WEB APPENDICES
CONTENTS:
A: Derivation of Geometric Mean Beam Length Relations
B. Radiative Transfer in Porous and Dispersed Media
C. Benchmark Solutions for Verification of Radiation Solutions
D. Flames, Luminous Flames, and Particle Radiation
Web Appendix-1
A: Derivation of Geometric Mean Beam Length Relations
The Geometric Mean Beam Lengths depend on both geometry and wavelength through the
definitions
Aj Fj  k t , j  k  
Ak

Aj
e  S cos  k cos  j
 S2
dAj dAk
Aj Fj  k  , j  k  Aj Fj k 1  t , j k 
(A-1)
(A-2)
The double integral in (A-1) must be evaluated for various orientations of surfaces Aj and Ak;
the result will depend on kλ. Derivations for some specific geometries are now considered.
A.1 Hemisphere to Differential Area at Center of its Base
Let Aj be the surface of a hemisphere of radius R, and dAk be a differential area at the center
of the base (Fig. A-1). Then (A-1) becomes, since S=R and  i = 0,
Aj dFj  dk t , j  dk  dAk 
Aj
Web Appendix-2
e R cos  k cos  0 
dAj
 R2
FIGURE A-1 Hemisphere filled with isothermal medium.
The convenient dAj is a ring element dAj = 2πR2 sin  k d  k, and the factors involving R can
be taken out of the integral. This gives
e R 2 R 2  2
cos k sin k dk  dAk e  R
Aj
 R 2 k 0
Aj dFj dk t , j dk  dAk 
With Aj dFj  dk  dAk Fdk  j and Fdk  j  1, , this reduces to
t , j dk  e R
Web Appendix-3
(A-3)
This especially simple relation is used later in the concept of mean beam length where
radiation from an actual volume of a medium is replaced by that from an equivalent
hemisphere.
A-2 Top of Right Circular Cylinder to Center of its Base
This geometry is in Fig. A-2. Since θj = θk = θ, the integral in Eq. (A-1) becomes, for the top
of the cylinder Aj radiating to the element dAk at the center of its base,
e S cos 2 
dAj
Aj
 S2
Aj dFj dk t , j dk  dAk 
(A-4)
Since  2  S 2  h2 , dAj  2 d  2 S dS. Then, using cos θ = h/S,
Aj dFj dk t , j dk  dAk 2h 2 
h
Web Appendix-4
R2  h2
e   S
dS
S3
(A-5)
FIGURE A-2 Geometry for exchange from top of gas-filled cylinder to center of its base.
Web Appendix-5
Now let λS = λ to obtain
Aj dFj dk t , j dk  dAk 2h 2 2 
  R2  h2
e 
 3
    h
d 
(A-6)
This integral can be expressed in terms of the exponential integral function defined in
Appendix D, by writing
  R 2  h2

e 
 3
   h

Letting      R 2  h 2



d   
  R 2  h2
  
e 
 3
a h
e 
  
 3
d   
d 
(A-7)
 and   h  , respectively, in the two integrals gives
1
  R 2  h2

2

1
0
 e  
R2  h2 
d 
1
1
e
  h  
2
0
  h 
d
The integral in (A-6) is then written in terms of the exponential integral function as
  R2  h2

e  
   h
 3
d  
1
  h 
2


R


E3   h  
E

h

1
3



2
h
2




 h ( R / h)  1 


 

1
2
so it can be readily evaluated for various values of the parameters R/h and λh.
Web Appendix-6
(A-8)
A-3 Side of Cylinder to Center of its Base
Let dAj be a ring around the wall of a cylinder as in Fig. A-3, and note that dAj = 2πRdz, cos
θk = z/S, cos θj = R/S, and z dz = SdS. Then (10-61) is written for the side of the cylinder to
dAk as
Aj dFj dk t , j dk  2dAk R 2 
R
Web Appendix-7
R 2  h2
e   S
dS
S3
(A-9)
FIGURE CD-A3 Geometry for exchange from side of gas-filled cylinder to center of its
base.
This is of the same form as (A-5), and gives the result
Web Appendix-8
Aj dF j  dk t , j  dk


2
2


1

1
R
R
 R 

2 
 2dAk   (  h) 
E3   h    
E3   h    1  
2
2
h


2


h


h
  h ( R / h)  1 
   h  R / h   


 



(A-10)
As for (A-8), this is readily evaluated for various values of R/h and λh.
A.4 Entire Sphere to Any Area on its Surface or to its Entire Surface
From Fig. A-4, since θk = θj let them be simply θ; then S = 2R cos θ. Starting with (CD-A4),
dAj cos θ/S2 is the solid angle by which dAj is viewed from dAk. The intersection of the solid
angle with a unit hemisphere shows that this equals 2π sin θ dθ. Then
Aj dFj dk t , j dk  dAk 
 /2
 0
Web Appendix-9
e  S 2 cos  sin  d 
2dAk
4R2

2R
S 0
e  S SdS
FIGURE A-4 Geometry for exchange from surface of gas-filled sphere to itself.
Integrating gives
Aj dFj  dk t , j  dk 
2dAk
1   2  R  1 e 2 R 
(2a R) 2 
which has a single parameter 2λR, the sphere optical diameter.
Web Appendix-10
(A-11)
Equation (A-11) is integrated over any finite area Ak to give t from the entire sphere
2
to Ak as Aj Fj k t , j k  2 Ak /  2  R   1   2  R  1 e2 R  . Since Fj–k = Ak/Aj, from (4-17),


t , j  k 
2
1   2  R  1 e 2 R 
(2  R) 2 
(A-12)
which also applies for the entire sphere to its entire surface.
A.5 Infinite Plate to Any Area on Parallel Plate
In Fig. A-5 consider on one plate an element dAk, and on the other plate a concentric ring
element dAj centered about the normal to dAk. The geometry is like that in Fig. A-2 for a ring
on the top of a cylinder to the center of its base. Then, from (A-6),
Aj dFj dk t , j dk  dAk 2(  D)
2


   D
e 
 3
d 
where λD is the optical spacing between the plates. By using the procedure leading to (A-8),
the integral is transformed to E3 (  D) / (  D) 2 . Then integrating over any finite area Ak as in
Fig. A-5 gives Aj Fj k t , j k  Ak 2 E3 (  D) . With Aj Fj  k  Ak Fk  j and Fk–j = 1, this reduces to
t , j  k  2 E3 (  D)
Web Appendix-11
(A-13)
FIGURE A-5 Isothermal layer of medium between infinite parallel plates.
A.6 Rectangle to a Directly Opposed Parallel Rectangle
Consider, as in Fig. A-6, the exchange from a rectangle to an area element on a directly
opposed parallel rectangle. The upper rectangle is divided into a circular region and a series
of partial rings of small width. The contribution from the circle of radius R to Aj dFj  dk t , j  dk
can be found from (A-6) and (A-8), for the top of a cylinder to the center of its base. For the
nth partial ring, let fn be the fraction it occupies of a full circular ring. Then, by use of (A-4),
the contribution of all the partial rings to Aj dFj  dk t , j  dk is approximated by
dAk 
m
Web Appendix-12
e   Sn
fn
 Sn2
2
D
e   Sn
2
R R
  2 Rn Rn  dAk 2 D  f n
n
 D2  Rn2  n n
 Sn 
FIGURE A-6 Geometry for exchange between two directly opposed parallel rectangles with
intervening translucent medium.
This evaluation of Aj dFj  dk t , j  dk is carried out for several area patches on Ak. This is usually
sufficient, so the integration over Ak can be performed as indicated by (10-62) to yield
Aj Fj  k t , j  k   Aj dFj  dk t , j  dk
Ak
EXAMPLE A-1 A nongray, absorbing-emitting plane layer with λ as in Fig. A-7 is
1.2 cm thick and is on top of an opaque diffuse-gray infinite plate. The plate
temperature has been raised suddenly, so the layer is still at its initial uniform
temperature. What is the net heat flux being lost from the plate-layer system? Neglect
Web Appendix-13
heat conduction effects in the layer, and neglect any reflections at the upper boundary
of the layer.
FIGURE A-7 Plane layer of nongray absorbing-emitting material.
The uniform environment above the layer acts as a black enclosure at Te.
Equation (10-77b) can be applied directly by integrating in two spectral bands and
using Eλ2 = 1. The geometric-mean transmittance factors are obtained from Eq. (A13)
as
t ,1  2 E3 (0.16 1.2)  0.7142, 0    6.5 m; t ,2  2 E3 (0.65  1.2)
 0.2974, 6.5 m     . This yields
q  q2  ε1t ,1 ( T14 F06.5T1   Te4 F06.5Te )
 (1  t ,1 )[1  (1  ε1 ) t ,1 ]( Tg4 F06.5Tg   Te4 F06.5Te )
 ε1 t ,2 [ T14 (1  F06.5T1 )   Te4 (1  F06.5Te )
 (1  t ,2 )[1  (1  ε1 ) t ,2 ][ Tg4 (1  F06.5Tg )   Te4 (1  F06.5Te )]
Inserting the values F0→6.5×630 = 0.4978, F0→6.5×550 = 0.3985 and F0→6.5×495 = 0.3220
yields q = 2954 W/m2.
Web Appendix-14
A.7 Geometric-Mean Beam Length for Spectral Band Enclosure Equations
For use in the spectral band enclosure equations (10-82), the absorption integral in (10-85)
must be evaluated between pairs of enclosure surfaces for the wavelength bands involved.
When more than a few bands absorb appreciably, the enclosure solution requires considerable
computational effort. A simplification was developed by Dunkle (1964) by assuming that the
integrated band absorption  l(S) is a linear function of path length. This has some physical
basis, as it holds exactly for a band of weak nonoverlapping lines [Eq. (9-16)]. Also, it is the
form of some of the effective bandwidths in Eq. (9-32). As shown in Dunkle by means of a
few examples, reasonable values for the energy exchange are obtained using this
approximation. Hence, let l in (10-87) have the linear form from (9-27) [note that the
bandwidth Δλl = ω in (9-27)]
l (S ) 
Al ( S ) Al ( S ) Al ( S ) Sc


 S
l

m

(A-14)
where Sc and δ are the line intensity and the spacing of the individual weak spectral lines, as
in Chap. 9.
Now define a mean path length Sk  j called the geometric-mean beam length, such
that αl evaluated from (A-14) by using S  Sk  j will equal  l ,k  j from the integral in (10-85).
After substituting l ,k  j   Sc /   Sk  j and l   Sc /   S into (10-85), the relation for Sk  j is
Sk  j  S j k 
Web Appendix-15
1
Ak Fk  j
 
Aj
Ak
cos  j cos  k
S
dAk dAj
(A-15)
which depends only on geometry. This integral is also obtained in Eq. (A-1) when λS is
small (optically thin limit). In Dunkle, Sk–j values are tabulated for parallel equal rectangles,
for rectangles at right angles, and for a differential sphere and a rectangle. Analytical
relations for rectangles are in Eqs. (A-16a,b). For directly opposed parallel equal rectangles
with sides of length a and b and spaced a distance c apart,
Sk  j Ak Fk  j
abc

