Numerical Simulation of a Thermoelectric Generator André Nuno Figueira van der Kellen Thesis to obtain the Master of Science Degree in Mechanical Engineering Supervisor: Prof. Pedro Jorge Martins Coelho Examination Committee Chairperson: Prof. Carlos Frederico Neves Bettencourt da Silva Supervisor: Prof. Pedro Jorge Martins Coelho Member of the Committee: Prof. Viriato Sérgio de Almeida Semião October 2020
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Numerical Simulation of a Thermoelectric Generator
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Numerical Simulation of a Thermoelectric Generator
André Nuno Figueira van der Kellen
Thesis to obtain the Master of Science Degree in
Mechanical Engineering
Supervisor: Prof. Pedro Jorge Martins Coelho
Examination Committee
Chairperson: Prof. Carlos Frederico Neves Bettencourt da SilvaSupervisor: Prof. Pedro Jorge Martins Coelho
Member of the Committee: Prof. Viriato Sérgio de Almeida Semião
October 2020
ii
Resumo
O fenomeno termoeletrico esta associado a conversao de calor em eletricidade e vice versa. Os instru-
mentos termoeletricos, com base no efeito de Seebeck, podem atuar como geradores, onde e produzida
potencia eletrica, ou como refrigeradores termoeletricos para remocao de calor.
Para avaliar como e que um sistema termoeletrico converte esta energia termica em energia eletrica,
sao utilizados modelos matematicos. Esta dissertacao apresenta tres modelos diferentes para estimar
o desempenho de um gerador termoeletrico: um modelo analıtico assumindo propriedades indepen-
dentes da temperatura e dois modelos em que as propriedades nao sao constantes, um analıtico e um
numerico.
Os resultados foram obtidos para dois modulos termoeletricos fabricados pela Hi-Z Technology, Inc.,
HZ-14 e HZ-20, e comparados com os dados de desempenho disponibilizados pelo Module Perfor-
mance Calculator. Com base no desempenho previsto pelos tres modelos descritos nesta dissertacao,
e importante considerar a influencia da temperatura nas propriedades dos materiais quando se analisa
o desempenho, como indicam os resultados. A hipotese de que as propriedades dos materiais sao con-
stantes, e razoavel para baixas temperaturas de operacao, no entanto, a temperaturas elevadas esta
premissa causara uma sobrevalorizacao do desempenho. Tanto as solucoes analıtica como numerica
do modelo nao-linear usado, revelaram uma boa correspondencia entre si e com os resultados obtidos
pelo Module Performance Calculator. A avaliacao do desempenho do modulo termoeletrico necessita
de considerar a variacao das propriedades com a temperatura, de maneira a obter resultados mais
precisos.
Palavras-chave: Termoeletricidade, gerador termoeletrico, efeito de Seebeck, recuperacao
de calor, energia termica
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Abstract
Thermoelectricity is associated with the conversion of heat to electricity and vice versa. Thermoelectric
devices, based on the Seebeck effect, can either act as generators, where electrical power is produced,
or as thermoelectric coolers for refrigeration.
To evaluate how well a thermoelectric system converts this thermal energy into electrical energy,
mathematical models are used. This thesis presents three different mathematical models to evaluate the
performance of a thermoelectric generator: one analytical model with the assumption of temperature-
independent properties and two models where the properties are not assumed to be constant, one
analytical and one numerical.
The results were obtained for two thermoelectric modules manufactured by Hi-Z Technology, Inc.,
HZ-14 and HZ-20, and compared with the performance data provided by the Module Performance Cal-
culator. Based on the predicted performance by the three models employed in this thesis, it is important
to consider the temperature-dependence of material properties in the analysis as the results show. The
assumption of constant material properties can be reasonable in lower temperatures of operation, where
the variation of the thermoelectric properties with temperature is negligible, but at higher temperatures
of operation this assumption causes overestimation of the performance. Both analytical and numerical
solutions of the non-linear models used have shown a good correspondence with each other and with
the data obtained from the Module Performance Calculator. The performance evaluation of the thermo-
electric modules needs to consider the temperature-dependence of the materials properties to obtain
A Cross-sectional area of the thermoelement (m−2).
E Electric field intensity (V m−1).
I Electric current intensity (A).
J Electric current density (A m−2).
L Thermoelement leg length (m).
n Number of thermocouples.
q Heat flux (W m−2).
R Electrical resistance (Ω).
T Temperature (K).
V Voltage (V).
Z Figure of merit.
xiii
Q Heat rate (W).
W Electrical power (W).
Subscripts
A Wire A.
B Wire B.
C Carnot.
c Cold-side.
E Eastern nodal point.
e East face.
e West face.
egg Eggcrate material.
h Hot-side.
i, j Computational indexes.
L Load.
mc Maximum conversion efficiency.
mod Module.
n N-type unit.
oc Open-circuit.
Ohm Ohmic voltage.
P Local nodal point.
p P-type unit.
Sbk Seebeck voltage.
W Western nodal point.
xiv
Chapter 1
Introduction
Thermoelectrics is defined as the science and technology associated with the thermoelectric generation
and refrigeration [1]. This allows for the conversion of thermal energy into electrical energy or vice versa.
