numerical simulation of a thermoelectric generator
TRANSCRIPT
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
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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
results with improved accuracy.
Keywords: Thermoelectricity, thermoelectric generator, Seebeck effect, waste heat, thermal
energy
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Contents
Resumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Concepts of Thermoelectricity 7
2.1 Thermoelectric Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.1 Seebeck Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.2 Peltier Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.3 Thomson Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.4 Kelvin Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Thermocouple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Thermoelectric Generator (TEG) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.1 Performance Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Thermoelectric Modules (TEM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4.1 Hi-Z Thermoelectric Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4.2 The Module Performance Calculator . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 Hi-Z TEM Modeling: Analytical Models 21
3.1 Conservation Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1.1 Energy Conservation equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1.2 Electric Potential Conservation equation . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 Simplified Linear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2.1 Performance Evaluation with the SLM . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3 Non-linear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
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3.3.1 Homotopy Perturbation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3.2 Application of the HPM to a thermoelement . . . . . . . . . . . . . . . . . . . . . . 27
4 Hi-Z TEM Modeling: Numerical Model 33
4.1 The Finite Volume Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.2 Application of the FVM to a thermoelement . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.2.1 Discretisation of the Energy conservation equation . . . . . . . . . . . . . . . . . . 34
4.2.2 Discretisation of the Seebeck potential conservation equation . . . . . . . . . . . . 37
4.2.3 Discretisation of the ohmic potential conservation equation . . . . . . . . . . . . . 38
5 Results and Discussion 39
5.1 Thermocouple Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.2 Thermoelectric Module Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.2.1 Contact Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.2.2 TEM Performance Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.2.3 Performance at matched load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.2.4 Temperature and electric potential distributions . . . . . . . . . . . . . . . . . . . . 51
6 Conclusions 63
6.1 Achievements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
References 67
A Technical Datasheets 71
A.1 HZ-14 Datasheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
A.2 HZ-20 Datasheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
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List of Tables
2.1 Coefficients of the polynomials that represent α, κ and ρ of Bi2Te3 material [19]. . . . . . 18
2.2 Geometric data of both Hi-Z modules [19]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1 Coefficients of the polynomials representing α, κ and ρ [3]. . . . . . . . . . . . . . . . . . 28
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List of Figures
1.1 A typical thermoelectric module [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1 Electron concentration in a conductor [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Schematic of a basic thermocouple subjected to a temperature difference at both ends [2]. 8
2.3 Representation of the Peltier and Thomson effects on a thermocouple [2]. . . . . . . . . . 9
2.4 A basic p-type and n-type thermocouple [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.5 Example of a n-type semiconductor of silicon doped with antimony [23]. . . . . . . . . . . 12
2.6 Example of a p-type semiconductor of silicon doped with boron [23]. . . . . . . . . . . . . 12
2.7 Electrical circuit representation for a TEG connected to a load and in open-circuit [2]. . . . 13
2.8 Configuration of a single stage thermoelectric module [1]. . . . . . . . . . . . . . . . . . . 16
2.9 A typical TEG power system with a representation of the thermal resistances involved. . . 17
2.10 HZ-14 TEM. The hot-side is on the left where the dots show the ”eggcrate” material and
the cold-side is on the right [25]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.11 User-interface of the Module Performance Calculator [19]. . . . . . . . . . . . . . . . . . . 20
3.1 Differential control volume in Cartesian coordinates [27]. . . . . . . . . . . . . . . . . . . . 22
3.2 One-dimensional representation of the heat balance considered for analysis. . . . . . . . 24
4.1 One-dimensional representation of the control volumes discretized for analysis [31] . . . . 35
4.2 Schematic of the grid and iterative solution procedure used for analysis [13]. . . . . . . . 37
5.1 Load curve for the HZ-14 thermocouples, operating between Th = 250oC and Tc = 50oC . 40
5.2 Voltage curve for the HZ-14 thermocouples, operating between Th = 250oC and Tc = 50oC 40
5.3 Power curve for the HZ-14 thermocouples, operating between Th = 250oC and Tc = 50oC 41
5.4 Efficiency curve for the HZ-14 thermocouples, operating between Th = 250oC and Tc =
50oC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.5 Thermoelectric properties as a function of temperature [3, 19]. . . . . . . . . . . . . . . . 44
5.6 Load curves for the HZ-14 thermoelectric module under different ∆T , for Tc = 50oC. . . . 46
5.7 Voltage curves for the HZ-14 thermoelectric module under different ∆T , for Tc = 50oC. . . 47
5.8 Power curves for the HZ-14 thermoelectric module under different ∆T , for Tc = 50oC. . . 48
5.9 Efficiency curves for the HZ-14 thermoelectric module under different ∆T , for Tc = 50oC. 49
5.10 Performance curves for the HZ-14 TEM at matched load conditions, as a function of ∆T . 50
xi
5.11 Load curves for the HZ-20 thermoelectric module under different ∆T , for Tc = 50oC. . . . 52
5.12 Voltage curves for the HZ-20 thermoelectric module under different ∆T , for Tc = 50oC. . . 53
5.13 Power curves for the HZ-20 thermoelectric module under different ∆T , for Tc = 50oC. . . 54
5.14 Efficiency curves for the HZ-20 thermoelectric module under different ∆T , for Tc = 50oC. 55
5.15 Performance curves for the HZ-20 TEM at matched load conditions, as a function of ∆T . 56
5.16 Seebeck potential distribution along the HZ-14 thermocouple legs. . . . . . . . . . . . . . 58
5.17 Ohmic potential distribution along the HZ-14 thermocouple legs. . . . . . . . . . . . . . . 59
5.18 Temperature distribution at matched load along the HZ-14 thermocouple legs. . . . . . . . 60
5.19 Non-dimensional temperature distribution at matched load along the HZ-14 thermocouple
legs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
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Nomenclature
Greek symbols
α Seebeck coefficient (V K−1).
η Efficiency.
θ Non-dimensional temperature.
κ Thermal conductivity coefficient (W m−1 K−1).
ξ Non-dimensional coordinate.
π Peltier coefficient (V).
ρ Electrical resistivity (Ω m).
σ Electrical conductivity (S m−1).
τ Thomson coefficient (V K−1).
Roman symbols
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.
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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.
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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
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[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.
α (V K−1) α1 α2 α3 α4 α5
n-type 7.42322× 10−5 −1.5018× 10−6 2.94× 10−9 −2.50× 10−12 1.36× 10−15
p-type −0.000255904 2.3184× 10−6 −3.18× 10−9 9.17× 10−13 −4.88× 10−16
κ (W m−1 K−1) κ1 κ2 κ3 κ4 κ5
n-type 1.425785 0.006514882 −0.00005162 1.12× 10−7 −7.60× 10−11
p-type 6.9245746 −0.05118914 0.000199588 −3.89× 10−7 3.04× 10−10
ρ (Ω m) ρ1 ρ2 ρ3 ρ4 ρ5
n-type −0.00195922× 10−2 1.79153× 10−7 −3.82× 10−10 4.92× 10−13 −2.98× 10−16
p-type −0.002849603× 10−2 1.96768× 10−7 −3.32× 10−10 3.47× 10−13 −1.90× 10−16
Table 2.1: Coefficients of the polynomials that represent α, κ and ρ of Bi2Te3 material [19].
18
HZ-14 and HZ-20 data
The geometry of the thermocouples within both the HZ-14 and HZ-20 is similar to figure 2.4, where only
conducting strips linking the n-type and p-type are represented. Each thermoelement has the same leg
length and cross-sectional area, which means
Lp = Ln = L (2.33)
Ap = An = A (2.34)
Since no heat exchangers or ceramic plates were considered in the performance calculations, the
only contact effects present will derive from the copper conducting strip. Copper is assumed to have a
very high thermal conductivity, therefore thermal resistance is negligible and so is the thermal contact
effect between the copper strip and the top surface of both the n-type and p-type units. Using the
notation described in figure 2.9 (b) we assume
TH = T1 (2.35)
TC = T2 (2.36)
The electrical contact effects and internal resistance of the copper strip are not negligible and need
to be quantified. Thermal conduction along the eggcrate also needs to be considered since it represents
a significant amount of the module surface as seen in 2.10.
Table 2.2 contains information regarding the number of couples n in each TEM, area of the surface of
the modules As (see Appendix A), cross-sectional area of each thermoelement A, leg length L, copper
conductor internal resistance Rc, contact electrical resistivity ρcon, eggcrate height hegg and eggcrate
thermal conductivity kegg.
