efficient design for thermoelectric generators in high-concentrated beam-split photovoltaic hybrid...

Efficient Design for ThermoelectricGenerators in High-ConcentratedBeam-Split Photovoltaic Hybrid

Systems

Jónsson, A. M.

Kovács, Á.

Kvedaravicius, E.

Martín, D.

Rincón M. J.

Thermal Energy and Process Engineering

Title: Efficient Design for Thermoelectric Generators inHigh-Concentrated Beam-Split Photovoltaic Hybrid Systems

Semester: 7th, Spring semester 2017Project period: 08/09/2017 - 15/12/2017ECTS: 10Project group: INTRO-TEPE 764Supervisors: Alireza Rezaniakolaei, Thomas Condra

Atli Már Jónsson

Ágnes Kovács

Edgaras Kvedaravicius

Diego Martín Aguilera

Mario Javier Rincón Pérez

Synopsis:

This report reviews the design of ThermoelectricGenerators (TEGs) taking part in Beam-Split HighConcentrated Photovoltaic (HCPV) systems. HCPVsystems are a concept regarding solar energy productionwhich reaches higher efficiency and power generationvalues than the current Photovoltaic (PV) systems. TheHCPV receiver unit is mainly composed by a solarconcentrator, a beam-splitter, a CPV cell and a TEGwhich transforms the excess heat directly into electricity.The system utilises the radiation more efficiently due tothe application of a spectrally selective beam-splitter.To achieve high and desirable levels of energy productionthe temperature gradient throughout the TEG has tooperate inside a optimum range of values, hence theimportance of an effective cooling technology appliedon a heat sink is crucial.

The parametric model designed in this study has

shown that desirable and expected values were achieved.

High efficient Bi2Te3 has been selected as a TE

semiconductor and heat sink geometry parameters have

been optimised to dissipate maximum heat flux utilising

forced convection.

Copies: 3Pages, total: 82Appendices: A - D

By signing this document, each member of the group conforms that all partic-ipated in the project work and thereby all members are collectively liable forthe content of the report. Furthermore, all group members conform that thereport does not include plagiarism.

PREFACE

This report was composed by a group of five students from the 7th semester pursuing aM.Sc. in Thermal Energy and Process Engineering at Aalborg University. The projectaims to model an efficient design for thermoelectric generators in high-concentrated beam-split photovoltaic hybrid systems.

The objective of this project is the design and optimisation of TEG unit which formspart of a HCPV beam-split system. A parametric model is developed to be used in orderto simulate different design options for the unit and evaluate its efficiency. Furthermore,the developed MATLAB model is attached as an annex to this report for its electronicversion.

Reading guide

Source Citation

The report conforms to the scientific standard of citing the sources used throughout. Thereport follows the Harvard citation method, where sources in the text are referenced witha number the following way: [Surname, Year]. This citation refers to the bibliography atthe end of the report, where books are listed with author, year, title, edition and publisher.Websites are listed with author, title, date and URL. The bibliography is alphabeticallyordered.

Figures, tables and formulas

Figures, tables and equations are numbered according to chapter and order of appearancetherein. Abbreviations are used throughout the references as Fig. Sect. and Eq. referringto Figure, Section and Equation respectively. Tables and formulas in the report arenumbered under which chapter they belong and which number in the sequence of tables,figures and formulas they are in chapter. As an example, “Figure 5.2” will be found inChapter 5 and it will be the second figure of this chapter. Equations numbers appear inparentheses and shifted to the right side of the document.

Appendix

Appendix is named according to the letters of the alphabet and can be found in the back ofthe report. The appendices contains subsidiary information, both theoretical and practicaland also extra results that are relevant to different parts of the main report, and can beused to support the explanations given throughout the report.

Annex

The annex with the model is available on a separate file on the report’s electronic version.This file is referred as “HCPV-TEG Model” and contains all the calculations underlyingthe report’s contents.

I

Nomenclature

Abbreviations

Abbreviation ExplanationAAU Aalborg UniversityBS Beam-splitterCTE Coefficient of Thermal ExpansionCFD Computational Fluid DynamicsCPV Concentrated PhotovoltaicDC Direct currentHVAC Heating, Ventilation and Air ConditioningEMF Electromotive ForceFEM Finite Element MethodHCPV High Concentrated PhotovoltaicHS Heat SinkPV PhotovoltaicSSBS Selective Solar Beam-SplitSTEG Solar Thermoelectric GeneratorTE ThermoelectricTEG Thermoelectric GeneratorUN United Nations

Symbol listMany “standard" symbols in thermophysics have been in use for decades. This reportattempts to keep many of these common symbols, giving alternatives wherever appropriate,and try to stay consistent throughout it.

First derivative in time dot rSecond derivative in time double dot rVector angular symbol ~rUnit vector hat rMatrix bold r

The following list of symbols is alphabetical-lowercase, then uppercase; Arabic, thenGreek letters:

Symbol Explanation Unitc Speed of light kms−1

f Frequency s−1

fapp Apparent friction −g Gravity at Earth’s surface ms−2

h Convective heat transfer coefficient Wm−2K−1

h Plank constant m2kgs−1

III

Aalborg University

Symbol Explanation Unit

k Thermal conductivity Wm−1K−1

m Fin geometric factor −m Volumetric flow rate m3/s

np Number of TEGs connected in parallel −ns Number of TEGs connected in series −p General pressure kgm−1s−2

p∞ Ambient pressure kgm−1s−2

t Fin’s thickness m

u x -axis velocity ms−1

v y-axis velocity ms−1

x+ Hydrodynamic entry length m

zT Figure of merit for a type of semiconductor −A General surface area m2

Ac Fin’s tip area m2

Ac,b Cross-sectional area at the base of the fin m2

Ap Corrected fin profile m2

Bi Biot’s number −Dh Hydraulic diameter m

E Emissive power Wm−2

G Solar irradiation Wm−2

Gref Reflection Wm−2

Gr Grashof number −I Current intensity A

Is Solar radiation intensity Wsr−1

J Electric current density Am−2

J Radiosity Wm−2

Kc Coefficient of contraction −Ke Coefficient of expansion −L General longitude m

Lc Corrected fin length m

N Fin number −Nu Nusselt number −P General power W

P Fin’s tip perimeter m

PJ Heat generated by Joule effect W

PT Absorbed/produced heat by Thomson effect W

Ra Rayleigh number −R General electrical/thermal resistance Ω/WK−1

Rcond Electrical resistance of a conductor Ω

IV


Rt,b Fin thermal resistance due to convection at thebase

WK−1

Rt,f Total fin thermal resistance WK−1

S Fin pitch m

U∞ Air velocity m/s

Uch Air velocity in the heat sink channel m/s

Q General heat flux W

QP Heat through TEG concerning Peltier effect W

Qa Heat flux dissipated by the fin W

Qch Heat flux from heat sink channels by radiation W

Qconc Concentrated heat flux W

Qcond Conduction heat flux W

Qconv Convection heat flux W

Qe Irradiative heat loss W

Qr Heat flux by radiation W

Qs Heat flux carried by solar radiation W

T General temperature K

Tm Mean temperature K

Ts Temperature at the surface of the heat sink K

T∞ Ambient temperature K

V General voltage V

VOC Open circuit voltage V

X Concentration ratio −ZT Figure of merit of combined semiconductors −α Seebeck coefficient V K−1

α Thermal diffusivity m2s−1

α Absorptivity −αpn Relative seebeck coefficient of a thermocouple V K−1

β Volumetric thermal expansion K−1

βref Constant absolute efficiency reduction %K−1

ε Emission coefficient −ε Emissivity −εf Fin effectiveness −η General efficiency −θ Excess temperature K

θb Excess temperature at fin’s base K

λ Wavelength nm

ν Kinematic viscosity m2s−1

πP Peltier coefficient V

V

Aalborg University


ρ Electrical resistivity Ωm

ρ Density kgm−3

ρ Reflectivity −ρf Convective fluid density kgm−3

σ Electrical conductivity Sm−1

σ Stefan-Boltzmann constant Wm−2K−4

τ Thomson coefficient V K−1

∆P Pressure drop Pa

∆T Increment in temperature K

∆x Heat sink base thickness m

Commonly used subscripts

Symbol Explanationc Cold side of the TEGCPV Concentrated Photovoltaic cellh Hot side of the TEGhT Total hot sideHS Heat sinkn n-type semiconductorload Connected load to the systemp p-type semiconductorpn Absolute semiconductorTEG Thermoelectric generator

VI

CONTENTS

1 Introduction and motivation 11.1 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Project limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Problem Analysis 52.1 Photovoltaics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Beam-splitter technology . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Thermoelectrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 Seebeck effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.2 Peltier effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.3 Thomson effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.4 Joule effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.5 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3.1 Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3.2 Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3.3 Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3.4 Natural Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3.4.1 Natural Convection in Heat Sinks . . . . . . . . . . . . . . 192.3.4.2 Fin performance . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4 Thermoelectric Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3 Model Design 273.1 Beam-Splitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 CPV cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.3 TEG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.3.1 Electrical Equivalent . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.3.2 Thermal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.4 Heat sink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.4.1 Operation range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.4.2 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.4.3 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4 Results and Discussion 434.1 Power production and efficiency . . . . . . . . . . . . . . . . . . . . . . . . . 434.2 Economic viability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5 Conclusion 535.1 Further work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Bibliography 55

VII

Aalborg University Contents

A Heat transfer coefficient calculations

B Cooling power calculation

C Convection methods plots

D Installation prices plots

VIII

INTRODUCTION AND

MOTIVATION 1Energy crisis and environment deterioration, such as global warming and climate change,are two major problems for the 21st century. Investing in clean and renewable energysources can significantly reduce the emission of greenhouse gasses and reliance on fossilfuels.

World’s energy demand is growing due to population growth and technical advancementsMoreover, in recent years the significance of renewable energy resources became moreimportant, however these technologies are still far away from taking over the economicleadership [He et al., 2015; Remeli et al., 2015].

Current scientific predictions state that, at the current level of CO2 emissions, globaltemperature will raise more than 2015 UN Climate Change Conference of Paris limit of2°C [UN, 2015], leading to Arctic and Polar melting and to catastrophic sea level raising.The biggest CO2 emission contribution is cause due to energy production and consumptionhence, in order to avoid future hazards, current research and development in alternativeand renewable energy production has a vast importance in this matter.

Figure 1.1: Despite ups and downs from year to year, global average surface temperatureis rising. By the beginning of the 21st century, Earth’s temperature was roughly0.5 degrees Celsius above the long-term (1951–1980) average. (NASA figure adapted fromGoddard Institute for Space Studies Surface Temperature Analysis) [NASA, 2016]

With regard to renewable energy, solar and wind power are the main sources which aimto lead and be widespread the upcoming years. Solar energy among other renewables isa promising and freely avalable energy source. It is the most abundant renewable energysource, the Sun emits it at the rate of 3.8 · 1023kWh, and approximately 1.8 · 1014kWh

intercepted by Earth. Studies reviled, that the global energy demand can be satisfied by

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Aalborg University 1. Introduction and motivation

utilizing solar energy, since it is freely available with no cost, and not exhaustible. [Kannanand Vakeesan, 2016]

Solar energy can be utilized in many forms, including photovoltaic solar cells, solarpanels, solar power plants. Solar radiation can be transformed into electricity byphotovoltaic systems, the process is done through Photovoltaic (PV) cells manufacturedwith semiconductor materials. However, the main issue of this technology is the lowefficiency in energy conversion from power generation. That lower efficiencies makes thistechnologies not competitive compared to fossil fuels.

To solve this issue, several new concepts have been introduced in recent years. Oneof them is the application of High Concentrated Photovoltaics (HCPV). In order to gethigher power output and efficiency per unit area of PV cell, optical concentrators are usedto concentrate solar radiation on PV cells.

