Numerical Simulation of a Thermoelectric Generator

André Nuno Figueira van der Kellen

Thesis to obtain the Master of Science Degree in

Mechanical Engineering

Supervisor: Prof. Pedro Jorge Martins Coelho

Examination Committee

Chairperson: Prof. Carlos Frederico Neves Bettencourt da SilvaSupervisor: Prof. Pedro Jorge Martins Coelho

Member of the Committee: Prof. Viriato Sérgio de Almeida Semião

October 2020

ii

Resumo

O fenomeno termoeletrico esta associado a conversao de calor em eletricidade e vice versa. Os instru-

mentos termoeletricos, com base no efeito de Seebeck, podem atuar como geradores, onde e produzida

potencia eletrica, ou como refrigeradores termoeletricos para remocao de calor.

Para avaliar como e que um sistema termoeletrico converte esta energia termica em energia eletrica,

sao utilizados modelos matematicos. Esta dissertacao apresenta tres modelos diferentes para estimar

o desempenho de um gerador termoeletrico: um modelo analıtico assumindo propriedades indepen-

dentes da temperatura e dois modelos em que as propriedades nao sao constantes, um analıtico e um

numerico.

Os resultados foram obtidos para dois modulos termoeletricos fabricados pela Hi-Z Technology, Inc.,

HZ-14 e HZ-20, e comparados com os dados de desempenho disponibilizados pelo Module Perfor-

mance Calculator. Com base no desempenho previsto pelos tres modelos descritos nesta dissertacao,

e importante considerar a influencia da temperatura nas propriedades dos materiais quando se analisa

o desempenho, como indicam os resultados. A hipotese de que as propriedades dos materiais sao con-

stantes, e razoavel para baixas temperaturas de operacao, no entanto, a temperaturas elevadas esta

premissa causara uma sobrevalorizacao do desempenho. Tanto as solucoes analıtica como numerica

do modelo nao-linear usado, revelaram uma boa correspondencia entre si e com os resultados obtidos

pelo Module Performance Calculator. A avaliacao do desempenho do modulo termoeletrico necessita

de considerar a variacao das propriedades com a temperatura, de maneira a obter resultados mais

precisos.

Palavras-chave: Termoeletricidade, gerador termoeletrico, efeito de Seebeck, recuperacao

de calor, energia termica

iii

iv

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

v

vi

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

vii

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

viii

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

ix

x

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

xii

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.

xiii

Q Heat rate (W).

W Electrical power (W).

Subscripts

A Wire A.

B Wire B.

C Carnot.

c Cold-side.

E Eastern nodal point.

e East face.

e West face.

egg Eggcrate material.

h Hot-side.

i, j Computational indexes.

L Load.

mc Maximum conversion efficiency.

mod Module.

n N-type unit.

oc Open-circuit.

Ohm Ohmic voltage.

P Local nodal point.

p P-type unit.

Sbk Seebeck voltage.

W Western nodal point.

xiv

Chapter 1

Introduction

Thermoelectrics is defined as the science and technology associated with the thermoelectric generation

and refrigeration [1]. This allows for the conversion of thermal energy into electrical energy or vice versa.

Thermoelectric devices have no moving parts and require no maintenance thus making them suitable for

a broad range of applications such as waste heat recovery from power plants and automotive vehicles

but also refrigeration and temperature control in electronic packages and medical instruments. It is very

important to understand the fundamental concepts of thermoelectricity in order to properly develop and

design such devices [2].

Chapter 1 begins with a motivation of the topic at hand in section 1.1, followed by an overview of

the topic and references of scientific work and research on thermoelectrics in section 1.2. Section

1.3 explicitly states the objectives set to be achieved with the study of the thermoelectric generator

considered. Finally, a brief outline of this thesis is presented in section 1.4.

1.1 Motivation

Fossil fuels are the main source of energy provided to all of the traditional technologies used currently

around the world. Whether it is in automobiles, industry, or other mankind’s activities, the resultant

emissions of the combustion of this type of fuels have a clear impact on the environment, and a very large

amount of this energy is wasted through the atmosphere. As such, the need to reduce wasted energy

and the environmental impact have been an increasing concern, thus leading to research for other

energy-producing alternatives. Directly converting this energy into electricity through a thermoelectric

generator (TEG) is considered an attractive solution to this problem [3].

Thermoelectric technology received considerable attention for the waste heat recovery in energy

conversion devices like internal combustion engines (ICE). There is plenty of scope for improvement

in this regard since only a third of the amount of fuel burnt in a conventional ICE is used to provide

mechanical power, the rest is wasted heat. The recovery of such heat and conversion to electricity

may be used for propulsion and to power the vehicle’s electrical components such as air conditioning,

lights, etc. Overall, by reducing the load on the alternator, the fuel efficiency of the system is improved

1

[4]. Coupling this factor with the TEG advantages, such as durability and quiet operation, no waste

production, and reliable power production in remote areas, make this alternative increasingly demanded

[3, 5, 6]. However, the automotive industry is just one example where a thermoelectric system can be

valuable. These devices may also be used in spacecraft for energy generation, recapture energy from

hot effluents of powerplant smokestacks, and harvest heat generated by photovoltaic cells. Although

these advantages should cause renewed interest in thermoelectric power, the coupling of heat and

electricity is weak since a great deal of thermal energy is required to generate a small amount of electrical

energy.