 1 

4  1  1 
 
 
 ln 
  ln 

 tan

   1   2 (   1   2    1   2 (  1   2 ) 



1 
1  2  1   2 1   

 
(A-16a)
where η = a/c, β = b/c and   1   2   2 . The Fk–j can be obtained from Factor 4 in
Appendix C. For rectangles ab and bc at right angles with a common edge b,
Sk  j Ak Fk  j
abc
1   (1  1   2 )  2   2  (1  1   2 )  2   2
  ln
 ln
 

 (1  1   2   2
 (1  1   2   2 )
1

[(1   2 )3/ 2  (1   2 )3/ 2  ( 2   2 )3/ 2  (1   2   2 )3/ 2 ]
3

 [ 1 2   2   2   2  1 2 ]


(A-16b)

2  2  2
1  
[ 1 2   2   2   2  1  2 ]   



3
 2  
where α = a/b and γ = c/b. The Fk–j is obtained from Factor 8 in book Appendix C. Results
for equal opposed parallel rectangles are in Fig. A-8, and values for equal parallel rectangles
and for rectangles at right angles are in Tables A-1 and A-2. Other Sk  j values are referenced
Web Appendix-16
by Hottel and Sarofim (1967) In Anderson and Hadvig (1989), values are obtained for a
medium in the space between two infinitely long coaxial cylinders.
FIGURE A-8 Geometric mean beam lengths for equal parallel rectangles [Dunkle (1964)].
For a medium at uniform conditions, the geometric-mean beam length can be used in
the effective-bandwidth correlations in Chap. 9 to obtain Al ( S ) . Using Δλl obtained in the
next paragraph yields  l  S   Al  S  / l from Eq. (12-54) and tl from 1   l . Then (10-82)
and (10-83) can be solved for ql,k for each wavelength band l. The total energies at each
surface k are found from a summation over all bands
qk 

absorbing
bands
Web Appendix-17
ql ,k l 

nonabsorbing
bands
ql ,k l
(A-17)
The wavelength span Δλl of each band is needed to carry out the solution. As
discussed after (9-24), this span can increase with path length. Edwards and coworkers
[Edwards and Nelson (1962), Edwards (1962), Edwards et al. (1967)] give recommended
spans for CO2 and H2O vapor; these values, in wave number units, are in Table A-3 for the
parallel-plate geometry. For other geometries, Edwards and Nelson give methods for
choosing approximate spans for CO2 and H2O bands. Briefly, the method is to use
approximate band spans based on the longest important mass path length in the geometry
being studied. With this in mind, the limits of Table A-3 are probably adequate for problems
involving CO2 and H2O vapor.
Web Appendix-18
TABLE A-1 Geometric mean beam-length ratios and configuration factors for parallel equal rectangles [Dunkle (1964)]
c/b
a/c
0
0
Sk  j / c 1.000
0.1
0.2
0.4
0.6
1.0
2.0
4.0
6.0
10.0
20.0
1.001
1.003
1.012
1.025
1.055
1.116
1.178
1.205
1.230
1.251
1.002
1.004
1.013
1.026
1.056
1.117
1.179
1.207
1.233
1.254
0.00316
0.00626
0.01207 0.01715
0.02492
0.03514
0.04210
0.04463
0.04671
0.04829
1.004
1.006
1.015
1.058
1.120
1.182
1.210
1.235
1.256
0.00626
0.01240
0.02391 0.03398
0.04941
0.06971
0.08353
0.08859
0.09271
0.09586
1.013
1.015
1.024
1.067
1.129
1.192
1.220
1.245
1.267
0.01207
0.02391
0.04614 0.06560
0.09554
0.13513
0.16219
0.17209
0.18021
0.18638
1.026
1.028
1.037
1.080
1.143
1.206
1.235
1.261
1.282
0.01715
0.03398
0.06560 0.09336
0.13627
0.19341
0.23271
0.24712
0.25896
0.26795
Fk–j
0.1
Sk  j / c 1.001
Fk–j
0.2
Sk  j / c 1.003
Fk–j
0.4
Sk  j / c 1.012
Fk–j
0.6
Sk  j / c 1.025
Fk–j
Web Appendix-19
1.028
1.037
1.050
1.0
Sk  j / c 1.055
Fk–j
2.0
Sk  j / c 1.116
Fk–j
4.0
Sk  j / c 1.178
Fk–j
6.0
Sk  j / c 1.205
Fk–j
10.0
Sk  j / c 1.230
Fk–j
20.0
Sk  j / c 1.251
Fk–j
Sk  j / c 1.272
Fk–j
1.056
1.058
1.067
0.02492
0.04941
1.117
1.110
1.175
1.242
1.272
1.300
1.324
0.09554 0.13627
0.19982
0.28588
0.34596
0.36813
0.38638
0.40026
1.120
1.129
1.175
1.246
1.323
1.359
1.393
1.421
0.03514
0.06971
0.13513 0.19341
0.28588
0.41525
0.50899
0.54421
0.57338
0.59563
1.179
1.182
1.192
1.242
1.323
1.416
1.461
1.505
1.543
0.04210
0.08353
0.16219 0.23271
0.34596
0.50899
0.63204
0.67954
0.71933
0.74990
1.207
1.210
1.220
1.272
1.359
1.461
1.513
1.564
1.609
0.04463
0.08859
0.17209 0.24712
0.36813
0.54421
0.67954
0.73258
0.77741
0.81204
1.233
1.235
1.245
1.300
1.393
1.505
1.564
1.624
1.680
0.04671
0.09271
0.18021 0.25896
0.38638
0.57338
0.71933
0.77741
0.82699
0.86563
1.254
1.256
1.267
1.324
1.421
1.543
1.609
1.680
1.748
0.04829
0.09586
0.18638 0.26795
0.40026
0.59563
0.74990
0.81204
0.86563
0.90785
1.274
1.277
1.289
1.349
1.452
1.584
1.660
1.745
1.832
0.04988
0.09902
0.19258 0.27698
0.41421
0.61803
0.78078
0.84713
0.90499
0.95125
Web Appendix-20
1.080
1.143
1.206
1.235
1.261
1.282
1.306
TABLE A-2 Configuration factors and mean beam-length functions for rectangles at right angles [Dunkle (1964)]
c/b
a/b
0.05
0.02 AkFk–j/b2
Ak Fk  j Sk  j / abc
0.05 AkFk–j/b2
Ak Fk  j Sk  j / abc
0.10 AkFk–j/b2
Ak Fk  j Sk  j / abc
0.10
0.20
0.4
0.6
1.0
2.0
4.0
6.0
10.0
20.0
∞
0.007982 0.008875 0.009323 0.009545 0.009589 0.009628 0.009648 0.009653 0.009655 0.009655 0.009655 0.009655
0.17840
0.12903 0.08298
0.04995
0.03587
0.02291
0.01263
0.006364 0.004288 0.002594 0.001305
0.014269 0.018601 0.02117
0.02243
0.02279
0.02304
0.02316
0.02320 0.02321
0.21146
0.18756 0.13834
0.08953
0.06627
0.04372
0.02364
0.01234 0.008342 0.005059 0.002549
0.02819 0.03622
0.04086
0.04229
0.04325
0.04376
0.04390 0.04393
0.20379 0.17742
0.12737
0.09795
0.06659
0.03676
0.01944 0.013184 0.008018 0.004049
0.05421
0.06859
0.07377
0.07744
0.07942
0.07999 0.08010
0.08015
0.18854
0.15900
0.13028
0.09337
0.05356
0.02890 0.01972
0.012047 0.006103
0.10013
0.11524
0.12770
0.13514
0.13736 0.13779
0.13801
0.13811
0.16255
0.14686
0.11517
0.07088
0.03903 0.02666
0.01697
0.008642
0.13888
0.16138
0.17657
0.18143 0.18239
0.18289
0.18311
0.20 AkFk–j/b2
Ak Fk  j Sk  j / abc
0.40 AkFk–j/b2
Ak Fk  j Sk  j / abc
0.60 AkFk–j/b2
Web Appendix-21
0.02321
0.04394
0.02321
0.04394
0.08018
0.02321
0.04395
0.08018
0.13814
0.18318
Ak Fk  j Sk  j / abc
0.14164
1.0 AkFk–j/b2
Ak Fk  j Sk  j / abc
2.0 AkFk–j/b2
Ak Fk  j Sk  j / abc
4.0 AkFk–j/b2
Ak Fk  j Sk  j / abc
6.0 AkFk–j/b2
Ak Fk  j Sk  j / abc
10.0 AkFk–j/b2
Ak Fk  j Sk  j / abc
20.0 AkFk–j/b2
0.11940
0.07830
0.04467 0.03109
0.02025
0.010366
0.20004
0.23285
0.24522 0.24783
0.24921
0.24980
0.11121
0.08137
0.04935 0.03502
0.02196
0.01175
0.29860
0.33462 0.34386
0.34916
0.35142
0.07086
0.04924 0.03670
0.02401
0.01325
0.40544 0.43104
0.44840
0.45708
0.04051 0.03284
0.02320
0.01300
0.46932
0.49986
0.51744
0.02832
0.02132
0.01272
0.5502
0.5876
0.01759
0.01146
0.6608
Ak Fk  j Sk  j / abc
0.008975
Web Appendix-22
0.25000
0.35222
0.46020
0.52368
0.6053
0.7156
TABLE A-3 Approximate band limits for parallel-plate geometry [Edwards and Nelson (1962),
Edwards (1962), Edwards et al. (1967)]
Band center
Band center η,
 m
cm−1
Gas
CO2
H2O
a
Band limits η, cm−1a
Lower
Upper
15
667
667 – ( A15 /1.78)
667 + ( A15 /1.78)
10.4
960
849
1013
9.4
1060
1013
1141
4.3
2350
2350 – ( A4.3 /1.78)
2430
2.7
3715
3715 – ( A2.7 /1.76)
3750
6.3
1600
1600 – ( A6.3 /1.6)
1600 + ( A6.3 /1.6)
2.7
3750
3750 – ( A2.7 /1.4)
3750 + ( A2.7 /1.4)
1.87
5350
4620
6200
1.38
7250
6200
8100
A are found for various bands as in Example 9-1. Terms such as A15 /1.78 are A / 2 1   g  from Eq.
(17) and Tables 1 and 2 of Edwards and Nelson (1962).
If all surface temperatures are specified in a problem, the results from Eq. (A-17)
complete the solution. If qk is specified for n surfaces and Tk for the remaining N–n surfaces, then
the n unknown surface temperatures are guessed, the equations are solved for all q, and then the
calculated qk are compared to the specified values. If they do not agree, new values of Tk for the
n surfaces are assumed and the calculation is repeated until the given and calculated qk agree for
Web Appendix-23
all Ak with specified qk. Equation (11-8), expressed as a sum over the wavelength bands, gives
the required energy input to the medium for the specified Tg.
EXAMPLE A-2 Two black parallel plates are D = 1 m apart. The plates are of width W
= 1 m and have infinite length normal to the cross section shown (Fig. A-9). Between the
plates is carbon dioxide gas at pCO2  1 atm and Tg = 1000 K. If plate 1 is at 2000 K and
plate 2 is at 500 K, find the energy flux that must be supplied to plate 2 to maintain its
temperature. The surroundings are at Te << 500 K.
FIGURE A-9 Example A-2 Figure
As shown by Fig. A-9, the geometry is a four-boundary enclosure formed by two
plates and two open planes. The open areas are perfectly absorbing (nonreflecting) and
Web Appendix-24
radiate no significant energy as the surrounding temperature is low. The energy flux
added to surface 2 is found by using the enclosure equation (10-72) where k = 2 and N =
4. All surfaces are black, ελ,j = 1, so Eq. (10-72) reduces to
4
4
j 1
j 1
 2 j q , j  [( 2 j  F2 j t ,2 j )Eb, j  F2 j ,2 j Eb,g ]
(A-18)
The self-view factor F2–2 = 0 and Eλb,3 = Eλb,4 ≈ 0, so this becomes
q ,2   F21t ,21 Eb,1  Eb,2  ( F21  ,2 1  F2 3  ,2 3  F2 4  ,2 4 ) Eb, g
(A-19)
To simplify the example, it is carried out by considering the entire wavelength region as a
single spectral band. To obtain the total energy supplied to plate 2, integrate over all
wavelengths to obtain

