Preface to the Fifth Edition .................................. xxi
List of Symbols ............................................. xxiii
Chapter 1 Introduction to Radiative Transfer .................... 1
1.1 Importance of Thermal Radiation in Engineering ........... 3
1.2 Thermal Energy Transfer .................................. 4
1.3 Thermal Radiative Transfer ............................... 6
1.4 Radiative Energy Exchange and Radiative Intensity ........ 8
1.4.1 Solid Angle ........................................ 9
1.4.2 Spectral Radiative Intensity ...................... 10
1.5 Characteristics of Emission ............................. 11
1.5.1 Perfect Emitter ................................... 13
1.5.2 Radiation Isotropy in a Black Enclosure ........... 13
1.5.3 Perfect Emitter in Each Direction and
Wavelength ........................................ 13
1.5.4 Total Radiation into Vacuum Is a Function
Only of Temperature ............................... 14
1.5.5 Blackbody Intensity and Its Directional
Independence ...................................... 14
1.5.6 Blackbody Emissive Power: Cosine-Law
Dependence ........................................ 16
1.5.7 Hemispherical Spectral Emissive Power of
a Blackbody ....................................... 16
1.5.8 Planck's Law: Spectral Distribution of
Emissive Power .................................... 17
1.5.9 Approximations for Blackbody Spectral
Distribution ...................................... 20
1.5.9.1 Wien's Formula ............................ 20
1.5.9.2 Rayleigh-Jeans Formula .................... 21
1.5.10 Wien's Displacement Law .......................... 21
1.5.11 Total Blackbody Intensity and Emissive
Power ............................................ 23
1.5.12 Blackbody Radiation within a Spectral
Band ............................................. 24
1.5.13 Summary of Blackbody Properties .................. 28
1.6 Radiative Energy Loss and Gain along a Line-of-Sight .... 31
1.6.1 Radiative Energy Loss due to Absorption
and Scattering .................................... 31
1.6.2 Mean Penetration Distance ......................... 32
1.6.3 Optical Thickness ................................. 33
1.6.4 Radiative Energy Gain due to Emission ............. 34
1.6.5 Radiative Energy Gain due to In-Scattering ........ 35
1.7 Radiative Transfer Equation ............................. 37
1.8 Radiative Transfer in Nonparticipating Enclosures ....... 38
1.9 Concluding Remarks and Historical Notes ................. 41
Homework .................................................... 42
Chapter 2 Definitions of Properties at Interfaces .............. 47
2.1 Introduction ............................................ 47
2.1.1 Nomenclature for Properties ....................... 49
2.1.2 Notation .......................................... 50
2.2 Emissivity .............................................. 50
2.2.1 Directional Spectral Emissivity
ελ(θ,φ,Τ).......................................... 51
2.2.2 Directional Total Emissivity ε(θ,φ,Τ) ............. 53
2.2.3 Hemispherical Spectral Emissivity
ελ(Τ).............................................. 54
2.2.4 Hemispherical Total Emissivity ε(Τ) ............... 55
2.3 Absorptivity ............................................ 59
2.3.1 Directional Spectral Absorptivity
αλ(θi,φi,Τ) ........................................ 59
2.3.2 Kirchhoff's Law ................................... 60
2.3.3 Directional Total Absorptivity
α(θi,φi,Τ) ... ..................................... 61
2.3.4 Kirchhoff's Law for Directional Total Properties .. 61
2.3.5 Hemispherical Spectral Absorptivity
αλ(Τ) ........... ................................. 62
2.3.6 Hemispherical Total Absorptivity α(Τ) ............. 63
2.3.7 Diffuse-Gray Surface .............................. 65
2.4 Reflectivity ............................................ 66
2.4.1 Spectral Reflectivities ........................... 66
2.4.1.1 Bidirectional Spectral Reflectivity
ρ(θr,φr,θi,φi) ............................. 66
2.4.1.2 Reciprocity for Bidirectional Spectral
Reflectivity .............................. 66
2.4.1.3 Directional Spectral Reflectivities ....... 68
2.4.1.4 Reciprocity for Directional Spectral
Reflectivity .............................. 68
2.4.1.5 Hemispherical Spectral
Reflectivity ρλ ........................... 69
2.4.1.6 Limiting Cases for Spectral Surfaces ...... 69
2.4.2 Total Reflectivities .............................. 71
2.4.2.1 Bidirectional Total Reflectivity
ρ(θr,φr,θi,φi) ............................. 71
2.4.2.2 Reciprocity for Bidirectional Total
Reflectivity .............................. 71
2.4.2.3 Directional Total Reflectivity
ρ(θi,φi)or ρ(θr,φr) ........................ 71
2.4.2.4 Reciprocity for Directional Total
Reflectivity .............................. 72
2.4.2.5 Hemispherical Total Reflectivity,ρ ........ 72
2.4.3 Summary of Restrictions on Reflectivity
Reciprocity Relations ............................. 72
2.5 Transmissivity at an Interface .......................... 72
2.5.1 Spectral Transmissivities ......................... 73
2.5.1.1 Bidirectional Spectral Transmissivity
τλ(θi,φi,θt,φt,Τ) .......................... 73
2.5.1.2 Directional Spectral Transmissivities
τλ(θi,φi) .................................. 73
2.5.1.3 Hemispherical Spectral Transmissivity