Recently, the development of nanotechnology has been critical in overcoming this technological chal-

lenge. New thermoelectric materials have been manufactured (most of them in laboratories) with higher

figures of merit (ZT ) and then used in R&D programs to convert waste heat into electricity in automotive

vehicles exhausts [5].

To predict and optimize the performance of thermoelectric systems, a correct mathematical model

for the analysis accompanied by a deep understanding of the heat and electrical current transfer phe-

nomena and the selection of the right materials according to the temperature range of operation is

indispensable.

1.2 State of the Art

In 1823, Thomas J. Seebeck reported results of experiments where a compass needle suffered a de-

flection if placed in a circuit of two dissimilar conductors when one of the junctions was heated. Seebeck

investigated this phenomenon in many other materials and arranged them in order of the product ασ,

where α is the Seebeck coefficient, expressed in volts per degree (V/K), and σ is the electrical conduc-

tivity [7]. As a result of his experiments, the first thermoelectric effect had been discovered.

The second thermoelectric effect was discovered in 1834 by Jean Peltier. Peltier observed the re-

verse effect when he noticed temperature changes on the thermocouple depending on the direction of

the current flow. Although both effects were proved to exist, they were very difficult to measure as a

property of the materials used [2, 7]. After the discoveries of both Seebeck and Peltier, slow progress

was made in the research of thermoelectric phenomena, and in 1850 interest in the topic was once

again due to the development of thermodynamics.

In 1851, William Thomson established a relationship between the Seebeck and the Peltier effect

which is the third thermoelectric effect, called the Thomson effect. Thomson discovered that in the pres-

ence of a temperature gradient between any two points of a conductor with current flowing, heat is ab-

sorbed or released, depending on the direction of the current and the conductor material [2]. Thomson’s

work also related the three thermoelectric effects thermodynamically, leading to important relationships

called the Kelvin Relationships.

Soon after these discoveries, the generation of electricity based on the thermoelectric phenomena

was considered. Edmund Altenkirch showed in the early 20th century that good thermoelectric materials

should have a high Seebeck coefficient to retain the heat at the junctions of the conductors and a low

2

electrical resistance to minimize the Joule heating. The non-dimensional figure of merit (ZT ) relates

these favorable properties [7]. Most materials researched at the time were mainly metal and metal

alloys, in which the ratio of thermal conductivity to electrical conductivity is a constant, therefore it is not

possible to adjust one parameter without affecting the other. The majority of metals exhibit small values

of α, resulting in low values of ZT . For many years ZT was limited to less than 1 and for many practical

applications, this value needs to be at least close to 2 [4, 7].

During the late 1930s, semiconductors started to be considered as alternatives to metals due to their

high Seebeck coefficient, and, with potential military applications in mind, the technology of thermo-

electricity began during World War II when the Soviet Union produced a 2-4 watt TEG. Major advances

in semiconductor technology and thermoelectric theory originated further development and research

in thermoelectric applications in the 1950s and 1960s, with large companies actively engaged in ther-

moelectric research including Whirlpool, Westinghouse, Bell Telephone, GE, Carrier, and others [1].

Recently, NASA reported conversion efficiencies of up to 15% for large temperature gradients. If similar

values of efficiency could be reproduced in smaller temperature gradients, like in automobiles exhausts,

for example, capturing about 5-10% of a vehicle’s waste heat could lead to a 3-6% reduction in fuel

consumption, which would be significant for both cost and emissions savings [4].

A typical modern thermoelectric module consists of several n-type and p-type semiconductors, form-

ing a thermocouple, connected electrically in series between two electrical conductors. The conductors

on the other hand are protected by an electrical insulator, usually ceramic plates, as figure 1.1 shows.

In p-type semiconductors the absence of electrons creates a positive charge and in the n-type semi-

conductors excess of electrons create a negative charge, thus creating a flow of electrons across the

junctions of the thermocouple. Provided a temperature difference is maintained across the module, the

device operates as a TEG and supplies electrical power to an external load, while if the electric current

passes through the module instead, heat is absorbed in one side and rejected at the other, acting as a

thermoelectric cooler (TEC) [7].

The increasingly growing interest of the scientific community in this technology led to several studies

to evaluate and predict the performance of such devices, using both analytical and numerical models

for the evaluation of the energy and electric potential transport equations. Fraisse et al. [8] compared

four different one-dimensional steady-state models to analyze the coefficient of performance (COP)

and efficiency as well as voltage and thermal/electrical power in a bismuth telluride Bi2Te3 material.

The standard simplified model introduced is based on an overall thermal balance to the thermocouple

considering constant thermoelectric properties estimated at the mean temperature T of the hot and

cold sides, Th and Tc, respectively. The second model, the improved simplified model, assumes a

non-constant Seebeck coefficient, thus taking into account the Thomson effect considering a constant

Thomson coefficient. The third model is based on a local energy balance and the temperature and heat

flux distribution along the thermoelement are derived assuming constant leg section and thermoelec-

tric coefficients. The fourth and final model evaluated is the electrical analogy model presented in [9].

