Continuum Theory and Modeling of Thermoelectric Elements

List of Contributors XIII

Preface XV

List of Frequently Used Symbols XVII

Glossary XIX

1 Thermodynamics and Thermoelectricity 1
Christophe Goupil, Henni Ouerdane, Knud Zabrocki, Wolfgang Seifert, Nicki F. Hinsche, and Eckhard Müller

1.1 Milestones of Thermoelectricity 1

1.1.1 Discovery of the Seebeck Effect 2

1.1.2 Discovery of the Peltier Effect 8

1.1.3 Discovery of the Thomson Effect 9

1.1.4 Magnus' Law 10

1.1.5 Early Performance Calculation of Thermoelectric Devices 11

1.1.6 First Evaluation of the Performance of a Thermoelectric Device by E. Altenkirch 11

1.1.7 Benedicks' Effect 12

1.1.8 The Bridgman Effect 13

1.1.9 Semiconductors as Thermoelectric Materials 14

1.1.10 Thermoelectric Applications –Excitement and Disappointment 1920–1970 15

1.1.11 Thermoelectric Industry – Niche Applications 1970–2000 16

1.1.12 New Concepts in Thermoelectricity 2000-Present 17

1.2 Galvanomagnetic and Thermomagnetic Effects 17

1.2.1 The Hall Coefficient 21

1.2.2 The Nernst Coefficient 21

1.2.3 The Ettingshausen Coefficient 21

1.2.4 The Righi–Leduc Coefficient 22

1.2.5 Devices Using Galvano- and Thermomagnetic Effects and the Corresponding Figure of Merit 22

1.3 Historical Notes on Thermodynamic Aspects 25

1.4 Basic Thermodynamic Engine 27

1.5 Thermodynamics of the Ideal Fermi Gas 28

1.5.1 The Ideal Fermi Gas 28

1.5.2 Electron Gas in a Thermoelectric Cell 29

1.5.3 Entropy Per Carrier 30

1.5.4 Equation of State of the Ideal Electron Gas 32

1.5.5 Temperature Dependence of the Chemical Potential μc(T) 34

1.6 Linear Nonequilibrium Thermodynamics 35

1.6.1 Forces and Fluxes 35

1.6.2 Linear Response and Reciprocal Relations 36

1.7 Forces and Fluxes in Thermoelectric Systems 37

1.7.1 Thermoelectric Effects 37

1.7.2 Forces, Fluxes, and Kinetic Coefficients 38

1.7.3 Energy Flux and Heat Flux 39

1.7.4 Thermoelectric Coefficients 40

1.7.5 The Entropy Per Carrier 41

1.7.6 Kinetic Coefficients and Transport Parameters 42

1.7.7 The Dimensionless Figure of Merit zT 43

1.8 Heat and Entropy 44

1.8.1 Volumetric Heat Production 45

1.8.2 Entropy Production Density 45

1.8.3 Heat Flux and the Peltier–Thomson Coefficient 46

1.8.4 The Peltier–Thomson Term 46

1.8.5 Local Energy Balance 47

1.9 The Thermoelectric Engine and Its Applications 48

1.10 Thermodynamics and Thermoelectric Potential 50

1.10.1 Relative Current, Dissipation Ratio, and Thermoelectric Potential 51

1.10.2 Local Reduced Efficiency and Thermoelectric Potential of TEG, TEC, and TEH 53

1.10.3 Thermoelectric Potential and Nonequilibrium Thermodynamics 56

2 Continuum Theory of TE Elements 75
Knud Zabrocki, Christophe Goupil, Henni Ouerdane, Yann Apertet, Wolfgang Seifert, and Eckhard Müller

2.1 Domenicali's Heat Balance Equation 75

2.1.1 Tensorial Character of Material Properties 75

2.1.2 Heat Balance and Source Terms 76

2.1.3 Spatial and Temperature Averaging of the Material Properties 79

2.2 Transferred Heat Balance 80

2.3 Ioffe's Description and Performance Parameters of CPM Devices 81

2.3.1 Single-Element Device 82

2.3.2 Performance Parameters of a Thermoelectric Element with Constant Material Properties 84

2.3.3 Inverse Performance Equations and Effective Device Figure of Merit 91

2.4 Maximum Power and Efficiency of a Thermogenerator Element 93

2.4.1 Load Resistance as Design Parameter for a Thermogenerator 96

2.4.2 Efficiency versus Power Approach 97

2.4.3 Constant Heat Input (CHI) Model 100

2.5 Temperature-Dependent Materials–Analytic Calculations 102

2.5.1 Inverse Temperature Dependence of the Thermal Conductivity 103

2.5.2 Inverse Temperature Dependence of the Thermal Conductivity and a Variable Electrical Resistivity 105

2.5.3 Constant Thomson Coefficient–Logarithmic Behavior of the Seebeck Coefficient 111

2.5.4 Algebraic and General Temperature Dependence 119

2.5.5 Constant Thomson Coefficient Combined with Linear Temperature Dependence of Resistivity 121

2.5.6 Linear Temperature Dependence of the Resistivity 123

2.5.7 Linear Temperature Dependence of the Thermal Conductivity 125

2.6 The Influence of Contacts and Contact Resistances on the TE Performance 125

2.6.1 Thermoelectric Element with Contacting Bridge 126

2.6.2 Numerical Example for the Influence of the Electrical Contact Resistance on the Performance 130

2.7 Dissipative Coupling between the TEG and the Heat Baths 133

2.7.1 Finite-Time Thermodynamics Optimization 133

2.7.2 Thermoelectric Generator Model 133

2.7.3 Thermal Flux and Electrical Current 135

2.7.4 Calculation of the Temperature Difference across the TEG 136

2.7.5 Maximization of Power and Efficiency with Fixed ZTm 137

2.8 Shaped Thermoelectric Elements 140

2.9 Other Influences on the Performance of TE Devices 144

2.9.1 Lateral Heat Losses, Convective and Radiative Heat Transfer 144

2.9.2 Anisotropic Thermoelectric Elements 147

3 Segmented Devices and Networking of TE Elements 157
Knud Zabrocki, Christophe Goupil, Henni Ouerdane, Eckhard Müller, and Wolfgang Seifert

