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Pirela R. et al. Novel Methodology for Characterization of Thermoelectric Modules and Materials
https://doi.org/10.47460/athenea.v5i15.72
Novel methodology for characterization of thermoelectric
modules and materials
Received (30/10/2023), Accepted (25/02/2024)
Abstract. - The document presents an innovative methodology that combines forced response and natural
response theories in thermoelectric materials and devices. It stands out for expressing the thermoelectric
figure of merit in terms of the ratio of two temperatures
, enabling comprehensive testing
and precise characterization of thermoelectric modules and materials, including measurements of thermal
conductance, electrical resistance, Seebeck coefficient, and figure of merit. Additionally, it addresses the
determination of thermal resistances and thermal capacitances related to thermal contacts, as well as the
derivation of characteristic time constants and angular frequencies. This approach, applicable to both
modular devices and individual samples, allows for the simultaneous measurement of all parameters on a
single sample. The experiments considered non-ideal contacts and non-adiabatic conditions at room
temperature
, enhancing the feasibility of in-situ characterization and positioning this
methodology as a key tool in thermoelectric research.
Keywords: thermoelectric characterization, Harman method, transient test method, thermoelectric time
constants, thermoelectric frequencies, complete response, figure of merit.
Metodología novedosa para la caracterización de módulos y materiales termoeléctricos
Resumen: El documento presenta una metodología innovadora que combina teorías de respuesta forzada
y respuesta natural en materiales y dispositivos termoeléctricos. Destaca por expresar la figura de mérito
termoeléctrica en términos de la relación de dos temperaturas
, permitiendo ensayos
completos y caracterizaciones precisas de módulos y materiales termoeléctricos, incluyendo mediciones de
conductancia térmica, resistencia eléctrica, coeficiente de Seebeck y figura de mérito. Además, aborda la
obtención de resistencias térmicas y capacitancias térmicas relacionadas con contactos térmicos, así como
la determinación de constantes de tiempo características y frecuencias angulares. Este enfoque, aplicable
tanto a dispositivos modulares como a muestras individuales, posibilita la medición simultánea de todos
los parámetros en una misma muestra. Los experimentos consideraron contactos no ideales y condiciones
no adiabáticas a temperatura ambiente T=300K mejorando la viabilidad de la caracterización in situ y
posicionando esta metodología como una herramienta clave en la investigación termoeléctrica.
Palabras clave: caracterización termoeléctrica, método de Harman, método de prueba transitorio,
constantes de tiempo termoeléctricas, frecuencias angulares termoeléctricas, figura de mérito.
Ronald Pirela
https://orcid.org/0000-0002-1411-6333
repirelalc@estudiante.unexpo.edu.ve
Alstom Ferroviaria, S.P.A.
IEEE Membership
Savigliano-Italia
Sergio Velásquez
https://orcid.org/0000-0002-3516-4430
svelasquez@unexpo.edu.ve
Universidad Experimental Politécnica
Antonio José de Sucre
Puerto Ordaz-Venezuela
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Pirela R. et al. Novel Methodology for Characterization of Thermoelectric Modules and Materials
I. INTRODUCTION
The Thermoelectric Modules and Thermoelectric Materials (TEMs) are solid-state power converters that
typically consist of p and n type semiconductor material arrangements, connected so that they are thermally
in parallel and electrically in series [1] [2] [3] [4]. TEMs are marketed for applications in the areas of cooling
and heating; as well as for the generation of electrical power, collecting solar energy and residual heat [5]
[6]. One of the arduous tasks that must be carried out in the thermoelectric field is the characterization of
thermoelectric materials and modules [7]. It is essential to obtain reliable measurements of global efficiency
to assess its technological and economic interest [8] [9]. Another challenge to meet is to determine the
performance of thermoelectric devices adequately and accurately [10]. The thermoelectric performance is
reduced to the determination of a single quantity called the figure of merit
and a way of expressing it is
presented in (1), where
is the average working temperature of the system,
is the electrical resistance
of the module,
is the thermal conductance to the vanishing electric current, and is the global Seebeck
coefficient characterizing the thermoelectric coupling between the electric current and the heat flux through
the TEM terminals. The figure of merit
is related to the theoretical maximum efficiency of a thermoelectric
generator that works between two thermal reservoirs, one at a hot temperature
and the other at a cold
temperature
, where
<
. The maximum efficiency
is determined by means of (2), in which
is the Carnot efficiency [11] [12].
