Thermoelectrics: Design and Materials

By HoSung Lee

Thermoelectrics: Design and Materials

HoSung Lee, Western Michigan University, USA

 

A comprehensive guide to the basic principles of thermoelectrics

 

Thermoelectrics plays an important role in energy conversion and electronic temperature control. The book comprehensively covers the basic physical principles of thermoelectrics as well as recent developments and design strategies of materials and devices.

The book is divided into two sections: the first section is concerned with design and begins with an introduction to the fast developing and multidisciplinary field of thermoelectrics. This section also covers thermoelectric generators and coolers (refrigerators) before examining optimal design with dimensional analysis. A number of applications are considered, including solar thermoelectric generators, thermoelectric air conditioners and refrigerators, thermoelectric coolers for electronic devices, thermoelectric compact heat exchangers, and biomedical thermoelectric energy harvesting systems. The second section focuses on materials, and covers the physics of electrons and phonons, theoretical modeling of thermoelectric transport properties, thermoelectric materials, and nanostructures.

 

Key features:

• Provides an introduction to a fast developing and interdisciplinary field.
• Includes detailed, fundamental theories.
• Offers a platform for advanced study.

 

Thermoelectrics: Design and Materials is a comprehensive reference ideal for engineering students, as well as researchers and practitioners working in thermodynamics.

 

Cover designed by Yujin Lee

• Language: English
• Publisher: Wiley
• Release date: Sep 12, 2016
• ISBN: 9781118848937

Book preview

CONTENTS

Cover

Title Page

Copyright

Dedication

Preface

Chapter 1: Introduction

1.1 Introduction

1.2 Thermoelectric Effect

1.3 The Figure of Merit

Problems

References

Chapter 2: Thermoelectric Generators

2.1 Ideal Equations

2.2 Performance Parameters of a Thermoelectric Module

2.3 Maximum Parameters for a Thermoelectric Module

2.4 Normalized Parameters

2.5 Effective Material Properties

Problems

Computer Assignment

References

Chapter 3: Thermoelectric Coolers

3.1 Ideal Equations

3.2 Maximum Parameters

3.3 Normalized Parameters

3.4 Effective Material Properties

Problems

Reference

Chapter 4: Optimal Design

4.1 Introduction

4.2 Optimal Design for Thermoelectric Generators

4.3 Optimal Design of Thermoelectric Coolers

Problems

References

Chapter 5: Thomson Effect, Exact Solution, and Compatibility Factor

5.1 Thermodynamics of Thomson Effect

5.2 Exact Solutions

5.3 Compatibility Factor

5.4 Thomson Effects

Problems

Projects

References

Chapter 6: Thermal and Electrical Contact Resistances for Micro and Macro Devices