They concluded that there were no significant differences between the simplified and improved models

in TEC and thermoelectric heater (TEH) modes, however, maximum efficiency was overestimated in the

3

Figure 1.1: A typical thermoelectric module [2].

simplified linear model when operating in TEG mode. Slightly more accuracy was observed when con-

sidering the Thomson effect in the calculations if the Seebeck coefficient is strongly thermal dependent.

It was also shown that the electrical analogy model agrees very well with the finite element model (FEM)

simulations performed.

Zhang [3] studied the effect of material temperature dependence when evaluating the performance

of a Hi-Z thermoelectric module [10] and solved the non-linear heat transport equation based on the

homotopy perturbation method as described in [5] and [11]. In both works, the homotopy perturba-

tion method solution is compared with other linear analytical solutions as well as the electrical analogy

method. The non-linear analytical solution provided consistent results with the electrical analogy method

and estimations of temperature variation along the thermoelement leg, absorbed power at the cold-end

when in TEC mode, and power output at the hot-end when in TEG mode. Marchenko [12] presented

a non-linear analytical solution for the heat transport equation based on the perturbation method, com-

paring this non-linear solution with five other proposed methods. Marchenko’s research showed that the

accepted accuracy for real-world applications is achieved with the quadratic approximation of the pertur-

bation method, requiring less computation than a traditional numerical integration of the differential heat

balance equation.

To account for irreversibilities in the process is also significant when modeling a thermoelectric device

as these can strongly influence power output. Shen et al. [13] studied thermal losses in two commercially

available TEG based on the side surface heat transfer effect, which represents the heat losses across the

side surfaces of each thermoelement leg due to convection in the air gaps between each thermocouple.

Temperature distributions for the n-type and p-type legs and output power and efficiency under different

values for the convective heat transfer coefficient h of air were obtained. As h increases the degree

of non-linearity in the temperature distributions along the n-type and p-type legs increases and the

4

temperature difference between the hot and cold junctions decreases, leading to reduced efficiency of

the device, while the existence of convection can either increase or reduce power output depending on

the leg length and material volume. A similar analysis was performed by H. Lee et al. [14] where the

model evaluated considered radiative losses and interfacial resistances inside the device besides the

convective losses and then compared to FEM results, showing the reduction in efficiency due to these

effects. The effect of the leg geometry in the performance of the device was also studied and it was

shown that the increase in leg spacing reduces the thermal resistance and increases heat flow and

power, but decreases efficiency. Kim [15] presented a study with internal thermal and electric interfacial

contact resistances modeled while also varying the n-type and p-type leg length, concluding that this

interfacial resistance cannot be neglected when the n-type and p-type units are sufficiently short.

Niu et al. [6] further studied the effect of leg geometry in the temperature gradient while developing

two three-dimensional numerical models based on different formulations and boundary conditions to

analyze heat and electricity transfer. It was shown that one model was more precise for power output

prediction and suitable for simulations with defined output voltage or current and the influence of the

leg cross-sectional area could also lead to significant improvement in power output. Also, smaller n-type

and p-type units could enhance efficiency and power density. Another three-dimensional model was also

studied by Bjørk et al. [16], taking into account radiative, convective, and conductive heat losses with

very detailed modeling. A 3-D finite element model developed by Liao et al. [17] was compared to results

obtained experimentally for the TEG1-127-1.4-1.6 TEG module [18], showing a maximum deviation of

6%, hence showing the accuracy of the models presently used to predict performance parameters for

either TEG or TEC modules.

Thermoelectric module performance evaluation can be done through a variety of models depending

on the type of application that is expected. Although several implemented models have simplifications

to make the computations easier to perform, a detailed evaluation of such a device needs to consider

non-linearities and irreversibilities to predict power or efficiency correctly, especially if working under

high-temperature differences between the heat source and heat sink.

1.3 Objectives

The objective of this master thesis is to develop a model able to calculate the performance of the com-

mercially available modules HZ-14 and HZ-20 by Hi-Z Technology, Inc. by comparing it with the available

data provided by the Module Performance Calculator [19], resorting to both analytical and numerical

methods.

From the reviewed literature, none of the works mentioned have attempted to evaluate Hi-Z ther-

moelectric modules by justifying the available data in [19] with a comprehensive numerical or analytical

model. The current thesis work attempts to provide that while also doing its analysis of the modules’

performance with the proposed methods.

Both HZ-14 and HZ-20 will be evaluated for different operating conditions and different temperature

differences and the results will be analyzed together with the data from [19].

5

1.4 Thesis Outline

The present master thesis work is divided from Chapter 1 to Chapter 6.

Chapter 1, the current chapter, introduces some motivation and context to the work developed in this

thesis. Also, a few examples of previous studies in the area of thermoelectricity are presented as part

of the state of the art. The objectives of the present master’s thesis work are defined in this chapter as

well.

Chapter 2 introduces the theoretical background needed to understand the fundamentals of the ther-

moelectric phenomena and to develop the necessary analytical and numerical tools to predict module

performance.

In Chapter 3 the analytical models used will be explained in detail followed by the numerical model

in Chapter 4.

Chapter 5 lists the obtained results using the models introduced in Chapter 3 and 4 together with the

data from the Module Performance Calculator. Here a discussion of the results is also presented.