(1)
(2)
The precise evaluation of
is far from simple, and several approaches can be applied, e.g.: measure ,
y
separately and then calculate
using (1). However, this method is quite inaccurate without great
experimental attention, since each measurement error for each parameter contributes to the accumulated
global error in the resulting
value [13] [14]. Other methods use different measurement systems for each
property. Often the three main properties are not measured on the same sample or in the same direction.
Also, these methods are time-consuming and susceptible to increasing the uncertainties in
. In the late
1950’s a method was presented by Harman for testing the resistance in alternating current and determining
the figure of merit of a thermoelectric material sample [15]. Also, the same methodology was used by
Harman, Cahn, and Logan for the measurement of thermal conductivity applying the Peltier Effect [16]. This
technique has many variations and it has been traditionally applied to both bulk modules and thin films.
The drawbacks are that it only works with small temperature differences and requires adiabatic boundary
conditions that can be difficult to satisfy [17] [18]. In the early 1990s a test methodology was developed by
Buist, this method is referred to as the transient test method [19], which is based on a similar concept but
some fundamental differences compared with Harman’s method which gave rise to improvement in
accuracy, and reproducibility [5] [20]. The fundamental similarity is that the two techniques cited above are
designed to solve the voltage components of a thermoelectric device and the fundamental difference is
that the Harman method does this by measuring the resistive component and the transient test method
measures the Seebeck component, whereas the novel methodology for characterization hat this moment
described is designed by determination of the thermal components, and it is intended to provide the
ultimate solution for measuring the thermal conductance, thermal conductivity and the figure of merit of
TEMs.
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Pirela R. et al. Novel Methodology for Characterization of Thermoelectric Modules and Materials
The new method presented in this paper provides a set of guidelines for the direct measurement of all
the parameters needed to characterize the thermoelectric properties of either materials and devices
under test, such as the thermoelectric resistance of the module
, thermal resistances related to
thermal contacts
, thermoelectric capacitance of the module
, thermal capacitances related to
thermal contacts
, thermal resistance of the thermoelectric material
and the capacitance of the
thermoelectric material
. As well the Seebeck coefficient , electrical resistance
, thermal
conductance
, electrical resistivity , thermal conductivity and figure of merit
. Additionally,
through this methodology is possible to determine the characteristic time constants and relaxation times,
as well as the characteristic angular frequencies.
This article is structured as follows: in section II the complete response of TEMs is presented, then in
section III the temperature and voltage stability are explained, obtaining as a result the characteristic
thermoelectric time constants
,
,
and the relaxation times, and it continues to section IV
developing thermoelectric modeling and equation derivations, where new equations of the figure of merit
are shown in V characterization configurations. Finally, the experimental results, implementation,
conclusions, and references. Furthermore, this research is framed within UNESCO's 2015-2030 agenda for
sustainable development, specifically objective number 7, entitled “Affordable and Clean Energy”, which
aims to improve access to clean energy through inclusive science, technology, and innovation systems
(STI).
II. COMPLETE RESPONSE OF THERMOELECTRIC MODULES AND MATERIALS
The temperature difference and voltage of TEMs generated due to the Peltier and Seebeck effects,
respectively [2] [1], have two components and there are two classical ways to break it down into two parts.
The first way is to divide it into “a forced response (independent source) and a natural response (stored
energy)”, and the second way is to divide it into “a steady-state response (permanent or stable part, this is
the behavior of the TEMs long after external excitation applied) and a transient response (temporary part,
which will extinguish with time)” [20].