However, by the use of a concentrator, the temperature in the PV receiver increasessignificantly which results in a decrease of its efficiency. Recently, the use of SolarThermoelectric Generators (STEGs) has gained wide attention in the conversion of solarenergy. For STEGs, the solar irradiation is initially converted into thermal energy byusing a solar absorber, and then this thermal energy is converted into electricity usingThermoelectric Generators (TEGs) based on the Seebeck effect for this specific application[Zhu et al., 2015; Lamba and Kaushik, 2016].

The HCPV concept consists in a combination of a PV cell with a TEG into a highconcentrated radiation receiver. However, to achieve a greater electricity output on theTEG, a high temperature gradient is necessary. In order to maintain that difference intemperature a heat sink is required to dissipate the excess heat from the TEG.

1.1 Problem statement

Current HCPV systems have the PV cell and the TEG connected thermally inseries. Therefore, all the radiation is concentrated directly to the PV cell. As thesesystems are restricted to temperature limitations in order to achieve a better efficiency(around 373-423 K) this temperature restriction directly influences the heat flux and thevoltage output of the TEG.

2

1.1. Problem statement

Figure 1.2: Schematic of a HCPV-TEG in series

Lately, other design alternatives for this systems have been developed, nevertheless,they are in a very early development stage. One of the approaches is the separation ofthe spectrum of the incoming solar radiation by a spectral beam-splitter. The splitterseparates short and long wavelengths and directs them towards the PV cell and the TEGrespectively. In order to yield the maximum power of this system, the CPV cell and theTEG are placed in parallel to point each beam towards the respective element as shownin Fig. 1.3.

PV materials produce energy according to the specific band gap energy, which is morereactive for short wavelengths (approximately the visible light spectrum 400-700 nm). Inaddition, shorter wavelengths produce a significantly lower amount of heat compared tothe infrared spectrum.

On the other hand, the long wavelength beam is directed towards the TEG and hencea greater gradient of temperature is achieved as so does the heat flux compared to theHCPV and TEG in-series design.

This report is focused on the net power produced by this particular design of HCPV-TEGsystem, aiming to model the power output and efficiency of the whole system. Furthermore,the output in the TEG is fully defined by the gradient of temperature throughout it, thusthe design of a effective heat sink is a determining factor.

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Aalborg University 1. Introduction and motivation

Figure 1.3: Schematic of HCPV-TEG with double beam-split

In order to model the whole beam-split HCPV-TEG system, each component is studiedthoroughly, analysing the physical effects that drive energy production and evaluating itsefficiency. A power generation from TEG is directly related to temperature differenceor Seebeck effect. The main focus is concentrated on modeling the HCPV-TEG systemwith beam-splitter and the attached heat sink (Fig.1.3), and calculating the power output,efficiency, and the cost of generated electricity.

1.2 Project limitations

In this study, simplifications and assumptions have been taken into considerationconcerning the beam-split HCPV-TEG system combined with the heat sink.

• The report only concerns a standard solar radiation with constant intensity.• In the report an ideal beam-splitter and concentrator was taken into consideration,

without any absorption in the material.• For CPV material, silicon (currently the most accessible material) was chosen for the

system, without any major research on the alternatives.• The emissivity coefficient of the CPV cell was considered as a constant value of 0.9.• The model of the system does not include the thermal analysis of the ceramic plate

on the surface of the TEG.• In the calculations no contact thermal resistance was considered.• The heat loss through convection and radiation inside the TEG was neglected for

the thermal analysis.• For the TEG and heat sink a commercially available size was chosen, without further

optimisation study.• Concerning boundary conditions at the heat sink fins, convection heat transfer from

the tip of the fin is considered.• Concerning convection in the heat sink, a steady state, laminar flow, two dimensional

and incompressible fluid with constant properties was considered.• In the report air was used as an external fluid for convection in the heat sink, no

other mediums were studied.• In the report optimisation on the prices of the system was not carried out.• Opaque medium is used, so transmissivity is not considered as fraction of the

irradiation that is transmitted in medium.

4

PROBLEM ANALYSIS 2In this chapter the diverse physical phenomena governing the system are detailed likewisethe effects concerning solar energy transformation into electricity. The current systems

and prototypes to achieve that energy conversion are again defined. Furthermore, athorough study with regard to thermoelectric materials is performed.

2.1 Photovoltaics

The photovoltaic systems produce electric power by directly transforming the solar energyinto electricity. The operation of PV cells are based on the photovoltaic effect, whichis defined as the creation of an electromotive force by the absorption of light in aninhomogeneous solid.

The photovoltaic effect: when the semiconductor material absorbs the solar radiation,the incident photons generate electrons and holes. The minimum energy that thephoton must have to create an electron-hole pair is the band gap energy. Each type ofsemiconductor has a different characteristic band gap energy which allows the absorptionof light most efficiently at a certain wavelength. The semiconductor contains a p-typeand an n-type material. The combination of a pair of p-n materials forms a junctionwhich separates the generated electrons and holes and allows the minority carriers (holesin n-type material and electrons in p-type material) to reach the region where they arein a majority. Hence, electrons then proceed to the electrode and contribute to thephotogenerated current [Rothwarf and Böer, 1975].

Figure 2.1: Structure of a single-junction solar cell. [El Chaar et al., 2011]

The efficiency of PV cells depends on the light absorbing materials and the temperature

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Aalborg University 2. Problem Analysis

of the cell. Moreover, the efficiency is limited by thermal relaxation of carriers created byphotons with energies above the band-gap and transmission losses of photons with energiesbelow the band-gap. The high temperature has a negative effect on the PV cell efficiencydue to the fact, that at higher temperature there is a higher chance for the generatedelectron-hole pairs to recombine. Several types of semiconductor materials are used inPV cells, including silicon, cadmium telluride (CdTe), copper indium gallium diselenide(CIGS). Recently new methods and concepts have been offered to increase the powerconversion efficiency, including high concentrated photovoltaic systems, tandem cells, andnanomaterials.

Figure 2.2: PV technologies[Jelle et al., 2012]]

High concentrated photovoltaic systems are a new concept to utilize solar energy moreefficiently. In order to get higher power conversion efficiency and power output perunit area, optical concentrators are used, which increase the intensity of incident beamradiation. However, the main issue of the high concentrated PV systems is the significantincrease in temperature which results in a decrease in PV cell efficiency and degradationof the PV material [Lamba and Kaushik, 2016].

In order to achieve high energy conversion efficiency in the CPV cell, it is important tokeep the cell operating at low temperatures, below 100°C. The HCPV cells can utilize theshort wavelength fraction of the solar radiation most efficiently, and the long wavelengthfraction turns into heat.

Silicon is the most commonly used material in PV technology. Amorphous(uncrystallized) silicon is the most popular thin-film technology with cell efficiencies of5-7 %, and low manufacturing costs compared to the crystalline silicon PV cells. In thisstructure the silicon atoms are randomly located from each other, and this randomness in

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2.2. Thermoelectrics

the atom’s structure has a major effect on the electron’s properties of the material causinga higher band-gap energy than crystalline silicon. The band gap wavelength for amorphoussilicon PV cells is 800 nm. Crystalline silicon offers higher energy conversion efficiency,around 14-19%, due to the crystalline structure, while still using only a small amount ofmaterial. The band gap wavelength of the crystalline silicon cells is 1170 nm.

2.1.1 Beam-splitter technology

PV cells do not utilize the long wavelength of the solar spectrum since it surpasses theband-gap and mostly turns that energy into heat. A new concept to solve this problem isthe application of the spectrally selective beam splitter (SSBS), where a selective absorberis inserted between the concentrator and the CPV cell. The solar radiation is focusedusing either a lens or a reflector arrangement. The role of the beam-splitter is to transmit(reflect) those parts of the solar spectrum that can be converted most efficiently in theHCPV cell while the incident long-wavelength radiation is transmitted to the hot side ofa thermoelectric generator.

The beam-splitter is constructed using thin-film layers of SiO2 and SiN4 deposited onN-BK7 glass. The number of layers varies between 20 and 200 depending on the type ofthe used PV material. One advantage of this concept is the usage of conventional PVmaterials, such as amorphous (a-Si) or microcristalline silicon (mc-Si). Alternatively whenthe HCPV and the TEG are not physically separated, and the hot side of the TEG isthermally connected to the PV cell, the operating temperature is limited by the PV cell,which is around 100 °C. At this temperature the heat flux is not optimal for the TEG,and at higher temperatures the cells may experience short-term degradation, such as theefficiency loss, and long-term degradation due to excess temperatures.

2.2 Thermoelectrics

Thermoelectricity is the science which studies the direct conversion of thermal energy intoelectricity and vice-versa. The first of the thermoelectric effects was discovered in 1821 byT.J. Seebeck who showed that an electromotive force could be generated by producing atemperature difference in the junction of two different electrical conductors. Thirteen yearsafter this discovery, J. Peltier observed a second thermoelectric effect: a current througha thermocouple produces a small-scale heating or cooling effect (depending on the currentdirection).

Later on, in 1855 W. Thomson (who later became Lord Kelvin) recognised that thosetwo effects are dependent by a third effect known as the Thomson effect.

Despite the past research and development carried on in this matter, thermoelectricityhas been severe restricted to materials development. It was only in the 1950s thatthe introduction of semiconductors allowed a practical use of these effects, however theefficiency yielded was particularly low. It was not until recent years that significantadvances are being made, at least on a laboratory scale. It is expected that this new

7


technology will soon serve as an efficient electric generator, increasing power productionand adding more usability to waste heat yielded [Goldsmid, 2010].

2.2.1 Seebeck effect

The Seebeck Effect consists on a conversion of a temperature difference directly into avoltage between the hot and the cold end of a semiconductor. A change of the Fermienergy is caused by the temperature gradient at the hot end of the semiconductor,which produces an electromotive force (EMF). Hence, this electromotive force generatesa potential difference proportional to the temperature difference. The Seebeck effect isexplained in the following equation, where Vs is the voltage between the hot side andthe cold side of the semiconductor, Tc is the cold side temperature, Th is the hot sidetemperature and α is the absolute Seebeck coefficient [Wolfgang and Stefan, 2009].

VTEG = −∫ Th

Tc

α(T )dT = −α(Th − Tc) (2.1)

The absolute Seebeck coefficient, which has the dimension of V/K, is assumed to beconstant. In Table 2.1 some typical Seebeck coefficients of metals and semiconductors arelisted.

In order to make use of the Seebeck effect, two legs of p and n-type materials withdifferent Seebeck coefficients must be connected to a thermoelectric generator by meansof a metal layer (usually copper or aluminium). The temperature difference between thehot and cold end yields a voltage between the two conductors, when the contact voltagebetween the two legs is insignificant. This effect is explained by the following equation,where Vpn is the voltage between the semiconductors, Tc is the cold side temperature,Th is the hot side temperature, αn is the absolute Seebeck coefficient of the n-typesemiconductor, αp is the absolute Seebeck coefficient of the p-type semiconductor andαpn is the relative Seebeck coefficient of the thermocouple [Wolfgang and Stefan, 2009].

VTEG =

∫ Th

Tc

αp(T )dT −∫ Th

Tc

αn(T )dT = αpn(Th − Tc) (2.2)

8


Metals α (µV K−1) Semiconductors α (µV K−1)

Antimony 47 Se 900Nichrome 25 Te 500Molybdenum 10 Si 440Cadmium 7.5 Ge 300Tungsten 7.5 n-type Bi2Te3 -230Gold 6.5 p-type Bi2xSbxTe3 300Silver 6.5 p-type Sb2Te3 185Copper 6.5 PbTe -180Rhodium 6.0 Pb3Ge39Se58 1670Tantalum 4.5 Pb6Ge36Se58 1410Lead 4.0 Pb9Ge33Se58 -1360Aluminium 3.5 Pb13Ge29Se58 -1710Carbon 3.0 Pb15Ge37Se58 -1990Mercury 0.6 SnSb4Te7 25Platinum 0 SnBi4Te7 120Sodium -2.0 SnBi3Sb1Te7 151Potassium -9.0 SnBi2.5Sb1.5Te7 110Nickel -15 SnBi3Sb1Te7 90Bismuth -72 PbBi4Te7 -53

Table 2.1: Seebeck coeffient of different metals and semiconductors Lasance [2006]

2.2.2 Peltier effect

The Peltier effect consists of the conversion of a current flow into a temperature differencebetween two metals or two semiconductors. This effect is the reverse of the Seebeck effectand expressed with the following formula, where I is the current flowing through the circuit,QP is the amount of heat per time unit produced and πP is the Peltier coefficient.