Finally in Chapter 6 an overview of the work developed with this thesis and some conclusions on the

achievements are stated along with some suggestions for future work.

6

Chapter 2

Concepts of Thermoelectricity

In this chapter the fundamental concepts regarding thermoelectricity are presented in order to under-

stand the models applied to evaluate the performance of the Hi-Z thermoelectric modules.

Starting with section 2.1, the three thermoelectric effects are presented with some mathematical

definitions. Section 2.2 describes a basic thermocouple configuration with some background definitions

concerning its materials. The performance parameters of a thermoelectric generator are presented

in section 2.3, and in the last section 2.4, a detailed explanation of a typical thermoelectric module

configuration is presented. In this section, material and geometric data used in the Hi-Z modules studied

are also referenced.

2.1 Thermoelectric Effects

Thermoelectricity deals with the direct conversion of heat into electricity employing the three thermo-

electric effects that manifest in the presence of a temperature difference across the surface of the ther-

moelectric module: the Seebeck effect, the Peltier effect, and the Thomson effect [20]. The interrela-

tionships between these three effects, the Kelvin Relationships, are of extreme important as they gather

together the three thermoelectric effects to get a unique and consistent description of thermoelectric

phenomena [21].

2.1.1 Seebeck Effect

In a thermoelectric material, when a temperature difference is applied across a conductor, the hot region

produces more free electrons and natural diffusion of these electrons occur from the hot region to the

cold region, as show in figure 2.1. The resultant electromotive force generates electric current flowing

against the temperature gradient. This is known as the Seebeck effect [2].

The concept of conversion of heat into electricity is even more evident when a thermocouple of two

dissimilar materials is subjected to a temperature difference. Provided this temperature difference is

maintained at the junctions of two wires joined at both ends, an electromotive force is produced, and

consequently electric current flows in a loop like figure 2.2 shows.

7

Figure 2.1: Electron concentration in a conductor [2].

Figure 2.2: Schematic of a basic thermocouple subjected to a temperature difference at both ends [2].

The electric field intensity vector ~E (V m−1) is related to the applied temperature gradient through the

Seebeck coefficient α (also called thermopower ), usually measured in µV/K. The sign of α is positive if

the electromotive force drives the electric current from the hot junction to the cold junction, and negative

if the current flows from the cold junction to the hot junction. ~E is defined as

~EA,B = αA,B∇TA,B (2.1)

where ~E represents the electric field intensity vector in wire A or wire B, αA,B and ∇TA,B represent the

Seebeck coefficient and the temperature gradient on wire A and wire B separately, respectively. Since~E = −∇V , the Seebeck potential VSbk (V) in each wire can be written as

∇VSbkA,B= −αA,B∇TA,B (2.2)

Depending on the sign of α, the Seebeck voltage will be negative or positive in each conductor. The

resultant voltage of the thermocouple is given by

VSbk = VSbkA − VSbkB (2.3)

The Seebeck potential represents the highest voltage possible in the circuit, which is equivalent to the

open-circuit voltage Voc. Although this potential difference is only a function of the hot-side temperature

(TH ) and cold-side temperature (TL), its distribution is indeed a function of the temperature distribution

along the conductors. This effect is not affected by either the Peltier or the Thomson effect, the latter

two thermal effects are present only when current flows in the circuit and are not voltages, whereas the

Seebeck effect exists if a temperature gradient is maintained whether current flows or not [7].

8

2.1.2 Peltier Effect

As an electric current flows across a junction between two wires of dissimilar materials, heat must be

continuously added or subtracted at the junction to keep its temperature constant as shown in figure 2.3.

This is known as the Peltier effect, which results of the change in the entropy of the electrical charge

carriers as they cross a junction. Hence, heat is either absorbed or released at the junctions and it is

proportional to the current flow. The heat QPeltier (W) in each wire is defined as

QPeltierA,B= πA,BI (2.4)

where πA,B (V) is the Peltier coefficient of each wire with positive or negative sign depending on the

direction of the electric current, and I represents the electric current intensity (A) across the junctions.

The Peltier coefficient π is the change in the reversible heat content at the junction of conductors A

and B when unit current flows across it in unit time [7]. Even though π can be expressed in volts, the

Peltier effect does not produce an electromotive force and it is a reversible process, which means that

the heating or cooling effect will produce electricity and, if electricity is supplied to the system instead of

a temperature difference, heating or cooling is produced with no energy being lost.

Figure 2.3: Representation of the Peltier and Thomson effects on a thermocouple [2].

2.1.3 Thomson Effect

The third thermoelectric effect is the Thomson effect and it represents the reversible change of heat

content within a conductor in a temperature gradient when an electric current passes through it [7]. As

seen on figure 2.3, heat is absorbed or released across the wire depending on the material and the

direction of the current. The Thomson heat per unit volume QThomson (W m−3) of each wire is related to

the temperature gradient by

QThomsonA,B= τA,B∇TA,B ~J (2.5)

where τA,B ,∇TA,B and ~J are the Thomson coefficient (V K−1), the temperature gradient and the current

density vector (A m−2) on wire A and wire B, respectively. The sign of τ is positive if heat is absorbed as

shown in wire A and negative if heat is released like in wire B. The Thomson coefficient is the reversible

change of the heat content within a conductor [7] and it is the only thermoelectric parameter directly

measured for individual materials.

