A Systems Approach to Managing the Complexities of Process
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1.1
1.2
1.3
1.4
1.5
1.6
1.4.1
1.4.2
1.5.1
1.5.2
1.5.3
1.5.4
1.5.5
1.5.6
1.5.7
1.5.8
1.5.9
1.5.10
1.5.11
1.5.12
1.5.13
1.6.1
1.6.2
1.6.3
1.6.4
1.6.5
1.7
1.8
1.9
2.2
1.5.9.1
1.5.9.2
2.2.2 Directional Total Emissivity ϵ(θ,ϕ,T ) ................................................53
2.2.3 Hemispherical Spectral Emissivity ϵ λ(T ) ..........................................54
2.2.4 Hemispherical Total Emissivity ϵ(T ) ................................................55
2.3 Absorptivity .....................................................................................................59
2.3.1 Directional Spectral Absorptivity αλ(θi , ϕi , T ) ...................................59
2.3.2 Kirchhoff’s Law ................................................................................60
2.3.3 Directional Total Absorptivity α (θi , ϕi , T ) .........................................61
2.3.4 Kirchhoff’s Law for Directional Total Properties .............................61
2.3.5 Hemispherical Spectral Absorptivity αλ(T ) ......................................62
2.3.6 Hemispherical Total Absorptivity α (T ).............................................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 Refelctivities ....................................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 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
2.5.1.2 Directional Spectral Transmissivities τλ(θ i, ϕi)...................73
2.5.1.3 Hemispherical Spectral Transmissivity τλ ........................74
2.5.2 Total Transmissivities ........................................................................74
2.5.2.1 Bidirectional Total Transmissivity τλ(θ i, ϕ
2.5.2.2 Directional Total Transmissivities τ(θ i, ϕi) .........................75
2.5.2.3 Hemispherical-Directional Total Transmissivity τ(θ t, ϕt) ................................................................................75
2.5.2.4 Hemispherical Total Transmissivity τ ..............................76
2.6 Relations among Reflectivity, Absorptivity, Emissivity, and Transmissivity ...........................................................................................76
3.1 Introduction
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
3.2.2.2
3.2.2.3
3.3
3.4
3.4.1
3.5.1
3.5.2
3.5.3
3.5.4
3.5.5
3.6
4.2.2.2
4.3.3
4.3.3.1
4.3.3.3
4.3.4
4.3.5
4.3.6
4.4
5.1
5.2
5.2.1
5.2.2
5.3
5.3.1.1
5.3.1.2
5.3.2
5.3.3
5.4.1.1
5.4.1.2
5.4.1.3
5.4.2
6.2.2
6.4
6.5
6.5.1
6.5.2
6.6
6.6.1
6.6.2
7.9
7.8.2
7.8.3
7.9.1
7.9.2
7.8.1.2
7.8.1.3
7.8.2.1
7.8.2.2 Accelerated
7.9.3
7.9.4
7.9.5
7.9.6
7.9.2.1
7.9.2.2
7.9.2.3
7.9.2.4
7.9.3.1
7.9.3.2
7.9.6.1
7.9.6.2
7.9.6.3
9.2.4
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 .................................... 50 0
10.4.1 Radiative Flux Vector 501
10.4.2 Divergence of Radiative Flux without Scattering (Absorption Alone) 50 4
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
10.7 Evaluation of Spectral Geometric-Mean Transmittance and Absorptance Factors ...............................................................................
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 1Chapter 1 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 dq r /dx = 0 between Opaque Diffuse-Gray Boundaries 547
11.4.5 Solution for Gray Medium with dq r /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 ...................... 56 4
11.6.4 One-Dimensional
11.6.5
11.8 Discussion of Solution Procedures
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
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 ............................................
12.3.1 Simplified Derivation of the Radiative Diffusion Approximation .................................................................................
12.3.2 General Radiation-Diffusion Relations in a Medium .....................
12.3.2.1 Rosseland Diffusion Equation for Local Radiative Flux ..................................................................................