The unification of the forced response (slow and fast perturbation) and natural response (absence of
perturbation) theories allows for a description of the complete response of thermoelectric modules and
materials; as well as allows the study and characterization of TEMs [21] [22] [23]. The equations used to
represent the complete response (or total response) of a thermoelectric module either to the abrupt
application of a DC voltage source on the electrical terminals or to the abrupt application of a temperature
differential on the thermal contacts are shown in (3) and (4), respectively; assuming that the thermoelectric
module represents a thermoelectric capacitor initially discharged and at ,
, and
.
(3)
(4)
A. Forced response of thermoelectric modules and materials
The forced response of TEMs can be obtained through the temperature difference or the voltage
generated by a TEM, from the corresponding Peltier and Seebeck effects, and are given by the mathematical
expressions (5) and (6) [22]. Where
is the temperature difference measured across the thermal
contacts of the TEM, is the Seebeck coefficient,
is the average temperature,
is the thermoelectric
resistance, is the electric current. Also,
,
and
, in
which
represents the external voltage source,
the Seebeck voltage,
the electrical resistance
of the TEM. And
is the thermoelectric time constant of the module.
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Pirela R. et al. Novel Methodology for Characterization of Thermoelectric Modules and Materials
(5)
(6)
Consequently, for the case where the thermal contacts are absent; i.e., the case where there is only the
presence of thermoelectric material, mathematical expressions (5) and (6) are reduced to expressions (7)
and (8), where
is the temperature differential generated at the thermal terminals by the thermoelectric
material,
is the thermal resistivity of the thermoelectric material and
is the characteristic
thermoelectric time constant of the thermoelectric material.
(7)
(8)
It should be noted that the forced response of TEMs expressed in the temperature difference is the
exponential increase of the Peltier effect and expressed in voltage is an exponential increase of the Seebeck
effect and complies with the theory of first-order electrical circuits [22].
B. Natural response of thermoelectric modules and materials
Since the theory of the natural response of TEMs, it is known that the temperature difference associated
with the thermoelectric capacitance of the thermoelectric material is
and is defined as
and the temperature difference related to the equivalent thermal capacitance of the thermal
contacts is given by
. Thus, the natural response of a thermoelectric module is defined by equations (9)
and (10) [23].
(9)
(10)
The formal mathematical expression for
is presented in (11).
(11)
Where,
and
are the roots and are called natural frequencies,
measured in Nepers per second (Np/s), because they are associated with the natural response of the
thermoelectric device;
is called the resonant frequency of the thermoelectric module, or more strictly
the undamped natural frequency of the TEM, expressed in radians per second (rad/s), and is the natural
frequency or damping factor, expressed in Nepers per second, and represents the frequency associated
with the thermal contacts and is also called
, therefore
. Further, considering that at ,
and
, where
is the thermal resistance of the contact cold side of the
TEM,
is the thermal contact resistance of the hot side of the TEM, and
is the short-circuit current
of the device in thermo generator mode; that is, the load
connected to the thermogenerator is zero
ohms. So, for
, we have that
. And the formal mathematical expression for
is as shown in (12).
(12)
Therefore, the natural response concerning the thermoelectric material is defined using the following
expressions (13) and (14).
(13)
(14)
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The time constant associated with the capacitance of the thermoelectric material
could be obtained at
a slightly longer time than expected. Considering that the amplitude is reduced by a factor
(to 36.8 %
of what it had), it can be achieved at a longer time to the relation
, where
is the characteristic
angular frequency related to the thermoelectric material, for which the case of an overdamped
thermoelectric circuit is considered when
. Therefore, the roots can be write as
, and
, where
and
is called the damping frequency. Both
and
are natural frequencies, because they help
determine the natural response of the TEM; while
is often called the undamped natural frequency,
is
called the damped natural frequency or offset frequency. Such a response has a time constant associated
with the thermoelectric capacitance
and a period of
, and as the amplitude is reduced by a
factor
(to 36.8 % of what it had), we then have two scales of time: measures the time it takes to
oscillate and
the time it takes to damp. The dimensionless quotient is represented in (15) [23].
(15)
It is important to highlight that the natural response of TEMs expressed in temperature is an exponential
drop in the temperature difference and satisfies the Newton's law of cooling, and the natural response of
TEMs expressed in voltage is an exponential Seebeck voltage drop and complies with second order electrical
circuits theory [23].