9

Like the Peltier effect, the Thomson effect is also not a voltage and it is reversible between heat and

electricity. There is another form of heat that arises in the presence of an electric current flowing, called

the Joule heating, which is always irreversible.

Joule Effect

The Joule effect describes the process where the energy of an electric current is converted into heat

as it flows through a resistance. When current flows in a material with finite electrical conductivity,

electric energy is converted to heat through resistive losses in the material [22]. Although it is not

a thermoelectric effect, Joule heating is present in thermoelectric systems and so it is necessary to

consider it when assessing the performance of such systems since it represents an irreversible heat

loss. The volumetric rate at which heat QJoule (W m−3) is generated due to a flowing electric current is

defined as

QJoule = ρ‖ ~J‖2 (2.6)

where ρ is the material electrical resistivity (Ω m) and ~J the current density vector passing through the

material (A m−2). In the case of thermoelectric materials, this effect is usually less relevant than the

Peltier or Thomson effects due to the low resistivity of the thermoelectric semiconductors, but it is a

function of the dimensions of the material and becomes significant for high values of ~J .

2.1.4 Kelvin Relationships

The Kelvin (or Thomson) Relationships were developed by William Thomson in 1854 by applying the first

and second laws of thermodynamics assuming the reversible and irreversible processes in thermoelec-

tricity are separable [2]. This provided with interrelationships between the three thermoelectric effects

very important to understand the phenomena.

The first Kelvin relation describes the Peltier coefficient π as a function of the Seebeck coefficient α

and Thomson coefficient τ

dπ

dT= α+ τ (2.7)

The second Kelvin relation links the Peltier coefficient to the Seebeck coefficient by the following

equation

π = αT (2.8)

where T is the local temperature. Introducing equation (2.8) in equation (2.7), the Thomson coefficient

can now be defined as a function of α

τ = Tdα

dT(2.9)

10

Both of the Kelvin relations rely on fundamental principles of physics. The second of Kelvin’s rela-

tions is associated with a specific case of Onsager’s reciprocal relations, which is based on microscopic

reversibility. On the other hand, the first Kelvin relation, regarding the way heat evolves in a thermo-

electric system, is often used as a convenient mathematical expression (2.9) relating the Seebeck and

Thomson coefficients [21].

By combining equation (2.8) with equation (2.4), it is possible to write the Peltier heat as a function

of the Seebeck effect only

QPeltierA,B= αA,BTA,BI (2.10)

Combining equations (2.9) and (2.5), the Thomson heat volumetric rate can now be expressed as

QThomsonA,B= TA,B

dαA,BdT

∇TA,B ~J (2.11)

2.2 Thermocouple

A modern thermocouple typically consists of p-type and n-type semiconductor materials. A basic rep-

resentation of a thermocouple of two dissimilar thermoelements can be seen in figure 2.4, where L is

the thermoelement leg length; A is the cross-sectional area of the leg; α, ρ and κ are the Seebeck

coefficient, electrical conductivity and thermal conductivity (W m−1 K−1) of the leg, respectively; Th is

the hot-side temperature and Tc the cold side temperature. Subscripts p and n refer to the p-type and

n-type material, respectively.

Figure 2.4: A basic p-type and n-type thermocouple [2].

Semiconductors are materials that have electrical properties between those of a conductor and an

insulator. Since the atoms are very closely grouped in a pure semiconductor material, few free elec-

trons are present in their atomic structure, but electrons are still able to flow. To improve the electrical

conductivity of semiconductors, certain ”impurities”, called donor or acceptor atoms, can be added to

the intrinsic material through a process called doping. With the doping process, it is possible to con-

trol the amount of ”impurities” to produce more free electrons or holes hence creating p-type or n-type

semiconductor materials [23].

11

N-type semiconductors

In n-type semiconductors, the intrinsic material is doped with donor atoms that donate electrons to

the basic semiconductor material. When stimulated by an external source, the electrons freed from

the intrinsic material are quickly substituted by the donated electrons from the doping agent, but some

electrons remain free, resulting in a doped semiconductor that is negatively charged. Since there are

more donor atoms than acceptor atoms, an n-type semiconductor material has more electrons than

holes, therefore, creating a negative pole [23].

Figure 2.5: Example of a n-type semiconductor of silicon doped with antimony [23].

P-type semiconductors

P-type semiconductors on the other hand are doped with acceptor atoms, which instead of donating

electrons, give the pure semiconductor material excess of positively charged atoms that leave holes in

the crystalline structure due to the lack of electrons in the ”impurity”. Free electrons around the hole

will move in order to fill it, however this action will leave another hole where the free electron was and

so on, giving the impression that the holes are moving through the crystalline structure of the material.

This continuous ”acceptance” of electrons by the acceptor atoms leave the semiconductor with excess

of holes compared to free electrons, resulting in a positive pole [23].

Figure 2.6: Example of a p-type semiconductor of silicon doped with boron [23].

12

2.3 Thermoelectric Generator (TEG)

The basic component of a thermoelectric generator is a thermocouple like the one represented in figure

2.4. In a TEG the n-type and p-type semiconductors are connected electrically in series by a conducting

strip, the most common material used is copper. Thermal energy from the heat source (Qh) is ab-

sorbed in the hot-end and converted into electrical energy, while heat is rejected at the cold-end of the

thermocouple (Qc).