q2   F21  t ,21 Eb ,1d    T24
0

  ( F21  ,21  F23  ,23  F2 4  ,2 4 ) Eb, g d 
0
By use of the definitions of total transmission and absorption factors,

t21 T14   t ,21Eb,1d 
0
Web Appendix-25

21 Tg4    ,21Eb, g d 
0
and so forth, q2 becomes
q2   T24  F21t21 T14  (F2121  F2323  F2424 ) Tg4
(A-20)
To determine t and  , the geometric-mean beam length is used. For opposing
rectangles (Fig. A-8) at an abscissa of 1.0 and on the curve for a length-to-spacing ratio
of ∞, the S21 / D  1.34 , so S21  1.34 m . To determine  21 , which determines the
emission of the gas, use the Lechner correlation (9-62) or Fig. 9-15) at a pressure of 1
atm, Le = 1.34 m (giving pLe = 138 bar-cm) and Tg = 1000 K. This gives 21  0.22 .
When obtaining t2 1 , note from (A-19) that the radiation in the t2 1 term is Eλb,1 and is
coming from wall 1. Therefore it has a spectral distribution different from that of the gas
radiation. To account for this nongray effect, (10-118) is used with ε+ evaluated at
pCO2 S21 (T1 / Tg )  1.34(2000 /1000)  2.68 atm  m =277 bar-cm and T1 = 2000 K. Then,
using the Lechner correlation or Fig. 9-15 (extrapolated) and Eq. (10-118) result in
t21  1  0.2( 12 )0.5  0.86 .
From
factor
3
in
Appendix
C
F21  [( D 2  W 2 )1/ 2  D] / W  2  1  0.414 . Then
the
F2–1
is
given
by
F23  F24  12 (1  0.414)  0.293 .
The  23   24 , and they remain to be found. For adjoint planes, as in the geometry for
Table 12-2, the following expression from Eq. (12) of Dunkle (1964) can be used,
obtained
Web Appendix-26
for
the
present
case
where
b
→∞,
a
=
1,
and
c
=
1:
S23  2ln 2 /( F23 )  2  0.347 /(  0.293)  0.753 m.
Using
Fig.
9-15
pS  0.753atm  m = 77.7 bar-cm and Tg= 1000 K gives  23   24  0.19 . Then the
desired result is
q2   T24  0.414(0.86) T14  (0.414  0.22  2  0.293  0.19) Tg4
 5.6704 1012 (5004  0.36  20004  0.20 10004 )
 33.4 W/cm2
Note that the largest contribution to q2 is by energy leaving surface 1 that reaches and is
absorbed by surface 2. Emission from the gas to surface 2, and emission from surface 2,
are small.
An alternative approach for this particular example is to note that the term
involving Tg in (A-20) is the flux received by surface 2 as a result of emission by the
entire gas. This can be calculated from (10-114) using the mean beam length. Then
q2   T24  F21t21 T14  ε g Tg4 . For this symmetric geometry the average flux from the
gas to one side of the enclosure is the same as that to the entire enclosure boundary.
Consequently,
the
mean
beam
length
can
be
obtained
from
(10-112)
as
Le  0.9(4)V / A  0.9(4)(1 m) 2 / 4 m  0.9 m. Then, from Fig. 9-15 at Tg = 1000 K and
pCO 2 Le  0.9 atm  m = 92.9 bar-cmthe εg = 0.20. This gives the same q2 as previously
calculated.
Web Appendix-27
B. RADIATIVE TRANSFER IN POROUS AND DISPERSED MEDIA
Radiative transfer in porous and dispersed media is reviewed in Tien and Drolen (1987),
Tien (1988), Dombrovsky (1996), Baillis-Doermann and Sacadura (1998), Howell (2000), and
Kamiuto (1999, 2008). These references provide extensive bibliographies. Many early models
neglected the effect of anisotropic and/or dependent scattering that have been found important in
some applications. If the porous structure is well defined, newer models have moved toward
numerical and statistical modeling approaches that incorporate the detailed porous structure and
its thermal properties. Considering radiative transfer among the structural elements provides
detailed radiative transfer results, but does not yield simplified engineering heat transfer
correlations. If the structural elements are translucent, refraction at the solid-fluid interfaces must
be considered as well as the external and internal reflections that occur at these interfaces (see
Sec. 17-5.3). Almost always, the structural elements are so closely spaced that dependent
scattering occurs (Chapters 15, 16). Independent scattering has been shown to fail for systems
with low porosity and for packed beds [Singh and Kaviany (1994), Coquard and Biallis (2004)],
and deviations from independent scattering theory have been found at porosities as high as 0.935.
A less detailed engineering approach has been to assume that the porous material can be
treated as a continuum. This depends on the minimum porous bed or medium dimension, L,
relative to the particle or pore diameter, Dp or Dpore, and the particle size parameter ξ = πDP/λ.
For most practical systems the porous material is many pore or particle diameters in extent
(L/Dpore or L/Dp > ~10), and the pores or particles are large relative to the wavelengths of the
important radiative energy (ξ > ~ 5). In this case, the porous medium may be treated as
Web Appendix-28
continuous, and the effective radiative properties are measured by averaging over the pore
structure. Traditional radiative transfer analysis methods for translucent media can then be
applied without using detailed element-to-element modeling. When temperature gradients are
large and/or the thermal conductivity of the elements is small, the assumption of isothermal
elements may not be accurate [Singh and Kaviany (1994); Robin et al. (2006)]. Combustion in
some liquid- and gas-fueled porous burners occurs within the porous structure, producing large
temperature gradients; temperatures within the porous medium can increase by over 1000 K
within a few pore diameters.
The energy equation for porous media analysis depends on the complexity of the
particular problem to be solved. To determine the energy transfer between the porous solid and a
flowing fluid, energy equations are written for both the solid and the fluid, with a convective
transfer term providing the coupling between the two equations. This approach permits
calculation of the differing bed and fluid temperatures. If the fluid is transparent (no absorption
or scattering) the radiative flux divergence is only in the equation for the absorbing and radiating
solid structure of the porous material. In this case, the energy equations (in one dimension)
become, for the fluid and solid phases,
uc p
dT f
dx

d  dT f 
kf
  q  hv (T f  Ts )  0
dx 
dx 
d  dTs  dqr
 hv (T f  Ts )  0
 ks

dx  dx  dx
Web Appendix-29
(B-1)
(B-2)
The hv is the volumetric convective heat transfer coefficient, and q is the volumetric energy
source in the fluid from combustion or other internal energy sources. The properties are effective
properties that depend on the structure of the solid and the flow configuration. If multiple species
are present, as for combustion, additional terms for species diffusion must be included in the
fluid-phase equation. The coupling of (B-1) and (B-2) through the convective transfer terms
shows that radiation affects both the solid and fluid temperature distributions even when a
radiation term is only in the solid energy equation.
If there is no fluid within the porous medium, a single energy equation is used for the
temperature distribution of the solid structure. Then Eqs. (B-1) and (B-2) reduce to the steady
form of (10-3) with no viscous dissipation. Alternatively, if the fluid flow rate is sufficiently
large, the volumetric heat transfer coefficient becomes large and the local fluid and solid
temperatures become essentially equal; then Eqs. (B-1) and (B-2) combine to give
 uc p
dT d 
dT  dqr
  keff
q 0

dx dx 
dx  dx
(B-3)
In this equation, keff is the effective thermal conductivity of the fluid-saturated porous medium,
and the other properties are corrected for the porosity and are on a per unit of total volume basis.
Most analyses of radiative transfer in porous media rely on solving the radiative transfer
equation (RTE) and it is necessary to measure or predict the effective continuum radiative
properties of the porous medium. This can be done by direct or indirect measurements, or by
predicting the properties using models of the geometrical structure and surface properties of the
Web Appendix-30
porous structural material. Most radiative property measurements are made by inferring the
detailed properties from measurement of radiative transmission or reflection by the porous
material. Measured and predicted properties of various packed beds are discussed in Howell
(2000), and the properties of open-cell foam insulation are studied in Baillis et al. (2000).
Haussener et al. (2009) use computerized tomography to determine the physical characteristics
of a packed bed of CaCO3 particles, and then use forward Monte Carlo to determine the spectral
scattering and absorption coefficients as well as the spectral phase function. Although the results
by Haussener et al. were independent of the reflectivity characteristics of the system boundaries
(specular or diffuse), they were strongly dependent on the assumed reflectivity characteristics of
the particles themselves. Lalich et al. (2009) examine the radiative properties of nanoporous
matrices composed of silica particles.
Solutions are difficult when the radiative mean free path is of the order of the overall bed
dimensions. Simplifying assumptions cannot be made, such as an optically thick medium, and in
this case a complete solution of the RTE may be required. The two-flux method is often used
when one-dimensional radiative transfer is assumed; however, if the particles in the bed are
absorbing, this method does not give satisfactory results [Singh and Kaviany (1994)]. If the solid
structure of the porous medium has a defined shape it is possible to use conventional surface-tosurface radiative interchange analysis. This is used in Antoniak et al. (1996) and Palmer et al.
(1996) with a Monte Carlo cell-by-cell analysis to simulate radiative transfer among arrays of
geometric shapes. Some research has tried to define an effective radiative conductivity that can
be combined with heat conduction. For use in optically thick systems with porosity between 0.4
and 0.5, composed of opaque solid spherical particles with surface emissivity εr and thermal
Web Appendix-31
conductivity of the solid ks, the radiative conductivity can be approximated by [Singh and
Kaviany (1991, 1994), Kaviany (1991)]:


kr  4 Dp T 3 0.5756ε r tan 1 1.5353(ks* )0.8011 / ε r   0.1843
(B-4)
where ks*  ks /(4Dp T 3 ) . This is reasonably accurate for both diffuse and specular reflecting
particles (the constants in (B-4) from Singh and Kaviany (1994) are specifically for diffuse
surfaces), and is weakly dependent on bed porosity. Coquard and Baillis (2004) treat beds of
specular and diffuse opaque spheres by Monte Carlo, and find good agreement with the
correlations of Singh and Kaviany.
B.1 Radiative Properties of Porous Structures and Materials
Radiatively porous materials find applications in filament-wound structures, ablative coatings for
spacecraft atmospheric entry, porous media combustion systems, packed and pebble beds for
combustion and nuclear reactors, and others.
With the exception of porous media used as absorbers from a very-high-temperature
external radiation source or for internal combustion or nuclear applications, most radiative
transfer porous media analyses require knowledge of the radiative properties dominated by
wavelengths in the infrared region of the spectrum. Internal medium temperatures are limited by
the material properties of the porous medium, and for applications where bed temperatures are of
the order of 1500K or below, the important radiative transfer will be at wavelengths of 2 m and
greater. For solar collectors or reentry ablation materials, the effective incident energy
Web Appendix-32
distribution is from very high temperature sources (up to 15,000 K in some reentry cases), and
the important incident radiation is at short wavelengths. In such a case, the spectral dependence
of the radiative properties must be considered.
RADIATIVE PROPERTY PREDICTION
If a simple geometry is representative of a porous material, it is possible to predict the equivalent
radiative properties of a homogeneous medium using radiative transfer analysis based on
surface-surface interactions.
Cell models
Most early models of radiative properties in porous materials are based on
simplified models of the structure. For example, random packings of spheres are assumed to
have characteristics of regular packings. This clearly allows much simpler analysis because of
geometric regularity. Usually, a unit cell of the regular packing is analyzed, and the
characteristics of this cell are then assumed to be repeated throughout the porous medium.
Radiation and conduction are assumed to occur in some parallel-series mode that is dependent on
the cell structure. Cell models often predict agreement for a limited range of parameters, but fail
outside of that range. Further, the radiative behavior of the cell components is usually assumed to
be governed by diffuse reflections, and this is often far from the case. Finally, most early cell
models did not recognize the importance of dependent scattering effects. Vortmeyer (1978) and
Baillis and Sacadura (2000), review cell models, and compare their predictions with experiment
and with each other. The models agree best with each other when the bed particles are radiatively
black, as might be expected because radiative transfer for each model becomes less dependent on
the surface reflective behavior.
Web Appendix-33
Measured Properties of the Equivalent Homogeneous Material. Because most treatments of
radiative transfer in porous media rely on solution of the RTE, it is necessary to measure or
predict the effective continuum radiative properties of the porous medium. Approaches to
finding these properties can be separated into two classes; direct or indirect measurement of the
required properties, or prediction of the properties from cell models and surface properties of the
porous material. The properties are dependent on the particle scale, material and geometry of the
particular porous material, and may be functions of temperature and wavelength. Depending on
the criteria for particle size and clearance relative to important wavelengths, dependent scattering
and absorption effects may also become important. The problem of adequately describing
particle geometry is the most daunting.
Most measurements of radiative properties are made by inferring the detailed properties
from measurements of transmission or reflection of radiation from the porous material. Inversion
of these measurements using methods outlined in Chap. 8 is then used to find the best set of
scattering coefficient, absorption coefficient and phase function that will predict the measured
effects.
Two problems are inherent in these measurements. First, inverse techniques are
susceptible to large uncertainties in the inferred values that depend on the experimental
uncertainty in the measured values, and there are questions of uniqueness as well (Chap. 8).
Second, the model of radiative transfer that is used in the inversion may have particular
assumptions embedded within it. For example, a simple phase function behavior (isotropic,
linearly anisotropic, etc.) may be assumed to simplify the radiation model used in the inversion.
Web Appendix-34
If this is done, then the resulting values of inferred absorption and scattering coefficient will
depend on that assumed type of phase function. It follows that the radiative model used in
describing radiation in a porous material must use the same assumptions, or the absorbing and
scattering coefficients will not be compatible. This may be overlooked when reported properties
are taken from the literature and then using them in an incompatible model of radiative transfer.
Dependent scattering For closely-spaced particles, fibers, or other bodies, the scattering and
absorption of radiation is affected by near-field radiation interactions (Chapter 16). As the
volume fraction occupied by particles increases (porosity decreases), these effects become more
important. Yamada et al. (1986) present experiments for particles with dependent scattering.
For spherical particles, Kaviany and Singh (1993) recommend that independent
scattering can be assumed when the criterion
C  0.1d p 

2
(B-4)
is met, where C is the interparticle clearance distance. This criterion can also be written as
 C  0.1 

2
(B-5)
where the clearance parameter C = C/ If this criterion is not met, near-field effects must be
considered. This result was derived for porosities typical of rhombohedral packing,   0.26, but
dependent scattering in packed beds remains an important effect for bed porosities as high as
0.935, and the effect is most pronounced for opaque particles. Both the porosity requirement and
Web Appendix-35
the relation between C and  [Eqs. (B-4) or (B-5)] must be satisfied before the assumption of
independent scattering can be used with confidence. Brewster and Tien (1982) provided the
criterion for independent scattering as
 
C
1   1/ 2
 0.905  (1   )1/ 2
(B-6)
where the independent scattering region (i.e., deviation of more than five percent from
independent scattering results) is demarcated when C/ = 0.5 is inserted into the equation [Tien
(1988)]. This relation is based on far-field interference effects.
For a packed bed of spherical particles with an average bed temperature near 500K,
Wien’s displacement law gives the wavelength at the peak of the blackbody spectrum as max =
C3/T = 2898(m K)/500K = 5.8 m. This wavelength splits the energy in the blackbody
spectrum such that 25 percent of the radiant power is at shorter wavelengths, and the remainder
is at longer wavelengths. This provides a reasonable criterion for an approximate determination
of whether dependent scattering will be important. If the bed is closely packed so that the
clearance distance can be taken near zero, then the particle diameter below which dependent
scattering becomes important according to Eq. (B-4) is dp=5max = 29 m. At the same
temperature, Eq. (B-6) with  = 0.26 gives the minimum particle diameter for assuming
dependent scattering as dp = 30.6 m. Lower temperature beds will have correspondingly greater
values of max and particle diameters in the dependent scattering region will be larger.
Web Appendix-36
Kaviany and Singh (1993) also present a relation for a scaling factor that gives the
relation between transmission through a packed bed of opaque particles with independent
scattering to the result assuming dependent scattering, and show that this result is only dependent
on the bed porosity (i.e., independent of particle radiative emissivity).
Closely-spaced parallel cylinders are of interest in applications such as in-situ curing by
an infrared source during filament winding and the determination of nanomaterial properties. Lee
(1994) and Chern et al. (1995c) determined the radiative properties of arrays of parallel cylinders
subject to normal irradiation and far-field dependent effects. They show the ranges of cylinder
volume fraction and size parameter where these effects become important for various values of
the fiber effective refractive index. Lee and Grzesik (1995) later extended the analysis to coated
and uncoated fibers subject to obliquely incident radiation. Chern et al. (1995b) show that
dependent scattering effects for parallel cylinders are important in treating radiative transfer
within filament-wound epoxy/glass-fiber composites because of the large fiber volume fraction
and small size parameter of the fibers typical of these systems. For fiber layers, the interaction
among wavelength, refractive index, fiber diameter, and fiber spacing is complex, and no simple
criterion is available for deciding whether dependent scattering will be important. However, for 
(based on cylinder diameter) > 2, the dependent scattering is relatively independent of the
refractive index of the fibers and the size parameter, and dependent scattering is present for
< 0.925. Yamada and Kurosaki (2000) examined the characteristics of fibers with
large size parameter.
Beds of spheres. Chen and Churchill (1963) measured the transmission of radiation through an
Web Appendix-37
isothermal bed of randomly packed equal diameter spheres, as well as beds of cylinders and
irregular grains. They included materials of glass, aluminum oxide, steel and silicon carbide.
They used these data to determine the bed effective scattering and absorption cross-sections
based on a two-flux model. Kamiuto et al. (1990) measured the angularly-dependent reflectance
of planar beds of glass and alumina spheres contained between glass plates due to normally
incident radiation. An inversion technique was used to find the extinction coefficient, albedo, and
Henyey-Greenstein asymmetry factor. Kudo et al. (1991a) measured the transmittance through a
randomly packed bed of polished ball bearings of uniform diameter, and observed significant
transmittance through the bed in regions near the duct surfaces, while the transmittance near the
bed centerline decreased rapidly with increasing bed thickness.
A porous medium composed of regularly or randomly packed equal-diameter spheres has
a well-determined size parameter,  = dp/ based on the diameter dp the spheres. The
absorption and scattering characteristics for such a medium are discussed in classic texts (Bohren
and Huffman, 1983). If the spheres are close-packed and have small diameters relative to the
important wavelengths, then the potential for near-field effects should be checked by using Eqs.
(B-4) or (B-5).
Ray tracing and Monte Carlo techniques have successfully predicted radiation transfer
through regular and random arrays of particles and fibers. Yang et al. (1983) derived the
statistical attributes of a randomly packed bed of identical cold (non-emitting) spheres
(distribution of number of contact points between spheres and the conditional distributions of
contact angles for spheres with a given number of contact points.) Distributions were generated
from a zero-momentum random packing model. The upper layer of spheres in a randomly
Web Appendix-38
packed container is not level in either real or simulated packed beds. Thus, defining the bed
thickness is somewhat arbitrary, especially for beds that are only a few sphere diameters in
depth. Wavy container boundaries were assumed so that the radial dependence of packing
characteristics present in rigid containers could be ignored. Yang et al. used the bed packing
distributions to construct a ray-tracing algorithm for specularly reflecting spheres, and computed
the fraction of diffuse incident radiation transmitted through the bed. No dependent scattering
effects were considered. Comparisons with the experimental results of Chen and Churchill
(1963) showed similar trends of transmittance vs. bed thickness, but not exact agreement.
When particle sizes are large such that geometric optics is applicable, Saatdjian (1987)
provides a model for the extinction behavior of uniformly dispersed spheres, and shows that an
extinction coefficient can be predicted that conforms to exponential attenuation in the limit of
small spheres. A porosity range of 0.70 < 
< 1.0 is covered by the theory. This work is
applicable to cases where the particles are separated such as in particle-laden flows or in
radiating-droplet space radiators where only radiation transfer governs interparticle energy
transfer.
Tien (1988) discusses all of the above references and others in detail, and shows the
comparisons of the various models with the experimental data for transmission through a bed of
spheres generated by Chen and Churchill (1963).
Kudo et al. (1991a) used a model similar to that of Yang for determining transmittance,
and added further experimental results for comparison. They simulated random packing by
Web Appendix-39
perturbing regular packing within a bounded region and then removing those spheres that
overlapped the boundary after perturbation. This produced a low-density region near the
boundary as observed in real packed beds. This was found to introduce a considerable increase in
bed radiative transmissivity. Both specularly and diffusely reflecting spheres were modeled.
Singh and Kaviany (1991, 1994) and Kaviany and Singh (1993) extended the Monte Carlo
analysis of packed beds of spheres to consider semitransparent and emitting spheres, and in
particular to show that dependent scattering effects are quite important even for quite large bed
porosities. Kaviany and Singh (1993) pointed out that the results of the earlier analyses by Yang
and Kudo did not account for the non-diffuse nature of the incident radiation in the Chen and
Churchill experiments, and thus tended to underpredict the experimental bed transmittance.
Xia and Strieder (1994a) derive the effective emissivity εR,eff of an isothermal semiinfinite slab of randomly-packed spheres in terms of the sphere solid surface emissivity εR,s.
They predict the values of εR,eff to lie within the bounds
ε R,s  0.3804ε R,s 1  ε R,s    ε R,eff  ε R ,s  0.6902 1  ε R ,s  
(B-7)
This relation is valid for εR,s > 0.6. Based on these results, an estimate for εR,eff is