I =QPπP

(2.3)

The Peltier effect is relevant when a thermoelectric module works as a TE heat pump,converting electric power into heat. In this configuration, an external power source suppliesan electrical excitation in the circuit, which leads to a heat transportation between thetwo sides of the thermocouple. However, in TE power generation this effect is consideredparasitic, because it increases the thermal conductivity in the semiconductors. Therefore,heat will be pumped from the hot side to the cold side, decreasing the temperature gradientand the electrical potential established across the device.

2.2.3 Thomson effect

The Thomson effect is defined as the generation or absorption of a reversible heat in acurrent-carrying homogeneous material when subjected to a temperature difference. Theheat absorbed or generated (PT ) is proportional to the temperature gradient across theconductor (∆T ) and the electrical current (I).

PT = τI∆T (2.4)

9


Where τ is the Thomson coefficient, with the dimension of V/K.This effect is usually neglected in the TE device modelling due to its minor influence

compared to the Joule effect. However, in the case of large temperature gradients„ it canhave a more significant influence in the performance of TE modules, and therefore it shouldnot be neglected.

2.2.4 Joule effect

The Joule effect is included in the heat balance of TE modules due to its correlationbetween heat and electrical energy. This effect consists of the generation of a heat (PJ)along the length of a conductor when an electric current (I ) is flowing through.

PJ = RcondI2 (2.5)

Where Rcond is the electrical resistance of the conductor.The Joule effect heating is independent of the direction of the current and thermody-

namically irreversible. Thus, it is always considered as a heat loss [Anamaria Man, 2016].

2.2.5 Efficiency

Recently, TE materials have showed low energy conversion efficiencies, highly limiting itsapplications. However, with the advent of new nano-manufacturing techniques, the interestin TE energy has increased in a wide range of study areas.

Figure 2.3: zT values vs T for diverse advanced TE materials [Baranowski et al., 2012]

The efficiency of TEGs is analysed using a local approach to generator efficiencywhich has been developed recently and greatly simplifies the analysis and optimisation[Snyder and Ursell, 2003]. The model states that efficiency of TEGs is governed by the

10


dimensionless figure of merit:

zT =α2T

kρ=α2σT

k(2.6)

Where α is the Seebeck coefficient, T is the absolute temperature and ρ, k and σ arerespectively the electrical resistivity and the thermal and electrical conductivity.

However, an actual TEG is constructed from two types of semiconductor materials,p-type and n-type. This different materials need to be taken into consideration for thecalculation of the whole device figure of merit (ZT ). For a good approximation of a p− ncouple the following formula is used [Anamaria Man, 2016]:

ZT =(αp − αn)2T[

(ρnkn)12 + (ρpkp)

12

]2 (2.7)

Figure 2.4: zT values vs k, σ and α [Snyder and Ursell, 2003]

Where ρp ρn are the electrical resistivity of the semiconductors. To achieve a highervoltage output, TE materials need to have a large power factor α2σ/k. As shown in Fig.2.4, the intercorrelated power factor parameters display a challenging control to accomplisha high ZT value. Moreover lower carrier concentration insulators and semiconductors aredefined by a large Seebeck coefficient, however they have low electrical conductivity sinceboth values are inversely proportional to each other.

Therefore, with regard to TE materials, the efficiency of a TEG is defined by thefollowing expression:

ηTEG =Th − TcTh

√1 + ZT − 1√

1 + ZT + Tc/Th(2.8)

11


2.3 Heat Transfer

2.3.1 Radiation

HCPVs based their energy production on radiation which can be present into diffuse anddirect states. On one hand, solar direct radiation (or sometimes called "beam radiation")is defined as solar radiation travelling in straight line down to the Earth’s surface. On theother hand, diffuse solar radiation is described as scattered direct radiation due to impactswith molecules and particles although it still reaches the Earth’s surface.

HCPVs are based on the use of optical devices that increase the light flux received onthe solar cell surface. These concentration level varies from 300 to 2000 suns.

Different kinds of concentrators are needed depending on the sky conditions, mainlysubdued by the amount of cloud cover. The portion of the sky that a concentrator canuse is inversely proportional to the concentration ratio. For instance, in cloudy regions flatand low-concentrator systems work in better conditions due to the scattered solar radiationwhereas in sunny regions, high-concentrator systems have better a performance [Unger,2009; Pérez-Higueras et al., 2011].

Figure 2.5: Radiation cycle on Earth’s surface [Bergman et al., 2011]

All matter that is greater in temperature than absolute zero emanates electromagneticwaves. Movements in the matter from either molecules or atoms, cause a transformationof thermal energy (generated by inter-atomic collisions) to electromagnetic energy. Thisprocess is generated when particles in the matter get charged and they create thermalmotion. That motion leads to both coupled electric and magnetic fields [Blundell andBlundell, 2009].

Electromagnetic radiation travels in the form of waves. The energy of the waves isproportional to their frequency, thus high frequency waves (short wavelengths) are moreenergetic than low frequency waves (long wavelengths) [Howell et al., 2010] as shown inFig. 2.6.

12

2.3. Heat Transfer

Figure 2.6: Solar energy distribution vs wavelength according to PV production [Biernatet al., 2013]

The main functions of radiative transfer are emission and absorption since they add orsubtract energy from matter. While emissivity is the energy in form of radiation that isbeing sent out of a body, absorptivity is the radiant energy captured by it. Furthermore,absorptivity in a body depends on factors concerning the emissivity and incident radiationquality [Howell et al., 2010].

Electromagnetic radiation does not need a medium to spread. When emanating fromthe surface, the thermal radiation depends on several properties according to Kirchhoff’slaw on thermal radiation [Howell et al., 2010], such as:

• Temperature difference• Spectral absorptivity of the surface• Spectral emissivity of the surface

At thermal equilibrium, matter can achieve the radiant characteristics of what is definedas a black body. A black body is a radiating body that at thermal equilibrium has perfectabsorptivity and emissivity for all wavelengths. However, when the emissivity of the bodyis less than unity and is not dependent on the wavelength matter behaves as a grey body.Planck’s law of black-body radiation can be applied to find the thermal radiation powerof a black body per unit area of radiating surface per unit of solid angle and per unitfrequency/wavelength (I) [Rybicki and Lightman, 2008].

I(f, T ) =2hf3

c21

ehfσT − 1

I(λ, T ) =β

λ51

ehcσTλ − 1

(2.9)

Where h is the Plank constant (6.625 · 10−34 J · s), c is the speed of light in vacuum(299 796 km/s), σ is the Stefan-Boltzmann constant (1.38 · 10−23 J/K), f is the elec-tromagnetic radiation frequency, T is the absolute temperature of the body, λ is thewavelength and constant β = 2hc).

13


For this analysis of radiation, the different heat fluxes taken place in a opaque mediumare studied.

Hence, in an opaque surface, the medium experiences no transmission. Where theemission and part of the irradiation are reflected from the opaque surface, as can be seenin Fig. 2.7 [Bergman et al., 2011].

Figure 2.7: How four distinct radiation fluxes react to opaque medium [Bergman et al.,2011]

To find the values on radiation fluxes E, G, Gref and J the following equations areapplied:

E = εσAT 4 (2.10)

α = 1− ρ (2.11)

J = E +Gref = E + ρG (2.12)

Where the emissive power E is the rate at which radiation is emitted from a surface perunit area, ε is the emissivity, that can measure how efficiently a surface emits energyrelative to a blackbody σ is the Stefan-Boltzmann constant, T is the absolute temperatureof the body, G is the irradiation rate at which radiation is incident upon a surface perunit area, J is defined as radiosity, which is the rate at which radiation leaves a surfaceper unit area, ρ is the reflectivity, the fraction of the irradiation that is reflected, α isthe absorptivity, the fraction of absorbed irradiation and the reflection Gref = ρG is theportion of the irradiation reflected from an opaque medium [Bergman et al., 2011].

2.3.2 Conduction

The driving force of any heat transfer is the temperature difference. A heat transfer byconduction is described by Fourier’s Law:

14

2.3. Heat Transfer

Figure 2.8: Conductionthrough a wall. [Cengeland Ghajar, 2015]

Qcond = −kAdTdx

(2.13)

It states that temperature distribution T across the wall islinear under steady conditions if assuming that area A andmaterial properties (thermal conductivity k) are constant.Same approach can be adopted in analyzing heat transfer fromTEG cold side to a heat sink, where two surfaces are in contact.

2.3.3 Convection

One of the most critical parameters regarding the system’s power output is the heat fluxthroughout the TEG. To obtain high temperature gradients and therefore a high heat flux,an optimum heat sink is design and attached to the cold side of the TEG.

Heat sink’s efficiency is dependent purely in geometrical parameters. Moreover, fre-quently fins are attached to the heat transfer surface in order to achieve the highestheat flux throughout the system. To obtain the heat transfer rate associated with fin’sconfigurations an unidimensional energy balance in a differential element is performed asshown in Fig. 2.9.

Figure 2.9: Energy balance of a fin’s differen-tial element [Bergman et al., 2011]

0 =∑

Q (2.14)

0 = Qx − Qx+dx − dQconv (2.15)

Applying Fourier’s law to fins of uniform cross-sectional area Eq. 2.15 the following formulais obtained:

d2T

dx2− hP

kAc(Ts − T∞) = 0 (2.16)

15


Where h is the heat transfer coefficient, κ is the thermal conductivity, P is the fin’s tipperimeter, Ac is the fin’s tip area, Ts is the temperature at the base of the fin and T∞ isthe infinity temperature of the medium. To simplify the previous equation’s parameter mand the excess temperature θ are defined as:

θ(x) = T (x)− Ts (2.17)

m2 =hP

kAc(2.18)

Hence Eq. 2.16 is reduced to:

d2θ

dx2−m2θ = 0 (2.19)

Eq. 2.19 is an homogeneous, linear, second order differential equation which generalsolution is of the form:

θ(x) = C1emx + C2e

−mx (2.20)

To evaluate both constants, it is necessary to establish appropriate boundary conditions.One at the base of the fin and other at the tip.

Four different physical situations might be taken into consideration with regard to thetip’s boundary condition: convection heat transfer from the tip fin, adiabatic tip, prescribedtemperature at the tip or infinite fin. To achieve the most accurate approximation in themodel, convection in the tip is considered. As a result, both boundary conditions yield thefollowing equations:

θ(x) = Tx − T∞ (2.21)

θ(0) = T0 − T∞ ≡ θb (2.22)

hθ(L) = −kAcdT

dx

∣∣∣∣x=L

(2.23)

As a result, the heat transfer is defined by Ac(x) and P (x) which are purely geometricalparameters which depend on the fins design. For a heat sink model with rectangular-constant section fins model, the following geometry is described:

16

2.3. Heat Transfer

Figure 2.10: Geometrical parameters of a rectangular uniform cross-sectional fin [Bergmanet al., 2011]

After some manipulation with the equations and boundary conditions the temperaturedistribution is defined as:

θ

θb=

cosh [m(L− x)] + hmk sinh [m(L− x)]

coshmL+ hmk sinhmL

(2.24)

While the dissipated power or dissipated heat flux by the fin is:

Qa = θb√hPkAc

sinhmL+ hmk coshmL

coshmL+ hmk sinhmL

(2.25)

This method makes the assumption that the average temperature has an approximatevalue to Ts therefore to accomplish highly approximate values Biot’s number in the finhas to be much less than unity, this number is defined as the ratio of the internal thermalresistance of a solid to the boundary layer thermal resistance.