Model HZ-14 HZ-20
n 49 71
As (m2) 33.76× 10−4 45.16× 10−4
A (m2) 14.7× 10−6 24.5× 10−6
L (m) 1.6× 10−3 2.7× 10−3
Rc (Ω) 2× 10−5 2× 10−5
ρcon (Ω m2) 3× 10−10 3× 10−10
hegg (m) 3× 10−3 5.8× 10−3
kegg (W m−1 K−1) 0.1 0.1
Table 2.2: Geometric data of both Hi-Z modules [19].
19
2.4.2 The Module Performance Calculator
The Module Performance Calculator [19] is a free program created by manufacturer Hi-Z Technology,
Inc., available to download on their website [10], that allows the user to visualize performance data of Hi-
Z thermoelectric modules. The inputs required by the user are the hot-side and cold-side temperatures
and the module type, which can be from one of the commercially available by Hi-Z Technology, Inc. or
a module with custom defined parameters. The resulting outputs are the performance curves and the
thermoelectric properties variation for the operating temperatures range as shown in figure 2.11.
Figure 2.11: User-interface of the Module Performance Calculator [19].
20
Chapter 3
Hi-Z TEM Modeling: Analytical Models
This chapter presents a detailed description of the analytical tools used to model the thermoelectric
modules. Starting with section 3.1, the conservation equations solved for analysis are presented. Sec-
tion 3.2 explains the first model used for analysis, the Simplified Linear Model, with the assumption of
temperature-independent properties. Here is also detailed how the concepts presented in Chapter 2 are
used to predict module performance with the Simplified Linear Model.
Following similar logic, section 3.3 introduces the non-linear analytical model used with the assump-
tion of temperature-dependent material properties. A brief mathematical description of the method used,
the homotopy perturbation method, is followed by its application to the non-linear analysis. Finally, mod-
ule performance evaluation with the method described is also explained.
3.1 Conservation Equations
3.1.1 Energy Conservation equation
One of the major objectives in a heat conduction related problem is the determination of the temperature
field in the medium, resulting from the imposed boundary conditions [27]. To obtain the temperature
distribution in the thermoelement leg, the energy conservation is applied to a generic differential control
volume like the one shown in figure 3.1. By solving the energy conservation equation, the resulting
temperature field can then be used to obtain the Seebeck and ohmic electric potential fields.
The energy generation term Eg represents a conversion process of some form of energy to thermal
energy and the term is positive if energy is generated (source) or negative if energy is consumed (sink).
Est is the energy storage term that refers to the change of thermal energy stored in the medium [27].
Applying the conservation equation to the heat balance equation of the differential control volume, the
following is obtained
∇ · (κ∇T ) + Eg = Est , (3.1a)
∇ · (κ∇T ) + Eg = ρcp∂T
∂t(3.1b)
21
Figure 3.1: Differential control volume in Cartesian coordinates [27].
Equation (3.1) is the general form of the heat diffusion equation [27] and allows the computation of
the temperature field T (x, y, z) in Cartesian coordinates. For steady-state operating conditions there is
no change in thermal energy stored, hence Est = 0. The energy source term (W m−3) consists of two
effects that arise when an electrical current flows through the leg of the n-type or p-type material: the
Thomson effect, caused by the temperature gradient on the material, and the Joule effect caused by the
electrical resistance of the thermocouple. Inserting equations (2.6) and (2.11) to account for the Joule
and Thomson effects, respectively, equation (3.1b) becomes
∇ · (κp,n∇Tp,n) + Tp,ndαp,ndT∇Tp,n · ~J + ρp,n‖ ~J‖2 = 0 , (3.2a)
∇ · (κp,n∇Tp,n) + Tp,n∇αp,n · ~J + ρp,n‖ ~J‖2 = 0 (3.2b)
Equation (3.2) represents the general conservation of energy equation for a generic control volume
of the n-type and p-type thermoelements without convective or radiative heat losses. For the TEM built
by Hi-Z Technology, Inc. this assumption is acceptable due to the presence of the ”eggcrate” material.
3.1.2 Electric Potential Conservation equation
Seebeck electric potential
By determining the temperature field using the conservation equation (3.2), the Seebeck potential can
now be derived. To calculate the Seebeck potential distribution in a thermoelement control volume, the
divergence is taken on both sides of equation (2.2) resulting
∇ · (∇VSbkp,n) = ∇ · (−αp,n∇Tp,n) , (3.3a)
∇ · (∇VSbkp,n) +∇ · (αp,n∇Tp,n) = 0 , (3.3b)
22
Equation (3.3) is the conservation equation of the Seebeck potential, where the only source for the
potential diffusion is the Seebeck effect.
Ohmic electric potential
To determine the output voltage of the system, the ohmic potential distribution along the thermoelements
of the module needs to be computed as well as shown by equation (2.17b). To calculate the potential
due to Ohm’s law, the conservation equation is also applied. Rearranging equation (2.14b), the vector
form of Ohm’s law can be presented as
− σp,n∇VOhmp,n= ~J (3.4)
where σ = 1ρ is the material electrical conductivity (S m−1). Based on the continuity of current, the
divergence of ~J is null and so
∇ · ~J = 0 (3.5)
Applying the conservation principle to equation (3.4), the conservation of ohmic potential is written
as
∇ · (σp,n∇VOhmp,n) = 0 (3.6)
where the diffusion of the electric potential is only due to the material electrical conductivity and there is
no source term.
3.2 Simplified Linear Model
The analytical model presented in this section solves the conservation equations by assuming constant
thermoelectric material properties. Since the temperature gradient between the heat source and heat
sink is applied only on the top and bottom surfaces of the thermoelectric module, the analysis can be
reduced to a one-dimensional problem as shown in figure 3.2.
For a one-dimensional problem, equation (3.2) reduces to
d
dx
(κp,n
dTp,ndx
)+ Tp,n
dαp,ndT
dTp,ndx
~Jx + ρp,n‖ ~Jx‖2 = 0 (3.7)
where the current density vector along the x direction, ~Jx, is defined as
‖ ~Jx‖ = J =I
A(3.8)
where I is the current intensity (A) and A is the cross-sectional area (m−2). Inserting equation (3.8) in
(3.7) and considering the thermoelectric properties do not change along the x coordinate, the resultant
energy conservation equation is
23
(a) Direction considered for analysis [13]. (b) Differential element [2].
Figure 3.2: One-dimensional representation of the heat balance considered for analysis.
κp,n
(d2Tp,ndx2
)+ρp,nI
2
A2= 0 (3.9)
where κ and ρ are the thermal conductivity and the electrical resistivity, respectively, evaluated by equa-
tion (2.31) and equation (2.32) at the average temperature T , defined by (2.23). The temperature profile
for the n-type and p-type legs can now be obtained by integrating equation (3.9) twice
Tp,n(x) = − qp,n2κp,n
x2 + C1x+ C2 (3.10)
where qp,n =ρp,nI
2
A2 is the volumetric heat rate produced by Joule effect and C1 and C2 are constants
of integration. To determine C1 and C2, the boundary conditions are applied, in this case as imposed
temperatures, where
T (x = 0) = Th (3.11)
T (x = L) = Tc (3.12)
Solving for C1 and C2, equation (3.10) yields
Tp,n(x) = − qp,n2κp,n
x2 +
(Tc − Th
L+qp,nL
2κp,n
)x+ Th (3.13)
To determine the Seebeck potential distribution, equation (3.3b) is also reduced to its one-dimensional
form as
d
dx
(dVSbkp,ndx
)+
d
dx
(αp,n
dTp,ndx
)= 0 (3.14)
Once again, properties are assumed to be constant along x and so the average Seebeck coefficient
αp,n is evaluated at T using equation (2.30) and inserted in (3.14), yielding
24
d2VSbkp,ndx2
+ αp,nd2Tp,ndx2
= 0 (3.15)
By inserting the second derivative of equation (3.10) and integrating twice like in (3.9), the analytical
Seebeck potential profile for constant properties is defined as
VSbkp,n(x) = αp,nqp,n
2κp,nx2 + C1x+ C2 (3.16)
To obtain the constants, boundary conditions for the Seebeck potential must be applied. A reference
potential of 0 V is imposed at the hot-side while on the cold-side an electric field is imposed due to the
temperature gradient [6]. As such, there are now two different types of boundary conditions: imposed
potential on the hot-side and imposed flux (gradient) on the cold-side.