12.3.2.2 Emissive Power Jump Boundary Condition in the Limit without Heat Conduction .............................
12.3.2.3 Gray Stagnant Medium between Parallel Gray Walls
12.3.2.4 Other Radiative Diffusion Solutions for Gray Media without Heat Conduction
12.4 Approximations for Combined Radiation and Conduction ...........................
12.4.1
12.4.2
12.5 Approximate
12.6 Use
12.6.2
13.2.6
13.3
13.3.3
13.3.4
13.3.4.1
13.3.4.2
13.3.5
13.4
13.5
13.6 Finite-Difference
13.6.1
13.7
13.7.1
13.7.2
13.7.3
13.8
13.8.3.1
13.9
13.11
13.9.3
15.6.3
16.2.2
16.2.3
16.4.1
16.4.2
17.5
17.5.1
17.5.2
17.5.3
17.5.3.1 Layer with Nondiffuse or Specular Surfaces
17.5.3.2
17.5.5 Emission from a Translucent Layer (n > 1) at Uniform Temperature with Specular or Diffuse Boundaries.........................
17.6.3.1
17.6.3.2
List of Symbols
This is a consolidated list of symbols for the entire text. Some symbols that are used in only a local development are defined where they are used and are not included here. The symbols used in radiative transfer have evolved from many different disciplines where radiation is important. This has led to the same quantity being defined by a variety of symbols, and to multiple quantities designated with the same symbol. The symbols listed here are typical of those used for engineering heat transfer, and follow where possible those adopted formally by the major heat transfer journals [Howell (1999)]. The study of radiative transfer combined with conduction and convection involves many types of applications, and hence requires definitions for a large number of different quantities and parameters. There is an insufficient number of convenient symbols that can be used, so some symbols must be used for multiple quantities. Attention has been devoted to making the particular definition clear from the context of its use. Some typical units have been indicated. Some care must be observed as quantities have multiple units, such as a spectral bandwidth that can be in terms of wavelength, wave number, or frequency; some of these are designated by (mu) meaning “multiple units.” A length could, for example, be in m, cm, μm, or other units, so that only a typical unit is shown. Some quantities are nondimensional; these are designated by (nd).
a quantity in reflectivity relations (nd); spacing between surfaces, m; thickness, m; coefficient in phase velocity of electromagnetic wave (nd)
a 0 autocorrelation distance of surface roughness, m
akj matrix elements (mu or nd)
a matrix of elements akj
a 1 inverse matrix (mu or nd)
A surface area, m 2 , absorptance of a translucent plane layer (nd)
A, B, C, D field amplitude coefficients
A r aspect ratio of rectangle (nd)
Alm coefficients in spherical harmonics expansion
Aij equivalent spectral line width (mu, wavelength, wave number, frequency)
AA l , equivalent spectral bandwidth (mu)
b spacing, m; width of a base, m; a dimension, m; coefficient in phase velocity of electromagnetic wave (nd); quantity in reflectivity relations (nd); pressure parameter in Table 9.2 (nd)
B pressure broadening parameter (nd); length dimension, m
B magnetic induction vector, Wb/m 2
c speed of electromagnetic radiation propagation in medium other than a vacuum, m/s
c 0 speed of electromagnetic radiation propagation in vacuum , m/s
c, cp, cv specific heat, J/(kg K)
C a coefficient or constant (mu or nd); clearance between particles, m; particle volume fraction (nd)
C1 constant in Planck’s spectral energy distribution (Table A.4), W · μm4/(m 2 · sr)
C2 constant in Planck’s spectral energy distribution (Table A.4), μm K
C3 constant in Wien’s displacement law (Table A.4), μm · K
Ci concentrations of the i components in a mixture (nd)
Ckj matrix elements (mu or nd)
C CO 2 , C H 2 O pressure-correction coefficients (nd)
d number of diffuse surfaces (nd); a dimension, m
dA* differential element on the same surface area as dA, m 2
D thickness of a layer or plate, m; a dimension, m; diameter of tube or hole, m; diameter of atom or molecule, m; number of dimensions (nd)
D f fractal dimension (nd)
Dp particle diameter, m
D electric displacement, C/m 2
e energy level of a quantized state or photon, J
E emissive power (usually with a subscript), W/m 2; amplitude of electric intensity wave, N/C; overall emittance of a translucent layer (nd); the quantity (1 − ϵ)/ϵ, (nd)