6.1 Modeling and Validation

6.2 Micro and Macro Thermoelectric Coolers

6.3 Micro and Macro Thermoelectric Generators

Computer Assignment

References

Chapter 7: Modeling of Thermoelectric Generators and Coolers With Heat Sinks

7.1 Modeling of Thermoelectric Generators With Heat Sinks

7.2 Plate Fin Heat Sinks

7.3 Modeling of Thermoelectric Coolers With Heat Sinks

Problems

References

Chapter 8: Applications

8.1 Exhaust Waste Heat Recovery

8.2 Solar Thermoelectric Generators

8.3 Automotive Thermoelectric Air Conditioner

Problems

References

Chapter 9: Crystal Structure

9.1 Atomic Mass

9.2 Unit Cells of a Crystal

9.3 Crystal Planes

Problems

Chapter 10: Physics of Electrons

10.1 Quantum Mechanics

10.2 Band Theory and Doping

Problems

References

Chapter 11: Density of States, Fermi Energy, and Energy Bands

11.1 Current and Energy Transport

11.2 Electron Density of States

11.3 Fermi-Dirac Distribution

11.4 Electron Concentration

11.5 Fermi Energy in Metals

11.6 Fermi Energy in Semiconductors

11.7 Energy Bands

Problems

References

Chapter 12: Thermoelectric Transport Properties for Electrons

12.1 Boltzmann Transport Equation

12.2 Simple Model of Metals

12.3 Power-Law Model for Semiconductors

12.4 Electron Relaxation Time

12.5 Multiband Effects

12.6 Nonparabolicity

Problems

References

Chapter 13: Phonons

13.1 Crystal Vibration

13.2 Specific Heat

13.3 Lattice Thermal Conductivity

Problems

References

Chapter 14: Low-Dimensional Nanostructures

14.1 Low-Dimensional Systems

Problems

References

Chapter 15: Generic Model of Bulk Silicon and Nanowires

15.1 Electron Density of States for Bulk and Nanowires

15.2 Carrier Concentrations for Two-band Model

15.3 Electron Transport Properties for Bulk and Nanowires

15.4 Electron Scattering Mechanisms

15.5 Lattice Thermal Conductivity

15.6 Phonon Relaxation Time

15.7 Input Data for Bulk Si and Nanowires

15.8 Bulk Si

15.9 Si Nanowires

Problems

References

Chapter 16: Theoretical Model of Thermoelectric Transport Properties

16.1 Introduction

16.2 Theoretical Equatons

16.3 Results and Discussion

16.4 Summary

Problems

References

Appendix A: Physical Properties

References

Appendix B: Optimal Dimensionless Parameters for TEGs with ZT2 = 1

Appendix C: ANSYS TEG Tutorial

Appendix D: Periodic Table

Appendix E: Thermoelectric Properties

References

Appendix F: Fermi Integral

Appendix G: Hall Factor

References

Appendix H: Conversion Factors

Index

End User License Agreement

List of Tables

Table 1.1

Table 2.1

Table 3.1

Table 3.2

Table 4.1

Table 4.2

Table E4.2

Table 5.1

Table 5.2

Table 6.1

Table 6.2

Table 6.3

Table 7.1

Table 7.2

Table 7.3

Table 7.4

Table 8.1

Table 8.2

Table 8.3

Table 8.4

Table 8.5

Table 8.6

Table 8.7

Table 8.8

Table 8.9

Table 8.10

Table 8.11

Table 8.12

Table 9.1

Table 10.1

Table 10.2

Table 10.3

Table 10.4

Table 10.5

Table 10.6

Table 15.1

Table 16.1

Table A.1

Table A.2

Table A.3

Table A.4

Table A.5

Table A.6

Table A.7

Table A.8

Table A.9

Table A.10

Table A.11

Table A.12

Table A.13

Table B.1

Table B.2

Table B.3

Table D-1

Table D-2

Table D-3

Table D-4

Table D-5

Table D-6

Table D-7

Table D-8

Table D-9

Table D-10

Table D-11

Table D-12

Table D-13

Table D-14

Table F-1

Table F-2

List of Illustrations

Figure 1.1

Figure 1.2

Figure 1.3

Figure 1.4

Figure 1.5

Figure 2.1

Figure 2.2

Figure 2.6

Figure 2.7

Figure 2.3

Figure 2.4

Figure 2.5

Figure P2-1

Figure P2-2

Figure 3.1

Figure 3.2

Figure 3.3

Figure 3.4

Figure 3.5

Figure 3.6

Figure 3.7

Figure P3-1

Figure 4.1

Figure 4.2

Figure 4.3

Figure 4.4

Figure 4.5

Figure 4.6

Figure 4.7

Figure 4.8

Figure 4.9

Figure 4.10

Figure 4.11

Figure 4.12

Figure 4.13

Figure 4.14

Figure 4.15

Figure E4.1-1

Figure E4.1-2

Figure E4.2

Figure P4.1

Figure P4.2

Figure 5.1

Figure 5.2

Figure 5.3

Figure 5.4

Figure 5.5

Figure 5.6

Figure 5.7

Figure 5.8

Figure 5.9

Figure 5.10

Figure 5.11

Figure 5.12

Figure 5.13

Figure 5.14

Figure 6.1

Figure 6.2

Figure 6.3

Figure 6.4

Figure 6.5

Figure 6.6

Figure 6.7

Figure 6.8

Figure 6.9

Figure 6.10

Figure 7.1

Figure 7.2

Figure 7.3

Figure 7.4

Figure 7.5

Figure 7.6

Figure 7.7

Figure 7.8

Figure 7.9

Figure 7.10

Figure 7.11

Figure 7.12

Figure 7.13

Figure 8.1

Figure 8.2

Figure 8.3

Figure 8.4

Figure 8.5

Figure 8.6

Figure 8.7