τλ ........................................ 74
2.5.2 Total Transmissivities ............................ 74
2.5.2.1 Bidirectional Total Transmissivity
τ(θi,φi,θt,φt) ............................. 75
2.5.2.2 Directional Total Transmissivities
τ(θi,φi) .................................. 75
2.5.2.3 Hemispherical-Directional Total
Transmissivity τ(θt,φt) ................... 75
2.5.2.4 Hemispherical Total Transmissivity τ ...... 76
2.6 Relations among Reflectivity, Absorptivity,
Emissivity, and Transmissivity .......................... 76
Homework .................................................... 80
Chapter 3 Radiative Properties of Opaque Materials ............. 87
3.1 Introduction ............................................ 87
3.2 Electromagnetic Wave Theory Predictions ................. 87
3.2.1 Dielectric Materials .............................. 88
3.2.1.1 Reflection and Refraction at the
Interface between Two Perfect
Dielectrics (k → 0) ....................... 88
3.2.1.2 Reflectivity .............................. 90
3.2.1.3 Emissivity ................................ 91
3.2.2 Radiative Properties of Metals .................... 94
3.2.2.1 Electromagnetic Relations for
Incidence on an Absorbing Medium .......... 94
3.2.2.2 Reflectivity and Emissivity Relations
for Metals (Large k) ...................... 95
3.2.2.3 Relations between Radiative Emission
and Electrical Properties ................ 100
3.3 Extensions of the Theory for Radiative Properties ...... 105
3.4 Measured Properties of Real Dielectric Materials ....... 106
3.4.1 Variation of Total Properties with Surface
Temperature ...................................... 110
3.4.1.1 Effect of Surface Roughness .............. 112
3.4.2 Properties of Semiconductors and
Superconductors .................................. 116
3.5 Measured Properties of Metals .......................... 117
3.5.1 Directional and Spectral Variations .............. 117
3.5.2 Effect of Surface Temperature .................... 118
3.5.3 Effect of Surface Roughness ...................... 120
3.5.4 Effect of Surface Impurities ..................... 125
3.5.5 Molten Metals .................................... 129
3.6 Selective and Directional Opaque Surfaces .............. 131
3.6.1 Characteristics of Solar Radiation ............... 131
3.6.1.1 Solar Constant ........................... 131
3.6.1.2 Solar Radiating Temperature .............. 131
3.6.2 Modification of Surface Spectral
Characteristics .................................. 131
3.6.3 Modification of Surface Directional
Characteristics .................................. 137
3.7 Concluding Remarks ..................................... 139
Homework ................................................... 139
Chapter 4 Configuration Factors for Diffuse Surfaces
with Uniform Radiosity .............................. 151
4.1 Radiative Transfer Equation for Surfaces Separated
by a Transparent Medium ................................ 151
4.1.1 Enclosures with Diffuse Surfaces ................. 152
4.1.2 Enclosures with Directional (Nondiffuse) and
Spectral(Nongray) Surfaces ....................... 153
4.2 Geometric Configuration Factors between Two Surfaces ... 153
4.2.1 Configuration Factor for Energy Exchange between
Diffuse Differential Areas ....................... 153
4.2.1.1 Reciprocity for Differential-Element
Configuration Factors .................... 155
4.2.1.2 Sample Configuration Factors between
Differential Elements .................... 155
4.2.2 Configuration Factor between a Differential Area
Element and a Finite Area ........................ 157
4.2.2.1 Reciprocity Relation for Configuration
Factor between Differential and Finite
Areas .................................... 159
4.2.2.2 Configuration Factors between Differential
and a Finite Areas ....................... 159
4.2.3 Configuration Factor and Reciprocity for
Two Finite Areas ................................. 161
4.3 Methods for Determining Configuration Factors .......... 163
4.3.1 Configuration-Factor Algebra ..................... 163
4.3.1.1 Configuration Factors Determined by Use
of Symmetry .............................. 167
4.3.2 Configuration-Factor Relations in Enclosures ..... 170
4.3.3 Techniques for Evaluating Configuration Factors .. 172
4.3.3.1 Hottel's Crossed-String Method ........... 172
4.3.3.2 Contour Integration ...................... 176
4.3.3.3 Differentiation of Known Factors ......... 182
4.3.4 Unit-Sphere and Hemicube Methods ................. 185
4.3.5 Direct Numerical Integration ..................... 187
4.3.6 Computer Programs for Evaluation of Configuration
Factors .......................................... 187
4.4 Constraints on Configuration Factor Accuracy ........... 188
4.5 Compilation of Known Configuration Factors and Their
References—Appendix C and Web Catalog .................. 189
Homework ................................................... 190
Chapter 5 Radiation Exchange in Enclosures Composed of
Black and/or Diffuse-Gray Surfaces .................. 205
5.1 Approximations and Restrictions for Analysis of
Enclosures with Black and/or Diffuse-Gray Surfaces ..... 205
5.2 Radiative Transfer for Black Surfaces .................. 206
5.2.1 Transfer Between Black Surfaces by Use
of Configuration Factors ......................... 208