 1  ε R,s  1  ε R,s  




0.6902

0.3803
ε

1


   1  ε R,s  0.3451  0.1902ε R ,s 
R ,s


2
2




ε R,eff  
(B-8)
Web Appendix-40
In a companion paper, Xia and Strieder (1994b) provide a more complex but more accurate
relation for the upper bound on bed emissivity. Equations (B-7) and (B-8) contain the
assumption of independent scattering, so the bed characteristics must conform to the constraints
of Eqs. (B4) or (B-5).
Argento and Bouvard (1996) revisited the work of Yang et al. (1983), and were able to
use Monte Carlo results evaluated in the bed interior, thus eliminating the problem of
indeterminate bed height in random packing.
Göbel et al. (1998) have further extended Monte Carlo analysis to account for the effect
of inhomogeneities inside nonopaque spherical particles on the radiative scattering from
individual spheres. Results are compared with the limiting cases of transparent refracting
spheres, and purely reflecting solid spheres. Independent scattering must be assumed in the
approach used, and the spheres are also assumed to be large compared with the wavelength of
the incident radiation.
Tancrez and Taine (2004) implemented a Monte Carlo technique for small /D for
recovering the absorption, scattering, and phase function characteristics of a dispersion of
overlapping opaque spheres.
Fibers and fiber layers. Glass-fiber insulations generally have very high porosities, so that even
though the fiber diameter is in the range where dependent scattering might be important, the fiber
Web Appendix-41
spacing is so large that dependent effects can be neglected. Tong et al. (1983) measured the
spectral transmittance of fibrous insulation layers using radiation from a 1300K blackbody
source. A two-flux and a linear anisotropic scattering model were used to invert the
transmittance data and find the effective radiative conductivity of the fiber layers for use in
multi-mode heat transfer calculations. White and Kumar (1990) considered interference affects
among parallel fibers.
Nicolau et al. (1994) investigated methods for experimentally determining the spectral
optical thickness, albedo, and four parameters describing a scattering phase function. They
present data for fiberglass insulation and silica fiber-cellulose insulation.
Kudo et al. (1995) and Li et al. (1996) analyzed transfer through a bed of randomly
oriented fibers, and a bed of fibers that are randomly oriented but lie in planes parallel to the bed
surface. Far-field scattering was assumed to occur from individual fibers. Comparison with
available experimental results for transmission through fiber beds was quite good.
The anisotropic extinction coefficient eof rigid fiber insulation with
various anisotropic fiber orientations was calculated for independent scattering by Marschall and
Milos (1997). The authors used basic electromagnetic theory to predict values of e*(T) and
compared the difference in radiative transfer that results from the use of the assumption of
isotropic properties in the radiative conductivity for the diffusion equation. Differences of over
20 percent were found in a number of cases. The results for the anisotropic case are shown to be
scalable within 5 percent of the exact calculation by superposing the results for a matrix of
anisotropic fibers and the results for fibers normal to the incident radiation. No data for e*(T) are
Web Appendix-42
given.
When fibers are coated with a thin metal surface, their radiative properties are greatly
changed. Dombrovsky (1998) has provided predictions of the scattering efficiency and extinction
coefficient of fibers with various coatings in the infrared and microwave regions of the spectrum.
All results are under the assumption of independent scattering. Other research on the properties
of woven fibers is in and Kumar and White (1995), and scattering from single cylinders is treated
in Radzevicius and Daniels (2000). Lee and Cunnington (1998, 2000) give reviews of analytical
methods, and also analyze combined radiation/conduction.
Open-Celled Materials. Reticulated ceramic foams have many high-temperature applications
and the reticulated structure allows long radiation paths. Howell et al. (1996) and Kamiuto
(2008) review the radiative properties of commercially available reticulated porous ceramic
foams. These materials have an open pore structure in a random dodecahedral matrix created by
the interconnected struts or webs that make up the foam, and it is difficult to specify a particle
shape or size that is representative of the material. Most researchers and manufacturers specify
an effective pore diameter, dpore, and the scale of the structure is described by dpore and the
porosity. The porosity is typically about 85%, and pore sizes range from approximately 25 to 2
pores per centimeter (ppcm) [65 to 8 pores per inch (ppi)]. This is well outside the range where
dependent scattering should be of importance. The foam composition is usually specified by the
manufacturer in terms of a base material (silicon carbide, silicon nitride, mullite, cordierite,
alumina, and zirconia ) plus a stabilizing binder (e.g., magnesia and yttrium).
Properties are usually specified by the composition and the manufacturer's specification
Web Appendix-43
of "pores per centimeter" or ppcm. However, the actual pore diameter varies even among
materials with the same specified ppcm supplied by the same manufacturer. Thus, when using
experimental property data, significant error may be introduced unless reliable property data are
available on exactly the material being modeled.
When the porous material is treated as a homogeneous absorbing and scattering medium,
the effective absorption and scattering coefficients are needed along with the scattering phase
function. Hsu and Howell (1992) used a two-flux radiation model to infer the effective radiative
extinction coefficient 
from experimental heat transfer measurements. They obtained the
result as a function of pore size for partially stabilized zirconia reticulated ceramic foam in the
form of a data correlation [0.3 mm < dpore < 1.5 mm]:
 = 1340-1540dpore+527dpore2
(B-9)
 is the mean (spectrally averaged for long wavelength radiation) extinction coefficient
in m-1 and dpore is the actual pore diameter in mm. The data are for 290<T< 890 K, although no
significant temperature dependence was observed. Additionally, they present a relation based on
geometric optics that well predicted the trend of the data for dpore > 0.6 mm:
 = (3/dpore)(1-)
(B-10)
where  is the porosity of the sample, which varied over a narrow range from 0.87 at large pore
diameters to 0.84 at the smallest diameters. The inversion method to obtain these correlations
Web Appendix-44
assumed isotropic scattering. It was not possible to determine independent values of albedo or
scattering or absorption coefficients.
Hale and Bohn (1993) measured scattered radiation from a sample of reticulated alumina
that resulted from an incident laser beam at 488 nm. They then matched Monte Carlo predictions
of the scattered radiation calculated from various values of extinction coefficient and scattering
albedo, and chose the values that best matched the experimental data for reticulated alumina
samples of 4, 8, 12 and 26 ppcm. They found that a scattering albedo of 0.999 and an assumed
isotropic scattering phase function reproduced the measured data for all pore sizes.
Hendricks and Howell (1994, 1996) measured the normal spectral transmittance and
normal-hemispherical reflectance of three sample thicknesses each of reticulated partially
stabilized zirconia and silicon carbide at pore sizes of 4, 8 and 26 ppcm for 400<<5000 nm.
They used an inverse (Levenberg-Marquardt) discrete ordinates method to find the spectrallydependent absorption and scattering coefficients as well as the constants appropriate for use in
either the Henyey-Greenstein approximate phase function or a composite isotropic/forward
scattering phase function.
Integration over wavelength of the spectral results of Hendricks and Howell provides
mean extinction coefficient data that can be compared with those of Hsu and Howell (1992).
The Hale and Bohn (1993) data are for the single laser-source wavelength of 488 nm. Hendricks
and Howell (1996) found that a modified geometrical optics relation fit the data for the
integrated extinction coefficient of both zirconia and silicon carbide. They recommend the
Web Appendix-45
relation
 = (4.4/dpore)(1-)
(B-11)
where  is in m-1 and dpore is the actual pore diameter in mm.
High-temperature experiments (1200-1400K) were performed by Mital et al. (1996) on
reticulated ceramic samples of mullite, silicon carbide, cordierite, and yttria-zirconia-alumina.
They reduced the data using a gray two-flux approximation assuming isotropic scattering.
Temperature dependence was found to be small. They present the relations for absorption and
scattering coefficients for these materials as
  (2   r )
 r  r
3
(1   )
2d pore
3
(1   )
2d pore
(B-12)
(B-13)
When summed to obtain the extinction coefficient, the result agrees with Eq. (B-10).
Fu et al. (1997) used a unit cell model to predict the extinction coefficient and single
scattering albedo of reticulated ceramics, and compared their results with available data.
Kamiuto and Matsushita (1998) predicted the extinction coefficient of various open-celled
structures simulated by cubic unit-cells, based on the work of Kamiuto (1997). They compared
the predictions with experimental measurements obtained by an inversion technique using
Web Appendix-46
emission data from a heated plane layer of porous material. They assumed a Henyey-Greenstein
phase function, and obtained good agreement between experiment and prediction for the
extinction coefficient and asymmetry factor for Ni-Cr and cordierite porous plates.
Moura et al. (1998) examined four generic experimental techniques that can be used to
generate data that can then be inverted to find the fundamental radiative properties of porous
materials. Loretz et al. (2006) compared the predictions of various models for the radiative
properties of metallic foams, and Coquard et al. (2009) used a detailed cell model to predict the
spectral transmittance and reflectance of expanded polyethylene foams, and showed good
agreement with experiment.
C. BENCHMARK SOLUTIONS FOR VERIFICATION OF RADIATION
SOLUTIONS
In this section, benchmark solutions are presented for comparison verification of numerical
codes. Besides the tables presented here, additional benchmark solutions are in Table 11-2;
Burns et al. (1995c); Hsu and Tan (1996); Hsu and Farmer (1997); Tan et al. (2000), and
Coelho et al. (2003).
TABLE C-1 Incident radiation 4 I and z-component of radiative flux qz in cube of side length
2c exposed to uniform diffuse incident radiation qi on bottom surface z = −c with
nonhomogeneous
gray
extinction
coefficient
 = (0.05/c) + (0.45/c) {[1−(x2/c2)]
[1−(y2/c2)][1−(z2/c2)]}2, isotropic scattering, and scattering albedo . Results tabulated along line
z, x = y = 0. Results are by numerical quadrature using 17 quadrature points (QM17) [Wu et al.
(1996)] (origin of coordinates is at cube center).
Web Appendix-47
4 I ( z ) / qi ( c)
qz ( z ) / qi (c)
z/(2c)
 = 1.0
 = 0.5
= 1.0
 = 0.5
−0.49529
2.0674
2.0205
0.9466
0.9746
−0.47534
1.9955
1.9427
0.9410
0.9691
−0.44012
1.8730
1.8107
0.9246
0.9525
−0.39076
1.7066
1.6302
0.8883
0.9150
−0.32884
1.5055
1.4120
0.8251
0.8478
−0.25635
1.2837
1.1745
0.7359
0.7508
−0.17562
1.0588
0.9416
0.6311
0.6355
−0.08924
0.8487
0.7345
0.5259
0,5197
0.00000
0.6667
0.5657
0.4325
0.4180
0.08924
0.5196
0.4374
0.3566
0.3374
0.17562
0.4081
0.3451
0.2987
0.2778
0.25635
0.3280
0.2813
0.2560
0.2357
0.32884
0.2729
0.2384
0.2252
0.2066
0.39076
0.2365
0.2100
0.2033
0.1867
0.44012
0.2131
0.1917
0.1882
0.1733
0.47534
0.1989
0.1803
0.1785
0.1647
0.49529
0.1914
0.1744
0.1734
0.1602
Web Appendix-48
TABLE C-2 Integrated intensity 4 I and surface heat flux distributions qz, in cylinder with
diameter 2ro = height zo exposed to uniform collimated incident flux qi = 1 on top surface z = 0
(positive z extends vertically downward) with nonabsorbing gray homogeneous isotropic
scattering, scattering coefficient σs, and optical thickness zo = szo = 0.25. Results tabulated
along radius 0 < r < ro at z1 and z2, and along axial position 0 < z < zo at r1 and r2. Results by
numerical quadrature using 17 quadrature points (QM17) [Hsu et al. (1999)]a
Web Appendix-49
r/ro
4 I ( r , z1 )
4 I ( r , z 2 )
qz ( r , 0)
4 I ( r1 ,  z )
4 I ( r 2 , z )
qr ( ro , z )
0.015625
1.08356
0.85828
0.04490 0.81969 0.015625 1.08356
1.04819
0.02810
0.078125
1.08344
0.85819
0.04484 0.81964 0.078125 1.09101
1.04359
0.03485
0.140625
1.08313
0.85790
0.04464 0.81944 0.140625 1.08377
1.03439
0.03913
0.203125
1.08260
0.85742
0.04434 0.81915 0.203125 1.07618
1.02194
0.04052
0.265625
1.08188
0.85678
0.04390 0.81873 0.265625 1.06548
1.00914
0.04168
0.328125
1.08096
0.85595
0.04337 0.81824 0.328125 1.05352
0.99588
0.04252
0.390625
1.07980
0.85490
0.04271 0.81761 0.390625 1.04008
0.98229
0.04302
0.453125
1.07844
0.85370
0.04194 0.81689 0.453125 1.02603
0.96805
0.04293
0.515625
1.07688
0.85233
0.04106 0.81609 0.515625 1.01082
0.95352
0.04252
Web Appendix-50
qz ( r , zo )
 z /  zo
0.578125
1.07509
0.85076
0.04006 0.81519 0.578125 0.99534
0.93892
0.04194
0.640625
1.07303
0.84899
0.03892 0.81417 0.640625 0.97851
0.92381
0.04088
0.703125
1.07059
0.84686
0.03749 0.81286 0.703125 0.96127
0.90835
0.03942
0.765625
1.06747
0.84405
0.03557 0.81106 0.765625 0.94304
0.89258
0.03771
0.828125
1.06363
0.84070
0.03334 0.80902 0.828125 0.92268
0.87656
0.03583
0.890625
1.05879
0.83654
0.03064 0.80660 0.890625 0.90140