VSbkp,n(x = 0) = 0 (3.17)
~Ep,n(x = L) = αp,ndTp,ndx
∣∣∣∣x=L
, (3.18a)
dVSbkp,ndx
∣∣∣∣x=L
= −αp,n(− qp,nκp,n
L+(Tc − Th)
L+qp,nL
2κp,n
), (3.18b)
Combining equations (3.17) and (3.18) with (3.16), the analytical profile for the Seebeck potential is
VSbkp,n(x) = αp,nqp,n
2κp,nx2 − αp,n
(Tc − Th
L+qp,nL
2κp,n
)x (3.19)
The ohmic potential distribution is also derived in similar fashion. Starting with the one-dimensional
conservation equation of (3.6) and knowing that σp,n = 1ρp,n
, it is written as
σp,nd2VOhmp,n
dx2= 0 (3.20)
Integrating twice yields
VOhmp,n(x) = C1x+ C2 (3.21)
The boundary conditions are similar to the Seebeck potential. A reference potential of 0 V is imposed
in the hot-side, while on the cold-side the current density J is imposed with equation (3.4). Therefore, at
the boundaries results
VOhmp,n(x = 0) = 0 (3.22)
dVOhmp,n
dx
∣∣∣∣x=L
= − J
σp,n(3.23)
After applying these boundary conditions, equation (3.21) is defined as
25
VOhmp,n(x) = − J
σp,nx (3.24)
Equations (3.13), (3.19) and (3.24) form the Simplified Linear Model that solves the linear differential
heat equation (3.9), which assumes constant thermoelectric properties, thus neglecting the Thomson
effect in the balance equation. Even if it is built upon some simplifications, this model can be used to
estimate the parameters in a certain range of operating temperatures as it will be shown in Chapter 5.
3.2.1 Performance Evaluation with the SLM
The performance parameters can be evaluated using the equations defined in section 2.3.1. Because
the reference electric potential is defined at 0 V in the hot-side, the voltage drop along the n-type and
p-type units can be evaluated using
V = (VSbkp |x=L − VSbkn |x=L)− (VOhmp|x=L + VOhmn
|x=L) (3.25)
The electrcial power generated at the load is computed through equation (2.18) and the load resis-
tance is obtained by (2.19b). To calculate the efficiency, the total heat rate needs to be computed. By
determining the heat flux at the hot-end on each thermoelement using equations (2.12) and (2.13), the
total heat rate coming from the hot source Qh (W) is then defined as
Qh = (qhp |x=0 − qhn |x=0)A , (3.26a)
Qh = (αp − αn)ThI −(κpA
dTpdx
∣∣∣∣x=0
− κnAdTndx
∣∣∣∣x=0
), (3.26b)
Applying equation (2.21) yields the conversion efficiency. Multiplying V , W and Qh by the number of
couples n results in the total output voltage, output power and efficiency of the module.
3.3 Non-linear Model
In this section, a non-linear model is presented in order to solve the non-linear heat transport equation
(3.9) by an approximate analytical solution based on the homotopy perturbation method (HPM) [3, 5, 11].
This method assumes non-constant thermoelectric properties for the Bi2Te3 n-type and p-type.
A brief explanation of the method follows, and then the application to the problem at hand is pre-
sented.
3.3.1 Homotopy Perturbation Method
Considering the generic non-linear differential equation [28]
A(u)− f(r) = 0, r ∈ Ω (3.27)
26
with the following boundary conditions
B
(u,∂u
∂n
)= 0, r ∈ Γ (3.28)
where A is a differential operator, B is the boundary operator, f(r) is a known function and Γ is the
boundary of domain Ω. A can be splitted into two parts L and N , where L is linear and N is non-linear.
Equation (3.27) can be rewritten as
L(u) +N(u)− f(r) = 0 (3.29)
Applying the homotopy technique [29], a homotopy v(r, p) : Ω× [0, 1]→ R is constructed as
H(v, p) = (1− p)[L(v)− L(u0)] + p[A(v)− f(r)] = 0 , (3.30a)
H(v, p) = L(v)− L(u0) + pL(u0) + p[N(v)− f(r)] = 0 (3.30b)
where p ∈ [0,1] is an embedding ”small” parameter, u0 and v are the initial approximation and the
solution of equation (3.27), respectively. As p → 1, (3.30b) becomes (3.27). The solution v can be
written as a power series in p [28]
v = v0 + pv1 + p2v2 + ... (3.31)
By inserting equation (3.31) in the homotopy function (3.30b) and collecting terms from the zeroth to
the n-th order of p, a system of n+1 equations is obtained. Solving the system for each term of v will
yield the n-th order approximation for the exact analytical solution of (3.27) as
v = v0 + v1 + v2 + ... (3.32)
3.3.2 Application of the HPM to a thermoelement
To find an approximate analytical solution to (3.7) by employing the HPM, dimensionless parameters are
used in order to simplify the problem. Starting by rearranging (3.7), it is possible to obtain [3, 5, 11]
κp,nAd2Tp,ndx2
+Aκp,ndT
(dTp,ndx
)2
− ITp,ndαp,ndT
dTp,ndx
+ρp,nI
2
A= 0 (3.33)
Zhang [3] proposes third-degree order polynomials for α, κ and ρ, based on the coefficients given in
table 3.1
α(T ) =
3∑i=0
αiTi (3.34)
κ(T ) =
3∑i=0
κiTi (3.35)
27
ρ(T ) =
3∑i=0
ρiTi (3.36)
α (V K−1) α0 α1 α2 α3
n-type 8.959× 10−6 −0.9272× 10−6 0.001075× 10−6 0.1292× 10−12
p-type −274.4× 10−6 2.422× 10−6 −0.003274× 10−6 0.5921× 10−12
κ (W m−1 K−1) κ0 κ1 κ2 κ3
n-type 3.237 −0.01243 2.124× 10−5 −0.9998× 10−8
p-type −2.363 0.03874 −12.43× 10−5 12.52× 10−8
ρ (Ω m) ρ0 ρ1 ρ2 ρ3
n-type −1.05× 10−5 0.9103× 10−7 −0.6429× 10−10 −0.1246× 10−13
p-type −2.245× 10−5 1.388× 10−7 −1.251× 10−10 0.2249× 10−13
Table 3.1: Coefficients of the polynomials representing α, κ and ρ [3].
The non-dimensional temperature profile and coordinate to be introduced are given, respectively, by
θp,n =Tp,n − TcTh − Tc
(3.37)
ξ =x
L(3.38)
Inserting equations (3.37) and (3.38) along with the polynomial expressions of α, κ and ρ in the heat
transport equation (3.33), the dimensionless governing equation becomes
f(θp,n)
(dθp,ndξ
)2
+ g(θp,n)d2θp,ndξ2
− h(θp,n)dθp,ndξ
+ q(θp,n) = 0 (3.39)
where the functions f(θ), g(θ), h(θ) and q(θ) are given by the following polynomials
f(θ) =
2∑i=0
fiθi (3.40)
g(θ) =
3∑i=0
giθi (3.41)
h(θ) =
3∑i=0
hiθi (3.42)
q(θ) =
3∑i=0
qiθi (3.43)
The coefficients of f(θ), g(θ), h(θ) and q(θ) are functions of the material properties, electric current
28
intensity, absolute temperatures and geometry of the legs, defined as
f0 = (3κ3T2c + 2κ2Tc + κ1)
(Th − Tc
L
)2
A , (3.44a)
f1 = (6κ3Tc + 2κ2)(Th − Tc)(Th − Tc
L
)2
A , (3.44b)
f2 = 3κ3(Th − Tc)2
(Th − Tc
L
)2
A (3.44c)
g0 = (κ0 + κ1Tc + κ2T2c + κ3T
3c )
(Th − TcL2
)A , (3.45a)
g1 = (3κ3T2c + 2κ2Tc + κ1)(Th − Tc)
(Th − TcL2
)A , (3.45b)
g2 = (3κ3Tc + κ2)(Th − Tc)2
(Th − TcL2
)A , (3.45c)
g3 = κ3(Th − Tc)3
(Th − TcL2
)A (3.45d)
h0 = (3α3T3c + 2α2T
2c + α1Tc)
(Th − Tc
L
)I , (3.46a)
h1 = (9α3T2c + 4α2Tc + α1)(Th − Tc)
(Th − Tc
L
)I , (3.46b)
h2 = (9α3Tc + 2α2)(Th − Tc)2
(Th − Tc
L
)I , (3.46c)
h3 = 3α3(Th − Tc)3
(Th − Tc
L
)I (3.46d)
q0 = (ρ3T3c + ρ2T
2c + ρ1Tc + ρ0)
(I2
A
), (3.47a)
q1 = (3ρ3T2c + 2ρ2Tc + ρ1)(Th − Tc)
(I2
A
), (3.47b)
q2 = (3ρ3Tc + ρ2)(Th − Tc)2
(I2
A
), (3.47c)
q3 = ρ3(Th − Tc)3
(I2
A
)(3.47d)
Equation (3.39) can be written in the form of (3.29), with the operators L and N and the known
function term f(r) defined as
29
L(θ) = g0d2θ
dξ2− h0
dθ
dξ+ q1θ (3.48)
N(θ) = (f0 + f1θ+ f2θ2)