3.1 Segmented Devices 157

3.1.1 Double-Segmented Element 160

3.1.2 Algorithm of Multisegmented Elements 164

3.2 Networks 169

3.2.1 Presentation 169

3.2.2 Useful Expressions 171

3.2.3 Discretization 171

3.2.4 Solution: General Millman Theorem 172

3.2.5 Implementation 173

3.2.6 Numerical Illustration 174

4 Transient Response and Green's Function Technique 177
Wolfgang Seifert, Knud Zabrocki, Steven Achilles, and Steffen Trimper

4.1 Quasi-Stationary Processes 179

4.2 Supercooling with a Transient Peltier Cooler 182

4.2.1 Steady-State Operation of a Thermoelectric Cooler 183

4.2.2 Important Parameters for the Supercooling Case 187

4.3 Transient Behavior of a Thermoelectric Generator 189

4.4 Dynamic Measurements of the Thermal Conductivity: Laser Flash Analysis 190

4.5 Dynamic Measurements of the Thermal Conductivity: Classical Ioffe Method 193

4.5.1 Theoretical Basis of Simple Ioffe Method 194

4.5.2 Laplace Transformation and Important Properties 197

4.5.3 Solution of the Classical Ioffe Method 198

4.5.4 Solution of the Temperatures in the s-Domain 199

4.5.5 Inverse Laplace Transformation 202

4.5.6 Inversion Theorem for the Laplace Transformation 203

4.5.7 Inversion of the Temperature Profiles 205

4.6 Green's Function Approach in Thermoelectricity 206

4.6.1 Continuity Equations 208

4.6.2 Green's Function Approach in the Steady State 208

4.6.3 One-Dimensional Green's Functions in the Steady State 209

4.6.4 Perturbative Approach to a Full Description (1D) 210

4.7 Linear Transient Approach 212

4.7.1 Relaxation Time 212

4.7.2 Transient Field Equations 213

4.7.3 Transient Linear Response Approximation 214

4.8 Time-Dependent Green's Function Approach 215

5 Compatibility 227
Wolfgang Seifert, G. Jeffrey Snyder, Eric S. Toberer, Volker Pluschke, Eckhard Müller, and Christophe Goupil

5.1 Relative Current Density and Compatibility Factors 227

5.2 Compatibility and Segmented Thermogenerators 229

5.3 Reduced Efficiencies and Self-Compatible Performance 232

5.3.1 Performance Integrals for Efficiency and COP 233

5.3.2 Local Efficiency Dependence on Current (TEG) 235

5.4 Power-Related Compatibility 238

5.5 Optimal Material Grading for Maximum Power Output 241

5.6 The Criterion "u = s" and Calculus of Variations 243

5.7 Self-Compatibility and Optimum Material Grading 246

5.8 Thermodynamic Aspects of Compatibility 249

5.9 Analytic Results for Self-Compatible TEG and TEC Elements 251

5.9.1 Performance of Self-Compatible TEG and TEC Elements 251

5.9.2 Self-Compatible Elements and Optimal Figure of Merit 253

5.9.3 Optimal Seebeck Coefficients for Self-Compatible Material 255

5.9.4 Temperature Profile for u = s Material 256

5.10 Thermoelectric Thomson Cooler 259

5.10.1 Cooling Performance 262

5.10.2 Thomson Cooler Phase Space 265

5.10.3 Performance Limits 266

5.10.4 Further Characteristics of Self-Compatible Material for Cooling 268

5.11 Compatibility Approach versus Device Optimization 276

6 Numerical Simulation 281
Knud Zabrocki and Wolfgang Seifert

6.1 Finite Difference Methods 284

6.2 Finite Volume Method 288

6.3 Finite Element Method 290

6.4 Performance Calculation of a TEG-A Case Study 291

6.4.1 Averages of the Material Properties 294

6.4.2 Processing Measured Material Properties 295

6.4.3 Different Averages and the Corresponding Performance Values 297

6.4.4 Power Factor and Figure of Merit 298

6.4.5 Optimal Performance Based on Averaged Material Properties 299

6.4.6 Comparison between CPM, FDM, and FEM Simulations 300

6.4.7 Calculation for a p-n Thermocouple 309

6.4.8 Adjustment of Cross-Sectional Areas 310

6.4.9 FEM Simulation 311

6.5 Nonlinear Material Parameters 315

6.5.1 Temperature-Dependent Material Properties 315

6.5.2 Temperature-Dependent Material Properties: Mathematica Model Using the Compatibility Approach 320

A Numerical Data and Illustrative Cases 322

A.1 Coefficients of the Polynomials of the Material Properties 322

A.2 Material Properties: p-Type Skutterudite and Averages 323

A.3 Material Properties: n-Type Skutterudite and Averages 324

A.4 Power Factor and Figure of Merit of the n-Type Material 325

A.5 Performance Parameters in dependence on the Current Density 326

A.5.1 Program code 326

A.5.2 Results 327

Index 335

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