Parameters
n Thermocouples
L=100 µm
H=5×10-3L=500 nm
h=40H=20 µm
D=0.05L=5 µm
ΔT=47°C
r=1
Table 2 Studies of the thermoelectric device.
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