Bi =ht

2k<< 1 (2.26)

2.3.4 Natural Convection

Natural convection is a heat transfer that is generated by fluid motion due to densitydifference in the fluid. As fluid density is extremely sensitive to temperature, it causeswarmer fluid rise upwards due to lower density. In other words this fluid motion is due tobuoyancy forces within the fluid.

Consider the heated vertical plate, which is immersed in quiescent fluid (fluid at rest),with surface temperature higher than fluids medium Ts > T∞, buoyancy forces induce anatural convection boundary layer that causes heated fluid rise. In Fig. 2.11 it is illustratedhow fluid velocity u is developing throughout boundary layer. At y = 0, due to no slipcondition velocity is 0, it also approaches zero when y →∞. [Bergman et al., 2011]

17


Figure 2.11: Velocity (u) and thermal (T ) boundary layer development on a heated verticalplate. [Bergman et al., 2011]

Assuming in Fig. 2.11 a steady state, laminar flow, two dimensional and incompressiblefluid with constant properties. In order to describe or predict velocity and temperaturefields, some equations must be satisfied: conservation of mass likewise energy and Newton’ssecond law of motion.

Using the assumptions previously described, x direction equation of motion is definedas:

u∂u

∂x+ v

∂u

∂y= −1

ρ

dp∞dx− g + ν

∂2u

∂y2(2.27)

By applying Boussinesq approximation and making simplifications in Eq. 2.27, differentx-momentum equation is obtained, where β represents volumetric thermal expansioncoefficient.

u∂u

∂x+ v

∂u

∂y= gβ(T − T∞) + ν

∂2u

∂y2(2.28)

The set of momentum, continuity and energy equations has to be solved simultaneouslyas it depends on temperature T .

For the isothermal vertical plates shown in Fig. 2.11, this set of equations is solved withthe following boundary conditions:

At y = 0 u (x,0) = 0, v (x,0) = 0, T (x,0) = Ts

At y →∞ u (x, ∞) → 0, v (x,∞) → 0, T (x,∞) → T∞

u∂u

∂x+ v

∂u

∂y= gβ(T − T∞) + ν

∂2u

∂y2(2.29)

∂u

∂x+∂v

∂y= 0 (2.30)

u∂T

∂x+ v

∂T

∂y= α

∂2T

∂y2(2.31)

18

2.3. Heat Transfer

Making energy and motion equations dimensionless, Grashof, Reynolds and Prandtlnumbers are introduced into these equations. Hence, empirical correlations for averageNusselt number are defined, which is further used to define the convection coefficient.

Nu =hLck

= C(GrLPr)n = CRanL (2.32)

Rayleigh number is defined as a product of Grashof and Prandlt numbers. The values of Cand n depend on geometry and flow. From this equation an average convection coefficientis defined and then applied to Newton’s law of cooling to find the heat transfer rate fromthe surface to the ambient.

Qconv = hA(Ts − T∞) (2.33)

2.3.4.1 Natural Convection in Heat Sinks

Bar-Cohen and Rohsenow have developed equations for the Nusselt number and optimumspacing based on experimental studies for the parallel vertical fins. Where either Ts or Qcis constant. In this project it is assumed that Qc is constant and therefore use followingequations for modelling heat sink under natural convection:

Ras =gβQc(S − t)4

kν2Pr

Sopt = 2.212

((S − t)4W

Ras

)0.2

+ t

Nu =h(S − t)

k=

[48

(Ras(S − t)/W )+

2.51

(Ras(S − t)/W )0.4

]−0.5

Figure 2.12: Finned Heat Sink

Where in Fig.2.12 fin thickness t, distance between fins S − t, fin pitch S, width is Wand length of fin is L.

To reach maximum heat transfer by natural convection in a heat sink with fixeddimensions, it is necessary to calculate optimum spacing between fins. Furthermore, as

19


previously was stated from these equations we can find convection coefficient and introduceit to Newtons cooling law to calculate rate of heat transfer.

Qc = h(2NLW )(Ts − T∞) (2.34)

Where N represents the number of fins in the heat sink. [Cengel and Ghajar,2015].Unfortunately, in this project optimization of heat sink for natural convection willnot be accomplished, but instead the results will be compared with the heat sink thats isoptimized for forced convection.

Thermal radiation is emitted from every body as long as the temperature is above0 Kelvin. It has a very strong effect on heat dissipation from heat sink under naturalconvection. In an article [Shabany, 2008] and master thesis of [Guitart Corominas, 2010]a simplified approach for radiative heat transfer calculations in heat sink proposed. Fromequation ??, it can be noticed that emissivity, area and temperature plays an importantrole in heat transfer by thermal radiation. Following equations from [Guitart Corominas,2010] describes thermal radiation in the U-channel heat sink:

Qr = (N − 1)Qch +Adσε(T4s − T 4

∞)

Qch =σ [(S − t) + 2L]W (T 4

s − T 4∞)

1−εε + 1

Fs−surr

Fs−surr = 1−

[2L(1 +W

2)0.5 − 1

]2LW + (1 +W

2)0.5 − 1

Ad = N(Wt+ 2Lt) + 2LW + ∆x(W + L)

W =W

S − tL =

L

S − tWhere Qr is total heat transfer by radiation, N number of fins in the heat sink, Qchheat transfer by radiation in the channel made by two adjacent fins, Ad is the radiativearea, Fs−surr is a view factor from the channel surface to the surroundings, W and L arenormalized fin width and length.

2.3.4.2 Fin performance

Fins are employed to increase the effective heat transfer area of a surface, however the finitself represent a conductive resistance thus there is not guarantee that the heat transferrate will increase. To analyse this assessment the fin effectiveness is defined as follows:

εf =Qf

hAc,bθb(2.35)

Where Ac,b is the cross-sectional area at the base of the fin. To achieve a useful design εfshould be as large as possible. Furthermore, experimental designs might not be justifiedwith values of εf << 2.

20

2.4. Thermoelectric Materials

Fin effectiveness may be likewise expressed in terms of the thermal resistance:

εf =Rt,bRt,f

(2.36)

Where Rt,b = (hAc,b)−1 is the fin resistance due to convection at the exposed base and

Rt,f is the whole fin thermal resistance.A distinct measure of fin performance is provided by the fin efficiency. The efficiency is

defined as the difference between the dissipation of energy taking into consideration thetemperature gradient due to conduction throughout the fin, and the maximum rate atwhich the fin could dissipate energy if the whole fin surface were at the base temperature.Therefore:

ηf =QfQmax

=Qf

hAfθb(2.37)

Performing an analysis, as the fin longitude (L) approaches higher values, likewise highervalues of fin efficiency are achieved. Furthermore, a correlation between fin geometry andefficiency is accomplished where:

mLc =

(2h

kAp

)1/2

L3/2c (2.38)

Figure 2.13: Experimental values of ηf vs L3/2c (h/kAp)

1/2 for different fin geometries[Bergman et al., 2011]

Where Lc is the corrected fin length and Ap is the corrected fin profile, both of whichmay vary depending on fin geometry as shown in Fig. 2.13.

In practice, the efficiency of most fins is above 90% to achieve an efficient heat sinkdesign [VDI-Gesellschaft, 1993].

2.4 Thermoelectric Materials

Semiconductor materials are amorphous or crystalline materials with specific properties.The electrical characteristics of semiconductors are related to their atomic structure.

21


Atoms in a semiconductor are either from the group IV of the periodic table, or thecombination of group III and group V, or combination of group II and group VI. Thismaterials act as insulators at lower temperatures and as conductors at higher temperatures[Mehta, 2008].

The bond structure of a semiconductor determines its properties. Each of the atomsin the lattice are connected by a covalent bond between their electrons. Although theseelectrons cannot move or exchange energy on its natural state, when they acquire enoughenergy to participate in conduction they are excited to a high energy state. When thisoccurs, the electrons are free to move throughout the crystal lattice and participate inconduction; otherwise the electron is bonded at low energy state.

There are two distinct energy levels for the electrons in which they cannot attain energyvalues. Furthermore, the minimum energy that an electron has to gain to break free is theband gap energy [Sze and Ng, 2006].

When an electron is exited, the space left behind by it (called a hole) allows the covalentbond to move from one electron to another. This fact generates a positive charge movingthrough the crystal lattice.

The properties of the material can be modified by varying the number of electronsand holes in the semiconductor by a technique called doping. In a doped material theconcentration of one type of carriers is higher than the other. Hence, atoms with onemore valence electrons than the semiconductor material result in an n-type semiconductor,where the carriers are electrons. Likewise, atoms with one less valence electrons result inp-type semiconductors, where the carriers are holes. Moreover, when two differently dopedmaterials are bonded together they create a heterojunction. That junction results in anexchange of electrons and holes between the n-type and p-type creating a potential barrier[Shur, 1996].

(a) Holes produced in a p-type semiconductor (b) Extra electrons produced in a n-typesemiconductor

Figure 2.14: n and p-type semiconductors at atomic level [Shur, 1996]

22


Semiconductor materials allow the pass of electric current more easily in one directionthen in other, besides they show variable resistance and sensitivity to heat and light. Theseproperties make semiconductors applicable in energy conversion such as in thermoelectricand photovoltaic devices [Sze and Ng, 2006].

Semiconductors are used as a TE material due to their high conversion efficiency and theirhigh zT values compared to metals and metal alloys. Before the discovery of semiconduc-tors, high conversion efficiency metals and metal alloys were used as TE materials sincethey had the highest efficiencies available at that time. Nevertheless, between 1950´s to60´s plenty of progress was made in the research for high performance rate of semicon-ductors as different components where fuse together trying to optimise their efficiencies.Hence, with the advent of nanotechnology, that research and experiments led the semicon-ductors to reach higher efficiencies for TE applications. The efficiency of TE materials isdefined by high electrical conductivity and Seebeck coefficient, with low thermal conduc-tivity. Although metal and metal alloys are good electrical conductors, their properties arenot close enough to compete for their use in thermoelectric and photovoltaic technologies[Rowe, 1995; Pei et al., 2012].

When looking for a favourable TE material candidate, the material is mainly subduedto a high zT value to yield high efficiency. According to [Schönecker et al., 2014], otherparameters are certainly important to consider when looking for a material in TE modulessuch as:

• Material shortage and price• Environmental impact• Reliability and handle for the workload

Most of the materials used nowadays in TEGs are a mixture of several semiconductormaterials. With regard on their synthesis, raw materials (previously chosen the finalproduct) are transformed into powder by different pulverisation techniques to, eventually,bend and melt them together. Subsequently, the material mixture is cooled down by athermal cooling procedure, where different cooling paths can be chosen to turn the liquidinto solid. This thermal cooling technique depends on the shape and size of the finalproduct pursued [Rowe, 2005; Belov et al., 2005].

In recent years, nanostructuring has been the most popular method used in solidificationof TE materials, aiming the final product to obtain the smallest grain size as possible.Generally, grain size in the material is deeply dependent on the figure of merit value: thesmaller the grain the higher zT .

Material is crushed into powder after it has been through solidification. Therefore,following the process shown in Fig. 2.15, the powder is shaped into ingots or discs withtypical sizes of 50 − 150 mm diameter and 3 − 50 mm thick. This process happens byhaving the powder pressed and sintered at high pressures and temperatures. Subsequently,

23


the ingot or disc are cut into dices with dimension between 2 − 10 mm in all directions.Assembly is the final step of the process, where thin layers of metal are deposited onopposite sides of the dices, while the other four sides of the dices are covered by a protectivecoating thus the material can achieve lower contact resistance and higher durability [Liet al., 2005].