Thermoelectric devices have no moving parts and require no maintenance thus making them suitable for
a broad range of applications such as waste heat recovery from power plants and automotive vehicles
but also refrigeration and temperature control in electronic packages and medical instruments. It is very
important to understand the fundamental concepts of thermoelectricity in order to properly develop and
design such devices [2].
Chapter 1 begins with a motivation of the topic at hand in section 1.1, followed by an overview of
the topic and references of scientific work and research on thermoelectrics in section 1.2. Section
1.3 explicitly states the objectives set to be achieved with the study of the thermoelectric generator
considered. Finally, a brief outline of this thesis is presented in section 1.4.
1.1 Motivation
Fossil fuels are the main source of energy provided to all of the traditional technologies used currently
around the world. Whether it is in automobiles, industry, or other mankind’s activities, the resultant
emissions of the combustion of this type of fuels have a clear impact on the environment, and a very large
amount of this energy is wasted through the atmosphere. As such, the need to reduce wasted energy
and the environmental impact have been an increasing concern, thus leading to research for other
energy-producing alternatives. Directly converting this energy into electricity through a thermoelectric
generator (TEG) is considered an attractive solution to this problem [3].
Thermoelectric technology received considerable attention for the waste heat recovery in energy
conversion devices like internal combustion engines (ICE). There is plenty of scope for improvement
in this regard since only a third of the amount of fuel burnt in a conventional ICE is used to provide
mechanical power, the rest is wasted heat. The recovery of such heat and conversion to electricity
may be used for propulsion and to power the vehicle’s electrical components such as air conditioning,
lights, etc. Overall, by reducing the load on the alternator, the fuel efficiency of the system is improved
1
[4]. Coupling this factor with the TEG advantages, such as durability and quiet operation, no waste
production, and reliable power production in remote areas, make this alternative increasingly demanded
[3, 5, 6]. However, the automotive industry is just one example where a thermoelectric system can be
valuable. These devices may also be used in spacecraft for energy generation, recapture energy from
hot effluents of powerplant smokestacks, and harvest heat generated by photovoltaic cells. Although
these advantages should cause renewed interest in thermoelectric power, the coupling of heat and
electricity is weak since a great deal of thermal energy is required to generate a small amount of electrical
energy.
Recently, the development of nanotechnology has been critical in overcoming this technological chal-
lenge. New thermoelectric materials have been manufactured (most of them in laboratories) with higher
figures of merit (ZT ) and then used in R&D programs to convert waste heat into electricity in automotive
vehicles exhausts [5].
To predict and optimize the performance of thermoelectric systems, a correct mathematical model
for the analysis accompanied by a deep understanding of the heat and electrical current transfer phe-
nomena and the selection of the right materials according to the temperature range of operation is
indispensable.
1.2 State of the Art
In 1823, Thomas J. Seebeck reported results of experiments where a compass needle suffered a de-
flection if placed in a circuit of two dissimilar conductors when one of the junctions was heated. Seebeck
investigated this phenomenon in many other materials and arranged them in order of the product ασ,
where α is the Seebeck coefficient, expressed in volts per degree (V/K), and σ is the electrical conduc-
tivity [7]. As a result of his experiments, the first thermoelectric effect had been discovered.
The second thermoelectric effect was discovered in 1834 by Jean Peltier. Peltier observed the re-
verse effect when he noticed temperature changes on the thermocouple depending on the direction of
the current flow. Although both effects were proved to exist, they were very difficult to measure as a
property of the materials used [2, 7]. After the discoveries of both Seebeck and Peltier, slow progress
was made in the research of thermoelectric phenomena, and in 1850 interest in the topic was once
again due to the development of thermodynamics.
In 1851, William Thomson established a relationship between the Seebeck and the Peltier effect
which is the third thermoelectric effect, called the Thomson effect. Thomson discovered that in the pres-
ence of a temperature gradient between any two points of a conductor with current flowing, heat is ab-
sorbed or released, depending on the direction of the current and the conductor material [2]. Thomson’s
work also related the three thermoelectric effects thermodynamically, leading to important relationships
called the Kelvin Relationships.
Soon after these discoveries, the generation of electricity based on the thermoelectric phenomena
was considered. Edmund Altenkirch showed in the early 20th century that good thermoelectric materials
should have a high Seebeck coefficient to retain the heat at the junctions of the conductors and a low
2
electrical resistance to minimize the Joule heating. The non-dimensional figure of merit (ZT ) relates
these favorable properties [7]. Most materials researched at the time were mainly metal and metal
alloys, in which the ratio of thermal conductivity to electrical conductivity is a constant, therefore it is not
possible to adjust one parameter without affecting the other. The majority of metals exhibit small values
of α, resulting in low values of ZT . For many years ZT was limited to less than 1 and for many practical
applications, this value needs to be at least close to 2 [4, 7].