A TEG delivers electrical power if connected to a load. Figure 2.7 schematizes a TEG when con-

nected to a load and in open-circuit.

(a) Single TEG circuit (b) TEG open-circuit

Figure 2.7: Electrical circuit representation for a TEG connected to a load and in open-circuit [2].

The heat in the hot junction of the TEG unit is absorbed through the semi-conductors as Peltier heat

and conduction heat. Employing the Peltier effect and Fourier’s law of conduction, the heat flux ~qh (W

m−2) in the p-type and n-type units, respectively, is defined as

~qp = αpT ~J − κp∇T (2.12)

~qn = −αnT ~J − κn∇T (2.13)

where α and κ are the Seebeck coefficient and the thermal conductivity, respectively, T is the local

temperature, ~J is the current density vector and ∇ the gradient operator.

2.3.1 Performance Parameters

Output Voltage

If the TEG is delivering power to a connected load, the n-type and p-type units’ electrical resistivity will

produce a drop in potential when the operating current passes through them according to Ohm’s Law. A

13

potential drop will also occur at the load. The vector form of the Ohm’s Law is defined as

~Ep,n = ρp,n ~J , (2.14a)

∇VOhmp,n= −ρp,n ~J (2.14b)

The total potential drop in the thermocouple will be

VOhm = VOhmp+ VOhmn

(2.15)

Looking at figure 2.7 (b), the open-circuit voltage Voc can be written as a function of the voltage drop

in the thermocouple VOhm and across the load V due to Ohm’s law by the following relation

Voc = VOhm + V (2.16)

Since the Seebeck voltage VSbk is equivalent to Voc, equation (2.16) can be written in the form

V = Voc − VOhm , (2.17a)

V = VSbk − VOhm (2.17b)

Equation (2.17b) leads to an important relationship for the output voltage V . The maximum output

voltage is achieved in open-circuit when Vmax = VSbk and will decrease linearly with increasing values

of I until VSbk = VOhm.

Output Power

The power generated at the load can be calculated as a function of the output voltage as

W = V I (2.18)

where W is the electrical power supplied (W). Since the output voltage is related to the load resistance

RL (Ω) by Ohm’s law, a more common representation of the electrical power is defined as

W = (IRL)I , (2.19a)

W = I2RL (2.19b)

Equation (2.19b) is a useful relationship that allows the computation of the load resistance RL as a

function of the output power, or vice-versa.

14

Figure of Merit

The performance of a n-type or p-type thermoelectric material is represented by a parameter called

figure of merit Z (K−1) that is defined as

Zp,n =α2p,n

ρp,nκp,n(2.20)

where α is the Seebeck coefficient, ρ and κ are the electrical resistivity and thermal conductivity, respec-

tively, and the subscripts p and n denote the p-type and n-type materials. Inspecting equation (2.20),

one can conclude that, in order to achieve satisfactory values of Z, the selected thermoelectric material

should present high values of α while exhibiting low values of κ and ρ. This makes sense since the de-

sirable material should be able to retain the highest amount of reversible heat possible at the junctions

(hence a high value of the Seebeck coefficient and, consequently, of the Peltier coefficient) while keeping

the irreversible heat losses by conduction and Joule heating to a minimum. Usually, the dimensionless

figure of merit ZT is presented and often used as a characteristic of the material.

However, the thermoelectric properties used to calculate Z are not temperature independent, so the

figure of merit tends to vary significantly depending on the temperature gradient applied to the system.

Still, the impact of the figure of merit in the conversion efficiency can be evaluated by looking at a material

with constant properties in the entire range of operating temperatures.

Conversion Efficiency

The TEG conversion efficiency is given by

η =W

Qh(2.21)

where W is the electrical power produced by the system and Qh is the total heat rate (W) supplied to the

system by an external source. For any heat engine operating between a heat source and a heat sink,

the maximum theoretical efficiency possible is defined by the Carnot efficiency represented as:

ηC = 1− TcTh

(2.22)

where Th (K) is the heat source absolute temperature and Tc (K) is the heat sink absolute temperature.

The average absolute temperature is defined as

T =Th + Tc

2(2.23)

Lee [2] derived equation (2.24) to represent the maximum conversion efficiency achievable by a TEG

with temperature-independent properties in steady-state, as a function of the Carnot efficiency and the

dimensionless figure of merit ZT

ηmc = ηC(1 + ZT )

12

(1 + ZT )12 + Tc

Th

(2.24)

15

From inspection, it is possible to conclude that the maximum conversion efficiency tends to the value

of Carnot efficiency ηC as Z tends to infinity, hence the necessity of achieving high values for the figure

of merit in order to improve performance of the TEG system. Although this relationship does not apply to

an actual thermoelectric system with temperature-dependent materials, the importance of the parameter

Z is crucial to improve the performance.

2.4 Thermoelectric Modules (TEM)

A thermoelectric module is composed of several thermocouples connected electrically in series and

thermally in parallel to increase the output voltage of the module. Each thermocouple is also connected

through a copper conducting strip. The module, however, must be electrically isolated from the heat

source and the heat sink while also allowing high thermal conductivity to minimize the temperature

difference between the heat source/sink and the thermocouple surface. Usually, alumina ceramic plates

are used for this purpose. Figure 2.10 schematizes the basic configuration of a TEM.