Figure 2.15: Typical synthesis route of thermoelectric materials [Skomedal, 2016]

Regarding TE materials development through history, great progress has been performedin high ZT thermoelectric materials research for the last 20 years, as can be seen in Fig.2.16 [Heremans et al., 2013].

Figure 2.16: Evolution of the maximum ZT over time. Materials for thermoelectric coolingare shown as blue dots and for thermoelectric power generation as red triangles. The blackdashed line guides the eye [Heremans et al., 2013].

One of the most important factors to take into consideration for TE materials is the costof the material and the environmental impact it could have on the nearest surroundings.To follow a sustainable and clean development in TE energy production, is essential totake into consideration that not all raw materials for TE semiconductors are good for theenvironment. Furthermore, some TE materials are even highly toxic [Hertwich, 2010]. Inaddition, with stricter laws regarding pollution in increasing number of countries, the prizeof selected materials has the tendency to surely raise. The cost of materials according to[Skomedal, 2016] can be seen in Table 2.2.

24


Materials Material cost ($/kg)

Ge 1800Te 100Co 30Sn 23Bi 20Si 15Cr 8.6Mg 3Mn 2.5Pb 2Fe 1

Table 2.2: Different costs per kilo of thermoelectric materials [Skomedal, 2016]

Nevertheless, the most critical factor in the whole selection process for a TE material,is that the chosen material has to be able to perform at the highest level. The materialmust be able to withstand constant thermal exposure, dealing with creep and chemicaldeterioration. One of the most popular materials used in TEGs to date is Bi2Te3,composed by bismuth and tellurium. Although there is a possibility of a tellurium shortagein the future, which could lead to higher costs for the material, in present time, a materialwith better TE performance rate has not been found [Tang et al., 2007; Sun et al., 2008].

Taking into deep consideration all previous factors studied, the material chosen for thesemiconductor in the TEG in this report is Bismuth Telluride (Bi2Te3), the properties ofBi2Te3 according to [Lide, 2004], where the properties and structure are:

• Molar mass M = 800.761 g/mol

• Density ρ = 7.85 g/cm3

• Melting point T = 586°C

Bismuth telluride (Bi2Te3) is in the material type Bismuthtelluride along with the chemicalcomposition of Sb2Te3 and Bi2Se3, as can be seen in 2.3 [Skomedal, 2016].

25


Material type BismuthtellurideChemical composition Bi2Te3, Sb2Te3, Bi2Se3Temperature range (K) 173 to 523Thermoelectric properties α = 100 - 250 µV/K

σ = 100 - 1000 S/mk = 1 - 2 Wm−1K−1

ZT ∈ [2, 3]Mechanical properties CTE = 14-20Stability and durability Stable in air up to 573 KSynthesis Monocrystalline for high efficiency

Multicrystalline for low priceMaterial cost ($/kg) Te = 100

Bi = 20Health and environmental impact Te is highly toxic

Table 2.3: The properties and different factors of the materials in Bismuthtelluride

The TE properties for Bi2Te3 are used with equations which establish the propertiesvalue for every possible temperature that the material is exposed to [Lamba and Kaushik,2016].

α = [αp − (−αn)] = 2(22224 + 930.6Tm − 0.9905T 2m)10−9 (2.39)

ρp = ρn = (5112 + 163.4Tm − 0.6279T 2m)10−10 (2.40)

kp = kn = (62605 + 277.7Tm − 0.4131T 2m)10−4 (2.41)

And where different values of the TE properties are plotted as follows in Fig. 2.17.

26


(a) α vs Tm (b) k vs Tm

(c) ρ vs Tm (d) ZT vs Tm

Figure 2.17: Chosen material TE properties

27

MODEL DESIGN 3In the previous chapter was stated that a HCPV-TEG system is regulated by a synergy of

thermoelectric and photovoltaic influences combined together to exploit its maximumcapabilities. In this chapter, each element of the system is analysed and implement into a

mathematical model in order to yield the maximum power output in the whole system.

3.1 Beam-Splitter

The spectrally selective beam-splitter is a device which filters the concentrated solarradiation and reflects the fraction to the HCPV cell where the energy conversion efficiencyof the PV is higher then of the TEG. Beam-splitters are designed for maximizing the poweroutput of a hybrid PV-TEG system.

The ideal beam-splitter has zero absorbance, otherwise the absorbed radiation is lostfor conversion, furthermore the temperature of the BS would significantly rise in theconcentrator system. The ideal reflectance spectrum of the BS is a step function with100% reflectance for wavelengths where the conversion of the PV is more efficient than ofthe TEG. The ideal transmittance is 100% for other wavelengths where the efficiency ofTEG is higher.

The wavelength range where the PV cell is more efficient is defined by the TEG efficiency,assuming that the efficiency of the TEG is not a function of wavelength.

Figure 3.1: Efficiency as function of wavelength [Skjølstrup and Søndergaard, 2016]

According to the ASTM G-173-03 standard the direct+circumsolar radiation has an

29

Aalborg University 3. Model Design

integrated power density of 850 W/m2 on the surface of Earth, which is defined for usingoptical concentrators. It includes the direct beam from the Sun and the circumsolarcomponent.

Figure 3.2: Spectral irradiance on the surface of Earth []

To calculate the heat flux from the solar radiation after the beam-splitter, the twowavelength values, those between the PV cell efficiency is higher then the TEG efficiency,are needed. This wavelength range is given by the graphs in Fig.3.1 for crystalline-Si PVcells. The power density reflected to the PV cell based on the two wavelength valuesapplying Planck’s law, Stefan-Boltzmann’s law, and the black-body radiation function canbe calculated.

Assuming an ideal beam-splitter the other part of the radiation is transmitted to thethermoelectric generator.

For 8% TEG efficiency the wavelength range what the beam-splitter reflects to the CPVcell is between 489 nm and 1054 nm for crystalline-silicon solar cell according to Fig.3.1.It means, that the integrated power density of the concentrated solar radiation betweenthese two values is reflected to the CPV cell.

According to Planck’s law the specific power density of the radiation for one specificwavelength is:

Eλ =C1

λ5 · exp C2

λ · T− 1

(3.1)

Where

• C1 = 3.74117 · 108W · µm4

m2

• C2 = 1.43854 · 104 µm ·K• T = 5780 K, surface temperature of the Sun

30

3.2. CPV cell

By applying Stefan-Boltzmann’s law, and integrating the specific power density betweenthe wavelengths 489 nm and 1054 nm, the power density of that wavelength range can becalculated as follows:

q =

∫ 1.054

0.489Eλ(λ, T ) · dλ (3.2)

The result is the power density of the radiation that leaves the sun between the two specificwavelength values: q = 24, 365, 163 W/m2

The integrated radiation intensity of the Sun is 63, 288, 535.9 W/m2. It means with theapplication of the beam-splitter, the 38.5 % of the radiation intensity is reflected to theCPV cell, and the rest, 61.5 % is transmitted to the thermoelectric generator.

3.2 CPV cell

The concept of concentrated photovoltaics is to increase the beam radiation what reachesthe PV cell and increase the power output as well.

In order to achieve that, several types of concentrators are introduced. Linearconcentrators such as linear Fresnel lenses/concentrators are feasible for mediumconcentration ratios, between 10-100 suns, and they are combined with single-axis trackers.Point concentrators, including parabolic dishes and heliostat-power systems are commonlyused in Selective Solar Beam-Split (SSBS) systems. This types are able to provide highconcentration ratios, over 100 suns, and they require double-axis trackers. The main issueconcerning high concentration ratios is the significant increase in temperature of the solarcells. In application active or passive cooling is needed.

An alternative arrangement is the SSBS-HCPV system, where the solar radiation is firstconcentrated and then split by spectral filters.

By applying the concentrator and an ideal filter, the used CPV cell absorbs the fractionof solar radiation with wavelengths close so the band-gap wavelength of the PV cell. Thechosen photovoltaic material is crystalline silicon due to its high conversion efficiency, andband gap wavelength of 1170 nm.

The solar cell can be modeled with an electrical equivalent as a current source with aparalell diode, shown on Fig.3.3.

Figure 3.3: Equivalent solar cell electrical model

31


Solar cells are connected in series and paralell combinations to achieve the desired volatgeand current levels. The CPV module can be characterized by the maximum power point,whert the product of maximum current and voltage has at its maximum value. The twolimiting components of the PV cell are the open-circuit voltage (Voc), and the short-circuitcurrent (Isc). This values determine the Fill Factor (FF) of the PV cell.

Figure 3.4: CPV cell in the system: incoming radiation from the beam-splitter yields anelectric current through the CPV cell, likewise the surface area emissivity reflects part ofthe incoming radiation

Performing an energy balance in the CPV system shown in Fig. 3.4, the electrical powergenerated by the system is defined by a set of 6 nonlinear equations following the nextgeneral energy balance:

QconcCPV = Qrad + QCPV + PCPV (3.3)

Where the heat absorbed by the concentrator is:

QconcCPV = XGACPVBSratioCPV (3.4)

Moreover, the radiation loss from the CPV cell to the ambient due to environmentalconditions effect is described as:

Qrad = εσACPV (T 4CPV − T 4

∞) (3.5)

In this report the emissivity coefficient for the CPV cell is considered constant with a valueof 0.9. Moreover, the power yielded by the CPV cell in the system is defined as:

PCPV = XGACPVBSratioCPV ηCPV (3.6)

Traditionally, the efficiency of photovoltaic cells has been symbolise with the followinglinear expression [Notton et al., 2005]:

ηCPV = ηTref

1− βref

[(TCPV + THS

2

)− Tref

](3.7)

In the previous equation ηTref refers to an analytical value of the efficiency at a referencetemperature Tref = 298K, which yields ηTref = 39.6%. Besides, a constant absolute

32

3.3. TEG

efficiency reduction coefficient of βref = −0.047%/K is defined in this report [AzurSpace,2016].

Furthermore, a conductive and convective heat transfer occur due to the heat sink whichdissipates heat to the ambient:

QCPV = kCPVACPV

(∂T

∂x

) ∣∣∣∣x=CPV −

(3.8)

QCPV = kHSACPV

(∂T

∂x

) ∣∣∣∣x=CPV +

(3.9)

QCPV = hAHS(Ts − T∞) (3.10)

3.3 TEG

A thermoelectric generator is a solid-state device with no moving parts which convertsthermal energy to electricity by using the Seebeck effect. TEGs are made of two ceramicsubstrates which are located on their top and bottom, where the top ceramic substrate isa heat absorber and the bottom one is a heat rejecter [Goldsmid, 2010].

Figure 3.5: Schematic the modelled TEG in this study

Specific semiconductors are used as thermoelectric materials due to their low thermal andhigh electrical conductivity. Metal alloys interconnect them to create p-n junction betweenn-type and p-type materials to create electric current. Nowadays, this semiconductordesign is also used in energy production driven by a gradient of temperature. With adifference in temperature a movement of electrons is created in the semiconductors. Theseexcited electrons are e− in n-type materials and holes, h+ in p-type materials. TEGs canalso be used in a reversed way, where a current comes from the negative charge on theleft end of TEG and goes through it, to plus charge on the right end. With this reversedmethod, the TEG becomes hot on the top and cold on the bottom [Neamen, 2003; El-Genket al., 2003].

The thermoelectric generator is made from different materials, depending on how muchheat is being absorbed. To get the best efficiency in the TEG, the semiconductor materialhas to be able to keep itself cool, while promoting a good flow of electrons.

33


The conversion of a temperature difference directly into a voltage, or vice versa, is calledthe TE phenomena. These phenomena are based on different TE effects: Seebeck, Peltierand Thomson. Furthermore, the Joule effect will be taken into account in the heat balance.

3.3.1 Electrical Equivalent

As previously stated, a TE module (Fig. 3.6) is formed by thermocouples of p-type andn-type semiconductors connected thermally in parallel and electrically in series. Thesemiconductors are linked together by copper string, in order to conduct the electricalcurrent, and located between two ceramic plates which work as thermal conductors andelectrical insulator.