(dθ
dξ
)2
+ (g1θ+ g2θ2 + g3θ
3)d2θ
dξ2− (h1θ+ h2θ
2 + h3θ3)dθ
dξ+ q2θ
2 + q3θ3 (3.49)
f(r) = −q0 (3.50)
Assuming θ can be approximated by equation (3.31), the first-order approximate solution of θ is
described as
θ ≡ v = v0 + pv1 (3.51)
Introducing equation (3.51) in equation (3.30b) and collecting terms at the zeroth and first order of p
yields
L(v0)− L(u0) = 0 (3.52)
L(v1) + L(u0) +N(v0)− f(r) = 0 (3.53)
Equation (3.52) gives v0 = u0 ≡ θ0, where θ0 is the initial approximation of θ that satisfies the
boundary conditions. Zhang [5, 11] defines the initial approximation θ0 as
θ0 = ξ2 + aξ(1− ξ) (3.54)
where a is obtained by satisfying the weak form solution of (3.29) for θ0
∫ 1
0
ξ(L(θ0) +N(θ0)− f(r))dξ = 0 (3.55)
and by satisfying the imposed temperature boundary conditions. In non-dimensional parameters it yields
[3, 5, 11]
θ(ξ = 0) = 0 (3.56)
θ(ξ = 1) = 1 (3.57)
Once a is determined, equation (3.53) can be rewritten as a function of v1 and a known polynomial
function R1(a, ξ)
g0d2v1
dξ2− h0
dv1
dξ+ q1v1 +R1(a, ξ) = 0 (3.58)
30
Equation (3.58) can be solved by finding the general and the particular solution of the non-homogeneous
linear ordinary differential equation [30]. The general solution can be determined by finding the roots of
the characteristic polynomial of the homogeneous equation
erξ(g0r2 − h0r + q1) = 0 (3.59)
With roots
r =h0 ±
√h2
0 − 4g0q1
2g0(3.60)
If the characteristic polynomial has two distinct real roots b and d, the general solution v1g is
v1g = C1ebξ + C2e
dξ (3.61)
For one repeated real root b, the general solution is instead
v1g = (C1 + C2ξ)ebξ (3.62)
If the roots form a pair of complex conjugate roots, v1g becomes of the form
v1g = ebξ(C1 cos dξ + C2 sin dξ) (3.63)
Where b = <(r) and d = =(r). To find the particular solution of (3.58), the method of undetermined
coefficients is applied [30]. Since R1(a, ξ) is a polynomial function, the particular solution v1p is given by
v1p =
M∑k=0
wkξk (3.64)
Where M is the polynomial degree of R1(a, ξ). The coefficients wk are determined by inserting v1p
in (3.58) and comparing with the coefficients of the same order of R1(a, ξ). The first-order approximate
solution of θ can then be formulated by
θ(ξ) = θ0 + v1g + v1p (3.65)
And the coefficients C1 and C2 can be obtained by applying the boundary conditions (3.56) and
(3.57). Once the non-dimensional temperature profile is determined it is possible to convert back to
dimensional parameters. Inserting (3.38) in (3.65), θ becomes a function of the x coordinate. Applying
(3.37), the temperature profile can be written as
Tp,n(x) = θp,n(x)(Th − Tc) + Tc (3.66)
Since the material properties are no longer constant, it is easier to evaluate the Seebeck voltage by
using equation (2.2). For a one-dimensional problem it is defined as
31
VSbkp,n(x) = −∫αp,n(Tp,n(x))dT (3.67)
Where α(T ) is described by equation (3.34). The ohmic potential is evaluated by (2.14b) along the x
coordinate, yielding
VOhmp,n(x) = −
∫ρp,n(Tp,n(x))dx (3.68)
Where ρp,n(Tp,n(x)) is obtained by combining the expression from (3.66) with (3.36). The perfor-
mance of the TEM is evaluated with the equations from 3.2.1, except the properties are now evaluated
using (3.34), (3.35) and (3.36).
32
Chapter 4
Hi-Z TEM Modeling: Numerical Model
The following chapter introduces the application of the Finite Volume Method (FVM) to solve the con-
servation equations described in Chapter 3. Section 4.1 describes the mathematical definitions inherent
to the proposed method and how it solves the conservation laws. Section 4.2 shows how the FVM is
applied to the problem at hand in detail while solving the conservation equations numerically.
4.1 The Finite Volume Method
The Finite Volume Method is based on the integration of the conservation equations of a defined physical
property φ. The domain that is being studied is divided in a finite number of control volumes and the
conservation laws are integrated for each control volume.
For a steady-state diffusion problem, the general transport equation for property φ can be written as
[31]
∇ · (Γ∇φ) + Sφ = 0 (4.1)
where Γ is the diffusion coefficient, φ is the property transported and Sφ the source terms of the equation.
The integration over a defined control volume like the one in figure 3.1, for the one-dimensional case,
results in
∫V
∇ · (Γ∇φ)dV +
∫V
SφdV = 0 (4.2)
The volume integral of the diffusive term defined in (4.2) can be rewritten using Gauss’s theorem
[31]. For a vector ~a, it states that
∫V
∇ · (~a)dV =
∫S
~n.~adS (4.3)
This means that the component of vector ~a in the direction ~n normal to the surface bounding the
volume V , is equal to the volume integral of the gradient of ~a. Hence, the rate of increase of a property
is equal to the net rate of increase due to the fluxes in and out across the control volume surfaces [31].
33
This means that by using the Finite Volume Method, the conservation principle is always respected.
For the source terms, the integral is evaluated as follows
∫V
SφdV = S∆V (4.4)
where S is the average value of source term Sφ over the control volume and ∆V is the volume.
4.2 Application of the FVM to a thermoelement
4.2.1 Discretisation of the Energy conservation equation
Considering the heat transport equation (3.2), the integration over a control volume is defined as
∫V
∇ · (κp,n∇Tp,n)dV +
∫V
Tp,n∇αp,n ~JdV +
∫V
ρp,n‖ ~J‖2dV = 0 (4.5)
Since the problem was considered to be one-dimensional, the gradients can be reduced to
∫V
d
dx
(κp,n
dTp,ndx
)dV +
∫V
ITp,nA
dαp,ndx
dV +
∫V
ρp,nI2
A2dV = 0 (4.6)
Using Gauss’s theorem, the diffusive term can be expressed as a surface integral and so the resultant
transport equation is
∫S
(κp,n
dTp,ndx
)· ~ndS +
∫V
ITp,nA
dαp,ndx
dV +
∫V
ρp,nI2
A2dV = 0 (4.7)
Along the x coordinate, the infinitesimal volume dV can be represented as
dV = dxdydz = dxdS = dxA (4.8)
The domain where the conservation laws are applied needs to be discretized into control volumes.
The boundaries of each control volume are positioned mid-way between adjacent nodes. It is common
practice to set up control volumes near the edge of the domain so that the physical boundaries coincide
with the control volume boundaries [31]. The usual convention of computational fluid dynamics (CFD)
methods is represented in figure 4.1.
The generic nodal point P is surrounded by its neighbour points W and E. The west face of P is
referred by w and the east face by e. The distances between each node and each face are shown in
figure 4.1 (b). The boundary conditions are defined by equations (3.11) and (3.12) and so
T (x = 0) = TA = Th (4.9)
T (x = L) = TB = Tc (4.10)
Heat conduction (diffusive flux) is positive in the direction of the negative temperature gradient, from
34
(a) One-dimensional grid.
(b) One-dimensional nodal point geometry.