During the late 1930s, semiconductors started to be considered as alternatives to metals due to their
high Seebeck coefficient, and, with potential military applications in mind, the technology of thermo-
electricity began during World War II when the Soviet Union produced a 2-4 watt TEG. Major advances
in semiconductor technology and thermoelectric theory originated further development and research
in thermoelectric applications in the 1950s and 1960s, with large companies actively engaged in ther-
moelectric research including Whirlpool, Westinghouse, Bell Telephone, GE, Carrier, and others [1].
Recently, NASA reported conversion efficiencies of up to 15% for large temperature gradients. If similar
values of efficiency could be reproduced in smaller temperature gradients, like in automobiles exhausts,
for example, capturing about 5-10% of a vehicle’s waste heat could lead to a 3-6% reduction in fuel
consumption, which would be significant for both cost and emissions savings [4].
A typical modern thermoelectric module consists of several n-type and p-type semiconductors, form-
ing a thermocouple, connected electrically in series between two electrical conductors. The conductors
on the other hand are protected by an electrical insulator, usually ceramic plates, as figure 1.1 shows.
In p-type semiconductors the absence of electrons creates a positive charge and in the n-type semi-
conductors excess of electrons create a negative charge, thus creating a flow of electrons across the
junctions of the thermocouple. Provided a temperature difference is maintained across the module, the
device operates as a TEG and supplies electrical power to an external load, while if the electric current
passes through the module instead, heat is absorbed in one side and rejected at the other, acting as a
thermoelectric cooler (TEC) [7].
The increasingly growing interest of the scientific community in this technology led to several studies
to evaluate and predict the performance of such devices, using both analytical and numerical models
for the evaluation of the energy and electric potential transport equations. Fraisse et al. [8] compared
four different one-dimensional steady-state models to analyze the coefficient of performance (COP)
and efficiency as well as voltage and thermal/electrical power in a bismuth telluride Bi2Te3 material.
The standard simplified model introduced is based on an overall thermal balance to the thermocouple
considering constant thermoelectric properties estimated at the mean temperature T of the hot and
cold sides, Th and Tc, respectively. The second model, the improved simplified model, assumes a
non-constant Seebeck coefficient, thus taking into account the Thomson effect considering a constant
Thomson coefficient. The third model is based on a local energy balance and the temperature and heat
flux distribution along the thermoelement are derived assuming constant leg section and thermoelec-
tric coefficients. The fourth and final model evaluated is the electrical analogy model presented in [9].
They concluded that there were no significant differences between the simplified and improved models
in TEC and thermoelectric heater (TEH) modes, however, maximum efficiency was overestimated in the
3
Figure 1.1: A typical thermoelectric module [2].
simplified linear model when operating in TEG mode. Slightly more accuracy was observed when con-
sidering the Thomson effect in the calculations if the Seebeck coefficient is strongly thermal dependent.
It was also shown that the electrical analogy model agrees very well with the finite element model (FEM)
simulations performed.
Zhang [3] studied the effect of material temperature dependence when evaluating the performance
of a Hi-Z thermoelectric module [10] and solved the non-linear heat transport equation based on the
homotopy perturbation method as described in [5] and [11]. In both works, the homotopy perturba-
tion method solution is compared with other linear analytical solutions as well as the electrical analogy
method. The non-linear analytical solution provided consistent results with the electrical analogy method
and estimations of temperature variation along the thermoelement leg, absorbed power at the cold-end
when in TEC mode, and power output at the hot-end when in TEG mode. Marchenko [12] presented
a non-linear analytical solution for the heat transport equation based on the perturbation method, com-
paring this non-linear solution with five other proposed methods. Marchenko’s research showed that the
accepted accuracy for real-world applications is achieved with the quadratic approximation of the pertur-
bation method, requiring less computation than a traditional numerical integration of the differential heat
balance equation.
To account for irreversibilities in the process is also significant when modeling a thermoelectric device
as these can strongly influence power output. Shen et al. [13] studied thermal losses in two commercially
available TEG based on the side surface heat transfer effect, which represents the heat losses across the
side surfaces of each thermoelement leg due to convection in the air gaps between each thermocouple.
Temperature distributions for the n-type and p-type legs and output power and efficiency under different
values for the convective heat transfer coefficient h of air were obtained. As h increases the degree
of non-linearity in the temperature distributions along the n-type and p-type legs increases and the
4
temperature difference between the hot and cold junctions decreases, leading to reduced efficiency of
the device, while the existence of convection can either increase or reduce power output depending on
the leg length and material volume. A similar analysis was performed by H. Lee et al. [14] where the
model evaluated considered radiative losses and interfacial resistances inside the device besides the
convective losses and then compared to FEM results, showing the reduction in efficiency due to these
effects. The effect of the leg geometry in the performance of the device was also studied and it was
shown that the increase in leg spacing reduces the thermal resistance and increases heat flow and
power, but decreases efficiency. Kim [15] presented a study with internal thermal and electric interfacial
contact resistances modeled while also varying the n-type and p-type leg length, concluding that this
interfacial resistance cannot be neglected when the n-type and p-type units are sufficiently short.