Figure 8.8

Figure 8.9

Figure 8.10

Figure 8.11

Figure 8.12

Figure 8.13

Figure 8.14

Figure 8.15

Figure 8.16

Figure 8.17

Figure 8.18

Figure 8.19

Figure 8.20

Figure 8.21

Figure 8.22

Figure 8.23

Figure 8.24

Figure 8.25

Figure 8.26

Figure 8.27

Figure 8.28

Figure 8.29

Figure 8.30

Figure 8.31

Figure 8.32

Figure 8.33

Figure 8.34

Figure 8.35

Figure 8.36

Figure 8.37

Figure 8.38

Figure 8.39

Figure 9.1

Figure 9.2

Figure 9.3

Figure 9.4

Figure 9.5

Figure 9.6

Figure 9.7

Figure 9.8

Figure 9.9

Figure 9.10

Figure 10.1

Figure 10.2

Figure 10.3

Figure 10.4

Figure 10.5

Figure 10.6

Figure 10.7

Figure 10.8

Figure 10.9

Figure 10.10

Figure 10.11

Figure 10.12

Figure 10.13

Figure 10.14

Figure 11.1

Figure 11.2

Figure 11.3

Figure 11.4

Figure 11.5

Figure 11.6

Figure 11.7

Figure 11.8

Figure 11.9

Figure 11.10

Figure 11.11

Figure 11.12

Figure 11.13

Figure 11.14

Figure 12.1

Figure 12.2

Figure 12.3

Figure 12.4

Figure P12.11

Figure 13.1

Figure 13.2

Figure 13.3

Figure 13.4

Figure 13.5

Figure 13.6

Figure 13.7

Figure 13.8

Figure 13.9

Figure 13.10

Figure 13.11

Figure 13.12

Figure 14.1

Figure 14.2

Figure 14.3

Figure 14.4

Figure 14.5

Figure 14.6

Figure 14.7

Figure 14.8

Figure 14.9

Figure 14.10

Figure 14.11

Figure 14.12

Figure 14.13

Figure 14.14

Figure 15.1

Figure 15.2

Figure 15.3

Figure 15.4

Figure 15.5

Figure 15.6

Figure 15.7

Figure 15.8

Figure 15.9

Figure 15.10

Figure 16.1

Figure 16.2

Figure 16.3

Figure 16.4

Figure 16.5

Figure 16.6

Figure 16.7

Figure 16.8

Figure 16.9

Figure 16.10

Figure 16.11

Figure 16.12

Figure 16.13

Figure 16.14

Figure 16.15

Figure 16.16

Figure 16.17

Figure 16.18

Figure 16.19

Figure 16.20

Figure 16.21

Figure B.1

Figure B.2

Figure E.1

Figure E.2

Figure E.3

Figure E.4

Figure E.5

Figure E.6

Figure E.7

Figure G.1

Thermoelectrics

Design and Materials

HoSung Lee

Western Michigan University, USA

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Library of Congress Cataloging-in-Publication Data

Names: Lee, HoSung, author.

Title: Thermoelectrics : design and materials / HoSung Lee.

Description: Chichester, UK ; Hoboken, NJ : John Wiley & Sons, 2016. | Includes bibliographical references and index.

Identifiers: LCCN 2016025876| ISBN 9781118848951 (cloth) | ISBN 9781118848937 (epub) | ISBN 9781118848920 (epdf)

Subjects: LCSH: Thermoelectric apparatus and appliances--Design and construction. | Thermoelectric materials.

Classification: LCC TK2950 .L44 2016 | DDC 621.31/243--dc23 LC record available at https://lccn.loc.gov/2016025876

A catalogue record for this book is available from the British Library.

Cover design by Yujin Lee

ISBN: 9781118848951

Dedication

For Young-Ae and Yujin.

Preface

This book is written as a senior undergraduate or first-year graduate textbook. Thermoelectrics is a study of the energy conversion between thermal energy and electrical energy in solid state matters. Thermoelectrics is an emerging field with comprehensive applications such as exhaust waste heat recovery, solar energy conversion, automotive air conditioner, deep-space exploration, electronic control and cooling, and medical instrumentation. Thermoelectrics involves multiple interdisciplinary fields: physics, chemistry, electronics, material sciences, nanotechnology, and mechanical engineering. Much of the theories and materials are still under development, mostly on the materials but minimally on the design. The author has taught the thermoelectrics courses in the past years with a mind that a textbook is necessary to put a spur on the development. However, the author experienced considerable difficulties, partly because of the need to make a selection from the existing material and partly because the customary exposition of many topics to be included does not possess the necessary physical clarity. It is realized that the author's own treatment still has many defects, which are desirable to correct in future editions. The author has an open mind and appreciation for any comments and defects that may be found in the book. Typically, design and materials are separate fields, but in thermoelectrics, the two fields are interrelated particularly when the size is small and the variation of temperature is large. Hence, this book includes the design and materials for future vigorous engagement.