Figure 4.1: One-dimensional representation of the control volumes discretized for analysis [31]
A to B. Since the normal vector ~n is positive for the same direction of x and negative in the reverse
direction, and the source terms (Thomson and Joule heat) are evaluated at the nodal point P , (4.7)
becomes
κeAedT
dx
∣∣∣∣e
− κwAwdT
dx
∣∣∣∣w
+
(IT
A
dα
dx
)P
dxA+
(ρI2
A2
)P
dxA = 0 (4.11)
where κeAedTdx |e is the heat rate by conduction leaving the control volume, κwAw dTdx |w is the heat rate
by conduction going in the control volume, ( ITAdαdx )
PdxA is the average Thomson heat rate at node P
and (ρI2
A2 )P dxA is the average Joule heat rate at node P . Subscripts w and e indicate, respectively,
the coefficients and the derivatives evaluated at the west and east face of P . However, to obtain useful
discretised equations for each node on the defined grid, the coefficients and the derivatives at the inter-
faces need to be known. A linear approximation of the properties at the interfaces is used to evaluate
fluxes at w and e, with a technique called finite central differencing [31]. Thus,
κe =κE + κP
2(4.12)
κw =κP + κW
2(4.13)
κeAedT
dx
∣∣∣∣e
= κeAe
(TE − TPδxPE
)(4.14)
κwAwdT
dx
∣∣∣∣w
= κwAw
(TP − TWδxWP
)(4.15)
Central differencing is also employed to evaluate the derivative of α at node P as
35
(IT
A
dα
dx
)P
dxA =ITPA
(αe − αwδxwe
)δxweA (4.16)
where the Seebeck coefficients at the interfaces, αe and αw, are determined as in (4.12) and (4.13),
respectively. For an equally spaced grid, the distances along x are
δxPE = δxWP = δxwe = ∆x =L
N(4.17)
δxwP = δxPe =∆x
2(4.18)
where L is the length of the thermoelement leg and N the number of control volumes of the grid. The
cross-sectional area of both the n-type and p-type legs is constant, meaning Ae = Aw = AP = A.
Hence, the final discretised governing heat equation for a nodal point P in the grid is
κeA
(TE − TP
∆x
)− κwA
(TP − TW
∆x
)+ ITP (αe − αw) +
ρPI2
A∆x = 0 (4.19)
For n control volumes on the p-type leg and m control volumes in the n-type leg, as represented
in figure 4.2 (a), systems of n and m equations are constructed, respectively. To close the systems
of equations, imposed temperatures at the boundaries are applied, resulting, at the first and n-th/m-th
control volume, respectively
κeA
(TE − TP
∆x
)− κhA
(TP − Th
∆x2
)+ ITP (αe − αh) +
ρPI2
A∆x = 0 (4.20)
κcA
(Tc − TP
∆x2
)− κwA
(TP − TW
∆x
)+ ITP (αc − αw) +
ρPI2
A∆x = 0 (4.21)
where the properties with subscript h are evaluated at Th and the properties with subscript c are eval-
uated at Tc. At the boundaries, the fluxes are evaluated using forward finite differences (hot-side) and
backward finite differences (cold-side) [31]. To solve the systems of equations and find the correct tem-
perature distribution in each leg, an iterative procedure like the one in figure 4.2 (b) is used since the
thermal conductivity and the Seebeck coefficient are functions of an unknown temperature distribution
in the material. An initial assumption for Ti,j is made in order to evaluate the thermoelectric properties
with equations (2.30), (2.31) and (2.32). Solving for Ti,j , the absolute error is then calculated as
∆T = |Ti,j(k)− Ti,j(k − 1)| (4.22)
where k is the number of iterations. If ∆T > 10−6, the new assumption for Ti,j is based on
Ti,j(k + 1) =Ti,j(k) + Ti,j(k − 1)
2(4.23)
The convergence criterion is met when ∆T ≤ 10−6. After the temperature field for each leg is
calculated, the FVM can then be used to evaluate the Seebeck and ohmic potential fields.
36
(a) N-type and p-type legs discretised points. (b) Iterative procedure flowchart.
Figure 4.2: Schematic of the grid and iterative solution procedure used for analysis [13].
4.2.2 Discretisation of the Seebeck potential conservation equation
To obtain the discretised equations for the Seebeck potential, the procedure introduced in 4.2.1 is applied
to (3.3b). By integrating over a control volume, the equation for the Seebeck potential in a nodal point P
is described as
AdVSbkdx
∣∣∣∣e
−AdVSbkdx
∣∣∣∣w
+ αeAdT
dx
∣∣∣∣e
− αwAdT
dx
∣∣∣∣w
= 0 , (4.24a)
A
(VSbkE − VSbkP
∆x
)−A
(VSbkP − VSbkW
∆x
)+ αeA
(TE − TP
∆x
)− αwA
(TP − TW
∆x
)= 0 (4.24b)
Once again to solve the system of equations, the boundary conditions described by (3.17) and (3.18)
need to be implemented. For the n-th/m-th control volume, the imposed electric field at the cold junction
is now written as
dVSbkdx
∣∣∣∣e
= −αedT
dx
∣∣∣∣e
(4.25)
As a result, the two equations to apply on the hot-side and cold-side are, respectively
A
(VSbkE − VSbkP
∆x
)−A
(VSbkP − VSbkh
∆x2
)+ αeA
(TE − TP
∆x
)− αhA
(TP − Th
∆x2
)= 0 (4.26)
−A(VSbkP − VSbkW
∆x
)− αwA
(TP − TW
∆x
)= 0 (4.27)
where αh is the Seebeck coefficient evaluated at the hot-side temperature Th and VSbkh is the reference
potential of 0 V at x = 0. The Seebeck electric potential at x = L, obtained after the potential field in
37
each leg is known, is calculated by solving (4.25) as
VSbkc = VSbk|x=L = VSbkP − αc(Tc − TP ) (4.28)
Where αc is the Seebeck coefficient of the material evaluated at the cold-side temperature Tc.
4.2.3 Discretisation of the ohmic potential conservation equation
Using the FVM, the ohmic electric potential distribution for a nodal point P is obtained from (3.6), thus
σeAdVOhmdx
∣∣∣∣e
− σwAdVOhmdx
∣∣∣∣w
= 0 , (4.29a)
σeA
(VOhmE
− VOhmP
∆x
)− σwA
(VOhmP
− VOhmW
∆x
)= 0 (4.29b)
For the n-th/m-th nodal point, the boundary condition at the cold junction is an imposed electric
current density, hence
dVOhmdx
∣∣∣∣e
= − J
σe= − I
Aσe(4.30)
At the boundaries, the discretised equations are of the form
σeA
(VOhmE
− VOhmP
∆x
)− σhA
(VOhmP
− VOhmh
∆x2
)= 0 (4.31)
− I − σwA(VOhmP
− VOhmW
∆x
)= 0 (4.32)
where σh is the electrical conductivity evaluated at Th and VOhmhis the reference potential of 0 V. The
ohmic potential at x = L can then be evaluated through (4.33) as
VOhmc= VOhm|x=L = VOhmP
− I
Aσc
∆x
2(4.33)
where σc is the electrical conductivity evaluated at Tc. The performance of the model is then evaluated
as presented in 3.2.1. With the FVM, the derivatives at x = 0 when computing Qh are discretised with
forward finite differences as
Qh = (αph − αnh)ThI −
(κphA
Tp(1)− Th∆x2
− κnhATn(1)− Th
∆x2
)(4.34)
where the subscript h denotes the properties at Th and Tp(1) and Tn(1) is the temperature at nodal point
P for the first control volume of p-type and n-type legs, respectively.
38
Chapter 5
Results and Discussion
This chapter presents the results for the performance of the modules using the models described in
Chapters 3 and 4. Section 5.1 presents the initial results obtained for thermocouple analysis, where the
eggcrate and the contact effects in the modules were disregarded. In section 5.2 it is shown how these
effects are considered in the analysis and the results are presented for both the HZ-14 and HZ-20 mod-
ules under different regimes of operation. The performance parameters are also evaluated as a function
of the temperature, and the distributions of temperature and electric potential along the thermoelements
legs are presented and compared for constant and non-constant thermoelectric properties. The results
presented in this chapter are discussed in detail.
5.1 Thermocouple Analysis
To test the models presented in Chapters 3 and 4, the contact effects in the TEM were not considered.
Hence, the initial analysis was performed to the thermocouple only, based in an ideal scenario where
there was no electrical contact resistivity between the copper conducting strips and the top and bottom
surfaces of both the n-type and p-type units, and there was no drop in potential due to the presence of
the copper conductor. Heat conduction through the eggcrate material was also not considered.
To validate these hypotheses, the models were applied to the HZ-14 module for a cold-side temper-
ature of Tc = 50oC and a hot-side temperature of Th = 250oC. The results are presented as four plotted
performance curves: the load curve in figure 5.1 showing the electrical power W as a function of the load
resistance RL; the voltage curve in figure 5.2 showing the voltage across the load V as a function of the
electric current intensity I; the power curve in figure 5.3 showing the electrical power W as a function of
the electric current intensity I; and the efficiency curve in figure 5.4 where the efficiency η is plotted as
a function of the electric current intensity I. The data from the Module Performance Calculator [19] for
the same conditions is also shown in each plot for comparison.