0.85956
0.03307
0.984375
1.04819
0.82775
0.02527 0.80204 0.984375 0.85828
0.82775
0.02462
a = /64,  =63 /64,  = /64,  =63 /64.
r1 ro
r2
ro
z1 zo
z2
zo
Web Appendix-51
D. FLAMES, LUMINOUS FLAMES, AND PARTICLE RADIATION
Combustion consists of chemical reactions in series and in parallel and involving various
intermediate species. The composition and concentration of these species cannot be predicted
very well unless knowledge is available of the flame reaction kinetics; this detailed knowledge is
not usually available or convenient to obtain. Because the flame radiation properties depend on
the distributions of temperature and species within the flame, a detailed prediction of radiation
from flames is not often possible from knowledge of only the combustible constituents and the
flame geometry. It is usually necessary to resort to empirical methods for predicting radiative
transfer in systems involving combustion.
Under certain conditions the constituents in a flame emit much more radiation in the visible
region than would be expected from their gaseous absorption coefficients. For example, the
typical almost transparent blue flame of a Bunsen burner can become a more highly emitting
yellow-orange flame by changing the fuel-air ratio. Such luminous emission is usually ascribed
to incandescent soot (hot carbon) particles formed because of incomplete combustion in
hydrocarbon flames. Alternatively, Echigo et al. (1967)
and others have advanced the
hypothesis, supported by some experimental facts, that luminous emission from some flames is
by emission from vibration-rotation bands of chemical species that appear during the combustion
process prior to the formation of soot particles. However, since soot formation is the most
widely accepted view, soot radiation will be emphasized in this discussion of luminous flames.
Web Appendix-52
D-1 Radiation from Nonluminous Flames
Radiation from the nonluminous portion of the combustion products is fairly well understood.
For this the complexities of the chemical reaction are not as important, since it is the gaseous end
products above the active burning region that are considered. Most instances are for hydrocarbon
combustion, and radiation is from the CO2 and H2O absorption bands in the infrared. For flames
a meter or more thick, as in commercial furnaces, the emission leaving the flame within the CO 2
and H2O vibration-rotation bands can approach blackbody emission in the band spectral regions.
The gas radiation properties in Chap. 9, and the methods in this chapter, can be used to compute
the radiative transfer. The analysis is greatly simplified if the medium is well mixed and can be
assumed isothermal. A nonisothermal medium can be divided into approximately isothermal
zones, and convection can be included if the circulation pattern in the combustion chamber is
known. A nonisothermal analysis with convection was carried out in Hottel and Sarofim (1965)
for cylindrical flames. In Dayan and Tien (1974), Edwards and Balakrishnan (1973), Modak
(1975, 1977), Taylor and Foster (1974), and Lefebvre (1984), radiation from various types of
nonluminous flames (laminar or turbulent, mixed or diffusion) is treated. The flame shape for an
open diffusion flame is considered in Annamali and Durbetaki (1975). The local absorption
coefficient in nonluminous flames is calculated in Grosshandler and Thurlow (1992) as a
function of mixture fraction and fuel composition. Modest (2005) reviews models for radiative
transfer in combustion gases.
When considering the radiation from flames, a characteristic parameter is the average
temperature of a well-mixed flame as a result of the addition of chemical energy. WellWeb Appendix-53
developed methods exist [Gaydon and Wolfhard (1979)] for computing the theoretical flame
temperature from thermodynamic data. The effect of preheating the fuel and/or oxidizer can be
included. An ideal theoretical flame temperature T is computed using energy conservation
assuming complete combustion, no dissociation of combustion products, and no heat losses. The
energy in the constituents supplied to the combustion process, plus the energy of combustion, is
equated to the energy of the combustion products to give,
T  Tref 
(energy in feed air and fuel above Tref )  (energy released by combustion)
(total mass of products)  (mean heat capacity of products)
(D-1)
Energy losses by radiation and other means, that would lower the flame temperature, are not
included. Methods for including these effects are in Gaydon and Wolfhard (1979); extinction of
a flame by radiative energy loss is analyzed in Ju et al. (2000). A list of ideal theoretical flame
temperatures (no radiation or other losses included) is in Table 12-5 for various hydrocarbon
flames, using data from Gaydon and Wolfhard and from Barnett and Hibbard (1957). Results for
complete combustion with dry air are shown, followed by calculated results modified to allow
for product dissociation and ionization. The latter are compared with experimental results. The
heats of combustion of the substances are also given. Extensive tabulations of similar data for
more than 200 hydrocarbons are in Barnett and Hibbard and in Perry et al. (2007). After its
average temperature has been estimated, the radiation emitted by a nonluminous flame can be
considered, as illustrated by an example.
EXAMPLE D-1 As a result of combustion of ethane in 100% excess air, the combustion
products are 4 mol of CO2, 6 mol of H2O vapor, 33.3 mol of air, and 26.3 mol of N2.
Web Appendix-54
Assume these products are in a cylindrical region 4 m high and 2 m in diameter, are
uniformly mixed at a theoretical flame temperature of 1853 K, and are at 1 atm pressure.
Compute the radiation from the gaseous region.
The partial pressure of each constituent is equal to its mole fraction:
pCO2  (4 / 69.6)(1 atm) = 0.0575 atm , and pH2O  (6 / 69.6)(1 atm) = 0.0862 atm . The gas
mean
beam
length
for
negligible
self-absorption
is,
from
(12-66),
Le,0  4V / A  4[ (22 / 4)]4 /[(2  4)  2 (22 / 4)]  1.6 m . To include self-absorption, a
correction factor of 0.9 is applied to give Le = 0.9(1.6) = 1.44 m. Then pCO2Le = 0.0575 ×
1.44 = 0.0828 atm · m = 8.54 bar-cm, and pH2OLe = 0.0862 × 1.44 = 0.124 atm · m = 12.8
bar-cm.. Using the Leckner correlations [Eq. (9-62)] at 1853 K gives εCO2 = 0.070 and
εH2O = 0.096 × 1.03 = 0.099. The 1.03 factor in εH2O is a correction for the partial
pressure of the water vapor being nonzero [Eq. (9-64)]. There is also a negative
correction from spectral overlap of the CO2 and H2O radiation bands. This is obtained
from Eq. (9-67) at the values of the parameters:
pH2O /( pCO2  pH2O )  0.0862 /(0.0575  0.0862)  0.60
pCO2 Le  pH2O Le  0.0828  0.124  0.207 atm  m = 20.9 bar-cm.
Web Appendix-55
The
correction
is
Δε
=
0.031.
Then
the
gas
emittance
is
ε g  ε CO  ε H O  ε  0.070  0.099  0.031  0.138 . The radiation from the gas region
2
2
at the theoretical flame temperature is,
Q  ε g A Tg4  0.138(10 )5.6704 108 (1853)4  2900 kW
TABLE 12-5 Heat of combustion and flame temperature for hydrocarbon fuels [Gaydon and
Wolfhard (1979); Barnett and Hibbard (1957); Lide (2008)]
Maximum flame temperature, K (combustion with
dry air at 298 K)
Heat of
Theoretical
Theoretical (with
combustion
(complete
dissociation and
kJ/kg
combustion)
ionization)
Carbon monoxide (CO)
10.1 × 104
1756
1388
Hydrogen (H2)
14.1 × 104
2400
2169
Methane (CH4)
5.53 × 104
2328
2191
2158
Ethane (C2H6)
5.19 × 104
2338
2244
2173
Propane (C4H8)
5.03 × 104
2629
2250
2203
n-Butane (C4H10)
4.95 × 104
2357
2248
2178
n-Pentane (C5H12)
4.53 × 104
2360
2250
Ethylene (C2H4)
5.03 × 104
2523
2375
2253
Propylene (C3H6)
4.89 × 104
2453
2323
2213
Fuel
Web Appendix-56
Experimental
Butylene (C4H8)
4.53 × 104
2431
Amylene (C5H10)
4.50 × 104
2477
Acetylene (C2H2)
5.00 × 104
2859
2607
Benzene (C6H6)
4.18 × 104
2484
2363
Toluene (C6H5CH3)
4.24 × 104
2460
2344
2306
2208
D-2 Radiation from and through Luminous Flames and Gases with Particles
Several factors complicate radiative transfer in a flame region that is actively burning. The
simultaneous production and loss of energy produces a temperature variation within the flame,
and thus variations of local emission and properties. Intermediate combustion products from the
complex reaction chemistry can significantly alter the radiation characteristics from those of the
final products. If soot forms in burning hydrocarbons, it is a very important radiating constituent.
Soot emits a continuous spectrum in the visible and infrared regions and can often double or
triple the radiation emitted by only the gaseous products. Soot also provides radiant absorption
and emission in the spectral regions between the gas absorption bands. A method for increasing
flame emission, if desired, is to promote slow initial mixing of the oxygen with the fuel so that
large amounts of soot form at the base of the flame. Ash particles in the combustion gases can
also contribute to absorption and emission [Viskanta and Mengüc (1987), Sarofim and Hottel
(1978), Boothroyd and Jones (1986), Marakis et al. (2000)], and can significantly scatter
radiation [Im and Ahluwalia (1993)].
Calculating the effect of soot on flame radiation requires knowledge of the soot
concentration and its distribution in the flame; this is a serious obstacle for predictive
calculations. The soot concentration and distribution depend on the type of fuel, the mixing of
Web Appendix-57
fuel and oxidant, and the flame temperature. This is illustrated by the experimental results and
calculations in Santoro et al. (1987) and Coelho and Carvalho (1995). A semi-empirical
correlation is developed in De Champlain et al. (1997) for trying to predict the amount of smoke
in the exhaust of a gas-turbine engine. The correlation is based on residence time of the fuel in
the combustor, flow rates, reaction rates, and other parameters. The correlation does not yet yield
generalized results, and further developments are continuing. Soot concentration in a propane
turbulent diffusion flame was predicted in Coelho and Carvalho (1995) using several models that
use basic flow and energy equations. The models are found to need further improvement to
obtain accurate predictions of soot formation.
If the soot concentration and distribution can be estimated from basic equations or from
observations for similar flames, another requirement for making radiative transfer calculations is
that the soot radiative properties can be specified. These properties are known only
approximately. If flames in both the laboratory and industry are included, the soot particles
produced in hydrocarbon flames generally range in diameter from 0.005 μm to more than 0.3 μm.
Typical diameters measured in Charalampopoulos and Felske (1987) were 0.02–0.7 μm. Soot
can be in the form of spherical particles, agglomerated masses, or long filaments. The
experimental determination of the physical form of the soot is difficult, as a probe used to gather
soot for photomicrographic analysis may cause agglomeration of particles or otherwise alter the
soot characteristics. The nucleation and growth of the soot particles are not well understood.
Some soot can be nucleated in less than a millisecond after the fuel enters the flame, and the rate
at which soot continues to form does not seem to be influenced much by the residence time of
the fuel in the flame. An unknown precipitation mechanism governs soot production. In typical
Web Appendix-58
gaseous diffusion flames, the volume of soot per total volume of combustion products has been
found experimentally to range from 10−8 to 10−5 [Sarofim and Hottel (1978), Santoro et al.
(1983, 1987), Ku and Shim (1991), Lee et al. (1984), Ang et al. (1988), Sato et al. (1969), and
Kunugi and Jinno (1966)]. The aggregation of soot into long clusters is examined in Köylü and
Faeth (1994) and Farias et al. (1995b), where structural details are given.
Along a path in a transparent carrier gas containing suspended soot, it has been found
experimentally that the attenuation of radiation obeys Bouguer’s law,
S
  ( S
I  ( S )  I  (0)e 0
*
) dS *
(D-1)
From Mie theory the soot radiative properties depend on the size parameter πD/λ (D is the
particle diameter) and the optical constants n and k of the particles, which depend on the soot
chemical composition. The n and k depend somewhat on λ, as shown later, but do not depend
strongly on temperature [Lee and Tien (1981), Howarth et al. (1966), Dalzell and Sarofim
(1969)]. At the temperatures in combustion systems, radiation is mostly in the wavelength range
of 1 μm and larger; hence, for small soot particles πD/λ is generally much less than 0.3. In this
range, Mie theory implies that the scattering cross section depends on (πD/λ)4 and that the
absorption cross section depends on πD/λ to the first power. Thus scattering is small compared
with absorption, and λ in (D-1) is the absorption coefficient of the soot rather than the extinction
coefficient. Then the spectral emittance of an isothermal volume composed of soot suspended
uniformly in a nonradiating carrier gas is
Web Appendix-59
ε   1  e   L
e
(D-2)
where Le is the mean beam length for the volume. Radiation by the carrier gas will be included
later.
EXPERIMENTAL
CORRELATION
OF
SOOT
SPECTRAL
ABSORPTION
COEFFICIENT A relatively simple empirical relation for the absorption coefficient λ, found
experimentally in some instances, is
   Ck  
(D-3)
where C is the soot volume concentration (volume of particles per unit volume of cloud), and k is
a constant. The λ is in μm for the numerical quantities given here. Hottel (1954) recommends
using
 