This book consists of two parts: design (Chapters 1 through 8) and materials (Chapters 9 through 16). The design covers the theoretical formulation, optimal design, experimental verification, modeling, and applications. The materials cover the physics of thermoelectrics for electrons and phonons, experimental verification, modeling, nanostructures, and thermoelectric materials. Each part can be suggestably used for a semester period (usually an introductory session for the subtle phenomena of thermoelectrics is given in the beginning of class) or two parts in a semester period by skimming some topics when students or readers are familiar with the topics. The author put significant effort into managing the contents in Part I with a fundamental heat transfer course as prerequisite and the contents in Part II with an introductory material science course. The author also attempted to provide detailed derivation of formulas so students or readers can have a conviction on studying thermoelectrics, as well as to provide detailed calculations, so that they can even build their own mathematical programs. Hence, many exercise problems at the end of chapter ask students or readers to provide Mathcad programs for the problem solutions.

I would like to acknowledge the suggestions and help provided by undergraduate and graduate students through classes and research projects. Special thanks are given to Dr. Alaa Attar, who is now professor at King AbdulAziz University, Saudi Arabia, for his help in measurements and computations. I am also indebted to Professor Emeritus Herman Merte Jr. for his lifetime inspiration on the preparation of the book. I am very grateful to Professor Emeritus Stanley L. Rajnak, who read the manuscript and made many useful comments.

HoSung Lee

Kalamazoo, Michigan

1

Introduction

1.1 Introduction

Thermoelectrics is literally associated with thermal and electrical phenomena. Thermoelectric processes can directly convert thermal energy into electrical energy or vice versa. A thermocouple uses the electrical potential (electromotive force) generated between two dissimilar wires to measure temperature. Basically, there are two devices: thermoelectric generators and thermoelectric coolers. These devices have no moving parts and require no maintenance. Thermoelectric generators have great potential for waste heat recovery from power plants and automotive vehicles. Such devices can also provide reliable power in remote areas such as in deep space and mountaintop telecommunication sites. Thermoelectric coolers provide refrigeration and temperature control in electronic packages and medical instruments. The science of thermoelectrics has become increasingly important with numerous applications. Since thermoelectricity was discovered in the early nineteenth century, there has not been much improvement in efficiency or materials until the recent development of nanotechnology, which has led to a remarkable improvement in performance. It is, thus, very important to understand the fundamentals of thermoelectrics for the development and the thermal design. We start with a brief history of thermoelectricity.

In 1821, Thomas J. Seebeck discovered that an electromotive force or a potential difference could be produced by a circuit made from two dissimilar wires when one of the junctions was heated. This is called the Seebeck effect.

Thirteen years later, in 1834, Jean Peltier discovered the reverse process—that the passage of an electric current through a thermocouple produces heating or cooling depending on its direction. This is called the Peltier effect. Although these two effects were demonstrated to exist, it was very difficult to measure each effect as a property of the material because the Seebeck effect is always associated with two dissimilar wires and the Peltier effect is always followed by the additional Joule heating that is heat generation due to the electrical resistance to the passage of a current. Joule heating was discovered in 1841 by James P. Joule.

In 1854, William Thomson (later Lord Kelvin) discovered that if a temperature difference exists between any two points of a current-carrying conductor, heat is either liberated or absorbed depending on the direction of current and material, which is in addition to the Peltier heating. This is called the Thomson effect. He also studied the relationships between these three effects thermodynamically, showing that the electrical Seebeck effect results from a combination of the thermal Peltier and Thomson effects. Although the Thomson effect itself is small compared with the other two, it leads to a very important and useful relationship, which is called the Kelvin relationship.