Observing each curve it is immediately possible to notice the difference between the results obtained
from the computations with the developed models and the data extracted from the Module Performance
Calculator [19]. For the performance parameters shown it is clear that neglecting the presence of contact
39
Figure 5.1: Load curve for the HZ-14 thermocouples, operating between Th = 250oC and Tc = 50oC
Figure 5.2: Voltage curve for the HZ-14 thermocouples, operating between Th = 250oC and Tc = 50oC
effects and the eggcrate material leads to a large overestimation of module performance. However, from
this initial testing it is possible to conclude that, as expected, the Simplified Linear Model predicts higher
module performance, increasing even more the overestimation, while the non-linear models predict lower
performance, since the non-linearity of materials properties is considered. The results obtained from the
homotopy perturbation method and the finite volume method also agree very well for predicting power,
voltage and efficiency.
40
Figure 5.3: Power curve for the HZ-14 thermocouples, operating between Th = 250oC and Tc = 50oC
Figure 5.4: Efficiency curve for the HZ-14 thermocouples, operating between Th = 250oC and Tc = 50oC
5.2 Thermoelectric Module Analysis
5.2.1 Contact Effects
To properly predict the thermoelectric module performance, the presence of the copper conducting strip
and the eggcrate material need to be considered. Both the potential drop in the conductor and the
potential drop due to the electrical contact resistivity can be evaluated by Ohm’s law. Since the reference
potential at the hot-side for both the Seebeck and ohmic potential is 0 V, the voltage drop across the
load, results in
V = (VSbkp |x=L − VSbkn |x=L)− (VOhmp |x=L + VOhmn |x=L)− I(2Rc +Rcon) (5.1)
41
where 2Rc is the total copper conductor internal resistance (Ω). The copper conductor makes contact
on both the top and bottom surfaces of the n-type and p-type materials, hence the total electrical contact
resistance Rcon is defined as
Rcon =4ρconA
(5.2)
where ρcon is the electrical contact resistivity. The values of Rc and ρcon are taken from table 2.2. From
the Module Performance Calculator [19] it was revealed that Hi-Z Technology, Inc. also quantifies these
effects by considering a 17% loss in the total Seebeck voltage of the module. Applying this in (5.4), the
effective voltage across the load is defined as
V = 0.83(VSbkp |x=L − VSbkn |x=L)− (VOhmp|x=L + VOhmn
|x=L)− I(2Rc +Rcon) (5.3)
The heat conducted through the eggcrate material is evaluated using Fourier’s law. Since it is a
thermal insulator material, a reasonable assumption is to consider its thermal conductivity κegg constant
since it is very low as presented in table 2.2. The heat rate absorbed by the eggcrate is defined as
Qegg =κeggAegg(Th − Tc)
hegg(5.4)
where hegg is the height of the eggcrate given in table 2.2 and Aegg is the cross-sectional area of the
eggcrate. Aegg is the total cross-sectional area of the thermoelectric module minus the total cross-
sectional area of all the thermoelements inside the module, thus
Aegg = As − 2nA (5.5)
where As is the surface area of the module from table 2.2 and 2n represents the number of thermoele-
ments in the module. Accounting for the heat absorbed by the eggcrate, the total heat rate absorbed by
the module is now written as
Qmodule = Qh + Qegg (5.6)
where Qh is evaluated by (3.26). The efficiency of the module is now given by
η =W
Qmodule(5.7)
Inspecting equations (5.3) and (5.7) it is easy to see why initial testing was overestimating the per-
formance of the module. Since the output voltage V and output electrical power W will be lower and the
total heat absorbed will be higher, lower power and lower efficiencies for the thermoelectric modules are
expected.
42
5.2.2 TEM Performance Curves
The analyses of the HZ-14 and HZ-20 thermoelectric modules using the three proposed models were
carried out for a fixed cold-side temperature of Tc = 50oC and different temperature differences ∆T =
Th − Tc: ∆T = 100oC, ∆T = 200oC and ∆T = 300oC. The polynomial functions used to evaluate the
thermoelectric properties for the SLM and FVM (equations (2.30), (2.31) and (2.32)) and to evaluate the
properties in the HPM (equations (3.34), (3.35) and (3.36)) are plotted as a function of the temperature in
figure 5.5. The same performance curves shown in section 5.1 were re-evaluated for the HZ-14 and HZ-
20 for the different values of ∆T . From figure 5.6 to figure 5.9 the predicted performance parameters for
the HZ-14 are displayed. In figure 5.10 these same parameters are plotted as a function of temperature
at matched load conditions. Figures 5.11 to 5.14 show the calculated performance parameters for the
HZ-20, and figure 5.15 represents the variation of the performance parameters with temperature at
matched load. It can be immediately seen that the results agree very well with the data from the Module
Performance Calculator [19] once the phenomena described in 5.2.1 are considered in the analysis.
In fact, the simplified linear model results match with the results obtained through a non-linear anal-
ysis with the homotopy perturbation method and finite volume method for a temperature difference of
100oC which can be useful to estimate the performance parameters at low ∆T , since the computations
are very simplified. As shown by figure 5.5, properties variation with temperature are not significant,
hence the evaluation of the thermoelectric properties at average temperature T is reasonable.
However when ∆T is raised to 200oC, the non-linearity of materials properties is even more evident.
Thermoelectric properties exhibit higher variations with increasing temperatures as represented in figure
5.5 and so the simplified linear model deviates further from the homotopy perturbation method and the
finite volume method prediction and is no longer a viable solution to predict module performance. Still,
both the HPM and the FVM present a very good correspondence with the results obtained from the
Module Performance Calculator [19], but in terms of efficiency prediction small differences can be seen
from figures 5.9 and 5.14. Inspecting the power and efficiency curves for a ∆T of 200oC, it is possible
to observe that the homotopy perturbation method gives slightly lower efficiency for lower values of I.
For ∆T = 300oC the differences between the models used and the Module Performance Calculator
are less subtle. The deviation that the SLM presents with respect to the HPM and the FVM is larger
since the non-linearities with temperature are even more pronounced as figure 5.5 shows. Voltage and
power prediction don’t agree as well at higher values of I as represented in figures 5.7 and 5.12, and
in figures 5.8 and 5.13, respectively, and the non-linear models underestimate the performance of the
module. Regarding the HPM, efficiency prediction at low values of I is even lower. This might be due to
the fact that the model used is working with a first-order approximation of equation (3.31).
However, some considerations regarding the Module Performance Calculator [19] must be made.
The Module Performance Calculator uses a discretised form of equation (2.2) to evaluate the Seebeck
potential with 20 points, for every ∆T specified by the user. As ∆T increases so does the non-linearity
of materials properties and so the number of points to discretise the equation should be larger in or-
der to increase accuracy, which could explain why the voltage and power curves agree better with the
proposed non-linear models for lower values of ∆T . Nonetheless, the major difference lies in efficiency
43
Figure 5.5: Thermoelectric properties as a function of temperature [3, 19].
44
prediction. The Module Performance Calculator calculates the heat going in the module with average
parameters, meaning the Peltier heat and the heat by conduction are evaluated as in the SLM. Obviously
this assumption is not reasonable when ∆T is 300oC or larger, which might explain the differences in
figures 5.9 and 5.14.
The voltage curves obtained can be explained by looking at figure 2.7 and equation (2.19). The
voltage drop in the load starts with the maximum possible voltage in the circuit which is the open-circuit
voltage (Seebeck voltage). As the electric current intensity increases, electric power is generated at
the load and ohmic voltage drop occurs at the thermocouple. However, because the reversible effects
(Peltier and Thomson heat) are directly proportional to I while the irreversible effect (Joule heat) is pro-
portional to I2, there is an optimum value of I in which the module should operate, hence the parabolic
variation displayed by the power and the efficiency curves. While the increase in power generated is
higher than the ohmic voltage drop in the thermocouple, the output power and efficiency will increase
until a maximum value of I. If I increases any further, the ohmic voltage drop and, consequently, the
irreversible Joule heating overcomes the increase in produced power at the load and the heat losses in-
crease will be larger than the increase in retained heat at the junctions, resulting in a decrease in power
and efficiency.
5.2.3 Performance at matched load
A thermoelectric module should always be operating at either maximum power or maximum efficiency.
As seen previously with the results obtained for the HZ-14 and HZ-20, the difference in efficiency when
operating at maximum power or maximum efficiency is very small, so usually it is preferable to work
at maximum power. When a thermoelectric module is operating at maximum power it is working at
matched load conditions. This means that, for a TEM to produce the maximum power it is capable of,
its internal resistance must be equal to the load resistance, thus
RL = R (5.8)
Under matched load conditions, the output power, voltage and efficiency of both the HZ-14 and HZ-20
TEM were evaluated using the SLM and the HPM solutions and compared with the Module Performance
Calculator [19], as a function of the temperature difference between heat source and heat sink, ∆T .