E n exponential integral (Appendix D), (nd)
Ew weighted error (nd)
Eff absorption efficiency for a grooved directional absorber (nd)
E electric field vector, V/m
f (ξ) frequency distribution of events occurring at ξ (nd)
F configuration factor (nd); objective function in optimization (mu); separation variable in Equation 11.39
F0 →λ fraction of total blackbody intensity or emissive power in spectral region 0 to λ (nd)
F transfer factors in enclosure (nd)
g gravitational acceleration, m/s2
g(κη) cumulative distribution function in k-distribution method, Equation 9.46
gggs , gas-gas and gas-surface direct exchange areas, m 2
g Weyl component of the dyadic Green’s function, m
G incident radiative flux onto a surface, W/m 2; Green’s function
G dyadic Green’s function, 1/m
h Planck’s constant, J·s (Table A.1); height dimension, m; convective heat transfer coefficient, W/(m 2 K); enthalpy, J/kg
hv volumetric heat transfer coefficient, W/(m 3 · K)
H wave amplitude of magnetic intensity, C/(m s); convection-radiation parameter (nd)
H magnetic field vector, C/(m · s)
I λ spectral radiation intensity, W/(m 2 μm sr)
I radiation intensity, W/(m 2 · sr)
i, j, k unit vectors in x, y, z coordinate directions (nd)
i number of increments (nd)
ˆ
I source function, W/m 2
Im imaginary part
J radiosity; outgoing radiative flux from a surface; auxiliary variational function (mu or nd); number of increments (nd)
J current density vector, A/m 2
Jr random current density vector, A/m 2
k thermal conductivity, W/(m K); extinction coefficient for electromagnetic radiation, m−1; wave vector () = ′ + ′′ kik rad/m
k B Boltzmann constant (Table A.1), J/K;
K dielectric constant () = ′ + ′′ KiK (nd); kernel of integral equation (mu or nd)
Kij finite element function defined in Equation 13.128
l a length, m, or a dimensionless length (nd)
l, m, n direction cosines for normal direction used in contour integration method (nd)
l1, l2 , l 3 direction cosines for rectangular coordinates designated as x1, x 2 , x 3 (nd)
lm mean penetration distance, m
L length dimension, m
L e mean beam length of gas volume, m
L
e,0
mean beam length for limit of very small absorption, m
L Ladenberg-Reiche function (nd)
M mass of a molecule or atom; molecular weight
Mkj minor of matrix element akj (mu or nd)
n index of refraction of a lossless material c 0/c (nd); ratio n 2 /n1 in a few equations (nd); index in summation (nd); sample index (nd); number of a surface (nd); pressure parameter in Table 9.2 (nd); ordinate directions in Sn approximation (nd); normal direction (nd)
n complex refractive index n − ik (nd)
n unit normal vector (nd)
N number of surfaces in an enclosure (nd); number of sample bundles per unit time, s−1; number of particles per unit volume, m−3; density of electromagnetic states, s/(m3 · rad)
Nc conduction-radiation parameter (nd)
Nx , Ny number of x and y grid points (nd)
Nu Nusselt number hD/k (nd)
p partial pressure of gas in mixture, atm
ˆ
p TM-polarized unit vector
P perimeter, m; probability density function (nd); total pressure of gas, atm
P, Q, R functions used in contour integration, m−1
P0 pressure of 1 atm
Pe effective broadening pressure (nd)
Pl m associated Legendre polynomials (nd)
Pr Prandtl number cp μ f /k (nd)
q energy flux, energy per unit area and per unit time, W/m 2
q internal energy generation per unit volume, W/m 3
qc energy per unit area per unit time resulting from heat conduction, W/m 2
ql radiative flux in a spectral band (mu)
qr net radiant energy per unit area per unit time leaving a surface element, W/m 2
qr radiative flux vector, W/m 2
Q energy per unit time, W; ray origin point
Qa absorption efficiency factor (nd)
Qs scattering efficiency factor (nd)
r radial coordinate, m; radius, m
re electrical resistivity, N · m 2 · s/C2 = Ω · m
rij Fresnel’s reflection coefficient at interface i-j
r position vector (mu or nd)
R radius, m; overall reflectance of translucent plane layer or group of multiple layers (nd); random number in range 0 to 1 (nd)
Re Reynolds number Du mρf /μ f (nd); real part
s λ scattering cross section, m 2
s unit vector in S direction (nd)
ˆ
s TE-polarized unit vector
s j γ γ surface-gas direct exchange area in zonal method
ssjk surface-surface direct-exchange area in zonal method