Figure 3.6: Basic configuration of a TE module

If a constant temperature difference is maintained across a TEG, the electrical equivalentcan be modelled as a DC source in series with an internal resistance, since the steady-stateelectrical parameters are linearly related by linear equations. In Fig. 3.6 it is shown theequivalent electric circuit in which VOC is the sum of the Seebeck voltages generated bythe thermocouples without load connection, Rteg is the internal resistance of the generator,Rload is the load connected to the generator and Iload is the current through the circuit.

34

3.3. TEG


At load connection, the current and the voltage across the load can be written as:

Vload = VOC −RTEGIload = RloadIload (3.11)

Iload =VOC

RTEG +Rload=

α∆T

RTEG +Rload(3.12)

where α is the absolute Seebeck coefficient of the thermocouples and ∆T is the temperaturedifference across the module.

Thus, the output power of the TE module (Pload) can be formulated as:

Pload = VloadIload = RloadI2load =

Rloadα2∆T 2

(RTEG +Rload)2(3.13)

Frequently more than one module are needed to provide the necessary power output forthe application. Thus, more modules are interconnected depending on the temperaturegradient availabilities and the load requirements. This TEG structure is formed of nsmodules connected in series and np in parallel. A generalised structure of TEG is illustratedin Fig. 3.8


For this configuration, it is assumed that every module has the same electricalcharacteristics and is subjected to the same thermal gradient. Consequently,the voltage

35


generated by the TEG, the current and the output power become:

Vload = VOC −RTEGIload = RloadIload (3.14)

Iload =VOC

RTEG +Rload=

nsα∆T

RTEG +Rload(3.15)

Pload = VloadIload = RloadI2load =

Rloadn2sα

2∆T 2

(RTEG +Rload)2(3.16)

where

Rgen =nsRTEGnp

(3.17)

The maximum power transfer theorem states that the maximum power produced bya TEG is achieved when the ratio m between the load the resistance and the internalmodule resistance equals unity. However, this relationship does not match the conditionof maximum efficiency. As a result, [Lauryn L. Baranowski, 2014] demonstrates that theTEG design can be optimized to achieve maximum power and maximum efficiency at thesame operating point when:

m =RloadRTEG

=√

1 + ZT (3.18)

Where ZT is the figure of merit of the semiconductor material:

ZT =

(Th + Tc

2

)α2

ρk(3.19)

3.3.2 Thermal Analysis

The heat flow and the current through the TEG are governed by its electrical conductivityσ and thermal conductivity k. On one hand, in order to achieve and maintain largertemperature gradients across the module, low thermal conductivity must be ensured thetemperatures of the heat exchangers. On the other hand, a high value of the electricalconductivity guarantees the minimum Joule heating [Anamaria Man, 2016].

At open-circuit condition, the heat flow is not affected by Peltier effect or Joule effectsince there is not current flowing through the device. When a load is connected, the currentthrough the TEG increase the heat loss due to Joule effect and alters the effective thermalresistance. In addition, as a result of the load current, Peltier effect pumps heat from thecold to the hot side of the device. Consequently, the temperature at the hot side decreasesand the temperature at the cold side increases, reducing the temperature gradient ∆T .This reduction can result in a large output power change since it is proportional to ∆T 2.

The typical thermal equivalent circuit of a TEG is illustrated in Fig. 3.9. At open-circuit condition the Fourier law (Q = K(Th − Tc)) governs the heat flow. At close-circuitcondition, it is not only governed by the Fourier law, but also by the Peltier (αITj) and theJoule (1/2I2RTEG) terms. Furthermore there is a small heat loss in the thermoelectric legsthrough convection and radiation (Qloss), which will be neglected for the thermal analysis.

36

3.3. TEG

Figure 3.9: Heat distribution throughout the TEG

Excluding the Thomson effect from the analysis, the equivalent flow balance equationsof the hot and cold junctions are:

Qh = αITh +K(Th − Tc)− 1/2I2RTEG (3.20)

Qc = αITc +K(Th − Tc) + 1/2I2RTEG (3.21)

where K = (kA)/l is the thermal conductance, k is the thermal conductivity, A is the areaand l the thickness of the module.

Thus, the efficiency of the TEG is the ratio of the electrical output power(Pteg = Qh − Qc) divided by the thermal power at the hot junction (Qh):

ηTEG =PTEG

Qh=

PTEG

PTEG + Qc= 1 +

PTEG

Qc(3.22)

Up to this point it is necessary to analyse how each part of the HCPV-TEG system workand what theory stands behind it. To further proceed with this system, every segmentof the structure was evaluated separately, but at the same time inter-connected with eachother.

• Transfer of heat flux (Qconc) from Sun irradiation to the hot side of the TEG.• Radiation heat flux (Qrad) from the TEG surface.• Transfer of heat flux (Qh) from Sun irradiation to the thermocouples of the TEG.• Conduction of cold side heat flux (Qc) through the TEG.• Conduction of cold side heat flux (Qc) through the heat sink.• Convection of heat flux (Qc) to surroundings.

This leads to three unknowns Th, Tc, Ts. Which are hot temperature of TEG, coldtemperature of TEG and surface temperature of heat sink. These temperatures are linkedtogether with the heat flux of cold side Qc that needs to be dissipated respectively. To

37


solve the unknown variables, a set of non-linear equations are developed and solved withNewton-Raphson’s method.

Figure 3.10: Heat transfer and energy bal-ances throughout the system segments

Qconc = XGATEGBSratioTEG (3.23)

Qh = Qconc − Qrad (3.24)

Qh = PTEG + Qc (3.25)

Qrad = εσATEG(T 4h − T 4

∞) (3.26)

Qc = kTEGATEG

(∂T

∂x

) ∣∣∣∣x=TEG−

(3.27)

Qc = kHSATEG

(∂T

∂x

) ∣∣∣∣x=TEG+

(3.28)

Qc = hAHS(Ts − T∞) (3.29)

In the first section of TEG system (Fig. 3.10) there is a heat flux Qh that comes from Sunirradiation and heats up the top layer of TEG by temperature of Th. This heat conductsthrough thermoelectric leg and produces power PTEG. At the cold side of TEG there isa cold side heat flux Qc and a lower temperature that is called Tc. In the middle section,there is a heat transfer by conduction from cold side of TEG Tc to the base of heat sinkTs. After all, the heat sink that is attached to the cold side of TEG has to dissipate Qcby means of convection.

3.4 Heat sink

A heat sink is a device that is used to increase energy efficiency and performance ofprimary devices. It absorbs heat from a hot side and dissipates it to the surroundingsby phenomena associated with the 2nd law of thermodynamics. Efficiency of a heat sink

38

3.4. Heat sink

is dependent on material properties of fins and surface area that is in contact with theconvective fluid. Larger surface area and higher flow velocity increases the rate of heattransfer to the surroundings. Unfortunately, both high values cannot be achieved at thesame time for a heat sink with fixed dimensions. This is due the fact that the surface areaincreases and the gaps between the fins respectively decrease which results in a decreasein the fluid velocity. Therefore, it is important to find a balance value where surface areaand flow velocity are optimized for a maximum efficiency. [Lu and Bailey, 2011], [Poltorak,2016].

Qx = −kAdTdx

(3.30)

Simplified Fourier’s law describes conduction process in one-dimension, where Qx is heattransferred by conduction, k is thermal conductivity constant which depends on material,A is surface area and T is temperature.

Moreover, Newton’s cooling law describes heat dissipation in the heat sink, h is theconvective heat transfer coefficient, Ts represents surface temperature and T∞ is ambienttemperature of the fluid.

Q = hA(Ts − T∞) (3.31)

It is always a challenge to find optimal heat sink architecture for specific purpose. Hence,a detailed analysis of a heat sink under forced and natural convection has to be performed.The most important point in this research is to find out how the fluid flow across a heatexchanger could improve total system’s efficiency while at the same time draining ownproduced energy for the moving fluid flow.

To design an effective heat sink all the previous information throughout this report hasto be taken into deep consideration. Up to this point, the modelling of a heat sink isdefined by three main characteristics.

• Operation range• Geometry• Materials

3.4.1 Operation range

With regard to the operation range of the HCPV-TEG, very hight temperatures are yieldedin the TEG due to the beam-splitter. Despite the fact that this hight temperatures boostenergy production in the TEG, a greatly high heat flux needs to be dissipated to achievedesirable efficiencies and energy outputs.

Diverse technologies and heat transfer techniques can be utilised to dissipate heat,however, in order to achieve maximum efficiencies with the TEG of the study case,desirable values of a convective heat transfer coefficient reach a magnitude of approximately150 W/m2K (Fig. 3.11).

39


Figure 3.11: Behaviour of the TEG in function of the convective heat transfer coefficient ofthe heat sink with a pre-set configuration: it is noted that for values of h > 150 W/m2Kthe power generation and efficiency of the TEG reach a steady maximum value.

Certainly heat transfer is closely related with the medium in which takes place. Diversefluids can be used to improve heat transfer rate, however, to achieve reasonable resultsconcerning costs and calculations simplicity, this report is mainly focus in the use of air atatmospheric pressure with the following constant characteristics:

T (K) P (atm) ρ( kgm3 ) k( W

mK ) Cp( JkgK ) µ · 105(Pa s) ν · 105(m

2

s ) α · 105(m2

s )

300 1 0.6158 0.04418 1044 2.934 4.765 6.872

Table 3.1: Properties of air [Cengel and Ghajar, 2015]

Thermophysical properties of air do not greatly change for the ranges of temperature ofthe study case hence constant values can be used without compromise the results.

The two main convective heat transfer methods are analysed in this report: forced andnatural convection. Both methods have their pros and cons in order to yield the maximumefficiency and power output in the system. On one hand, natural convection yields lowervalues of h but does not require any external source of energy to dissipate heat. On theother hand, forced convection reaches higher values of h, however it requires an externalcooling power to set the external fluid in motion.

Natural convection can be optimised regarding the geometry of the heat sink, howevera notable inconvenient is the accurate calculation of the forced convective heat transfercoefficient without the use of computational Finite Element Methods (FEM). In this report,a set of correlations for a plain-plaque fin are taken to define h as accurate as possible.These correlations can be seen in the following formulae. Further information is stated in

40

3.4. Heat sink

App. A regarding these calculations.

Nu = 2NuL = 0.906Re1/2L Pr1/3 Laminar boundary layer (3.32)

Nu =5

4NuL = 0.0385Re

4/5L Pr1/3 Turbulent boundary layer (3.33)

Nu = (0.039Re4/5L − 755)Pr1/3 Mixed conditions (3.34)

With regard on the Reynolds number in the heat sink, a hydraulic diameter has to bedefined considering heat sink geometry. Hence:

Figure 3.12: Heat sink geometry and hy-draulic diameter

Dh =4A

P=L− (S − t)S − t+ L

(3.35)

Simulating the TEG behaviour with a prefixed heat sink geometry for both convectionmethods, it can be achieved that forced convection outmatches natural convection withradiation emissivity as shown in Fig. 3.13. However, in terms of cooling power needed,further studies must be conducted.

(a) Power yielded by the TEG vs concentrationrate

(b) Efficiency of the TEG vs concentration rate

Figure 3.13: Power and efficiency of the TEG for diverse convection methods in air atatmospheric pressure: Forced convection, natural convection with U∞ = 5ms−1 andnatural convection with radiation

Although forced convection requires an external cooling power, the method generatesmuch higher values of h and, therefore higher power output. That cooling power isgenerated by a fan attached on the top part of the heat sink in order to yield a flowvelocity throughout it. According to [Loh and Chou, 2004; Kays and London, 1984],

41


cooling power in heat sinks is defined by the following expression:

∆P =(4fappx

+ +Kc +Ke

)(1

2ρfU

2ch

)(3.36)

Where ∆P is the pressure drop, fapp is the apparent friction, x+ is the hydrodynamicentry length, Kc and Ke are the coefficient of contraction and expansion respectively, ρfis the convective fluid density and Uch is the flow velocity in the channel. Furthermore thecooling power is defined as:

Pcooling =∆Pm

ηfan(3.37)

Where m is the volumetric flow rate through the channel and ηfan = 0.25 is the efficiencyof the fan used [AAVIDThermalloy, 2016]. Further information regarding cooling powercalculations is described in App. B.