5.2.2 Radiation Exchange in a Black Enclosure .......... 208
5.3 Radiation Between Finite Diffuse-Gray Areas ............ 211
5.3.1 Net-Radiation Method for Enclosures .............. 211
5.3.1.1 System of Equations Relating Surface
Heating Q and Surface Temperature T ...... 218
5.3.1.2 Solution Method in Terms of Radiosity J .. 222
5.3.2 Enclosure Analysis in Terms of Energy Absorbed
at Surface ....................................... 224
5.3.3 Enclosure Analysis by Use of Transfer Factors .... 225
5.3.4 Matrix Inversion for Enclosure Equations ......... 226
5.4 Radiation Analysis Using Infinitesimal Areas ........... 231
5.4.1 Generalized Net-Radiation Method Using
Infinitesimal Areas .............................. 231
5.4.1.1 Relations between Surface Temperature T
and Surface Heat Flux q .................. 234
5.4.1.2 Solution Method in Terms of Outgoing
Radiative Flux J ......................... 235
5.4.1.3 Special Case When Imposed Heat Flux
q Is Specified for All Surfaces .......... 235
5.4.2 Methods for Solving Integral Equations ........... 243
5.4.2.1 Numerical Integration .................... 244
5.4.2.2 Analytical Solutions ..................... 245
5.4.2.3 Exact Solution of Integral Equation for
Radiation from a Spherical Cavity ........ 246
5.4.3 General Boundary Conditions that Provide Inverse
Problems ......................................... 248
5.5 Computer Programs for Enclosure Analysis ............... 248
Homework ................................................... 249
Chapter 6 Exchange of Thermal Radiation among Nondiffuse
Nongray Surfaces .................................... 269
6.1 Introduction ........................................... 269
6.2 Enclosure Theory for Diffuse Nongray Surfaces .......... 269
6.2.1 Parallel-Plate Geometry .......................... 271
6.2.2 Spectral and Finite Spectral Band Relations for
an Enclosure ..................................... 274
6.2.3 Semigray Approximations .......................... 275
6.3 Directional-Gray Surfaces .............................. 276
6.4 Surfaces with Directionally and Spectrally Dependent
Properties ............................................. 281
6.5 Radiation Exchange in Enclosures with Some Specularly
Reflecting Surfaces .................................... 289
6.5.1 Some Situations with Simple Geometries ........... 289
6.5.2 Ray Tracing and the Construction of Images ....... 293
6.5.3 Radiative Transfer by Means of Simple Specular
Surfaces for Diffuse Energy Leaving a Surface .... 294
6.5.4 Configuration-Factor Reciprocity for Specular
Surfaces; Specular Exchange Factors .............. 299
6.6 Net-Radiation Method in Enclosures Having Specular
and Diffuse Reflecting Surfaces ........................ 304
6.6.1 Enclosures with Planar Surfaces .................. 304
6.6.2 Curved Specular Reflecting Surfaces .............. 311
6.7 Multiple Radiation Shields ............................. 314
6.8 Concluding Remarks ..................................... 317
Homework ................................................... 319
Chapter 7 Radiation Combined with Conduction and
Convection at Boundaries ............................ 337
7.1 Introduction ........................................... 337
7.2 Energy Relations and Boundary Conditions ............... 338
7.2.1 General Relations ................................ 338
7.2.2 Uncoupled and Coupled Energy Transfer Modes ...... 340
7.2.3 Control Volume Approach for One- or Two-
Dimensional Conduction along Thin Walls .......... 341
7.3 Radiation Transfer with Conduction Boundary
Conditions ............................................. 342
7.3.1 Thin Fins with One- or Two-Dimensional
Conduction ....................................... 342
7.3.1.1 One-Dimensional Heat Flow ................ 342
7.3.1.2 Two-Dimensional Heat Flow ................ 347
7.3.2 Multidimensional and Transient Heat Conduction
with Radiation ................................... 349
7.4 Radiation with Convection and Conduction ............... 350
7.4.1 Thin Radiating Fins with Convection .............. 350
7.4.2 Channel Flows .................................... 352
7.4.3 Free Convection with Radiation ................... 356
7.5 Numerical Solution Methods ............................. 359
7.6 Numerical Integration Methods for Use with Enclosure
Equations .............................................. 360
7.6.1 Trapezoidal Rule ................................. 360
7.6.2 Simpson's Rule ................................... 362
7.6.3 Other Integration Methods ........................ 363
7.7 Numerical Formulations for Combined-Mode Energy
Transfer ............................................... 363
7.7.1 Finite-Difference Formulation .................... 364
7.7.2 Finite-Element Method Formulation ................ 369
7.7.2.1 Shape Function ........................... 370
7.7.2.2 Galerkin Form for the Energy Equation .... 371
7.8 Numerical Solution Techniques .......................... 374
7.8.1 Successive Substitution Methods .................. 375
7.8.1.1 Simple Successive Substitution (SSS) ..... 375
7.8.1.2 Successive Underrelaxation (SUR) ......... 375
7.8.1.3 Regulated Successive Underrelaxation
(RSUR) ................................... 375
7.8.2 Newton-Raphson-Based Methods for Nonlinear
Problems ......................................... 376