Niu et al. [6] further studied the effect of leg geometry in the temperature gradient while developing
two three-dimensional numerical models based on different formulations and boundary conditions to
analyze heat and electricity transfer. It was shown that one model was more precise for power output
prediction and suitable for simulations with defined output voltage or current and the influence of the
leg cross-sectional area could also lead to significant improvement in power output. Also, smaller n-type
and p-type units could enhance efficiency and power density. Another three-dimensional model was also
studied by Bjørk et al. [16], taking into account radiative, convective, and conductive heat losses with
very detailed modeling. A 3-D finite element model developed by Liao et al. [17] was compared to results
obtained experimentally for the TEG1-127-1.4-1.6 TEG module [18], showing a maximum deviation of
6%, hence showing the accuracy of the models presently used to predict performance parameters for
either TEG or TEC modules.
Thermoelectric module performance evaluation can be done through a variety of models depending
on the type of application that is expected. Although several implemented models have simplifications
to make the computations easier to perform, a detailed evaluation of such a device needs to consider
non-linearities and irreversibilities to predict power or efficiency correctly, especially if working under
high-temperature differences between the heat source and heat sink.
1.3 Objectives
The objective of this master thesis is to develop a model able to calculate the performance of the com-
mercially available modules HZ-14 and HZ-20 by Hi-Z Technology, Inc. by comparing it with the available
data provided by the Module Performance Calculator [19], resorting to both analytical and numerical
methods.
From the reviewed literature, none of the works mentioned have attempted to evaluate Hi-Z ther-
moelectric modules by justifying the available data in [19] with a comprehensive numerical or analytical
model. The current thesis work attempts to provide that while also doing its analysis of the modules’
performance with the proposed methods.
Both HZ-14 and HZ-20 will be evaluated for different operating conditions and different temperature
differences and the results will be analyzed together with the data from [19].
5
1.4 Thesis Outline
The present master thesis work is divided from Chapter 1 to Chapter 6.
Chapter 1, the current chapter, introduces some motivation and context to the work developed in this
thesis. Also, a few examples of previous studies in the area of thermoelectricity are presented as part
of the state of the art. The objectives of the present master’s thesis work are defined in this chapter as
well.
Chapter 2 introduces the theoretical background needed to understand the fundamentals of the ther-
moelectric phenomena and to develop the necessary analytical and numerical tools to predict module
performance.
In Chapter 3 the analytical models used will be explained in detail followed by the numerical model
in Chapter 4.
Chapter 5 lists the obtained results using the models introduced in Chapter 3 and 4 together with the
data from the Module Performance Calculator. Here a discussion of the results is also presented.
Finally in Chapter 6 an overview of the work developed with this thesis and some conclusions on the
achievements are stated along with some suggestions for future work.
6
Chapter 2
Concepts of Thermoelectricity
In this chapter the fundamental concepts regarding thermoelectricity are presented in order to under-
stand the models applied to evaluate the performance of the Hi-Z thermoelectric modules.
Starting with section 2.1, the three thermoelectric effects are presented with some mathematical
definitions. Section 2.2 describes a basic thermocouple configuration with some background definitions
concerning its materials. The performance parameters of a thermoelectric generator are presented
in section 2.3, and in the last section 2.4, a detailed explanation of a typical thermoelectric module
configuration is presented. In this section, material and geometric data used in the Hi-Z modules studied
are also referenced.
2.1 Thermoelectric Effects
Thermoelectricity deals with the direct conversion of heat into electricity employing the three thermo-
electric effects that manifest in the presence of a temperature difference across the surface of the ther-
moelectric module: the Seebeck effect, the Peltier effect, and the Thomson effect [20]. The interrela-
tionships between these three effects, the Kelvin Relationships, are of extreme important as they gather
together the three thermoelectric effects to get a unique and consistent description of thermoelectric
phenomena [21].
2.1.1 Seebeck Effect
In a thermoelectric material, when a temperature difference is applied across a conductor, the hot region
produces more free electrons and natural diffusion of these electrons occur from the hot region to the
cold region, as show in figure 2.1. The resultant electromotive force generates electric current flowing
against the temperature gradient. This is known as the Seebeck effect [2].
The concept of conversion of heat into electricity is even more evident when a thermocouple of two
dissimilar materials is subjected to a temperature difference. Provided this temperature difference is
maintained at the junctions of two wires joined at both ends, an electromotive force is produced, and
consequently electric current flows in a loop like figure 2.2 shows.
7
Figure 2.1: Electron concentration in a conductor [2].
Figure 2.2: Schematic of a basic thermocouple subjected to a temperature difference at both ends [2].
The electric field intensity vector ~E (V m−1) is related to the applied temperature gradient through the
Seebeck coefficient α (also called thermopower ), usually measured in µV/K. The sign of α is positive if
the electromotive force drives the electric current from the hot junction to the cold junction, and negative
if the current flows from the cold junction to the hot junction. ~E is defined as
~EA,B = αA,B∇TA,B (2.1)
where ~E represents the electric field intensity vector in wire A or wire B, αA,B and ∇TA,B represent the
Seebeck coefficient and the temperature gradient on wire A and wire B separately, respectively. Since~E = −∇V , the Seebeck potential VSbk (V) in each wire can be written as
∇VSbkA,B= −αA,B∇TA,B (2.2)
Depending on the sign of α, the Seebeck voltage will be negative or positive in each conductor. The
resultant voltage of the thermocouple is given by
VSbk = VSbkA − VSbkB (2.3)
The Seebeck potential represents the highest voltage possible in the circuit, which is equivalent to the
open-circuit voltage Voc. Although this potential difference is only a function of the hot-side temperature
(TH ) and cold-side temperature (TL), its distribution is indeed a function of the temperature distribution
along the conductors. This effect is not affected by either the Peltier or the Thomson effect, the latter
two thermal effects are present only when current flows in the circuit and are not voltages, whereas the
Seebeck effect exists if a temperature gradient is maintained whether current flows or not [7].