3.4.2 Geometry

The most important feature regarding the design is the geometry since, due to performancevs costs, the materials chosen for this type of applications are reduced to copper, aluminiumor iron. Furthermore, the heat flux and temperature range applies a restriction regardingmaterial selection to obtain desirable values of thermal conductivity.

Figure 3.14: Dimensionless approach to fin length design [Lienhard, 2011]

Concerning heat sink geometry and system’s modelling, constant-section-rectangularfins are chosen to obtain a heat flux prediction as accurate as possible together with thereduction of possible design options available with other types of fins.

42

3.4. Heat sink

The heat sink fin parameters to optimise are reduced to fin length (L), width (w),thickness (t), pitch (S), number (N) as shown in Fig. 3.12. Moreover, the heat sink area(A), width (W ) and base thickness (∆x) are also taken into consideration. The heat sinkis mainly design to operate within the range of parameters of the TEG, hence its area isdefined by the TEG’s plaque area which has to be equivalent to each other.

ATEG = AHS (3.38)

The chosen TEG yields a square plaque area which is also applied to the heat sink baseas shown in Eq. 3.39.

AHS = W 2 (3.39)

Hence, fin number, pitch, thickness and width are also related as follows:

N =w − tS

+ 1 (3.40)

To ensure that the heat sink is not over-dimensioned, it is necessary to calculate appropriatefin length that would dissipate previously calculated heat Qc. This results in cost effectiveheat sink design. The optimum fin length is achieved when mL value is equal to 2.65[Bergman et al., 2011]. This value is defined with the dimensionless analysis in Fig. 3.14and describes the fin length in which the temperature at the tip is quasi-equal to thetemperature of the convective fluid. This can be calculated by following correlations:

m =

√Ph

wtk(3.41)

mL = 2.65 (3.42)

In these equations P is fin tip perimeter, h convective coefficient, w fin width, t finthickness, k thermal conductivity of fin material and L - length of fin. Therefore, with thestudy performed in this report, the fin longitude achieves an optimum value of L = 0.087 m

as shown in Fig. 3.15

(a) PTEG vs L With X = 100 (b) PTEG vs L With X = 500

Figure 3.15: Power generation of the TEG in function on the fin longitude with differentheat transfer methods. Note that, for forced convection, for values of mL > 2.65 andL > 0.087 m the fin is over designed, hence it does not increase significantly the powergeneration in the system

43


3.4.3 Materials

Traditional materials such as aluminium and Copper are mostly used in heat sinkmanufacturing. Depending on applications, aluminium alloys are lighter, have betterstrength to weight ratio and are easily custom manufactured and cheaper than Copper.Therefore they are commonly used in many kind of industries, such as telecommunications,automotive, military, medical, and electronic industries.

Surface finish of the material is also an important part of adding efficiency to the heatsink. It influences radiation emissivity, since shiny metals tend to absorb and radiate onlya small amount of heat, while matt black radiates highly. In natural convection blackmatt surface is more efficient than pure shiny metals, while in forced convection a layer ofcoating is counterproductive.

Materials in heat sinks are more favourable if they have high thermal conductivityvalues. The most commonly used material in heat sinks is aluminium, where aluminiumalloys 6060, 6061 and 6063 have thermal conductivity values that ranges from 166 to201 W/mK [Poltorak, 2016].

On the other hand, copper has around 60% higher thermal conductivity than aluminiumalloys, hence this material is attractive by heat sink manufacturers. However, using coppercomes with 4-6 times higher costs and densities, which is the major factor for choosingaluminium as a heat sink material. Copper heat sinks are used in the following applications:

• Power Plants• Solar Thermal Water Systems• HVAC Systems• Gas Water Heaters• Forced Air and Cooling Systems

Therefore the material chosen for the heat sink is 6063 aluminium alloy. This material ischosen due to be the most common aluminium extrusion, its high thermal conductivity of200 W/mK and the a density of 2.68 g/cm3. The 6063 aluminium alloy has 97.75% +/−0.25 % of aluminium in the element [Poltorak, 2016].

44

RESULTS AND DISCUSSION 4Throughout this report, different design alternatives for a beam-split HCPV-TEG system

have been taken into deep consideration. Furthermore, a mathematical model has beendeveloped simulating the system’s behaviour in order to evaluate the influence of the

diverse parameters governing the power production. Critical parameters are considered toevaluate economic viability of the system compared with current HCPV-TEG systems.

Subsequently a thorough study of the model has been performed where results are exposed.

4.1 Power production and efficiency

To recapitulate the whole study performed, a beam-splitted HCPV-TEG system harvestsdirect sun radiation into DC electricity. To carry out that process, the system is composedby a solar concentrator which intensifies and redirects direct solar radiation into a narrowarea. Subsequently, the concentrated radiation is separated depending on its wavelengthby a beam-splitter; short wavelengths are directed to a CPV cell and long wavelengths(which generate higher temperatures) to a TEG in order to yield the maximum power andefficiency of each system. Moreover, forced convection is used on a heat sink attached tothe system with regard to dissipate the excess heat throughout.

Figure 4.1: Whole system schematic: red hue colours represent higher temperatures whileblue hue colours represent lower temperatures

The system certainly has five main characteristics that cause higher power and efficiency:• High heat flux throughout the system• Low temperatures in the CPV cell• Large differences of temperature between the TEG plaques• Efficient materials• High heat dissipation in the heat sink

45

Aalborg University 4. Results and Discussion

Higher solar intensities can be achieved depending, indeed, on the location’s radiation andthe concentration rate X. The higher the concentration rate, the higher the incident heatflux in the systems.

Low temperatures in the CPV cell and large ∆T between TEG plaques are definedby the designation of the range of wavelengths by the beam-splitter. Short wavelengthsgenerate lower temperatures than long ones.

With regard to materials, high efficiency Bi2Te3 thermoelectric material has been cho-sen for the TEG. For a medium temperature between the plaques of Tm ∈ [250, 450] K, thefigure of merit range of values of this material ZT ∈ [2, 3] (Fig. 2.17d). Likewise aluminiumhas been chosen for the heat sink due to its high thermal properties (k = 205 Wm−1K−1)and its low price.

In order to generate the results, all the previous studies performed in this reporthave defined the necessary parameters with the current background and development inthermoelectricity and photovoltaic technologies. In Table 4.1 a summary of the parameterschosen is represented:

Description Symbol Value Units

Solar irradiation G 850 Wm−2

Beam-splitter ratio of the TEG BSratioTEG 61.5 %Beam-splitter ratio of the CPV BSratioCPV 38.5 %Constant absolute efficiency reduction βref -0.047 %K−1

Temperature of reference (CPV cell) Tref 298 KEfficiency of reference (CPV cell) ηTref 39.6 %Thermal conductivity of the CPV cell kCPV 60 Wm−1K−1

CPV cell thickness lCPV 0.005 mCPV cell area ACPV 0.0016 m2

TEG thickness lTEG 0.01 mTEG area ATEG 0.0016 m2

HS area AHS 0.0032 m2

HS base thickness ∆x 0.005 mHS width W 0.04 mHS thermal conductivity kHS 205 Wm−1K−1

Fin thickness t 0.001 mFin longitude L 0.087 mFin pitch S 0.005 mFin number N 9 −Fan efficiency ηfan 25 %

Table 4.1: Parameters and values utilised in the study

Furthermore, constant air properties are considered as shown in Table 4.2:

46

4.1. Power production and efficiency

T (K) P (atm) ρ( kgm3 ) k( W

mK ) Cp( JkgK ) µ · 105(Pa s) ν · 105( ,

2

s ) α · 105(m2

s )

300 1 0.6158 0.04418 1044 2.934 4.765 6.872

Table 4.2: Properties of air utilised in the study

With regard to TE materials, dynamic behaviour of TE properties is considered infunction of the mean temperature Tm:

α = [αp − (−αn)] = 2(22224 + 930.6Tm − 0.9905T 2m)10−9

ρp = ρn = (5112 + 163.4Tm − 0.6279T 2m)10−10

kp = kn = (62605 + 277.7Tm − 0.4131T 2m)10−4

Additionally, the governing equations of the system are solved following Newton-Raphson’smethod for non-linear equations with the previous assumptions and values adopted. Theseequations are henceforth restated for the sake of good.

Solar concentrator:

Q = XGAconc

CPV cell:

QconcCPV = QradCPV + PCPV + QCPV

QconcCPV = XGACPVBSratioCPV

QCPV = kCPVACPVTCPV − THSCPV

lCPV

QradCPV = εσACPV (T 4CPV − T 4

∞)

PCPV = XGACPVBSratioCPV ηCPV

ηCPV = ηTref

1− βref

[(TCPV + THSCPV

2

)− Tref

]QCPV = kHSACPV

THS − Ts∆x

QCPV = hAHS(Ts − T∞)

47


TEG:

QconcTEG = XGATEGBSratioTEG

Qh = QconcTEG − QradTEGQh = PTEG + Qc

QradTEG = εσATEG(T 4h − T 4

∞)

Qc = kTEGATEGTh − Tc

l

Qc = kHSATEGTh − Tc

∆x

Qc = hAHS(Ts − T∞)

PTEG = ηTEGQh

ηTEG =Th − TcTh

√1 + ZT − 1√

1 + ZT + Tc/Th

ZT =

(Th + Tc

2

)α2

ρk

Hence, the whole system is yielding a power following the next formula:

PT = PTEG + PCPV − Pcooling (4.1)

Where PT is the total system power. Furthermore, note in both figures that very lowvalues of efficiency and power generation are generated at lower concentration rates forforced convection. These values are due to the necessary cooling power subtracted fromthe system.

Moreover, system’s total efficiency ηT is defined in Eq. 4.2:

ηT =PT

QhT=

PTEG + PCPV − PcoolingGX[ATEGBSratioTEG +ACPV (BSratioCPV )]

(4.2)

Therefore, for the study case the following results are produced:

48


(a) Power and efficiency vs solar concentrationof the TEG

(b) Power and efficiency vs solar concentrationof the CPV cell

(c) Power and efficiency vs solar concentration of the whole system

Figure 4.2: Power and efficiencies vs concentration rate of the system with forcedconvection and U∞ = 4.4 m/s

Nevertheless, utilising forced convection, the TEG is very sensible to changes when theflow velocity (U∞) varies due to the heat dissipation on the heat sink. These TEG changescan be visualised in Fig. 4.3a and 4.3b.

49


(a) TEG power generation vs solar concentration vs flow velocity

(b) TEG efficiency vs solar concentration vs flow velocity

Figure 4.3: TEG sensitivity analysis with the flow velocity

Finally, with the impact of the dynamic flow velocity, the whole system production yieldthe following results:

50


(a) Whole system power generation vs solar concentration vs flow velocity

(b) Whole system efficiency vs solar concentration vs flow velocity

Figure 4.4: Whole system analysis with dynamic flow velocity

51


4.2 Economic viability

An analysis of the rapid investment costs of the High-Concentrated Beam-SplittedPhotovoltaic Hybrid System has been considered. Therefore, the cost of the maincomponents of the system have been selected in order to study its variation by changingthe concentration rate and the flow velocity of the cooling air.