Figure 2.8: Configuration of a single stage thermoelectric module [1].

The number of thermocouples in a TEM is defined by n. The performance parameters of a TEM can

be obtained simply by multiplying the parameters of a single TEG by the total number of couples as

Wmod = nW (2.25)

Vmod = nV (2.26)

Qhmod= nQh (2.27)

The parameters not influenced by the number of thermocouples in a module are the current intensity

and the conversion efficiency

Imod = I (2.28)

ηmod =nW

nQh= η (2.29)

16

Naturally, due to the presence of the conducting strips and ceramic plates, contact effects occur. A

good TEM design must include electrical and thermal contact resistances (not shown in figure 2.9) which

will negatively impact the output voltage produced and, consequently, the electrical power delivered to

the connected load [24].

(a) Thermoelectric power generation system [13]. (b) Thermal resistance network [13].

Figure 2.9: A typical TEG power system with a representation of the thermal resistances involved.

2.4.1 Hi-Z Thermoelectric Modules

The Hi-Z thermoelectric modules convert low grade, waste heat into electricity that is intended to target

the waste heat market. The modules provided by Hi-Z Technology, Inc. use bismuth telluride Bi2Te3

based alloys as thermoelectric materials, with high efficiency at most waste heat temperatures and high

strength to endure rugged applications [10, 25]. The TEG couples inside the modules are electrically and

thermally insulated by a special frame called an ”eggcrate” that fills the air gaps between thermocouples,

which is manufactured through injection molding to make the TEM less expansive to fabricate [26].

The TEM considered for the analysis present in this master thesis were the HZ-14 and the HZ-20.

The data available in [19] is in accordance with the datasheets in Appendices A.1 and A.2. Performance

evaluation by Hi-Z Technology, Inc. for these modules did not contemplate the presence of heat exchang-

ers and ceramic plates, and so the given temperatures in the data sheets and the Module Performance

Calculator [19] were assumed to be on the module surface.

To predict Hi-Z modules’ performance three main components must be taken into consideration:

thermocouple, copper strips and the ”eggcrate”. Results that will be shown in Chapter 5 considering

only the thermocouple in the analysis are compared with results obtained for the three main components

together.

17

Figure 2.10: HZ-14 TEM. The hot-side is on the left where the dots show the ”eggcrate” material andthe cold-side is on the right [25].

Bi2Te3 data

The bismuth telluride alloys used in the Hi-Z modules are strongly temperature-dependent and so to

correctly predict the performance this needs to be taken into account in the analysis. The Seebeck

coeffcient, thermal conductivity and electrical resistivity are represented by a fourth degree polynomial

as a function of temperature, respectively as

α(T ) = α1 + α2T + α3T2 + α4T

3 + α5T4 (2.30)

κ(T ) = κ1 + κ2T + κ3T2 + κ4T

3 + κ5T4 (2.31)

ρ(T ) = ρ1 + ρ2T + ρ3T2 + ρ4T

3 + ρ5T4 (2.32)

where each of the coefficients in each polynomial is given in the table below.

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

References

[1] D. M. Rowe. Thermoelectrics Handbook: Macro to Nano. CRC Press, 2006. ISBN: 0-8493-2264-2.

[2] H.S. Lee. Thermal Design: Heat Sinks, Thermoelectrics, Heat Pipes, Compact Heat Exchangers,

and Solar Cells. John Wiley & Sons, Inc., 2010. ISBN: 978-0-470-49662-6.

[3] T. Zhang. New thinking on modeling of thermoelectric devices. Applied Energy, 168:65–74, April

2016. https://doi.org/10.1016/j.apenergy.2016.01.057.

[4] T. Caillat and P. Ball. Thermoelectric heat recovery could boost auto fuel economy. MRS Bulletin,

38:446–447, June 2013. https://doi.org/10.1557/mrs.2013.138.

[5] T. Zhang. Effects of temperature-dependent material properties on temperature variation in

a thermoelement. Journal of Electronic Materials, 44(10):3612–3620, October 2015. DOI:

10.1007/s11664-015-3875-5.

[6] Z. Niu, S. Yu, H. Diao, Q. Li, K. Jiao, Q. Du, H. Tian, and G. Shu. Elucidating modeling aspects of

thermoelectric generator. International Journal of Heat and Mass Transfer, 85:12–32, June 2015.

https://doi.org/10.1016/j.ijheatmasstransfer.2015.01.107.

[7] D. M. Rowe. CRC Handbook of Thermoelectrics. CRC Press LLC, 1995. ISBN: 0-8493-0146-7.

[8] G. Fraisse, J. Ramouse, D. Sgorlon, and C. Goupil. Comparison of different modeling approaches

for thermoelectric elements. Energy Conversion and Management, 65:351–356, January 2013.

https://doi.org/10.1016/j.enconman.2012.08.022.

[9] G. Fraisse, M. Lazard, C. Goupil, and J. Y. Serrat. Study of a thermoelement’s behavior through a

modelling based on electrical analogy. International Journal of Heat and Mass Transfer, 53:3503–

3512, August 2010. https://doi.org/10.1016/j.ijheatmasstransfer.2010.04.011.