7.8.2.1 Modified Newton-Raphson (MNR) ............ 376
7.8.2.2 Accelerated Newton-Raphson (ANR) ......... 377
7.8.3 Applications of the Numerical Methods ............ 377
7.9 Monte Carlo Method ..................................... 379
7.9.1 Definition of Monte Carlo Method ................. 379
7.9.2 Fundamentals of the Method ....................... 379
7.9.2.1 Random Walk .............................. 379
7.9.2.2 Choosing from Probability Distributions .. 380
7.9.2.3 Random Numbers ........................... 383
7.9.2.4 Evaluation of Uncertainty ................ 384
7.9.3 Application to Thermal Radiative Transfer ........ 385
7.9.3.1 Model of the Radiative Exchange Process .. 385
7.9.3.2 Useful Functions ......................... 390
7.9.4 Forward Monte Carlo .............................. 390
7.9.5 Reverse Monte Carlo .............................. 393
7.9.6 Results for Radiative Transfer ................... 396
7.9.6.1 Literature on Radiation Exchange
among Surfaces ........................... 396
7.9.6.2 Radiative Transmission through the Inside
of a Channel ............................. 397
7.9.6.3 Extension to Directional and Spectral
Surfaces ................................. 398
7.9.6.4 Application of Monte Carlo Methods
to Combined-Mode Problems ................ 399
7.10 Concluding Remarks .................................... 399
7.10.1 Verification ................................... 399
7.10.2 Validation ..................................... 400
7.10.3 Uncertainty Quantification ..................... 400
Homework ................................................... 400
Chapter 8 Inverse Problems in Radiative Heat Transfer ......... 421
8.1 Introduction to Inverse Problems ....................... 421
8.1.1 Inverse Design and Data Analysis ................. 422
8.1.1.1 Direct Inverse Solutions ................. 423
8.2 General Inverse Solution Methods ....................... 425
8.2.1 Regularization ................................... 426
8.2.2 Optimization ..................................... 428
8.2.2.1 Deterministic (Quasi-Newton) Approach .... 429
8.2.3 Metaheuristic Approaches ......................... 429
8.2.3.1 Simulated Annealing ...................... 429
8.3 Comparison of Methods for a Particular Problem ......... 430
8.3.1 Solution by Direct Inversion ..................... 432
8.3.1.1 TSVD Solution Method ..................... 432
8.3.1.2 Tikhonov Solution Method ................. 432
8.3.1.3 CGR Solution ............................. 432
8.3.2 Optimization Techniques .......................... 433
8.3.3 Metaheuristic Results ............................ 433
8.3.3.1 Simulated Annealing ...................... 433
8.3.4 Comparison of Selected Results ................... 434
8.4 Application of Metaheuristic Methods ................... 435
8.5 Unresolved Problems .................................... 435
8.6 Inverse Problems Involving Participating Media ......... 436
8.7 Concluding Remarks ..................................... 436
Homework ................................................... 437
Chapter 9 Absorption and Emission in Participating
Media ............................................... 441
9.1 Introduction ........................................... 441
9.2 Spectral Lines and Bands for Absorption and Emission
of Gases ............................................... 444
9.2.1 Physical Mechanisms .............................. 444
9.2.2 Condition of Local Thermodynamic
Equilibrium (LTE) ................................ 447
9.2.3 Spectral Line Broadening ......................... 448
9.2.3.1 Natural Broadening ....................... 449
9.2.3.2 Doppler Broadening ....................... 450
9.2.3.3 Collision Broadening and Narrowing ....... 450
9.2.3.4 Stark Broadening ......................... 451
9.2.4 Absorption or Emission by a Single Spectral
Line ............................................. 452
9.2.4.1 Property Definitions for a Path in a
Uniform Absorbing and Emitting Medium .... 452
9.2.4.2 Weak Lines ............................... 453
9.2.4.3 Relations for Lorentz Lines .............. 454
9.2.5 Band Absorption .................................. 455
9.2.5.1 Band Structure ........................... 455
9.2.5.2 Types of Band Models ..................... 455
9.2.5.3 Spectral Line-by-Line Databases .......... 458
9.3 Band Models and Correlations for Gas Absorption
and Emission ........................................... 458
9.3.1 Narrow-Band Models ............................... 458
9.3.1.1 Elsasser Model ........................... 458
9.3.1.2 Goody Model .............................. 460
9.3.1.3 Malkmus Model ............................ 460
9.3.2 Wide-Band Models and Correlations ................ 460
9.3.3 Contemporary Band Correlations ................... 467
9.3.3.1 k-Distribution Method .................... 467
9.3.3.2 Correlated-k Method ...................... 468
9.3.3.3 Weighted Sum of Gray Gases ............... 473
9.4 Total Gas-Total Emittance Correlations ................. 479
9.5 Mean Absorption Coefficients ........................... 484
9.5.1 Planck Mean Absorption Coefficient ............... 484
9.5.2 Rosseland Mean Absorption Coefficient ............ 484
9.5.3 Patch Mean Absorption Coefficient ................ 484
9.6 True Absorption Coefficient ............................ 485
