Chapter 1. Introduction
1.1 Symbols .................................................... 1
1.2 Thermphotovoltaic (TPV) Energy Conversion Concept .......... 3
1.3 A Short History of TPV Energy Conversion ................... 3
1.4 TPV Applications ........................................... 5
1.5 Propagation of Electromagnetic Waves ....................... 6
1.5.1 Plane Wave Solution to Maxwell's Equations .......... 6
1.5.2 Energy Flux for Plane Electromagnetic Waves ........ 14
1.5.3 Boundary Conditions at an Interface ................ 17
1.5.4 The Law of Reflection and Snell's Law of
Refraction ......................................... 20
1.5.5 Reflectivity and Transmissivity at an Interface .... 25
1.5.6 Connections between Electromagnetic Theory and
Radiation Transfer Theory .......................... 34
1.6 Introduction to Radiation Transfer ........................ 35
1.6.1 Radiation Intensity ................................ 35
1.6.2 Blackbody .......................................... 38
1.6.3 Blackbody Spectral Emissive Power .................. 39
1.6.4 Blackbody Total Emissive Power ..................... 42
1.6.5 Equations for Radiation Energy Transfer ............ 44
1.6.6 Energy Conservation Including Radiation ............ 47
1.7 Optical Properties ........................................ 50
1.7.1 Emittance and Absorptance .......................... 51
1.7.2 Hemispherical Spectral and Hemispherical Total
Reflectivity ....................................... 55
1.7.3 Independence of Emitted (Absorbed), Reflected,
and Transmitted Radiation .......................... 57
1.8 Radiation Energy Balance for One Dimensional Model ........ 63
1.9 Emittance of a Metal into a Dielectric .................... 66
1.10 Summary ................................................... 70
References ................................................ 71
Problems .................................................. 72
Chapter 2. Maximum Efficiency and Power Density for TPV
Energy Conversion
2.1 Symbols ................................................... 77
2.2 Maximum TPV Efficiency .................................... 79
2.3 Maximum TPV Efficiency for Constant Emitter Emittance
and PV Cell Reflectance ................................... 83
2.4 Ideal TPV System .......................................... 84
2.5 Approximation of Selective Emitter and Filter TPV
Systems ................................................... 86
2.6 Power Output .............................................. 89
2.7 Summary ................................................... 91
References ................................................ 92
Problems .................................................. 92
Chapter 3. Emitter Performances
3.1 Symbols ................................................... 95
3.2 Gray Body Emitters ........................................ 97
3.3 Selective Emitters ........................................ 98
3.3.1 Rare Earth Selective Emitters ...................... 99
3.3.2 Other Selective Emitters .......................... 102
3.4 Extinction Coefficient and Optical Depth ................. 104
3.5 Extinction Coefficients of Rare Earth Selective
Emitters ................................................. 105
3.6 Coupled Energy Equation and Radiation Transfer Equation
for a Solid Material ..................................... 108
3.7 One Dimensional Radiation Transfer Equations ............. 108
3.7.1 One Dimensional Source Function Equation .......... 112
3.7.2 One Dimensional Radiation Flux .................... 113
3.7.3 No Scattering Medium .............................. 114
3.8 Spectral Emittance for Planar Emitter .................... 115
3.8.1 No Scattering Spectral Emittance .................. 128
3.8.2 No Scattering, Linear Temperature Variation
Spectral Emittance ................................ 131
3.8.3 Importance of Temperature Change Across Planar
Emitter ........................................... 140
3.8.4 Effect of Scattering on Spectral Emittance of
a Planar Emitter .................................. 142
3.9 Cylindrical Emitter ...................................... 146
3.10 Emitter Performance ...................................... 153
3.10.1 Gray Body Emitter Performance ..................... 154
3.10.2 Selective Emitter Performance ..................... 157
3.10.3 Cylindrical Selective Emitter Performance ......... 157
3.10.4 Planar Selective Emitter Performance .............. 164
3.11 Comparison of Selective Emitters and Gray Body Emitters .. 170
3.12 Summary .................................................. 172
References ............................................... 174
Problems ................................................. 175
Chapter 4. Optical Filters for Thermophotovoltaics
4.1 Symbols .................................................. 179
4.2 Filter Performance Parameters ............................ 181
4.3 Interference Filters ..................................... 183
4.3.1 Introduction ...................................... 183
4.3.2 Interference ...................................... 183
4.3.3 Interference Filter Model ......................... 185
4.3.4 Reflectance, Transmittance, and Absorptance ....... 193
4.3.5 Single Film System ................................ 198
4.3.6 Many Layer System for Φi = Nπ or Φi = Nπ/2
and N is an Odd Integer ........................... 210
4.3.7 Equivalent Layer Procedure ........................ 214
4.3.8 Interference Filter with Embedded Metallic Layer .. 222
4.3.9 Interference Filter Performance for Angles of
Incidence Greater than Zero ....................... 229
4.4 Plasma Filters ........................................... 232
4.4.1 Drude Model ....................................... 232
4.4.2 Reflectance, Transmittance, and Absorptance of
a Plasma Filter ................................... 244
4.4.3 Efficiency and Total Transmittance, Reflectance,
and Absorptance of a Plasma Filter ................ 249
4.5 Combined Interference-Plasma Filter ...................... 253
4.6 Resonant Array Filters ................................... 260