The mechanisms of thermoelectricity were not understood well until the discovery of electrons at the end of the nineteenth century. Now it is known that solar energy, an electric field, or thermal energy can liberate some electrons from their atomic binding, even at room temperature, moving them (from the valence band to the conduction band of a conductor) where the electrons are free to move. This is the reason why we have electrostatics everywhere. However, when a temperature difference across a conductor is applied as shown in Figure 1.1, the hot region of the conductor produces more free electrons, and diffusion of these electrons (charge carriers including holes) naturally occurs from the hot region to the cold region. On the other hand, the electron distribution provokes an electric field, which also causes the electrons to move from the hot region to the cold region via the Coulomb forces. Hence, an electromotive force (emf) is generated in a way that an electric current flows against the temperature gradient. As mentioned, the reverse is also true. If a current is applied to the conductor, electrons move and interestingly carry thermal energy. Therefore, a heat flow occurs in the opposite direction of the current, which is also shown in Figure 1.1.

Figure depicting the electron concentrations in a thermoelectric material.

Figure 1.1 Electron concentrations in a thermoelectric material

In many applications, a number of thermocouples, each of which consists of p-type and n-type semiconductor elements, are connected electrically in series and thermally in parallel by sandwiching them between two high–thermal conductivity but low–electrical conductivity ceramic plates to form a module, which is shown in Figure 1.2.

Figure depicting the cutaway of a typical thermoelectric module, where n-type semiconductor, p-type semiconductor, electrical insulator, (ceramic), and electrical conductor (copper) are depicted.

Figure 1.2 Cutaway of a typical thermoelectric module

Consider two wires made from different metals joined at both ends, as shown in Figure 1.3, forming a close circuit. Ordinarily, nothing will happen. However, when one of the junctions is heated, something interesting happens. Current flows continuously in the circuit. this is the Seebeck effect. The circuit that incorporates both thermal and electrical effects is called a thermoelectric circuit. A thermocouple uses the Seebeck effect to measure temperature, and the effect forms the basis of a thermoelectric generator.

Figure depicting the thermocouple where wire A is joined at both ends to wire B and a voltmeter is inserted in wire B. The junction temperatures are Tc and Th.

Figure 1.3 Thermocouple

In 1834, Jean Peltier discovered the reverse of the Seebeck effect by demonstrating that cooling can take place by applying a current across the junction. The heat pumping is possible without a refrigerator or compressor. The thermal energy can convert to electrical energy without turbine or engines.

There are some advantages of thermoelectric devices despite their low thermal efficiency. There are no moving parts in the device; therefore, there is less potential for failure in operation. Controllability of heating and cooling is very attractive in many applications such as lasers, optical detectors, medical instruments, and microelectronics.

1.2 Thermoelectric Effect

The thermoelectric effect consists of three effects: the Seebeck effect, the Peltier effect, and the Thomson effect.

1.2.1 Seebeck Effect

The Seebeck effect is the conversion of a temperature difference into an electric current. As shown in Figure 1.3, wire A is joined at both ends to wire B and a voltmeter is inserted in wire B. Suppose that a temperature difference is imposed between two junctions; then, it will generally be found that a potential difference or voltage V will appear on the voltmeter. The potential difference is proportional to the temperature difference. The potential difference V is

(1.1) equation

where ΔT = Th − Tc and ; is called the Seebeck coefficient (also called the thermopower), which is usually measured in μV/K. The sign of α is positive if the emf tends to drive an electric current through wire A from the hot junction to the cold junction, as shown in Figure 1.3. In practice, one rarely measures the absolute Seebeck coefficient because the voltage meter always reads the relative Seebeck coefficient between wires A and B. The absolute Seebeck coefficient can be calculated from the Thomson coefficient.

1.2.2 Peltier Effect

When current flows across a junction between two different wires, it is found that heat must be continuously added or subtracted at the junction in order to keep its temperature constant, which is illustrated in Figure 1.4. The heat is proportional to the current flow and changes sign when the current is reversed. Thus, the Peltier heat absorbed or liberated is

(1.2) equation

where πAB is the Peltier coefficient and the sign of πAB is positive if the junction at which the current enters wire A is heated and the junction at which the current leaves wire A is cooled. The Peltier heating or cooling is reversible between heat and electricity. This means that heating (or cooling) will produce electricity and electricity will produce heating (or cooling) without a loss of energy.

Figure depicting a schematic for the Peltier effect and the Thomson effect.

Figure 1.4 Schematic for the Peltier effect and the Thomson effect

1.2.3 Thomson Effect

When current flows as shown in Figure 1.4, heat is absorbed in wire A due to the negative temperature gradient and liberated in wire B due to the positive temperature gradient, which is experimental observation [1], depending on the material. The Thomson heat is proportional to both the electric current and the temperature gradient, which is schematically shown in Figure 1.4. Thus, the Thomson heat absorbed or liberated across a wire is

(1.3) equation

where τ is the Thomson coefficient. The Thomson coefficient is unique among the three thermoelectric coefficients because it is the only thermoelectric coefficient directly measurable for individual materials. There is other form of heat, called Joule heating (I²R), which is irreversible and is always generated as current flows in a wire. The Thomson heat is reversible between heat and electricity.