9.7 Radiative Properties of Translucent Liquids
and Solids ............................................. 485
Homework ................................................... 491
Chapter 10 Radiative Transfer Relations in Simple
Systems ............................................ 493
10.1 Introduction .......................................... 493
10.2 Energy Equation and Boundary Conditions for a
Translucent Medium with Radiation ..................... 494
10.3 Radiative Transfer and Source Function Equations ...... 495
10.3.1 Radiative Transfer Equation .................... 495
10.3.2 Source Function Equation ....................... 497
10.4 Radiative Flux and Its Divergence within a Medium ..... 500
10.4.1 Radiative Flux Vector .......................... 501
10.4.2 Divergence of Radiative Flux without
Scattering (Absorption Alone) .................. 504
10.4.3 Divergence of Radiative Flux Including
Scattering ..................................... 505
10.5 Summary of Relations for Radiative Transfer in
Absorbing, Emitting, and Scattering Media ............. 507
10.5.1 Energy Equation ................................ 507
10.5.2 Radiative Energy Source ........................ 507
10.5.3 Source Function ................................ 507
10.5.4 Radiative Transfer Equation .................... 508
10.5.5 Relations for a Gray Medium .................... 508
10.6 Net-Radiation Method for Enclosures Filled with an
Isothermal Medium of Uniform Composition .............. 509
10.6.1 Definitions of Spectral Geometric-Mean
Transmission and Absorption Factors ............. 511
10.6.1.1 Definitions of Spectral Geometric-Mean
Transmission and Absorption Factors .... 511
10.6.2 Matrix of Enclosure-Theory Equations ........... 512
10.6.3 Energy Balance in a Medium ..................... 513
10.6.4 Spectral Band Equations for an Enclosure ....... 515
10.6.5 Gray Medium in a Gray Enclosure ................ 516
10.7 Evaluation of Spectral Geometric-Mean Transmittance
and Absorptance Factors ............................... 518
10.8 Mean Beam-Length Approximation for Spectral Radiation
from an Entire Volume of a Medium to All or Part of
Its Boundary .......................................... 518
10.8.1 Mean Beam Length for a Medium between Parallel
Plates Radiating to Area on Plate .............. 519
10.8.2 Mean Beam Length for Sphere of Medium Radiating
to Any Area on Its Boundary .................... 520
10.8.3 Radiation from Entire Medium Volume to Its
Entire Boundary for Optically Thin Medium ...... 520
10.8.4 Correction to Mean Beam Length When Medium
Is Not Optically Thin .......................... 521
10.9 Exchange of Total Radiation in an Enclosure
by Use of Mean Beam Length ............................ 526
10.9.1 Total Radiation from Entire Medium Volume
to All or Part of its Boundary ................. 527
10.9.2 Exchange between Entire Medium Volume and
Emitting Boundary .............................. 527
Homework ................................................... 529
Chapter 11 Energy Transfer in Plane Layers and
Multidimensional Geometries: Participating Media
with and without Conduction ........................ 535
11.1 Introduction .......................................... 535
11.2 Equations for Radiative Intensity, Flux, Flux
Divergence,and Source Function in a Plane Layer ....... 535
11.2.1 Radiative Transfer Equation and Radiative
Intensity for a Plane Layer .................... 535
11.2.2 Local Radiative Flux in a Plane Layer .......... 537
11.2.3 Divergence of the Radiative Flux—Radiative
Energy Source .................................. 538
11.2.4 Equation for the Source Function in
a Plane Layer .................................. 539
11.2.5 Relations for Isotropic Scattering ............. 539
11.2.6 Diffuse Boundary Fluxes for a Plane Layer
with Isotropic Scattering ...................... 541
11.3 Gray Plane Layer of Absorbing and Emitting Medium
with Isotropic Scattering ............................. 541
11.4 Gray Plane Layer in Radiative Equilibrium ............. 545
11.4.1 Energy Equation ................................ 545
11.4.2 Absorbing Gray Medium in Radiative Equilibrium
with Isotropic Scattering ...................... 546
11.4.3 Isotropically Scattering Medium with Zero
Absorption ..................................... 546
11.4.4 Gray Medium with dqr/dx = 0 between Opaque
Diffuse-Gray Boundaries ........................ 547
11.4.5 Solution for Gray Medium with dqr/dx = 0 between
Black or Diffuse-Gray Walls at Specified
Temperatures ................................... 548
11.4.5.1 Gray Medium between Black Walls ....... 548
11.4.5.2 Gray Medium between Diffuse-Gray
Walls ................................. 551
11.5 Radiation Combined with Conduction .................... 552
11.5.1 Energy Balance ................................. 554
11.5.2 Plane Layer with Conduction and Radiation ...... 554
11.5.2.1 Absorbing-Emitting Medium without
Scattering ............................ 554
11.5.2.2 Absorbing-Emitting Medium with
Scattering ............................ 556
11.6 Multidimensional Radiation in a Participating Gray
Medium with Isotropic Scattering ...................... 559
11.6.1 Radiation Relations in Three Dimensions ........ 559
11.6.2 Two-Dimensional Transfer in a
Rectangular Region ............................. 561