Thus the total cost of the system is calculated as follows:

CT = Ch + CBS + CCPV + CTEG + 2CHS (4.3)

CT,TEG = Ch/2 + CBS/2 + CTEG + CHS (4.4)

CT,CPV = Ch/2 + CBS/2 + CCPV + CHS (4.5)

Ch = chABSX (4.6)

Where Ch is the price of the heliostats, CBS is the price of the beam-splitter, CCPV is theprice of the CPV cell, CT,CPV is the total price of the CPV subsystem, CTEG is the priceof the TEG, CT,TEG is the total price of the TEG subsystem, CHS is the price of the heatsink including the fan, ch is the price per unit area of the heliostats, and ABS is the areaof the beam-splitter. The values utilised in the study are shown in Table 4.3.

Description Symbol Value Units

Price per unit area of the heliostats ch 167.41 $/m2

Price of 40 mm x 40 mm of the CPV cell CCPV 338 $Price of 40 mm x 40 mm of the TEG CTEG 25.3 $Price of 40 mm x 40 mm of the heat sink CHS 22 $Price of the beam-splitter CBS 139 $Area of the beam-splitter ABS 0.0016 m2

Table 4.3: Values utilised in the study, according to [Lovegrove K., 2012] [GmbH,2016a],[GmbH, 2016b], [Limited, 2016], [Optometrics, 2016]

In order to study the effect of concentration rate on the costs, given a value of airvelocity, figures 4.5, 4.6 and 4.7 are shown.

52

4.2. Economic viability

Figure 4.5: TEG cost vs solar concentration ratio (U∞ = 4.4 m/s)

Figure 4.6: CPV cost vs solar concentration ratio (U∞ = 4.4 m/s)

53


Figure 4.7: CPV-TEG cost vs solar concentration ratio (U∞ = 4.4 m/s)

According to 4.5, 4.6 and 4.7, the minimum cost of the TEG is 3.75 $/W at aconcentration rate of 620. On the other hand, the costs of the CPV cell and the costsof the hybrid system decrease with the concentration rate down to a minimum value of 2.6$/W and 3.02 $/W , respectively.

54

CONCLUSION 5In this final chapter, reactions and outcomes are exposed and discuss. Results yielded in

this study are now summarise and all important findings and answers to the problemstatement are defined.

According to the model results, the power output of the beam-split system is higher thanthe conventional stand-alone CPV system for the same amount of irradiation.

The results have shown that for a high efficient TE material as Bi2Te3 used in the TEG,the power generation surpasses current HCPV-TEG systems.

Due to the separation of the concentrated beam, the system achieves higher efficiencies,maximizing CPV cell and TEG performance. Furthermore, critical parameters regardingpower generation have been identified and exploited.

The system is subdued to a value of solar concentration, which defines the amount ofheat flux throughout it. Consequently the maximum efficiency and power generated donot correlate to the same value of concentration rate: higher values of solar concentrationyield more power besides a decrease of efficiency. Nevertheless, the system is limited toa high value of solar concentration in which the high temperatures will deteriorate TEmaterials and overall efficiencies.

The distribution of temperatures throughout the system is critically defined by theincident heat flux and the heat dissipation capabilities of the convection in the heat sink.Furthermore, maintaining the system cooled down causes a higher power generation inboth subsystems besides keeping them form critical temperature values. Therefore, as theconvective heat transfer coefficient is directly related to the TEG power output, the mainimpact on h comes from the air flow velocity generated by an external fan.

Nevertheless, for low concentration rates and high flow velocities (therefore high valuesof cooling power), the power output does not increase with respect to flow velocity.

With regard to installation costs, from values of solar concentration of X = [600, 800]

onwards, the price per watt does not decrease significantly. Moreover, to achieve highersolar concentrations the investment on heliostats increase likewise the system’s area.Hence, reaching higher concentration values with the current materials is not particularlyefficient.

5.1 Further work

In this last section of the report, possible future research and development of the studymade is presented. This topics are out of scope in this report, however they are certainlyrelevant for the overall topic.

55

Aalborg University 5. Conclusion

Prototyping is one of the most important parts that allows to verify the model and makea conclusion based on data gathered from experiments. To be confident that the resultsacquired from the model that was developed during this project are correct - laboratorywork is required. After reproducing same conditions in the lab as the ones were used in themodel some deviations are expected. That is were deeper analysis will be vital in terms ofmodel improvements and error reduction. The main reason would be due to assumptionthat was taken into consideration during this project, as linear CPV cell efficiency (ηCPV ),constant thermophysical properties of air, heat transfer and cooling power assumptions forcalculations...

In this project, cooling was done by air and physical properties were kept constantat 300 K and 1 atm. Although air properties do not have significant shift with smalltemperature increments, surely it induces minimal errors in the model. In the futureworks, it would be reasonable to implement correlations of thermophysical properties ofair in terms of temperature into the model. In addition, it would be likewise valuable toobtain the results of power generation and efficiency when the model is under the influenceof different coolants such as water.

56

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61

HEAT TRANSFER

COEFFICIENT CALCULATIONS ADefining an accurate convective heat transfer coefficient is one of the more complexcalculations regarding heat transfer. Without the utilisation of a FEM in the model,precise assumptions have to be made in order to accurately describe a h value in the heatsink.

The heat sink modelled in the study case has constant section rectangular-plain-plaquefins. A fluid flow set in motion by the fan is passing by them constantly, defining thecharacteristics of forced convection.

To simplify the model, it is assumed that the convective heat transfer coefficient oneach fin is approximately equal to the heat transfer on a finite plain-plaque with incidentparallel flow as shown in Fig. A.1 with the following assumptions:

• Incompressible flow• Constant heat flux throughout the plaque• Constant fluid propierties• Negligible viscous dissipation• Steady-state conditions• Neglected radiation

Figure A.1: Plain plaque convection case of study

The convective heat transfer coefficient is defined by the dimensionless Nusselt number asfollows:

Nu =hL

kf= +

∂T ∗

∂y∗

∣∣∣∣y∗=0

(A.1)

Where h is the convective heat transfer coefficient, Lc is the characteristic longitude of thesystem and kf is the thermal conductivity of the medium; ∗ terms implies for a prescribedgeometry. Likewise, Nusselt number is defined as a function of Reynolds and Prandtlnumbers evaluated in the given geometry of the system:

Nu = f(x∗, ReL, P r) (A.2)

Form knowledge of Nu, the local convection coefficient h may be found. Furthermore,since the average heat transfer coefficient is obtained by integrating over the whole plaque,it is independent of the spatial variable x∗. Hence Nusselt number is dependent on:

Nu =hL

kf= f(ReL, P r) (A.3)

Convective heat transmission takes place in the thermal boundary layer which is definedby the Reynolds number along the plaque:

Rex∗ =U∞ρx

∗

µ(A.4)

ReL =U∞ρLcµ

(A.5)

The boundary layer can assume 2 different flow states: laminar and turbulent flows. Eachstate is described by the local value of Reynolds number which reaches its critical value atRec = 5 ·105. Therefore, the boundary layer describes a laminar flow at values Rex∗ < Rec

and turbulent flow when Rex∗ > Rec.With the previous assumptions made, the boundary layer equations are reduced to:

∂u

∂x+∂v

∂y= 0 (A.6)

u∂u

∂x+ v

∂u

∂y= ν

∂2u

∂y2(A.7)

u∂T

∂x+ v

∂T

∂y= α

∂2T

∂y2(A.8)

The hydrodynamic and heat transfer solution of these equations follow the method ofBlasius [Blasius, 1908; Schlichting and Gersten, 2003] which yields analytical results forthe assumptions made. Therefore, the local Nusselt number for laminar flow is describedas follows:

Nux∗ = 0.453Re1/2x∗ Pr

1/3 Pr & 0.6 (A.9)

Nu = 2NuL = 0.906Re1/2L Pr1/3 (A.10)

For turbulent flow:

Nux∗ = 0.0308Re4/5x∗ Pr

1/3 0.6 . Pr . 60 (A.11)

Nu =5

4NuL = 0.0385Re

4/5L Pr1/3 (A.12)

While for mixed conditions (xc > 0.1L):

Nu = (0.039Re4/5L − 755)Pr1/3 5 · 105 .Re . 107 (A.13)

0.6 .Pr . 50 (A.14)

Although the equations of this section are suitable for most engineering calculations, inpractice they rarely provide exact values for the convection coefficients. Conditions varyaccording to free stream turbulence and surface roughness, and errors as large as 25% maybe incurred by using the expressions [Bergman et al., 2011].

COOLING POWER

CALCULATION BPressure drop across heat sink is one of the key variables that govern the thermalperformance of the heat sink in forced convection environment. There are several analyticalmethods to estimate the heat sink pressure drop, however correctly selecting one that canrepresent the reality over a range of airflow found in typical electronics cooling applicationis difficult.

In this study, the final cooling power can be defined as function of the flow velocity andthe pressure drop yielded by the fan and the heat sink.

Pcooling = f(U∞,∆P ) =∆Pm

ηfan(B.1)

Moreover, following the study of [Loh and Chou, 2004] the pressure drop from classicalFlemmings and Darcy equations is defined as:

∆P =(4fappx

+ +Kc +Ke

)(1

2ρfU

2ch

)(B.2)

In order to calculate each parameter of the equation the hydraulic diameter of the channeland the Reynolds number are defined as follows

Figure B.1: Heat sink geometry and hy-draulic diameter

Dh =4A

P=L− (S − t)S − t+ L

(B.3)

Rech =UchDh

νf(B.4)

Where νf is the kinematic viscosity of the fluid. Moreover, the hydrodynamic entry lengthand contraction and expansion coefficients are described as:

x+ =L

DhRech(B.5)

Kc = 0.42(1− λ2) (B.6)

Ke = (1− λ2)2 (B.7)

Where λ is an experimental geometric factor defined as follows:

λ = 1−(Nt

w

)(B.8)

The apparent friction factor fapp, which is closely related to the hydrodynamic entry length,is described by the following formula:

fapp =1

Rech

[(3.44√x+

)2

+ (fRech)2

](B.9)

Furthermore, for a fully developed laminar flow friction factor f is obtained by the studypublished in [Kays and London, 1984] as follows:

f =1

Rech

(24− 32.53β + 46.721β2 − 40.829β3 + 22.954β4 − 6.089β5

)(B.10)

Where

β =∆xHSL

(B.11)

In this report, the convective fluid chosen is air at atmospheric pressure with a density ofρ = 0.6158kgm−3, hence all parameters to calculate ∆P are described. Subsequently themass flow rate and area of the channel are calculated as follows:

Ach = L(S − t)(N − 1) (B.12)

m = AchUch (B.13)

Finally, defining the efficiency of the fan as constant with a value of ηfan = 0.25

[AAVIDThermalloy, 2016] the final cooling power is calculated with Eq. B.

CONVECTION METHODS

PLOTS CThis appendix shows diverse figures utilised in the report to determine the best heattransfer method, comparing natural convection, natural convection with emissivityradiation and forced convection.

(a) P vs X of the TEG (b) η vs X of the TEG

(c) P vs X of the CPV cell (d) η vs X of the CPV cell

Figure C.1: Power and efficiencies vs concentration rate of TEG and CPV cell for differentheat dissipation methods and U∞ = 4.4 m/s for forced convection

(a) P vs X of the whole system

(b) η vs X of the whole system

Figure C.2: Power and efficiency vs concentration rate of the whole system for differentheat dissipation methods and U∞ = 4.4 m/s for forced convection.

INSTALLATION PRICES

PLOTS DThis appendix shows diverse figures which confront values of installation price ($/W ),concentration rate (X) and flow velocity through the heat sink (m/s).

Referencing Sect. 4.2, the costs of TEG, CPV and CPV-TEG system varies as follows:

Figure D.1: TEG cost vs solar concentration vs flow velocity

Figure D.2: CPV cost vs solar concentration vs flow velocity

Figure D.3: CPV-TEG cost vs solar concentration vs flow velocity

According to the figures D.1, D.2 and D.3 the lower the concentration ratio and higherthe air flow velocity are, the higher the cost is. This raise in cost is more significant inTEG than in CPV cell.

efficient design for thermoelectric generators in high-concentrated beam-split photovoltaic hybrid...

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