Ck1
 0.95
(D-4)
for λ > 0.8 μm. In some experiments, Siddall and McGrath (1963) also found the functional
relation of (D-4) to hold approximately. They give, in the range λ = 1-7 μm, the following mean
values of α:
Source of soot
Mean α for λ = 1–7 μm
Amyl acetate
0.89, 1.04
Web Appendix-60
Avtur kerosene
0.77
Benzene
0.94, 0.95
Candle
0.93
Furnace samples
0.96, 1.14, 1.25
Petrotherm
1.06
Propane
1.00
In Köylü and Faeth (1996) the extinction by soot was examined for acetylene, propylene,
ethylene, and propane. The exponent was found as α = 0.83 ± 0.08 for all of these fuels. This was
reasonable for the spectral range λ = 0.514–5.2 μm.
In Siddall and McGrath (1963) the data were inspected in more detail to see whether α
had a functional variation with λ that would provide a more accurate correlation than using a
constant α. In some instances α took the form
  c1  c2 ln 
where c1 and c2 are positive constants. Examples are in Fig D-1. In other cases, a more general
polynomial was required to express α as a function of λ, Fig. D-2. Thus, as a generalization of
(D-4) for the infrared region,
 
Web Appendix-61
Ckl
  ( )
(D-5)
and letting α be a constant is only an approximation.
FIGURE D-1 Experimental values of the exponent  plotted against  for cases where  varies
approximately linearly with ln [Siddall and McGrath (1963)].
Web Appendix-62
FIGURE D-2 Experimental values of the exponent  plotted against  for cases where  does
not vary linearly with ln [Siddall and McGrath (1963)].
In the visible range experimental data led to the recommended form [Hottel (1954)]
 
Ck2
 1.39
(D-6)
for the wavelength region around λ = 0.6 μm (λ ≈ 0.3 – 0.8 μm).
ELECTROMAGNETIC-THEORY PREDICTION OF SOOT SPECTRAL ABSORPTION
To examine more fundamentally the absorption coefficient of a soot cloud in a nonabsorbing gas,
Web Appendix-63
electromagnetic theory is used [Charalampoulos and Felske (1987), Lee and Tien (1981), Dalzell
and Sarofim (1969), Siddall and McGrath (1963), Stull and Plass (1960), Hawksley (1952),
Felske and Tien (1973, 1977), Buckius and Tien (1977)]. By analogy to Eq. (1-46b), for particles
all of the same size, the absorption coefficient is
   Q Ap N
(D-7)
where N is the number of particles per unit volume and Ap is the particle projected area (= πD2/4,
as particles are assumed spherical). The Qλ is the spectral absorption efficiency factor, which is
the ratio of the spectral absorption cross section to the physical cross section of the particle (ratio
of energy absorbed to that incident on the particle). For small πD/λ, the Mie equations give Qλ
for a small absorbing sphere as
Q  24
D
nk
D
 24
F ( )
2
2
2
2 2
 (n  k  2)  4n k

(D-8)
where n and k are functions of λ. Then, from (D-7),

C

36

F ( ) 
36
nk
 (n  k  2)2  4n 2 k 2
2
2
(D-9)
where C = NπD3/6 is the volume of particles per unit volume of cloud. The λ/C can be
evaluated if the n and k of soot are known as functions of λ.
Web Appendix-64
In Dalzell and Sarofim (1969) the n and k of acetylene and propane soot were measured
by collecting soot and compressing it on a brass plate. The n and k are deduced from theory for
reflection from an interface in conjunction with measurements of the reflected intensity of
polarized light; values are in Table D-1. Using these values, λ/C was evaluated from (D-9),
yielding the points in Fig. D-3. Although λ/C decreases with λ, as expected from (D-3), an
approximate curve fit by straight lines on the logarithmic plot yields exponents on λ somewhat
different from those of (D-4) and (D-5). Similar results for the spectral absorption coefficient are
in Sivathanu et al. (1993); they also have a weak dependence on temperature, in agreement with
findings by previous investigators.
The form of the exponent α(λ) predicted by Mie theory is examined in the infrared region
by equating (D-5) and (D-9),

k1
 ( )