1.2.4 Thomson (or Kelvin) Relationships

The interrelationships between the three thermoelectric effects are important in order to understand the basic phenomena. In 1854, Thomson [2] studied the relationships thermodynamically and provided two relationships as shown in Equations (1.4) and (1.5) by applying the first and second laws of thermodynamics with the assumption that the reversible and irreversible processes in thermoelectricity are separable. The necessity for the assumption remained an objection to the theory until the advent of the new thermodynamics. The Thomson effect is relatively small compared with the Peltier effect, but it plays an important role in deducing the Thomson relationships. These relationships were later completely confirmed by experiments (See Chapter 5 for details).

(1.4) equation

(1.5) equation

Equation (1.4) leads to the very useful Peltier cooling in Equation (1.2) as

(1.6) equation

where T is the temperature at a junction between two different materials and the dot above the heat Q indicates the amount of heat transported per unit time.

1.3 The Figure of Merit

The performance of thermoelectric devices is measured by the figure of merit (Z), with units 1/K:

(1.7) equation

where

The dimensionless figure of merit is defined by ZT, where T is the absolute temperature. There is no fundamental limit on ZT, but for decades it was limited to values around ZT 1 in existing devices. The larger the value of ZT, the greater is the energy conversion efficiency of the material. The quantity of is defined as the power factor. Therefore, both the Seebeck coefficient α and electrical conductivity σ must be large, while the thermal conductivity k must be minimized. This well-known interdependence among the physical properties makes it challenging to develop strategies for improving a material's ZT.

1.3.1 New-Generation Thermoelectrics

Although Seebeck observed thermoelectric phenomena in 1821 and Altenkirch defined Equation (1.7) in 1911, it took several decades to develop the first functioning devices in the 1950s and 1960s. They are now called the first-generation thermoelectrics with an average of Z ∼ 1.0. Devices made of them can operate at ∼5% conversion efficiency. After several more decades of stagnancy, new theoretical ideas relating to size effects on thermoelectric properties in the 1990s stimulated new experimental research that eventually led to significant advances in the following decade. Although the theoretical ideas were originally about prediction on raising the power factor, the experimental breakthroughs were achieved by significantly decreasing the lattice thermal conductivity. Among a wide variety of research approaches, one has emerged, which has led to a near doubling of ZT at high temperatures and defines the second generation of bulk thermoelectric materials with ZT in the range of 1.3–1.7. This approach uses nanoscale precipitates and composition inhomogeneities to dramatically suppress the lattice thermal conductivity. These second-generation materials are expected to eventually produce power-generation devices with conversion efficiencies of 11–15% [3].

Third-generation bulk thermoelectrics have been under development recently, which integrate many cutting-edge ZT-enhancing approaches simultaneously, namely, enhancement of Seebeck coefficients through valence band convergence, retention of the carrier mobility through band energy offset minimization between matrix and precipitates, and reduction of the lattice thermal conductivity through all length-scale lattice disorder and nanoscale endotaxial precipitates to mesoscale grain boundaries and interfaces. This third generation of bulk thermoelectrics exhibits high ZT, ranging from 1.8 to 2.2, depending on the temperature difference, and a consequent predicted device conversion efficiency increase to ∼15–20% [3].

Table 1.1 shows the thermoelectric properties of bulk nanocomposite semiconductors. Figure 1.5 shows the dimensionless figures of merit for the materials in Table 1.1.

Table 1.1 Thermoelectric Properties of Single Crystal and Bulk Nanocomposite Semiconductors

A graphical representation for dimensionless figures of merit for various thermoelectric materials, where ZT is plotted on the y-axis on a scale of 0–2.5 and temperature (K) on the x-axis on a scale of 200–1.3×103.

Figure 1.5 Dimensionless figures of merit for various nanocomposite thermoelectric materials

Problems

1.1 Briefly describe the thermoelectric effect.

1.2 Describe the dimensionless figure of merit and why it is important in thermoelectric design.

1.3 Describe the Thomson relations.

References

1. Nettleton, H.R. The Thomson effect. Proceedings of the Physical Society of London. 1922.

2. Thomson, W., Account of researches in thermo-electricity. Proceedings of the Royal Society of London, 1854. 7: p. 49–58.