4.6.1 Transmission Line Theory .......................... 261
4.6.2 Transmission Line Equivalent Circuit for
Resonant Array Filter ............................. 266
4.6.3 Metallic Mesh Filter .............................. 270
4.7 Spectral Control Using a Back Surface Reflector (BSR) .... 277
4.7.1 Efficiency of a Back Surface Reflector (BSR)
for Spectral Control .............................. 277
4.8 Summary .................................................. 283
References ............................................... 284
Problems ................................................. 286
Chapter 5. Photovoltaic Cells
5.1 Symbols .................................................. 291
5.2 Energy Bands (Kronig-Penney Model) and Current
in Semiconductors ........................................ 294
5.3 Density of Electrons and Holes and Mass Action Law ....... 302
5.4 Transport Equations ...................................... 310
5.5 Generation and Recombination of Electrons and Holes ...... 313
5.5.1 Generation of Electrons and Holes ................. 313
5.5.2 Recombination of Electrons and Holes .............. 316
5.6 p-n Junction ............................................. 320
5.7 Current-Voltage Relation for an Ideal Junction in the
Dark ..................................................... 323
5.7.1 Assumptions for Ideal p-n Junction ................ 324
5.7.2 Current-Voltage Relation for Infinite Neutral
Regions ........................................... 326
5.7.3 Current-Voltage Relation for Finite Neutral
Regions ........................................... 330
5.7.4 Depletion Region Contribution to Current and
High-Injection Effects ............................ 336
5.8 Ideality Factor and Empirical Current-Voltage Relation
of p-n Junction in the Dark .............................. 338
5.9 Current-Voltage Relation for an Ideal p-n Junction
Under Illumination ....................................... 339
5.9.1 Electron Current Density in p Region .............. 340
5.9.2 Hole Current Density in n Region .................. 349
5.9.3 Current Generation in Depletion Region ............ 357
5.9.4 Current-Voltage Relation .......................... 359
5.10 Quantum Efficiency and Spectral Response ................. 362
5.11 Equivalent Circuit for PV Cells .......................... 368
5.12 PV Cell Efficiency and Power Output ...................... 374
5.13 Summary .................................................. 388
References ............................................... 391
Problems ................................................. 392
Chapter 6. Governing Equations for Radiation Fluxes in
Optical Cavity
6.1 Symbols .................................................. 395
6.2 Radiation Transfer Theory ................................ 397
6.2.1 Radiation Transfer for Uniform Intensity .......... 397
6.2.2 View-Factors for TPV Systems ...................... 399
6.2.3 Optical Properties of Components .................. 405
6.2.4 Energy Balance on a Component of a TPV System ..... 410
6.3 Radiation Energy Transfer in Planar TPV System ........... 413
6.4 Radiation Energy Transfer in Cylindrical TPV System ...... 417
6.5 Efficiency of TPV Systems ................................ 422
6.5.1 Overall Efficiency ................................ 422
6.5.2 Thermal Efficiency ................................ 422
6.5.3 Cavity Efficiency ................................. 422
6.5.4 Photovoltaic Efficiency ........................... 423
6.6 Summary .................................................. 423
References ............................................... 424
Problems ................................................. 424
Chapter 7. Radiation Losses in Optical Cavity
7.1 Symbols .................................................. 427
7.2 Cavity Efficiency for Planar Filter and Selective
Emitter TPV Systems without a Window ..................... 428
7.3 Cavity Efficiency for Cylindrical Filter and Selective
Emitter TPV Systems without a Window ..................... 440
7.4 Cavity Efficiency for TPV Systems with Reflectivity End
Caps ..................................................... 446
7.4.1 Development of Radiation Transfer Equations ....... 446
7.4.2 Radiation Transfer Equations for TPV Systems
with Close Coupled Emitter-Window and Filter-PV
Cells ............................................. 450
7.4.3 Cavity Efficiency ................................. 454
7.5 Summary .................................................. 457
Problems ................................................. 458
Chapter 8. TPV System Performance
8.1 Symbols .................................................. 461
8.2 TPV System Model ......................................... 462
8.3 Radiation Transfer Equations ............................. 463
8.4 Solution Method for TPV System Model ..................... 468
8.5 Results of TPV System Model for Hypothetical System ...... 472
8.5.1 Importance of Radiation Leakage ................... 474
8.5.2 Importance of Filter Absorptance .................. 474
8.6 TPV System with Selective Emitter and Back Surface
Reflector (BSR) .......................................... 476
8.6.1 Dependence of TPV Performance upon Input Power .... 479
8.7 Importance of PV Array Temperature on TPV Performance .... 483
8.8 Review of Radiation Transfer Method ...................... 484
8.9 Summary .................................................. 485
Problems ................................................. 485
Appendices
Appendix A - Exponential Integrals ....................... 491
Appendix В - Coupled Energy and Radiation Transfer
Equations ................................... 495
Appendix С - 2 × 2 Matrix Algebra ........................ 499
Appendix D - Mathematica Program for Multi-layer
Interference Filter ......................... 501
Appendix E - Quantum Mechanics ........................... 503
Appendix F - Mathematica Program for Planar Geometry
TPV Model ................................... 509
Index ......................................................... 513
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