[10] Hi-z technology, inc. https://hi-z.com/. Accessed on: September 4th 2020.

[11] T. Zhang. Analytical solution of nonlinear thermoelectric heat transport equation using homotopy

perturbation method. Journal of Computational Intelligence and Electronic Systems, 4:59–68,

March 2015. https://doi.org/10.1166/jcies.2015.1115.

[12] O. V. Marchenko. Performance modeling of thermoelectric devices by perturba-

tion method. International Journal of Thermal Sciences, 129:334–342, July 2018.

https://doi.org/10.1016/j.ijthermalsci.2018.03.006.

67

[13] Z.-G. Shen, S.-Y. Wu, L. Xiao, and G. Yin. Theoretical modeling of thermoelectric generator with

particular emphasis on the effect of side surface heat transfer. Energy, 95:367–379, January 2016.

https://doi.org/10.1016/j.energy.2015.12.005.

[14] H. Lee, J. Sharp, D. Stokes, M. Pearson, and S. Priya. Modeling and analysis of the effect of

thermal losses on thermoelectric generator performance using effective properties. Applied Energy,

211:987–996, February 2018. https://doi.org/10.1016/j.apenergy.2017.11.096.

[15] C. N. Kim. Development of a numerical method for the performance analysis of thermoelectric

generators with thermal and electric contact resistance. Applied Thermal Engineering, 130:408–

417, February 2018. https://doi.org/10.1016/j.applthermaleng.2017.10.158.

[16] R. Bjørk, D. V. Christensen, D. Eriksen, and N. Pryds. Analysis of the internal heat losses in a

thermoelectric generator. International Journal of Thermal Sciences, 85:12–20, November 2014.

https://doi.org/10.1016/j.ijthermalsci.2014.06.003.

[17] M. Liao, Z. He, C. Jiang, X. Fan, Y. Li, and F. Qi. A three-dimensional model for thermoelectric

generator and the influence of peltier effect on the performance and heat transfer. Applied Thermal

Engineering, 133:493–500, March 2018. https://doi.org/10.1016/j.applthermaleng.2018.01.080.

[18] Teg1-127-1.4-1.6 peltier module. https://www.alibaba.com/product-detail/TEG1-127-1-4-1-

6 60764466453.html. Accessed on: September 4th 2020.

[19] Module performance calculator. https://hi-z.com/performance-calculator/. Accessed on: May 6th

2020.

[20] M. A. Karri. Modeling of an automotive exhaust thermoelectric generator. Master’s thesis, Clarkson

University, 2005.

[21] Y. Apertet and C. Goupil. On the fundamental aspect of the first kelvin’s relation in

thermoelectricity. International Journal of Thermal Sciences, 104:225–227, June 2016.

https://doi.org/10.1016/j.ijthermalsci.2016.01.009.

[22] Comsol, inc. https://www.comsol.com/multiphysics/the-joule-heating-effect. Accessed on: Septem-

ber 4th 2020.

[23] Electronic tutorials: Semiconductor basics. https://www.electronics-tutorials.ws/diode/diode 1.html.

Accessed on: September 5th 2020.

[24] B. V. K. Reddy, M. Barry, J. Li, and M. K. Chyu. Convective heat transfer and contact resistances

effects on performance of conventional and composite. Journal of Heat Transfer, 136:101401,

October 2014. https://doi.org/10.1115/1.4028021.

[25] F. A. Leavitt, N. B. Elsner, and J. C. Bass. Use, application and testing of hi-z thermoelectric

modules. http://www.everredtronics.com/files/useandapplications.pdf. Accessed on: September

6th 2020.

68

[26] A. S. Kusch, J. C. Bass, S. Ghamaty, and N. B. Elsner. Thermoelectric development

at hi-z technology. 20th International Conference on Thermoelectrics, February 2001.

https://doi.org/10.1109/ICT.2001.979922.

[27] F. P. Incropera, D. P. DeWitt, T. L. Bergman, and A. S. Lavine. Fundamentals of Heat and Mass

Transfer. John Wiley & Sons, Inc., 2007. ISBN: 978-0-471-45728-2.

[28] J.-H. He. Homotopy perturbation technique. Computer Methods in Applied Mechanics and Engi-

neering, 178:257–262, August 1999. https://doi.org/10.1016/S0045-7825(99)00018-3.

[29] S.-J. Liao. An approximate solution technique not depending on small parameters: A

special example. International Journal of Non-Linear Mechanics, 30:371–380, May 1995.

https://doi.org/10.1016/0020-7462(94)00054-E.

[30] Z. S. Tseng. Notes on second order ode - part 1. http://www.personal.psu.edu/sxt104/class/Math251/Notes-

2nd%20order%20ODE%20pt1.pdf. Accessed on: August 3rd 2020.

[31] H. K. Versteeg and W. Malalasekera. An Introduction to Computational Fluid Dynamics: The Finite

Volume Method. Prentice Hall, 2007. ISBN: 978-0-13-127498-3.

69

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, info@hi-z.com, 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, info@hi-z.com, 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, info@hi-z.com, 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, info@hi-z.com, 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.