C. Temperature and voltage stability
The thermoelectric time constant
is obtained from the forced response (step response) of TEMs, from
the temperature difference on the faces of the module (Peltier effect), from the temperatures of either
any of the thermal contacts
,
or electric potencial
generated between the positive and
negative terminals of the TEMs (Seebeck effect) [22]. The time constant
, corresponds to the inverse of
the characteristic thermoelectric angular frequency of the TEMs
and is the time required for
,
,
y
to increase by a factor of or of its final value; that is, they take
to
reach their steady state, when no change occurs over time [22]. Taking into account that,
.
The time constant associated with the capacitance of the thermal contacts
is obtained from the natural
response of the TEMs, specifically from the temperature of the thermal contact related to the cold side [23].
For the case where the thermal contact of the cold side is equal to the thermal contact of the hot side, the
time constant
is obtained experimentally from the temperature measurement on the face of the cold
side of the device
and is determined considering that the amplitude is increased by a factor
(36.8 %
of the amplitude that it would have), so
, where
. The damping factor
determines the rate at which the response is damped. The time it takes for the temperature of the cold side
to rise is given by the decay of factor . So the thermal capacitor will be fully discharged after five time
constants. In other words, the capacitor associated with the thermal contact on the cold side of the TEM
takes
to reach its final state [23].
The thermoelectric time constant related to the thermoelectric material
is also obtained from the
natural response. From equation (15) an expression is obtained for the characteristic angular frequency
related to the thermoelectric material which is known as
, and is given by the following mathematical
expression in (16).
(16)
The time that takes the amplitude of the temperature difference between the faces of the TEMs (either
or
) to decay is given by the time constant
is obtained through the following equation (17)
[23].
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Pirela R. et al. Novel Methodology for Characterization of Thermoelectric Modules and Materials
D. Thermoelectric modeling and equation derivation
Thermoelectric coefficients and parameters of TEMs
Using the theory of forced response, it is possible to obtain the thermoelectric resistance
using (5)
[22]. Considering that, for the forced response, at , the temperature difference
and that after
it will reach its steady state, then
, and it can be expressed in a more
compact way if it is taken into consideration that for a time greater than
,
tends to zero,
then an expression for
is shown in (18).
(18)
Using the theory of natural response, the equivalent resistance of the thermal contacts is obtained
and is found through (11) [23]. Equation (11) shows that
is the result of the temperature contribution
of each thermal contact and
is the result of the sum of the thermal resistances of the thermal contacts.
Assuming that the thermal resistances corresponding to the contacts are equal,
, that the
equivalent resistance of the contacts is
and
, then
.
Taking into account that, at , the maximum value of
is obtained. Additionally, that
and
, so
. Therefore, the thermal resistance related to thermal contacts can be
written as in (19) [22] [23]. The electric resistance of TEMs
is found using (20) [3] [4]. And, to find the
electric resistivity it is recommended to use the equation (22), which correlates
and the geometry of
the specimen under test.
(19)
(20)
Also, using the theory of natural response, the thermal conductance
is obtained, considering the
equation (12). With the presence of the sine and cosine functions it is trivial that the natural response for
this case is exponentially damped and oscillatory in nature. Considering that, at the maximum value
of the temperature difference
; also that
,
and
; that is, a
term in (12) contributes neither to the temperature difference
nor to the Seebeck voltage
.
Therefore, the thermal resistance of the thermoelectric material is
and knowing that the
thermal conductance is equal to the inverse of the thermal resistance; that is,
, then the thermal
conductance of the thermometric material is given by
. Hence, the thermal conductance
of the thermoelectric material can be written as in (21), keeping in mind that
. Furthermore,
from the forced response is possible to find that
.
(21)
The electric resistivity is obtained using (20) by multiplying
per
, where and are the length and
area of the specimen under test, obtaining the equation (22). Also, the thermal conductivity of the
thermoelectric material is obtained through (21) by multiplying
per
, obtaining the equation (23).