Several values for Tc were considered. The plots obtained yield important results regarding not only the
differences between the applied models, but also how these parameters vary with different temperature
gradients in the system. The results for the HZ-14 are represented in figure 5.10, and figure 5.15 show
the results for the HZ-20.
Looking at figures 5.10 and 5.15 it is seen that both the SLM and the HPM agree very well with
the data from [19] until approximately a ∆T of 150oC for all the values of Tc considered, for power and
voltage prediction. For values higher than ∆T = 150oC, the SLM begins to overestimate the perfor-
mance parameters since the non-linearity of materials properties become stronger and so the results
exhibit differences when compared to the HPM and the Module Performance Calculator [19]. However,
45
Figure 5.6: Load curves for the HZ-14 thermoelectric module under different ∆T , for Tc = 50oC.
46
Figure 5.7: Voltage curves for the HZ-14 thermoelectric module under different ∆T , for Tc = 50oC.
47
Figure 5.8: Power curves for the HZ-14 thermoelectric module under different ∆T , for Tc = 50oC.
48
Figure 5.9: Efficiency curves for the HZ-14 thermoelectric module under different ∆T , for Tc = 50oC.
49
Figure 5.10: Performance curves for the HZ-14 TEM at matched load conditions, as a function of ∆T .
50
for matched load conditions, the HPM agrees well with the data from [19] with some not so significant
deviations when 250oC ≤ ∆T ≤ 300oC, due to the reasons mentioned in section 5.2.2
The efficiency at matched load is, as expected, where major differences are encountered between
the results. Up to a ∆T of 50oC there is a good match between the results obtained with the analytical
models and the results from [19]. For higher temperature differences, both models deviate further from
the Module Performance Calculator [19]. For high values of T , the thermal conductivity of the materials
increases significantly and the Seebeck coefficient begins to decrease slightly (figure 5.5). Introducing
an average κ and α to evaluate the heat rate absorbed by the module at high temperatures will introduce
a large error since the value differs very much from the evaluated temperatures at Th, as computed by
(3.26). Still, the absolute error between the HPM and [19] is not very large and so the HPM offers a
more accurate prediction.
The curves obtained for matched load operating conditions show that the modules produces more
electrial power and are more efficient for lower values of Tc under the same ∆T , which can be explained
by looking at the variation of the Seebeck coefficient and thermal conductivity from figure 5.5. For
Tc = 40oC, the properties obtained until the maximum ∆T = 300oC is achieved, result in higher Seebeck
coefficients and lower thermal conductivities for the n-type and p-type materials, meaning the Seebeck
voltage produced will be larger and the heat lost by conduction is less. Hence performance of the
module is better for lower values of Tc when comparing for the same value of ∆T . The bismuth telluride
thermoelectric properties deteriorate quickly for larger values of temperature, thus for applications with
larger values of temperature than those presented here, other materials should be considered.
5.2.4 Temperature and electric potential distributions
Since the operating condition of interest of the modules is at matched load, temperature and electric
potential distributions were evaluated for the value of I correspondent to Wmax for the temperatures
considered in section 5.2. The distributions were obtained for the HZ-14 thermoelectric module, with the
HZ-20 yielding similar results. The reason why the HZ-20 has increased performance is because the
number of thermocouples n in the module is larger (table 2.2).
The Seebeck potential, ohmic potential, temperature and normalized temperature distributions are
presented in figures 5.16, 5.17, 5.18 and 5.19, respectively. A side-by-side comparison between the
distributions obtained assuming constant and non-constant thermoelectric properties is presented for
each distribution. The Seebeck potential distribution was evaluated for I = 0, i.e. open-circuit. The FVM
solution is presented in both cases.
For Th = 150oC and Tc = 50oC there are no significant differences between the distributions obtained
for the Seebeck potential in figure 5.16, since α does not vary much as seen in figure 5.5. Nevertheless,
the presence of the Thomson effect can be observed in the p-type material, where the temperature
gradient is higher than the n-type material. In figure 5.5 it can be seen that the Seebeck coefficient
for the p-type unit is decreasing from 150oC to 50oC, which means Thomson heat is being released
by the p-type unit. For the n-type material however, α is increasing, which means Thomson heat is
51
Figure 5.11: Load curves for the HZ-20 thermoelectric module under different ∆T , for Tc = 50oC.
52
Figure 5.12: Voltage curves for the HZ-20 thermoelectric module under different ∆T , for Tc = 50oC.
53
Figure 5.13: Power curves for the HZ-20 thermoelectric module under different ∆T , for Tc = 50oC.
54
Figure 5.14: Efficiency curves for the HZ-20 thermoelectric module under different ∆T , for Tc = 50oC.
55
Figure 5.15: Performance curves for the HZ-20 TEM at matched load conditions, as a function of ∆T .
56
being absorbed, leading to lower temperature gradients as seen in 5.18. The ohmic potential however
shows some difference in the distributions, since the gradient of ρ in the considered temperature range
is significant. The ohmic potential drop in the n-type leg is higher than the drop in the p-type leg since
the electrical resistivity is larger for this temperature range. The temperature distribution is also similar,
with the Thomson and Joule effect having little influence on the results. The normalized temperature
profile compares the initial approximation θ0 with the first-order approximation θ obtained through the
HPM, and for low ∆T the difference is very small.
When the hot-side temperature is increased to Th = 250oC, the Seebeck distribution for the p-type
leg exhibits a more non-linear behavior than the n-type leg, since its α variation with T is more significant.
Also the calculated Seebeck potential drop is slightly smaller. For the ohmic potential, the drop is higher
in the p-type when non-constant properties are taken into account, since ρp is now larger than ρn. For
constant properties, ρp = ρn, hence the similar distribution and ohmic potential drop for the p-type and
n-type. The temperature distribution for constant properties suffers no significant changes since no
Thomson effect is involved, but its effect is now stronger in the temperature gradient obtained for the p-
type and n-type for non-constant properties. Once again, the gradient in α is greater for the p-type, thus
the Thomson effect is more pronounced. By looking at figure 5.5, it is possible to see that the Seebeck
coefficient from the p-type is increasing when Th decreases from 250oC to ≈ 150oC, and is decreasing
for the n-type unit, which means the p-type is absorbing heat and the n-type is releasing heat due to
the Thomson effect. By decreasing the temperature further, the Seebeck coefficient of the p-type leg
starts to decrease and the Seebeck coefficient of the n-type leg starts to increase, which means that
now the p-type is releasing Thomson heat and the n-type is absorbing Thomson heat. This is the reason
the temperature gradient in the p-type unit decreases and then increases along the leg length. For the
n-type this effect, although reversed, is negligible. The normalized temperature profile shown in figure
5.19 reflects the effect of the Thomson heat on the HPM solution, where the difference between the
zeroth and first-order approximation for the p-type leg is more evident.
The non-linearities presented are even stronger for Th = 350oC and Tc = 50oC, thus increasing the
variation for each parameter along the leg length in the distributions. However, it is now clear that the
HPM slightly deviates from the FVM solution because the values obtained with the polynomials proposed
by Zhang [3] to evaluate α, κ and ρ differ significantly from the values obtained with the polynomials used
by Hi-Z Technology, Inc. (equations (2.30), (2.31) and (2.32)), as represented in figure 5.5.
The three models presented are useful to predict performance depending on the application. The
SLM offers a fast and efficient way to estimate performance at lower temperatures with simplified equa-
tions that are easy to implement. The non-linear approximated analytical solution based on the HPM, of-
fers a fast and straight-forward model to evaluate the performance for non-constant properties with more
advanced equations while offering reduced computation time compared to the FVM, with the disadvan-
tage that at high temperatures, power and efficiency can be poorly estimated for low current intensities.
The numerical solution obtained with the FVM presents consistent results for all values of I. However,
for large numerical grids, the computation time increases considerably, since the number of equations
to solve increases as well.
57
Figure 5.16: Seebeck potential distribution along the HZ-14 thermocouple legs.
58
Figure 5.17: Ohmic potential distribution along the HZ-14 thermocouple legs.
59
Figure 5.18: Temperature distribution at matched load along the HZ-14 thermocouple legs.
60
Figure 5.19: Non-dimensional temperature distribution at matched load along the HZ-14 thermocouplelegs.