8
2.1.2 Peltier Effect
As an electric current flows across a junction between two wires of dissimilar materials, heat must be
continuously added or subtracted at the junction to keep its temperature constant as shown in figure 2.3.
This is known as the Peltier effect, which results of the change in the entropy of the electrical charge
carriers as they cross a junction. Hence, heat is either absorbed or released at the junctions and it is
proportional to the current flow. The heat QPeltier (W) in each wire is defined as
QPeltierA,B= πA,BI (2.4)
where πA,B (V) is the Peltier coefficient of each wire with positive or negative sign depending on the
direction of the electric current, and I represents the electric current intensity (A) across the junctions.
The Peltier coefficient π is the change in the reversible heat content at the junction of conductors A
and B when unit current flows across it in unit time [7]. Even though π can be expressed in volts, the
Peltier effect does not produce an electromotive force and it is a reversible process, which means that
the heating or cooling effect will produce electricity and, if electricity is supplied to the system instead of
a temperature difference, heating or cooling is produced with no energy being lost.
Figure 2.3: Representation of the Peltier and Thomson effects on a thermocouple [2].
2.1.3 Thomson Effect
The third thermoelectric effect is the Thomson effect and it represents the reversible change of heat
content within a conductor in a temperature gradient when an electric current passes through it [7]. As
seen on figure 2.3, heat is absorbed or released across the wire depending on the material and the
direction of the current. The Thomson heat per unit volume QThomson (W m−3) of each wire is related to
the temperature gradient by
QThomsonA,B= τA,B∇TA,B ~J (2.5)
where τA,B ,∇TA,B and ~J are the Thomson coefficient (V K−1), the temperature gradient and the current
density vector (A m−2) on wire A and wire B, respectively. The sign of τ is positive if heat is absorbed as
shown in wire A and negative if heat is released like in wire B. The Thomson coefficient is the reversible
change of the heat content within a conductor [7] and it is the only thermoelectric parameter directly
measured for individual materials.
9
Like the Peltier effect, the Thomson effect is also not a voltage and it is reversible between heat and
electricity. There is another form of heat that arises in the presence of an electric current flowing, called
the Joule heating, which is always irreversible.
Joule Effect
The Joule effect describes the process where the energy of an electric current is converted into heat
as it flows through a resistance. When current flows in a material with finite electrical conductivity,
electric energy is converted to heat through resistive losses in the material [22]. Although it is not
a thermoelectric effect, Joule heating is present in thermoelectric systems and so it is necessary to
consider it when assessing the performance of such systems since it represents an irreversible heat
loss. The volumetric rate at which heat QJoule (W m−3) is generated due to a flowing electric current is
defined as
QJoule = ρ‖ ~J‖2 (2.6)
where ρ is the material electrical resistivity (Ω m) and ~J the current density vector passing through the
material (A m−2). In the case of thermoelectric materials, this effect is usually less relevant than the
Peltier or Thomson effects due to the low resistivity of the thermoelectric semiconductors, but it is a
function of the dimensions of the material and becomes significant for high values of ~J .
2.1.4 Kelvin Relationships
The Kelvin (or Thomson) Relationships were developed by William Thomson in 1854 by applying the first
and second laws of thermodynamics assuming the reversible and irreversible processes in thermoelec-
tricity are separable [2]. This provided with interrelationships between the three thermoelectric effects
very important to understand the phenomena.
The first Kelvin relation describes the Peltier coefficient π as a function of the Seebeck coefficient α
and Thomson coefficient τ
dπ
dT= α+ τ (2.7)
The second Kelvin relation links the Peltier coefficient to the Seebeck coefficient by the following
equation
π = αT (2.8)
where T is the local temperature. Introducing equation (2.8) in equation (2.7), the Thomson coefficient
can now be defined as a function of α
τ = Tdα
dT(2.9)
10
Both of the Kelvin relations rely on fundamental principles of physics. The second of Kelvin’s rela-
tions is associated with a specific case of Onsager’s reciprocal relations, which is based on microscopic
reversibility. On the other hand, the first Kelvin relation, regarding the way heat evolves in a thermo-
electric system, is often used as a convenient mathematical expression (2.9) relating the Seebeck and
Thomson coefficients [21].