11.6.3 Rectangular Region with Conduction and
Radiation ...................................... 564
11.6.4 One-Dimensional Transfer in a Cylindrical
Region ......................................... 566
11.6.5 Additional Information on Nonplanar and
Multidimensional Geometries .................... 568
11.7 Transient Solutions Including Conduction .............. 569
11.8 Discussion of Solution Procedures ..................... 572
11.8.1 Simultaneous Solution of Energy and Radiative
Transfer Relations ............................. 572
11.8.2 Outline of Solution Methods for the Radiative
Transfer Equation .............................. 573
11.8.2.1 Solution Methods for the Differential
RTE ................................... 573
11.8.2.2 Solution Methods for the Integral
RTE ................................... 574
Homework ................................................... 575
Chapter 12 Optically Thin and Thick Limits
for Radiative Transfer in Participating Media....... 581
12.1 Introduction .......................................... 581
12.2 Optically Thin and Cold Media ......................... 581
12.2.1 Nearly Transparent Optically Thin Medium ....... 582
12.2.2 Optically Thin Media with Cold Boundaries or
Small Incident Radiation; the Emission
Approximation .................................. 585
12.2.3 Cold Medium with Weak Scattering ............... 587
12.3 Optically Thick Medium : Radiative Diffusion .......... 587
12.3.1 Simplified Derivation of the Radiative
Diffusion Approximation ........................ 588
12.3.2 General Radiation-Diffusion Relations
in a Medium .................................... 591
12.3.2.1 Rosseland Diffusion Equation for
Local Radiative Flux .................. 591
12.3.2.2 Emissive Power Jump Boundary Condition
in the Limit without Heat Conduction .. 592
12.3.2.3 Gray Stagnant Medium between Parallel
Gray Walls ............................ 593
12.3.2.4 Other Radiative Diffusion Solutions
for Gray Media without Heat
Conduction ............................ 595
12.4 Approximations for Combined Radiation and Conduction .. 599
12.4.1 Addition of Energy Transfer by Radiation
and Conduction ................................. 599
12.4.2 Diffusion Method for Combined Radiation and
Conduction ..................................... 600
12.5 Approximate Solutions for Combined Radiation,
Conduction, and Convection in a Boundary Layer ........ 606
12.5.1 Optically Thin Thermal Layer ................... 606
12.5.2 Optically Thick Thermal Layer .................. 607
12.6 Use of Mean Absorption Coefficients ................... 609
12.6.1 Definitions of Mean Absorption Coefficients .... 609
12.6.2 Approximate Solutions of the Radiative Transfer
Equations Using Mean Absorption Coefficients ... 610
12.7 Curtis-Godson Approximation ........................... 611
Homework ................................................... 614
Chapter 13 Solution of Radiative Transfer in
Participating Media ................................ 619
13.1 Introduction .......................................... 619
13.2 Differential Methods .................................. 619
13.2.1 Milne-Eddington (Differential) Approximation ... 619
13.2.2 General Spherical Harmonics (PN) Method ........ 623
13.2.3 Boundary Conditions for the PN Method .......... 628
13.2.4 PN Method for Radiation Combined with
Heat Conduction ................................ 633
13.2.5 Simplified PN (SPN) Method .............. ...... 636
13.2.5.1 SP1 Solution .......................... 637
13.2.5.2 SP1 Boundary Conditions ............... 638
13.2.5.3 Higher-Order Solutions ................ 638
13.5.2.1 SP3 Solution .......................... 639
13.2.6 Boundary Conditions for Higher-Order
SPN Solutions ........................... 640
13.3 Discrete Ordinates (SN) Method ........................ 640
13.3.1 Two-Flux Method: The Schuster-Schwarzschild
Approximation .................................. 640
13.3.2 Radiative Transfer Equation with Discrete
Ordinates Method ............................... 644
13.3.3 Boundary Conditions for the Discrete
Ordinates Method ............................... 644
13.3.4 Control Volume Method for Discrete Ordinates
Numerical Solution ............................. 645
13.3.4.1 Relations for Two-Dimensional
Rectangular Coordinates ............... 646
13.3.4.2 Relations for Three-Dimensional
Rectangular Coordinates ............... 648
13.3.5 Results Using Discrete Ordinates ............... 652
13.4 Other Methods That Depend on Angular Discretization ... 655
13.4.1 Discrete Transfer Method ....................... 655
13.4.2 Finite Volume Method ........................... 656
13.4.3 Boundary Element Method ........................ 657
13.5 Numerical Solution Methods for Combined Radiation,
Conduction, and Convection in Participating Media ..... 657
13.6 Finite-Difference Methods ............................. 658
13.6.1 Energy Equation for Combined Radiation
and Conduction ................................. 658
13.6.2 Radiation and Conduction in a Plane Layer ...... 660
13.6.3 Radiation and Conduction in a Two-Dimensional
Rectangular Region ............................. 662
13.7 Finite-Element Method (FEM) ........................... 666
13.7.1 FEM for Radiative Equilibrium
(No Conduction and/or Convection) .............. 666