61
62
Chapter 6
Conclusions
6.1 Achievements
The work presented in this master thesis consisted in the development of three different mathematical
models to evaluate the performance of two commeracially available thermoelectric modules from Hi-Z
Technology Inc., the HZ-14 and the HZ-20. The results obtained were validated with the data available
from the Module Performance Calculator [19] developed by Hi-Z. The first model used, the Simplified
Linear Model, assumed constant thermoelectric properties for the Bi2Te3 based alloy semiconductor
materials. The second model, a non-linear approximated analytical solution based on the homotopy
perturbation method, neglects the assumption of constant thermoelectric properties for the material and
instead considers that the properties depend on the temperature. The third model presents a numerical
solution based on the finite volume method, where the non-linear heat governing equation also assumes
non-constant material properties.
Initially, the analyses were done to the modules neglecting contact effects and the presence of the
eggcrate material, in order to test the models implemented. The initial results suggested that this as-
sumption overestimates the performance of both the HZ-14 and the HZ-20, which means that the pres-
ence of the copper conducting strips and the eggcrate has a negative impact on the performance of the
modules. However, once these effects were taken into account in the analyses, the models presented
in this dissertation were able to match with the curves obtained from [19].
The models proposed in this thesis were validated by comparing data from [19] with results obtained
from the SLM, HPM and FVM to evaluate the performance of the HZ-14 and HZ-20 thermoelectric
modules. It was shown that, up to a temperature difference of 100oC, the assumption of constant ther-
moelectric properties can provide a reasonable estimate of module performance, since the calculated
properties variation with temperature is reasonably small for this range of temperatures. However, the
thermoelectric modules studied yield poor performance results for such low temperature gradients, and
so it is convenient to work with larger temperature differences between the heat source and the heat
sink. By doing so, thermoelectric properties for the Bi2Te3 n-type and p-type semiconductors exhibit
stronger non-linear behaviour, which means the assumption of constant properties no longer holds and
63
the SLM does not predict module performance correctly.
For larger temperature gradients, the non-linear models applied yield more accurate results for per-
formance prediction. By looking at the performance curves obtained with both non-linear solutions, it is
clear that the non-linear approximated analytical solution based on the HPM and the non-linear numer-
ical solution based on the FVM agree very well. The results obtained for these two different methods
show good correspondence with the results from the Module Performance Calculator [19] for low and
for high temperature differences. Nevertheless, at sufficiently high temperature gradients, specifically for
∆T greater than 200oC, the results obtained with the non-linear models differ significantly from [19], es-
pecially when calculating efficiency. As mentioned in sections 5.2.2 and 5.2.3, the assumptions for which
the data from the Module Performance Calculator [19] is based on, explain the obtained differences. In
[19], the Seebeck voltage conservation equation is discretised in 20 points for every ∆T specified by
the user, and for sufficiently high values of ∆T , a discretisation with 20 points is not enough, hence the
better correspondence with the non-linear models at lower ∆T . For the case of the efficiency, where the
solutions obtained differ more from the Module Performance Calculator [19] at high temperature gradi-
ents, the difference is due to the fact that both the Peltier heat rate and the conduction heat rate going
into the system are evaluated at the average temperature T in [19]. From figure 5.5 it is shown that
the properties have higher gradients at high temperatures, and so the values of properties evaluated at
high hot-side temperatures (Th) are very different than the values for properties evaluated at the average
temperature.
The HPM does, however, present some advantages over the FVM. It is a very fast and efficient
method and easy to implement, but the poor accuracy at low values of I is a problem. On the other
hand, the FVM presents a high computational time especially when working with refined grids or multi-
dimensional problems, since the number of equations to solve will grow accordingly. However, for all
values of I obtained, the FVM provides consistent results.
The temperature and electric potential distributions obtained with the three mathematical models give
a better indication of the non-linear behavior of the materials properties. Even for low temperatures, there
are clear differences between the temperature distributions obtained. Since the Seebeck coefficient of
the p-type leg has a stronger dependence on temperature than the Seebeck coefficient of the n-type
leg, the Thomson effect is more evident in the p-type as represented in figure 5.18. This effect is always
neglected with the SLM, and at higher temperatures it strongly affects the temperature distributions. For
the Seebeck potential, the results obtained with the three methods are similar for low temperatures,
where the SLM gives a good estimation of the distribution. As expected, the p-type offers a positive
voltage drop (positive α) and the n-type shows a negative Seebeck voltage drop (negative α). At ∆T =
300oC, the non-linear distribution differs considerably from the linear distribution and even the voltage
drop is different. For the ohmic potential distribution, it is possible to see that the ohmic voltage drop is
similar for the three models in the studied temperature range, however, since the parameter ρ presents
large gradients with temperature, the difference in the distributions obtained are clear even for low ∆T .
The FVM and HPM solutions differ slightly for high temperatures because the values obtained by the
polynomials proposed by Zhang [3] to evaluate α, κ and ρ, deviate considerably from the values obtained
64
by the polynomials presented in [19].
In general, the results obtained in this thesis work were satisfactory. Three different detailed mathe-
matical models were studied for a one-dimensional analysis and can be applied to study other TEM. The
models presented in this dissertation can provide good estimations for temperature, electric potential,
output power, output voltage and efficiency of any TEM.
6.2 Future Work
Future works could be developed based on the models developed here. It would be interesting to esti-
mate the performance based on higher orders of approximation of the HPM and compare the results to
check if efficiency and power at lower values of I is more accurate. The one-dimensional approximation
could also be expanded to a three-dimensional approach, and see if the results obtained with the HPM
and the FVM also showed good correspondence. The performance obtained with a 3-D model could
also present an interesting comparison for the one-dimensional approximation. Convective and radia-
tive losses could be considered in the heat transport equation to see how much performance is affected.
This analysis could provide significant results to test the assumption of neglecting these losses in a Hi-Z
TEM due to the presence of the eggcrate material.
In most real life applications, the heat source temperature and the heat sink temperature can vary
with time and so a steady-state analysis does not apply. An interesting analysis of the TEM could be
carried out in transient regime and study the behavior of the module under this conditions. The module
needs time to stabilize when subjected to a certain temperature gradient, so varying the temperatures
affects the performance of the module, which means the module would not be always be working at the
desired point of operation. A transient analysis coupled with the non-linear models presented here could
provide an interesting time history regime of operation of the module based on a real life application.
Further studies could also be carried out considering the entire thermoelectric generator system,
with ceramic plates and heat exchangers coupled with the module. Naturally, since none of these were
considered in the analyses presented, other mathematical considerations would have to be applied in
order to modulate the plates and heat exchangers. This would provide additional contact effects, mainly
thermal contact resistances. A temperature difference would be created between the surface of the
module and the temperature in the heat exchanger, which means that the temperature at the surface of
the module is no longer the heat source temperature, thus indicating lower performance. A computa-
tional fluid dynamics analysis to the heat exchanger, assuming an exhaust gas flow is the heat source,
together with the non-linear heat transport equation model of the thermoelectric module would provide
a complete study of a TEG system and show how the performance of the module can be evaluated
for different conditions of the heat source and the heat sink. To build an even more accurate model,
this analysis could be performed in transient regime and see how the time-dependent flow would affect
the performance of the module. This could be applied to a real life application automotive application,
where the TEM is used to convert the waste heat from the exhaust gas to electricity and the exhaust
flow conditions depend on the engine regime of operation.
65
An interesting topic to also explore is the geometry of the module and the geometry of the coupled
heat exchanger. A parametric study of both could be carried out assuming non-constant properties and
search for the optimal geometry that provides best power and efficiency delivery.
66
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70
Appendix A
Technical Datasheets
A.1 HZ-14 Datasheet
71
14 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
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
Internal Resistance Th=250°C, Tc=50°C 0.15 0.16 0.17 Ω
T = 25°C 0.09 0.1 0.11 Ω
Current Th=250°C, Tc=50°C @matched load 8.0 9.0 10.0 Amps
Th=250°C, Tc=50°C @short circuit 16 18 20 Amps
Heat Flux Th=250°C, Tc=50°C @matched load 330 350 370 Watts
Th=250°C, Tc=50°C open circuit 200 210 220 Watts
Heat Flux Density Th=250°C, Tc=50°C @matched load 9 10 11 W/cm2
Mass 48 49 50 grams
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
Internal Resistance Th=250°C, Tc=50°C 0.24 0.25 0.26 Ω
T = 25°C 0.14 0.15 0.16 Ω
Current Th=250°C, Tc=50°C @matched load 9.0 9.5 10.0 Amps
Th=250°C, Tc=50°C @short circuit 18.0 19.0 20.0 Amps
Heat Flux Th=250°C, Tc=50°C @matched load 450 475 500 Watts
Th=250°C, Tc=50°C open circuit 310 325 340 Watts
Heat Flux Density Th=250°C, Tc=50°C @matched load 9 10 11 W/cm2
Mass 101 102 104 grams
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.