By combining equation (2.8) with equation (2.4), it is possible to write the Peltier heat as a function
of the Seebeck effect only
QPeltierA,B= αA,BTA,BI (2.10)
Combining equations (2.9) and (2.5), the Thomson heat volumetric rate can now be expressed as
QThomsonA,B= TA,B
dαA,BdT
∇TA,B ~J (2.11)
2.2 Thermocouple
A modern thermocouple typically consists of p-type and n-type semiconductor materials. A basic rep-
resentation of a thermocouple of two dissimilar thermoelements can be seen in figure 2.4, where L is
the thermoelement leg length; A is the cross-sectional area of the leg; α, ρ and κ are the Seebeck
coefficient, electrical conductivity and thermal conductivity (W m−1 K−1) of the leg, respectively; Th is
the hot-side temperature and Tc the cold side temperature. Subscripts p and n refer to the p-type and
n-type material, respectively.
Figure 2.4: A basic p-type and n-type thermocouple [2].
Semiconductors are materials that have electrical properties between those of a conductor and an
insulator. Since the atoms are very closely grouped in a pure semiconductor material, few free elec-
trons are present in their atomic structure, but electrons are still able to flow. To improve the electrical
conductivity of semiconductors, certain ”impurities”, called donor or acceptor atoms, can be added to
the intrinsic material through a process called doping. With the doping process, it is possible to con-
trol the amount of ”impurities” to produce more free electrons or holes hence creating p-type or n-type
semiconductor materials [23].
11
N-type semiconductors
In n-type semiconductors, the intrinsic material is doped with donor atoms that donate electrons to
the basic semiconductor material. When stimulated by an external source, the electrons freed from
the intrinsic material are quickly substituted by the donated electrons from the doping agent, but some
electrons remain free, resulting in a doped semiconductor that is negatively charged. Since there are
more donor atoms than acceptor atoms, an n-type semiconductor material has more electrons than
holes, therefore, creating a negative pole [23].
Figure 2.5: Example of a n-type semiconductor of silicon doped with antimony [23].
P-type semiconductors
P-type semiconductors on the other hand are doped with acceptor atoms, which instead of donating
electrons, give the pure semiconductor material excess of positively charged atoms that leave holes in
the crystalline structure due to the lack of electrons in the ”impurity”. Free electrons around the hole
will move in order to fill it, however this action will leave another hole where the free electron was and
so on, giving the impression that the holes are moving through the crystalline structure of the material.
This continuous ”acceptance” of electrons by the acceptor atoms leave the semiconductor with excess
of holes compared to free electrons, resulting in a positive pole [23].
Figure 2.6: Example of a p-type semiconductor of silicon doped with boron [23].
12
2.3 Thermoelectric Generator (TEG)
The basic component of a thermoelectric generator is a thermocouple like the one represented in figure
2.4. In a TEG the n-type and p-type semiconductors are connected electrically in series by a conducting
strip, the most common material used is copper. Thermal energy from the heat source (Qh) is ab-
sorbed in the hot-end and converted into electrical energy, while heat is rejected at the cold-end of the
thermocouple (Qc).
A TEG delivers electrical power if connected to a load. Figure 2.7 schematizes a TEG when con-
nected to a load and in open-circuit.
(a) Single TEG circuit (b) TEG open-circuit
Figure 2.7: Electrical circuit representation for a TEG connected to a load and in open-circuit [2].
The heat in the hot junction of the TEG unit is absorbed through the semi-conductors as Peltier heat
and conduction heat. Employing the Peltier effect and Fourier’s law of conduction, the heat flux ~qh (W
m−2) in the p-type and n-type units, respectively, is defined as
~qp = αpT ~J − κp∇T (2.12)
~qn = −αnT ~J − κn∇T (2.13)
where α and κ are the Seebeck coefficient and the thermal conductivity, respectively, T is the local
temperature, ~J is the current density vector and ∇ the gradient operator.
2.3.1 Performance Parameters
Output Voltage
If the TEG is delivering power to a connected load, the n-type and p-type units’ electrical resistivity will
produce a drop in potential when the operating current passes through them according to Ohm’s Law. A
13
potential drop will also occur at the load. The vector form of the Ohm’s Law is defined as
~Ep,n = ρp,n ~J , (2.14a)
∇VOhmp,n= −ρp,n ~J (2.14b)
The total potential drop in the thermocouple will be
VOhm = VOhmp+ VOhmn
(2.15)
Looking at figure 2.7 (b), the open-circuit voltage Voc can be written as a function of the voltage drop
in the thermocouple VOhm and across the load V due to Ohm’s law by the following relation
Voc = VOhm + V (2.16)
Since the Seebeck voltage VSbk is equivalent to Voc, equation (2.16) can be written in the form
V = Voc − VOhm , (2.17a)
V = VSbk − VOhm (2.17b)
Equation (2.17b) leads to an important relationship for the output voltage V . The maximum output
voltage is achieved in open-circuit when Vmax = VSbk and will decrease linearly with increasing values
of I until VSbk = VOhm.
Output Power
The power generated at the load can be calculated as a function of the output voltage as
W = V I (2.18)
where W is the electrical power supplied (W). Since the output voltage is related to the load resistance
RL (Ω) by Ohm’s law, a more common representation of the electrical power is defined as
W = (IRL)I , (2.19a)
W = I2RL (2.19b)
Equation (2.19b) is a useful relationship that allows the computation of the load resistance RL as a
function of the output power, or vice-versa.