S coordinate along path of radiation, m; distance between two locations or areas, m; surface, m 2; number of sample energy bundles per unit time, s−1
S Poynting vector, W/m 2; energy per unit area and time, W/m 2
S dimensionless internal energy source (nd)
Sc collisional line intensity (mu)
SdV number of energy bundles absorbed per unit time in volume dV (nd)
Sij spectral line intensity (mu)
S kj geometric-mean beam length from Ak to Aj, m
Sn two-dimensional radiation integral functions (Appendix D) (nd); singular values from matrix decomposition
Sr dimensionless radiative heat source (nd)
St Stanton number, Nu/(Re · Pr) (nd)
t time, s
tij Fresnel’s transmission coefficient at interface i-j
� t dimensionless time (nd)
t (S ) transmittance of a medium (nd)
t jk geometric-mean transmittance (nd)
T absolute temperature, K; overall transmittance of a plane layer or group of multiple layers (nd)
Tl mean transmission in a spectral band (nd)
Tw1, Tw2 temperatures of walls 1 and 2, K
u fluid velocity, m/s; the variable Χα /ω (nd); energy density, J/m 3
uk spectral band parameter hc ηk /kT (nd)
uum , mean fluid velocity, m/s
u, v velocity, m/s
U, V orthogonal matrices resulting from singular value decomposition
U total number of unknowns for an enclosure (nd); radiant energy density, J/m 3
U(x, y) approximate solution in finite element method (mu)
Uv spectral radiant energy density, J/(m 3 μm)
V volume, m 3; voltage signal, V = N/C
Vγ volume of element γ in zoning method, m 3
w width, m; energy carried by sample Monte Carlo bundle, J; weighting factors (nd)
W weighting function in finite element method (nd); width dimension, m
x, y, z coordinates in cartesian system, m
X coordinate, m, or dimensionless coordinate (nd); mass path length, g/m 2
X, Y, Z optical or dimensionless coordinates (nd)
Yl m normalized spherical harmonics (nd)
z height of surface roughness, m
Greek symbols
α absorptivity (nd); thermal diffusivity, m 2 /s; coefficient in soot scattering correlations
α(S ) absorptance of a medium (nd)
α , α0 band parameters in Tables 9.2 to 9.4, m 2 /(g · cm); regularization parameter in Tikhonov regularization
α , β, γ direction cosines (nd)
α , δ, γ angles measured from normal direction in contour integration method, rad
α jk geometric-mean absorptance, m−1
β extinction or attenuation coefficient κ + σs, m−1; angle in x-y plane, rad; coefficient of volume expansion, K−1; the parameter πγc /δ (nd)
βR Rosseland mean attenuation coefficient, m−1
ßi coefficients in shape function in finite-element method (nd)
γ electrical permittivity, C2 /(N·m 2); polynomial coefficients (nd); half-width of a spectral line (mu)
γ 2 variance in a statistical solution (nd)
Γ number of gas elements (nd)
Γ factors in Gebhart’s method (fraction of energy leaving one surface that is absorbed by another (nd); separation variable in Equation 11.39; function in integral equation (Equation 11.49), W/m 2
δ propagation angle in medium, rad; boundary layer thickness, m; average spacing between lines in absorption band (mu); penetration distance of evanescent waves, m
δkj Kronecker delta; = 1 when j = k; = 0 when j ≠ k
δ () ′ ′′ rr Dirac delta function
Δ distance above a radiating body
Δϵ correction for spectral overlap (nd)
Δφ intermediate function in alternating direction implicit method (mu)
ϵ emissivity of a surface (nd)
ϵ(S ) emittance of a medium (nd)
ϵ h eddy diffusivity for turbulent flow, m 2 /s
ϵ
ρ porosity (nd)
ζ arbitrary direction (nd); the quantity C2 / λT (nd)
η fin efficiency (nd); Blasius similarity variable (nd); wave number, l/ λ , m−1
θ polar or cone angle measured from normal of surface, rad
θo scattering angle, rad
Θ dimensionless temperature, T(σ/qmax)1/4; separation variable in Equation 11.39; mean energy of a Planck oscillator, J
ϑ dimensionless temperature T/Tref (nd)
κ absorption coefficient, m−1
κe effective mean absorption coefficient, m−1
κi incident mean absorption coefficient; absorption coefficients in weighted-sum-ofgray-gases emittance model, m−1
κP Planck mean absorption coefficient, m−1
κR Rosseland mean absorption coefficient
κλ spectral absorption coefficient, m−1
λ wavelength, m
λ m wavelength in a medium other than vacuum, m
μ magnetic permeability, N/A 2; dimensionless fin conduction parameter (nd); the quantity cosθ (nd); the quantity SSc / δ (mu)