13.7.2 Radiation with Conduction and/or Convection .... 668
13.7.3 Results from Finite-Element Analyses ........... 669
13.8 Zonal Method .......................................... 672
13.8.1 Exchange Area Relations ........................ 672
13.8.2 Zonal Formulation for Radiative Equilibrium .... 674
13.8.3 Developments for the Zonal Method .............. 676
13.8.3.1 Smoothing of Exchange Area Sets ....... 676
13.8.3.2 Other Formulations of the Zonal
Method ................................ 677
13.8.4 Numerical Results from Zone Method ............. 677
13.9 Monte Carlo Technique for Radiatively Participating
Media ................................................. 680
13.9.1 Discussion of the Computational Method ......... 681
13.9.2 Monte Carlo Results for Radiation through
Gray Gases ..................................... 685
13.9.2.1 Infinite Parallel Walls ............... 685
13.9.2.2 Cylindrical Geometry .................. 686
13.9.3 Consideration of Radiative Property
Variations ..................................... 686
13.9.4 Parallel Processing and Other Computational
Improvements ................................... 687
13.9.4.1 Monte Carlo in Combined-Mode
Problems .............................. 689
13.9.5 Reverse Monte Carlo in Participating Media ..... 689
13.10 Numerical Boundary Conditions and Additional
Solution Methods ..................................... 691
13.10.1 Boundary Condition for Numerical Solutions ... 691
13.10.2 Exponential Kernel Approximation ............. 692
13.10.3 Reduction of the Integral Order .............. 694
13.10.4 YIX Method ................................... 695
13.10.5 Additional Information on Numerical Methods .. 696
13.11 Results for Combined Convection, Conduction,
and Radiation ........................................ 697
13.11.1 Forced Convection Channel Flows .............. 698
13.11.2 Free Convection Flow, Heat Transfer,
and Stability ................................ 703
13.11.3 Radiative Transfer in Porous Media and
Packed Beds .................................. 706
13.11.4 Additional Topics with Combined Radiation,
Conduction, and Convection ................... 706
13.12 Benchmark Solutions for Computational Validation ..... 707
13.13 Inverse Problems Involving Participating Media ....... 709
13.14 Solution Using Commercially Available and
Other Codes .......................................... 709
13.15 Verification, Validation, and Uncertainty
Quantification ....................................... 709
Homework ................................................... 710
Chapter 14 Electromagnetic Wave Theory ........................ 725
14.1 Introduction .......................................... 725
14.2 EM-Wave Equations ..................................... 726
14.3 Wave Propagation in a Medium .......................... 727
14.3.1 EM-Wave Propagation in Perfect Dielectric
Media .......................................... 727
14.3.2 Wave Propagation in Isotropic Media with Finite
Electrical Conductivity ........................ 731
14.3.3 Energy of an EM Wave ........................... 732
14.4 Laws of Reflection and Refraction ..................... 733
14.4.1 Reflection and Refraction at the Interface
between Perfect Dielectrics (k → 0) ............ 733
14.4.2 Reflection and Refraction at the Interface
of an Absorbing Medium (k ≠ 0) ................. 738
14.5 Amplitude and Scattering Matrices ..................... 741
14.6 EM-Wave Theory and the Radiative Transfer Equation .... 744
Homework ................................................... 745
Chapter 15 Absorption and Scattering by Particles
and Agglomerates ................................... 747
15.1 Introduction .......................................... 747
15.2 Absorption and Scattering: Definitions ................ 749
15.2.1 Background ..................................... 749
15.2.2 Absorption and Scattering Coefficients,
Cross Sections, Efficiencies ................... 749
15.2.3 Scattering Phase Function ...................... 751
15.3 Scattering by Large Spherical Particles ............... 754
15.3.1 Scattering by a Large Specularly Reflecting
Sphere ......................................... 755
15.3.2 Reflection from a Large Diffuse Sphere ......... 757
15.3.3 Large Ideal Dielectric Sphere with n ≈ 1 ....... 759
15.3.4 Diffraction from a Large Sphere ................ 760
15.3.5 Geometric Optics Approximation ................. 760
15.4 Scattering by Small Particles ......................... 764
15.4.1 Rayleigh Scattering by Small Spheres ........... 764
15.4.2 Scattering Cross Section for Rayleigh
Scattering ..................................... 764
15.4.3 Phase Function for Rayleigh Scattering ......... 766
15.5 Lorenz-Mie Theory for Spherical Particles ............. 767
15.5.1 Formulation for Homogeneous and Stratified
Spherical Particles ............................ 767
15.5.2 Cross Sections for Specific Cases .............. 771
15.6 Prediction of Properties for Irregularly Shaped
Particles ............................................. 773
15.6.1 Integral and Differential Formulations ......... 773
15.6.2 T-matrix Approach .............................. 773
15.6.3 Discrete Dipole Approximation .................. 775
15.6.4 Finite Element Method .......................... 777
15.6.5 Finite Difference Time Domain Method ........... 777