(22)
(23)
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Pirela R. et al. Novel Methodology for Characterization of Thermoelectric Modules and Materials
Figure of Merit
As it is known, the figure of merit is calculated using (1). However, by substituting (21) into (1), a new
expression is found for the figure of merit and is written utilizing the mathematical equation (24).
(24)
Solving
from equation (9) it is found that
and substituting into (24) is obtained
(25). Since
is the result of the temperature contribution of each thermal contact,
and
,
where
and
, more expressions are found as shown in (26), (27), (28), and (29).
(25)
(26)
(27)
(28)
(29)
The new expression for the figure of merit of TEM obtained in this research; e.g.,
has the
form of the Harman equation
, but the
equation obtained herein is expressed as a function
of temperatures and the one obtained by T. C. Harman is expressed as a function of voltages [15]. Equations
(25), (26), and (27) for the figure of merit reciprocally have the form of the equation
,
obtained by R. Buist [19]. Furthermore, the second term on the right-hand side of equation (28) corresponds
to the expression obtained by A. F. Ioffe [12], to measure the maximum performance
of the TEM; that
is,
, where
and
.
III. CHARACTERIZATION CONFIGURATIONS
A. Test Configuration for Thermoelectric Modules
Through Figure 1 (a), four different test configurations for a thermoelectric module are illustrated. The
“Suspended” configuration employs the four probes test technique. The method used to adhere the
thermocouples to the thermal contacts is simply to apply a tiny dab of thermal paste where the
thermocouple junctions are positioned and can be held in place under compression with the use of thermal
insulation tape [5] [19]. The “Suspended”, “Heat Sink” and “Assembly” configurations are essentially the same
because these configurations employ the four probes test technique. The "Thermocouples" configuration
provides the ultimate solution in simplicity, connections, and speed of testing. This configuration is
especially recommended when TEMs are used as thermogenerators. The main parameter measured using
this setup is
. This is enough to ensure the production quality of thermoelectric modules.
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(a)
(b)
Fig 1. Characterization configurations for thermoelectric modules (a). Characterization configurations for
thermoelectric pellets (b).
This test is very effective for screening samples before more rigorous testing. Additionally, with this
configuration, the characteristic thermoelectric times and relaxation times can be determined, using
,
, and
. Furthermore, since a ºC error in the absolute temperature,
, produces an error of less than
1%, thus reducing the uncertainty of the measurement and the figure of merit, and
is measurable directly
with precision and versatility. The advantage of this configuration for the thermoelectric module testing is
that the thermocouples will not be in contact with the active circuit and voltage pick-up will not be a factor.
B. Test Configuration for a Thermoelectric Pellet Sample
Fig. 1 (b), illustrates four different test configurations for a thermoelectric pellet sample. The first
configuration is a suspended sample using the “Four Probes” technique. This test method is necessary
whenever the contact resistance is unknown or significant compared to the resistance of the thermoelectric
pellet. The difficulties with this configuration are that: (a) current flow through the pellet can be disturbed
by the presence of the thermocouples; (b) voltage pick-up in the probes can result in significant errors in
the thermocouple readings; (c) precise measurements of the probe separation are usually very difficult to
obtain; and (d) the voltage and temperature planes are affected by the probes and, therefore, are not nearly
as well defined at the probes as they are at the opposite ends where high-conductivity end caps are applied.
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The “Two Probes” setup is suggested for most thermoelectric materials where good contacts are relatively
easy to achieve. However, care must be taken to place the current and thermocouples on opposite edges
of the end caps to avoid voltage pick-up across the thermocouples. Essentially, the thermocouple should
not be placed in a position on the end cap where the current lines intersect. The "Heat Sink" configuration
is practically the same as the "Four-Probe" one concerning instrumentation and connections. The
"Thermocouples " is recommended to measure
,
, and the relaxation time of the material; e.g.,
.
IV. EXPERIMENTAL RESULTS
To illustrate the characterization and testing of TEMs a commercially available thermoelectric module is
considered as a sample, specifically, the Kryotherm TB-127-1.4-1.2, used by Lineykin and Ben-Yaakov [3] [4];
as well as used by Y. Apertet and H. Ouerdane [11]. The parameters at are presented in Table 1.