3. Zhao, L.-D., V.P. Dravid, and M.G. Kanatzidis, The panoscopic approach to high performance thermoelectrics. Energy & Environmental Science, 2014. 7(1): p. 251.

4. Jeon, H.-W., et al., Electrical and thermoelectrical properties of undoped Bi2Te3-SbeTe3 and Bi2Te3-Sb2Te3-Sb2Se3 single crystals. Journal of Physics and Chemistry of Solids, 1991. 52(4): p. 579–585.

5. Poudel, B., et al., High-thermoelectric performance of nanostructured bismuth antimony telluride bulk alloys. Science, 2008. 320(5876): p. 634–8.

6. Yan, X., et al., Experimental studies on anisotropic thermoelectric properties and structures of n-type Bi2Te2.7Se0.3. Nano Letters, 2010. 10(9): p. 3373–8.

7. Biswas, K., et al., High-performance bulk thermoelectrics with all-scale hierarchical architectures. Nature, 2012. 489(7416): p. 414–8.

8. Dismukes, J.P., et al., Thermal and electrical properties of heavily doped Ge-Si alloys up to 1300°K. Journal of Applied Physics, 1964. 35(10): p. 2899.

9. Wang, X.W., et al., Enhanced thermoelectric figure of merit in nanostructured n-type silicon germanium bulk alloy. Applied Physics Letters, 2008. 93(19): p. 193121.

10. Caillat, T., A. Borshchevsky, and J.P. Fleurial, Properties of single crystalline semiconducting CoSb3. Journal of Applied Physics, 1996. 80(8): p. 4442.

11. Tang, X., et al., Synthesis and thermoelectric properties of p-type- and n-type-filled skutterudite R[sub y]M[sub x]Co[sub 4−x]Sb[sub 12](R:Ce,Ba,Y;M:Fe, Ni). Journal of Applied Physics, 2005. 97(9): p. 093712.

12. Brown, S.R., et al., Yb14MnSb11: New high efficiency thermoelectric material for power generation. Chemistry of Materials, 2006. 18: p. 1873–1877.

13. May, A.F., J.-P. Fleurial, and G.J. Snyder, Optimizing thermoelectric efficiency in La3−xTe4via Yb substitution. Chemistry of Materials, 2010. 22(9): p. 2995–2999.

2

Thermoelectric Generators

2.1 Ideal Equations

In 1821, Thomas J. Seebeck discovered that an electromotive force or potential difference could be produced by a circuit made from two dissimilar wires when one junction was heated. This is called the Seebeck effect. In 1834, Jean Peltier discovered the reverse process—that the passage of an electric current through a thermocouple produces heating or cooling depending on its direction [1]. This is called the Peltier effect (or Peltier cooling). In 1854, William Thomson discovered that if a temperature difference exists between any two points of a current-carrying conductor, heat is either absorbed or liberated depending on the direction of current and material [2]. This is called the Thomson effect (or Thomson heat). These three effects are called the thermoelectric effects.

Let us consider a non–uniformly heated thermoelectric material. For an isotropic substance, the continuity equation for a constant current gives

(2.1) equation

The electric field is affected by the current density and the temperature gradient . The relationships are known as Ohm's law and the Seebeck effect [3]. The electric field is then expressed as

(2.2) equation

where ρ is the electrical resistivity. The heat flux is also affected by both the field and the temperature gradient . However, their coefficients were not readily attainable at that time. Thomson in 1854 arrived at the relationship assuming that thermoelectric phenomena and thermal conduction are independent [2]. Later, Onsager [4] supported that relationship by presenting the reciprocal principle, which was experimentally proved. The Thomson relationship and the Onsager's principle yielded a formula for the heat flow density vector (heat flux),

(2.3) equation

This is the most important equation in thermoelectrics (will be discussed later in detail). The general heat diffusion equation is given by

(2.4) equation

For steady state, we have

(2.5) equation

where is expressed by [3]

(2.6) equation

Substituting Equations (2.3) and (2.6) into Equation (2.5) yields

(2.7) equation

The Thomson coefficient τ, originally obtained from the Thomson relations, is defined by

(2.8) equation

In Equation (2.7), the first term is the thermal conduction, the second term is the Joule heating, and the third term is the Thomson heat. Note that if the Seebeck coefficient α is independent of temperature, the Thomson coefficient τ becomes 0 and then the Thomson heat is absent. The two equations, (2.3) and (2.7), govern thermoelectric phenomena.