14
Figure of Merit
The performance of a n-type or p-type thermoelectric material is represented by a parameter called
figure of merit Z (K−1) that is defined as
Zp,n =α2p,n
ρp,nκp,n(2.20)
where α is the Seebeck coefficient, ρ and κ are the electrical resistivity and thermal conductivity, respec-
tively, and the subscripts p and n denote the p-type and n-type materials. Inspecting equation (2.20),
one can conclude that, in order to achieve satisfactory values of Z, the selected thermoelectric material
should present high values of α while exhibiting low values of κ and ρ. This makes sense since the de-
sirable material should be able to retain the highest amount of reversible heat possible at the junctions
(hence a high value of the Seebeck coefficient and, consequently, of the Peltier coefficient) while keeping
the irreversible heat losses by conduction and Joule heating to a minimum. Usually, the dimensionless
figure of merit ZT is presented and often used as a characteristic of the material.
However, the thermoelectric properties used to calculate Z are not temperature independent, so the
figure of merit tends to vary significantly depending on the temperature gradient applied to the system.
Still, the impact of the figure of merit in the conversion efficiency can be evaluated by looking at a material
with constant properties in the entire range of operating temperatures.
Conversion Efficiency
The TEG conversion efficiency is given by
η =W
Qh(2.21)
where W is the electrical power produced by the system and Qh is the total heat rate (W) supplied to the
system by an external source. For any heat engine operating between a heat source and a heat sink,
the maximum theoretical efficiency possible is defined by the Carnot efficiency represented as:
ηC = 1− TcTh
(2.22)
where Th (K) is the heat source absolute temperature and Tc (K) is the heat sink absolute temperature.
The average absolute temperature is defined as
T =Th + Tc
2(2.23)
Lee [2] derived equation (2.24) to represent the maximum conversion efficiency achievable by a TEG
with temperature-independent properties in steady-state, as a function of the Carnot efficiency and the
dimensionless figure of merit ZT
ηmc = ηC(1 + ZT )
12
(1 + ZT )12 + Tc
Th
(2.24)
15
From inspection, it is possible to conclude that the maximum conversion efficiency tends to the value
of Carnot efficiency ηC as Z tends to infinity, hence the necessity of achieving high values for the figure
of merit in order to improve performance of the TEG system. Although this relationship does not apply to
an actual thermoelectric system with temperature-dependent materials, the importance of the parameter
Z is crucial to improve the performance.
2.4 Thermoelectric Modules (TEM)
A thermoelectric module is composed of several thermocouples connected electrically in series and
thermally in parallel to increase the output voltage of the module. Each thermocouple is also connected
through a copper conducting strip. The module, however, must be electrically isolated from the heat
source and the heat sink while also allowing high thermal conductivity to minimize the temperature
difference between the heat source/sink and the thermocouple surface. Usually, alumina ceramic plates
are used for this purpose. Figure 2.10 schematizes the basic configuration of a TEM.
Figure 2.8: Configuration of a single stage thermoelectric module [1].
The number of thermocouples in a TEM is defined by n. The performance parameters of a TEM can
be obtained simply by multiplying the parameters of a single TEG by the total number of couples as
Wmod = nW (2.25)
Vmod = nV (2.26)
Qhmod= nQh (2.27)
The parameters not influenced by the number of thermocouples in a module are the current intensity
and the conversion efficiency
Imod = I (2.28)
ηmod =nW
nQh= η (2.29)
16
Naturally, due to the presence of the conducting strips and ceramic plates, contact effects occur. A
good TEM design must include electrical and thermal contact resistances (not shown in figure 2.9) which
will negatively impact the output voltage produced and, consequently, the electrical power delivered to
the connected load [24].
(a) Thermoelectric power generation system [13]. (b) Thermal resistance network [13].
Figure 2.9: A typical TEG power system with a representation of the thermal resistances involved.
2.4.1 Hi-Z Thermoelectric Modules
The Hi-Z thermoelectric modules convert low grade, waste heat into electricity that is intended to target
the waste heat market. The modules provided by Hi-Z Technology, Inc. use bismuth telluride Bi2Te3
based alloys as thermoelectric materials, with high efficiency at most waste heat temperatures and high
strength to endure rugged applications [10, 25]. The TEG couples inside the modules are electrically and
thermally insulated by a special frame called an ”eggcrate” that fills the air gaps between thermocouples,
which is manufactured through injection molding to make the TEM less expansive to fabricate [26].
The TEM considered for the analysis present in this master thesis were the HZ-14 and the HZ-20.
The data available in [19] is in accordance with the datasheets in Appendices A.1 and A.2. Performance
evaluation by Hi-Z Technology, Inc. for these modules did not contemplate the presence of heat exchang-
ers and ceramic plates, and so the given temperatures in the data sheets and the Module Performance
Calculator [19] were assumed to be on the module surface.
To predict Hi-Z modules’ performance three main components must be taken into consideration:
thermocouple, copper strips and the ”eggcrate”. Results that will be shown in Chapter 5 considering
only the thermocouple in the analysis are compared with results obtained for the three main components
together.