And the Kryotherm TB-127-1.0-1.2 module was also tested [25].
Table1. Parameters at ,
,
,
,
(Tolerance:
+/- 10 %),
.
The Fig. 2 (a), (b), (c), and (d) show the test data taken on TB-127-1.0-1.2 and TB-127-1.4-1.2 using the
“Suspended” and “Thermocouples” configuration without being thermally insulated; that is, in non-adiabatic
conditions. The environment was at room temperature
, to which the module thermal contacts
were exposed. For the temperature measurements, two special type K thermocouples were used, with an
error (Special Limits Error) of either +/- or . The measurement process used is similar to
the one proposed by Buist in [20]. The results obtained for TB-127-1.4-1.2 at are shown in the
Table 2, and can be compared with Table I. Additionally, analyzing the data taken on both TEM has been
possible to distinguish that using the same power supply values setup (voltage, current, and time step) [25],
either the TB-127-1.0-1.2 and the TB-127-1.4-1.2 showed different thermoelectric parameters. The errors
are associated with the offset of thermocouples and the time step of the power source.
Table 2. Parameters at ,
,
,
,
(Tolerance: +/- 10 %),
.
0.394
2.871
1.592
0.336
0.633
0.0508
0.768
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(a)
(b)
(c)
(d)
Fig. 2. Electric resistance characterization data taken on TB-127-1.0-1.2 and TB-127-1.4-1.2 (a).
Thermoelectric conductance characterization data taken on TB-127-1.0-1.2 and TB-127-1.4-1.2 (b). Seebeck
coefficient characterization data taken on TB-127-1.0-1.2 and TB-127-1.4-1.2 (c). Figure of merit
characterization data taken on TB-127-1.0-1.2 and TB-127-1.4-1.2 (d).
CONCLUSIONS
The unification of the forced response and natural response theories allows us to describe the complete
response of TEMs. From the complete response of TEMs, it has been possible to create the basis to develop
and describe in detail a novel methodology for the characterization and testing of thermoelectric materials,
pellets, wafers, ingots, modules, and systems. The present method is proven to be fast, accurate, highly
appropriate to apply, and conveniently more cost-effective. Using an integrated measurement system that
is capable of solving simultaneous the voltage and temperature components of TEMs with high speed and
high resolution. Moreover, this methodology provides all the advantageous features of Harman's and
Transient’s methods. Numerous expressions for the figure of merit expressed as a function of the
temperatures were found similar to Harman's and Buist’s equations. The methodology provided a direct
measurement of the figure of merit with exactness and reproducibility. The subsequent computations yield
values for the Seebeck coefficient, electrical resistance, thermal conductance, as well as the famous
thermoelectric transport coefficients and .
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Sergio Rafael Velásquez Guzmán - Coautor, received the B.S.
degree in Electronic Engineering, from the UNEXPO, in 2008. M.S.
degree in Education from UPEL in 2011, an M.S. degree in Electronic
Engineering from UNEXPO, in 2012, an MBA degree from UNY in 2014,
a Doctor of Education degree in 2015 from UPEL, and a Doctor of
Engineering Sciences from UNEXPO in 2019. He is a type B Research
Professor accredited by the MINCYT in Venezuela. Currently, he is in
charge of the Research and Postgraduate Department of the UNEXPO
Vice-Rectorate, Puerto Ordaz, Venezuela.
Ronald Edgar Pirela La Cruz was born in Maracaibo, Venezuela in 1983.
He received a B.S. degree in Electronic Engineering, a Specialist degree
in Digital Telecommunications, and an M.S. degree in Electronic
Engineering from the UNEXPO in 2007, 2013 and 2020, respectively. Also,
he is currently pursuing a Ph.D. degree in Engineering Sciences at
UNEXPO. Currently, he is the Train Lab Engineer in charge of the Train
Lab for trains powered by hydrogen, Coradia Stream Project in Alstom
Ferroviaria S.P.A., Savigliano, Italia.