Consider a steady-state one-dimensional thermoelectric generator module as shown in Figure 2.1. The module consists of many p-type and n-type thermocouples, where one thermocouple (unicouple) with a circuit is shown in Figure 2.2. We assume that the electrical and thermal contact resistances are negligible, the Seebeck coefficient is independent of temperature, and the radiation and convection at the surfaces of the elements are negligible. Then, Equation (2.7) reduces to

Figure depicting the cutaway of a thermoelectric generator module, where n-type semiconductor, p-type semiconductor, electrical insulator, (ceramic), and electrical conductor (copper) are depicted.

Figure 2.1 Cutaway of a thermoelectric generator module

Figure depicting the p- and n-type unit thermocouple for a thermoelectric generator.

Figure 2.2 The p- and n-type unit thermocouple for a thermoelectric generator

(2.9) equation

The solution for the temperature gradient with two boundary conditions ( and ) in Figure 2.2 is

(2.10) equation

Equation (2.3) is expressed in terms of p-type and n-type thermoelements.

(2.11)

equation

where is the rate of heat absorbed at the hot junction in Figure 2.2 and n is the number of thermocouples. Substituting Equation (2.10) into Equation (2.11) gives

(2.12)

equation

Finally, the heat absorbed at the hot junction with temperature Th is expressed as

(2.13) equation

where

(2.14) equation

(2.15) equation

(2.16) equation

R is the electrical resistance and K is the thermal conductance. If we assume that the p-type and n-type thermocouples are similar, we have that R = ρL/A and K = kA/L, where ρ = ρp + ρn and k = kp + kn. Equation (2.13) is called the ideal equation and has been widely used in science and industry. The rate of heat liberated at the cold junction is given by

(2.17) equation

From the first law of thermodynamics for the thermoelectric module, the power output is . The total power output is then expressed in terms of the internal properties as

(2.18) equation

However, the total power output in Figure 2.2 can be defined by an external load resistance as

(2.19) equation

Equating Equations (2.18) and (2.19) with gives the total voltage as

(2.20) equation

2.2 Performance Parameters of a Thermoelectric Module

From Equation (2.20), the electrical current for the module is obtained as

(2.21) equation

Note that the current I is independent of the number of thermocouples. Inserting this into Equation (2.20) gives the voltage across the module by

(2.22) equation

Inserting Equation (2.21) in Equation (2.19) gives the power output as

(2.23) equation

The conversion (or thermal) efficiency is defined as the ratio of the power output over the heat absorbed at the hot junction:

(2.24) equation

Inserting Equations (2.13) and (2.23) into Equation (2.24) gives an expression for the conversion efficiency:

(2.25)

equation

where the average temperature is defined as . It is noted that the Carnot cycle efficiency is .

2.3 Maximum Parameters for a Thermoelectric Module

Because the maximum current inherently occurs in a short circuit where in Equation (2.21), the maximum current for the module is

(2.26) equation

The maximum voltage inherently occurs in an open circuit where I = 0 in Equation (2.20). The maximum voltage is

(2.27) equation

The maximum power output is attained by differentiating the power output in Equation (2.23) with respect to the ratio of the load resistance to the internal resistance and setting it to 0. The result yields a relationship of , which leads to the maximum power output as

(2.28) equation

The maximum conversion efficiency can be obtained by differentiating the conversion efficiency in Equation (2.25) with respect to the ratio of the load resistance to the internal resistance and setting it to zero. The result yields a relationship of . Then, the maximum conversion efficiency is

(2.29) equation

There are a total of four essential maximum parameters: , , , and . However, there is also the maximum power efficiency. The maximum power efficiency is obtained by letting in Equation (2.25). The maximum power efficiency is

(2.30)

equation

Note there are two thermal efficiencies: the maximum power efficiency and the maximum conversion efficiency .

2.4 Normalized Parameters

If we divide the actual values by the maximum values, we can normalize the characteristics of a thermoelectric generator. The normalized power output can be obtained by dividing Equation (2.23) by Equation (2.28), which leads to

(2.31) equation

Equations (2.21) and (2.26) give the normalized currents as

(2.32) equation

Equations