μf fluid viscosity, kg/m·s
ν frequency, cc c m 00 0 1 // /s Hz λλ λ == = ,
ξ length coordinate, m; parameter πD/ λ for scattering (nd); parameter SSij /2πγc for equivalent line width
ξ, η dimensionless coordinates (nd)
ξC clearance parameter for particle separation criteria, πC/ λ
ρ reflectivity (nd); gas density, kg/m 3
ρe electric charge density, C/m 3
ρij reflectivity at interface i-j
ρf density of a fluid, kg/m 3
ρM density of a material, kg/m 3
ρs specular reflectivity, (nd)
ρ* , ρ0 distances between points, m or (nd)
σ Stefan-Boltzmann’s constant, Equation 1.27 and Table A.4, W/(m 2 K4)
σs scattering coefficient, m−1
σ0 root-mean-square height of surface roughness, m
τ roughness correlation length, m; optical thickness (nd); transmittance (Chapter 17) (nd)
τD optical thickness for path length D (nd)
ϕ
circumferential or azimuthal angle, rad; dimensionless function, Equation 11.55 (nd)
Φ scattering phase function (nd); shape function in finite-element method (nd); function in integral equation (mu or nd); function in Equation 9.33d (nd)
Φd viscous dissipation function, J/(kg · m 2)
χ angle of refraction, rad
ψ dimensionless heat flux (nd); stream function
ψ1 temperature jump coefficient (nd)
ψ(3) pentagamma function (nd)
ψb dimensionless energy flux for black walls (nd)
ψ function in Equation 9.33c (nd)
ω albedo for scattering (nd); angular frequency, rad/s; width of spectral band (mu); band width parameter, cm−1
ω o parameter in Tables 9.2 to 9.4, cm−1
Ω solid angle, sr
Ωi incident solid angle, sr
F transfer factor, Equation 5.41
Subscripts
α absorbed; absorption; absorber; apparent value
α0, α1, … coefficients
abs absorbed
A property of surface A
b on a base surface; at base of a fin; bottom
b, black blackbody condition
bi-d bidirectional
c evaluated at cutoff wavelength; corrected values; collision broadening; at a collector (absorber) plate; cylinder; cross section
cond conduction
c coating
CO2 carbon dioxide
d disk
d, dif diffuse
d1, d 2 evaluated at differential elements d1, d 2
d-h directional hemispherical
D Doppler broadening
e emitted or emitting; entering; environment; element of area; energy input; electrical; effective value
eq at thermal equilibrium
evan evanescent wave
E electric
f fluid
fc free convection
fd fully developed
F final
g gas
h hemispherical
H magnetic
H 2O water vapor
i incident; inner; incoming
i, j energy states
I initial
j, k property of surface Aj or Ak
l spectral band; layer index
L long wavelength region
LO longitudinal optical
m mean value; in a medium; maximum value; metal
m, m ′ outgoing and incoming angular directions
m, n number of identical semitransparent plates in a system
mc metal on cold side
mh metal on hot side
mP evaluated at midpoint
max corresponding to maximum energy; maximum value; maximum refraction angle
min minimum
M maximum value; material
n normal direction; natural broadening
nd nondiffuse
N, S, E, W directions in Figures 13.8 and 13.9
o outer; outgoing; evaluated in vacuum
p projected; particle
prop propagating wave
P Planck mean value; perimeter; point in discrete ordinates method
r reflected; reduced temperature; reservoir; radiative
rad radiation
ref reference value
R radiator; radiating source; Rosseland mean value; radiative
s surface of a sphere; sun; scattering; surroundings; source; solid; specular
sol solar
sub substrate
S short wavelength region
t transmitted; top
TE transverse electric
TM transverse magnetic
TO transverse optical
u uniform conditions
w wall; window
x, y, z components in x, y, z directions
η wave number dependent
λ wavelength (spectrally) dependent
Δλ for a wavelength band Δλ
λ1 → λ2 in wavelength region from λ1 to λ2
λT evaluated at λT
ν frequency dependent
ω angular frequency dependent
0 in vacuum
1, 2 surface or medium 1 or 2
⊥ perpendicular component
|| parallel component
∩ hemisphere of solid angles
Superscripts
i inside of an interface
n nth time interval
o inlet value; outside of an interface
s specular exchange factor
(0), (1), (2) zeroth-, first-, or second-order term; designation for moments
+ along directions having positive cos θ along directions having negative cos θ (overbar) averaged over all incident or outgoing directions; mean value; complex value
~ dimensionless quantity
* complex conjugate