17
Figure 2.10: HZ-14 TEM. The hot-side is on the left where the dots show the ”eggcrate” material andthe cold-side is on the right [25].
Bi2Te3 data
The bismuth telluride alloys used in the Hi-Z modules are strongly temperature-dependent and so to
correctly predict the performance this needs to be taken into account in the analysis. The Seebeck
coeffcient, thermal conductivity and electrical resistivity are represented by a fourth degree polynomial
as a function of temperature, respectively as
α(T ) = α1 + α2T + α3T2 + α4T
3 + α5T4 (2.30)
κ(T ) = κ1 + κ2T + κ3T2 + κ4T
3 + κ5T4 (2.31)
ρ(T ) = ρ1 + ρ2T + ρ3T2 + ρ4T
3 + ρ5T4 (2.32)
where each of the coefficients in each polynomial is given in the table below.
Suite 7400, 7606 Miramar Road, San Diego, CA 92126, Tel 858.695.6660, [email protected], www.hi-z.com
FEATURES
Produces > 14 watts of power (Th=250°C, Tc=50°C)
Intermittent Operation beyond 350°C
Intermittent Power up to 25 watts
Rugged Construction (no ceramic, no
solders, fiber reinforced construction makes module tolerant to abuse)
Long life (> 10 years when properly used)
98 couples (Bi,Sb)2(Te,Se)3)
Produce 10mW @ ΔT=5°C
Some modules may have braided copper leads
DESCRIPTION The HZ-14 module is designed to generate power and is able to tolerate intermittent temperatures up to 350°C, but for maximum life expectancy it should not exceed 250°C. These high temperature properties are made possible by the bonded metal conductors that eliminate the presence of all solders. While the module is optimized for waste heat recovery, its reversible properties make it suitable as a thermoelectric cooler, especially for high temperature applications where sensitive electronic equipment must be cooled to below the ambient temperatures.
14 watt module Data Sheet
Suite 7400, 7606 Miramar Road, San Diego, CA 92126, Tel 858.695.6660, [email protected], www.hi-z.com
Thermal and Electrical Characteristics Parameter Conditions min typ max units
Power Th=250°C, Tc=50°C @matched load 14.0 15 16 Watts
Open Circuit Voltage Th=250°C, Tc=50°C 2.8 3.0 3.2 Volts
Matched load Voltage Th=250°C, Tc=50°C 1.4 1.5 1.6 Volts
Stated temperatures are assumed to be on the module surface and not the heat exchangers. Module surfaces are conductive and require the use of an insulator when used against metal heat exchangers. Ceramic wafers with thermal grease provide optimum performance.
Recommended mounting pressure is 100 to 200 psi uniformly distributed over the module surface.
All statements, technical information and recommendations contained herein are based on tests Hi-Z believes to be reliable, but the accuracy or completeness is not guaranteed. Neither seller nor manufacturer shall be liable for any injury, loss or damage including but not limited to special, incidental or consequential damages arising out of the use or the inability to use the product. Before using, user shall determine the suitability of the product for its intended use, and user assumes all risk and liability whatsoever in connection therewith. No statement or recommendation contained herein shall have any force or effect without a signed agreement by all parties.
A.2 HZ-20 Datasheet
74
20 watt module Data Sheet
Thermoelectric
Materials • Devices • Systems
Suite 7400, 7606 Miramar Road, San Diego, CA 92126, Tel 858.695.6660, [email protected], www.hi-z.com
FEATURES
Produce more than 20 watts of power (Th=250°C, Tc=50°C)
Intermittent Operation beyond 350°C
Intermittent Power up to 30 watts
Rugged Construction (no ceramic, no
solders, fiber reinforced construction makes module tolerant to abuse)
Long life (> than 10 years when properly
used)
71 couples (Bi,Sb)2(Te,Se)3)
Produce 15mW @ ΔT=5°C
DESCRIPTION This module is designed specifically for the generation of power and is able to tolerate intermittent temperatures exceeding 350°C but for maximum life expectancy it should not exceed 250°C. These high temperature properties are made possible by the bonded metal conductors that eliminate the presence of all solders. While the module is optimized for waste heat recovery, its reversible properties make it suitable as a thermoelectric cooler, especially for high temperature applications where sensitive electronic equipment must be cooled to below the ambient temperatures.
20 watt module Data Sheet
Suite 7400, 7606 Miramar Road, San Diego, CA 92126, Tel 858.695.6660, [email protected], www.hi-z.com
Thermal and Electrical Characteristics Parameter Conditions min typ max units
Power Th=250°C, Tc=50°C @matched load 20.0 21.0 22.0 Watts
Open Circuit Voltage Th=250°C, Tc=50°C 4.2 4.5 4.8 Volts
Matched load Voltage Th=250°C, Tc=50°C 2.1 2.25 2.4 Volts
Stated temperatures are assumed to be on the module surface and not the heat exchangers.
Module surfaces are conductive and require the use of an insulator when used against metal heat exchangers. Ceramic wafers with thermal grease provide optimum performance.
Recommended mounting pressure is 100 to 200 psi uniformly distributed over the module surface.