15.7 Approximate Anisotropic Scattering Phase Functions .... 778
15.7.1 Forward-Scattering Phase Function .............. 779
15.7.1.1 Linear-Anisotropic Phase Function ..... 779
15.7.1.2 Delta-Eddington Phase Function ........ 779
15.7.1.3 Henyey-Greenstein Phase Function ...... 779
15.8 Dependent Absorption and Scattering ................... 781
Homework ................................................... 785
Chapter 16 Near-Field Thermal Radiation ....................... 787
16.1 Introduction .......................................... 787
16.2 Electromagnetic Treatment of Thermal Radiation
and Basic Concepts .................................... 789
16.2.1 Near-Field Thermal Radiation versus Far-Field
Thermal Radiation .............................. 789
16.2.2 Electromagnetic Description of Thermal
Radiation ...................................... 790
16.2.3 Near-Field Radiative Heat Flux ................. 793
16.2.4 Density of Electromagnetic States .............. 794
16.2.5 Spatial and Temporal Coherence of Thermal
Radiation ...................................... 794
16.3 Evanescent and Surface Waves .......................... 795
16.3.1 Evanescent Waves and Total Internal
Reflection ..................................... 795
16.3.2 Surface Waves .................................. 797
16.4 Near-Field Radiative Heat Flux Calculations ........... 800
16.4.1 Near-Field Radiative Heat Flux in One-
Dimensional Layered Medium ..................... 801
16.4.2 Near-Field Radiative Heat Transfer between
Two Bulk Materials Separated by a Vacuum Gap ... 804
16.5 Experimental Studies of Near-Field Thermal
Radiation ............................................. 808
16.5.1 Historical Overview ............................ 809
16.5.2 Experimental Determination of Near-Field
Radiative Transfer Coefficient ................. 810
16.5.3 Near-Field Effects on Radiative Properties ..... 811
16.6 Concluding Remarks .................................... 813
Homework ................................................... 815
Chapter 17 Radiative Effects in Translucent Solids,
Windows, and Coatings .............................. 817
17.1 Introduction .......................................... 817
17.2 Transmission, Absorption, and Reflection of Windows ... 818
17.2.1 Single Partially Transmitting Layer with
Thickness D >> λ
(No Wave Interference Effects) ................. 819
17.2.1.1 Ray-Tracing Method .................... 819
17.2.1.2 Net-Radiation Method .................. 820
17.2.2 Multiple Parallel Windows ...................... 822
17.2.3 Transmission through Multiple Parallel
Glass Plates ................................... 823
17.2.4 Interaction of Transmitting Plates with
Absorbing Plate ................................ 824
17.3 Enclosure Analysis with Partially Transparent
Windows ............................................... 827
17.4 Effects of Coatings or Thin Films on Surfaces ......... 829
17.4.1 Coating without Wave Interference Effects ...... 829
17.4.1.1 Nonabsorbing Dielectric Coating on
Nonabsorbing Dielectric Substrate ..... 829
17.4.1.2 Absorbing Coating on Metal Substrate .. 831
17.4.2 Thin Film with Wave Interference Effects ....... 831
17.4.2.1 Nonabsorbing Dielectric Thin Film on
Nonabsorbing Dielectric Substrate ..... 831
17.4.2.2 Absorbing Thin Film on a Metal
Substrate ............................. 835
17.4.3 Films with Partial Coherence ................... 836
17.5 Refractive Index Effects on Radiation in a
Participating Medium .................................. 836
17.5.1 Effect of Refractive Index on Intensity
Crossing an Interface .......................... 837
17.5.2 Effect of Angle for Total Reflection ........... 838
17.5.3 Interface Conditions for Radiation Analysis
in a Plane Layer ............................... 839
17.5.3.1 Layer with Nondiffuse or Specular
Surfaces .............................. 839
17.5.3.2 Diffuse Surfaces ...................... 842
17.5.5 Emission from a Translucent Layer (n > 1)
at Uniform Temperature with Specular or Diffuse
Boundaries ..................................... 843
17.6 Multiple Participating Layers with Heat Conduction .... 845
17.6.1 Formulation for Multiple Participating
Plane Layers ................................... 846
17.6.2 Translucent Layer on a Metal Wall .............. 848
17.6.3 Composite of Two Translucent Layers ............ 850
17.6.3.1 Temperature Distribution Relations
from Energy Equation .................. 850
17.6.3.2 Relations for Radiative Flux .......... 853
17.6.3.3 Equation for the Source Function ...... 854
17.6.3.4 Solution Procedure and Typical
Results ............................... 854
17.7 Light Pipes and Fiber Optics .......................... 857
Homework ................................................... 859
Appendix A: Conversion Factors, Radiation Constants, and
Blackbody Functions ........................................... 867
Appendix B: Radiative Properties .............................. 873
Appendix C: Catalog of Selected Configuration Factors ......... 881
Appendix D: Exponential Integral Relations and Two-Dimensional
Radiation Functions ........................................... 889
Appendix E: List of References ................................ 895
Index ......................................................... 951
|