36

F ( )
(D-10)
By evaluating this relation at λ = 1, the constant k1 for the infrared region is k1 = 36πF(1). Then
 ( )   F (1) / F ( ) , and solving for α(λ) gives
 ( )  1 
ln[ F (1) / F ( )]
ln 
(D-11)
The optical properties of soot are used in F(1) and F(λ) to evaluate α(λ). This was done in
Siddall and McGrath (1963) using the properties of a baked electrode carbon at 2250 K. The
Web Appendix-65
results are in Fig. D-4. The trend is the same as in the experimental curves of Fig. D-1, but α(λ)
values are larger than the experimental values. They are also larger than the average value of
0.95 recommended in (D-4). The discrepancy is probably partly due to the optical properties of
the baked electrode carbon being different from those of soot. This is further discussed in
Kunitomo and Sato (1970), where optical properties of carbonaceous materials more like real
soot are given, and good comparisons of predictions with experiment are obtained in some
instances.
Web Appendix-66
FIGURE D-3 Ratio of spectral absorption coefficient to particle volume concentration for soot
particles.
FIGURE D-4 Calculated variation of the exponent  with wavelength, using properties of baked
electrode carbon at 2250 K [Siddall and McGrath (1963)].
Web Appendix-67
TABLE D-2 Optical constants of acetylene and propane soots [Dalzell and Sarofim (1969)]
Acetylene soot
Wavelength
λ, μm
Index
Extinction
refraction, n coefficient, 
Propane soot
Absorption
Absorption
coefficient per
coefficient per
particle volume
particle volume
fraction λ/C,
μm−1
Index of
Extinction
refraction, n coefficient, 
fraction λ/C,
μm−1
0.4358
1.56
0.46
9.37
1.57
0.46
9.29
0.4500
1.56
0.48
9.45
1.56
0.50
9.83
0.5500
1.56
0.46
7.42
1.57
0.53
8.44
0.6500
1.57
0.44
5.96
1.56
0.52
7.07
0.8065
1.57
0.46
5.02
1.57
0.49
5.34
2.5
2.31
1.26
1.97
2.04
1.15
2.34
3.0
2.62
1.62
1.44
2.21
1.23
1.75
4.0
2.74
1.64
0.998
2.38
1.44
1.24
5.0
2.88
1.82
0.747
2.07
1.72
1.30
6.0
3.22
1.84
0.505
2.62
1.67
0.727
7.0
3.49
2.17
0.383
3.05
1.91
0.484
8.5
4.22
3.46
0.213
3.26
2.10
0.357
10.0
4.80
3.82
0.143
3.48
2.46
0.271
In Felske et al. (1984) the compressed powder technique was used to obtain n and κ for
propane soot in the infrared region. When soot is compressed, the layer adjacent to the surface
Web Appendix-68
has a finite void fraction that may influence the reflection characteristics. In Felske et al., the
soot was compressed with a pressure of 2000 atm and the surface reflections conformed to the
Fresnel equations. Electron micrographs were used examine the surface void fraction and a
correction for it was made to give the results in Table D-2. The n values do not vary as much
with λ as those in Table D-1, and the κ values are smaller. The compressed soot technique yields
a specular reflection for infrared radiation, but in the visible region the surface may not be
optically smooth. For the visible region an in situ laser light-scattering technique is described in
Charalampoulos and Felske (1987). Using this method for a methane-oxygen flame, the n at λ =
0.488 μm (blue line) was found as n  n  ik  1.60  i0.59 . This was compared with n = 1.57 –
i0.56 from Kunugi and Jinno (1966) and n = 1.90 – i0.55 from Lee and Tien (1981). Additional
information from in situ measurements in premixed flat flames of methane, propane, and
ethylene is in Habib and Vervisch (1988). In Köylü and Faeth (1996) the n that gave good
agreement with data was n = 1.54 – i0.48 at λ = 0.514 μm. Spectral values of n and k for soot
for λ = 0.4 to 20 μm are predicted in Selamet (1992) from dispersion theory.
TOTAL EMITTANCE OF SOOT CLOUD A total emittance can be found for a path S
through an isothermal cloud of suspended soot having a uniform concentration. The emittance
given now accounts only for the soot absorption coefficient because the suspending gas is nonemitting. The effect of an emitting suspending gas is included later. The total emittance is found
from (9-12b) as
ε (T , S ) 
Web Appendix-69
1
T 4


0
Eb (T )[1  e(  / C )CS ]d 
(D-12)
By using λ/C from (D-9) the ε(T, CS) can be evaluated numerically, and is in Fig. D-5 (dashed
lines) for propane soot, using properties in Table D-1. By integrating over a distribution of
particle sizes, it was found in Siddall and McGrath (1963) that individual particle sizes were
unimportant, and for a fixed T and S the ε depends only on the soot volume concentration. An
electron oscillator model for soot optical constants was developed in Lee and Tien (1981). The
constants were used in the Rayleigh equation for the absorption coefficient, and the integration in
(D-12) then carried out. The results for two temperatures are in Fig. D-5, and they are somewhat
below those in Fig. D-6.
TABLE D-2 Complex refractive indices (n = n – ik) of propane soot corrected for surface layer
void fraction of 0.18 [Felske et al. (1984)]
λ(μm)
n
k
2.0
2.33
0.78
3.0
2.31
0.71
4.0
2.36
0.66
4.5
2.37
0.71
5.0
2.39
0.66
6.0
2.40
0.80
7.0
2.49
0.85
8.0
2.66
0.93
9.0
2.63
0.92
10.0
2.62
1.00
Web Appendix-70
With certain assumptions a convenient analytical expression can be derived for the total
emittance of a soot cloud in a nonemitting gas. From (D-9), if n and k are weak functions of λ,
then λ/C =
λ, where
is a constant depending on the type of soot. This is an
approximation, as the exponent α can deviate appreciably from 1. However, in some instances
this was found to be a good approximation; for example, for the soot data in Felske et al. (1984),
Habib and Vervisch (1988), Selamet (1992), and Liebert and Hibbard (1970), and in Fig. D-3 for
λ up to about 5 μm. The coefficient k was found as λ/C at λ = 1 μm, and Fig. D-3 gave k
(propane) ≈ 4.9 and k (acetylene) ≈ 4. For soot from coal flames, k has been found as 3.7 to 7.5;
for oil flames k ≈ 6.3 [Gray and Muller (1974)]. The value k = 5 was used for the calculations in
Felske and Charalampoulos (1982).
The above approximation for λ is inserted into (D-12). Also, for the wavelength regions
of interest
here, the
Eλb can be approximated quite well by Wien’s
formula,
Eb  2 C1 /  5 exp(C2 / T )  . Equation (D-12) then becomes

ε (T , CS ) 

0
(2 C1 /  5 )eC2 / T (1  e kCS /  )d 


0
(2 C1 /  5 )eC2 / T d 
where, to be consistent in the approximation, Wien’s formula is also used in the denominator.
For the integral


0

 5e /  d  , where 
 4   3e d . Then
0
Web Appendix-71
is independent of λ, the substitution η =  /λ yields

ε (T , CS ) 
[(C2 / T )4  (C2 / T  kCS ) 4 ]  3e d
(C2 / T )
4


0
0
3 
 e d
 1
1
[1  (kCS / C2 )T ]4
(D-13)
With k = 4.9 for propane soot, a comparison with values from the numerical integration in
Dalzell and Sarofim (1969) is in Table D-3 and further results are in Fig. D-6. A comparison
with the results of Lee and Tien (1981) is in Fig. D-5. From Table D-3, Eq. (D-13) is shown to
be a useful approximation. The k/C2 = 4.9/(0.01439 m · K) = 341 m−1 · K−1 for this soot. Sarofim
and Hottel (1978) recommend k/C2 = 350 m−1 · K−1 as a mean value for all types of soot, and
this coefficient does not vary significantly over the temperature range of interest in combustion
chambers.
FIGURE D-5 Total emittance of soot particles in the Rayleigh limit [Lee and Tien (1981)].
Web Appendix-72
FIGURE D-6 Total emittance of soot suspensions as a function of temperature and volume
fraction-path length product for propane soot [Dalzell and Sarofim (1969)].
A somewhat more complicated expression results from not using the Wien formula in (D12). Letting λ = Ck/λ and using (1-13) gives

ε (T , CS )  1 

0
Web Appendix-73
Eb e kCS /  d 
T 4
 1
2 C1  e kCS /  d 
 T 4 0  5 (eC2 / T  1)
If z = 1 + kCST/C2 and t = C2/λT, this becomes
2 C1  t 3etz
ε (T , CS )  1 
dt
 C24 0 1  et
The integral is the pentagamma function  (3) ( z ) , for which tabulated values are available. Then,
with (1-27), ε becomes
ε (T , CS )  1 
15

4

 (3) 1 

kCST 

C2 
(D-14)
Results are in Table D-3 (using k = 4.9); they are not better than those from (D-13). Yuen and
Tien (1977) show that (D-14) can be approximated by ε (T , CS )  1  exp(36k CS T / C2 ) .
TABLE D-3 Comparison of approximate suspension emittance for propane soot with values
from Dalzell and Sarofim (1969)
Suspension emittance ε
Concentration–path
length product CS,
μm
Suspension
Dalzell and
temperature T, K Equation (12-99) Sarofim (1969) Equation (12-100)
0.01
1000
0.0135
0.013
0.0127
0.10
1000
0.125
0.125
0.120
1.00
1000
0.690
0.64
0.671
0.01
2000
0.0268
0.030
0.0254
Web Appendix-74
0.10
2000
0.232
0.250
0.223
1.00
2000
0.875
0.90
0.857
Stull and Plass (1960) give results for the scattering coefficient of soot that show, in
agreement with the Mie theory, that scattering usually has little effect on emittance in the
wavelength range that contains significant energy at hydrocarbon combustion temperatures.
Erickson et al. (1964) experimentally studied scattering from a luminous benzene-air flame.
Their results agree with the predictions of Stull and Plass if the soot particles are taken to be of
two predominant diameters. This indicates that particles approximately 250 Å in diameter are
formed along with agglomerated particles with an equivalent diameter of 1850 Å. These sizes
were observed by gathering soot with a probe and using electron microscopy. A comparison of
some of the experimental results of Erickson et al. with the analysis of Stull and Plass (1960) is
in Fig. D-7. Similar figures are in Köylü and Faeth (1993) for soot aggregates in straight chains
and fractal shapes. Fractal descriptions provide a useful indication of the morphology of
agglomerated soot particles [Manickavasagam and Mengüc (1997), Charalampoulos (1992),
Charalampoulos and Chang (1991), Farias et al. (1995a)]. The agglomerated particles obey the
relation
 Rg 
N  kf 
 D 
 p
Df
(D-15)
where N is the number of spheres of diameter Dp comprising the aggregate, Rg is the radius of
gyration of the aggregate, kf is a constant to be determined, and Df is the fractal dimension. Most
Web Appendix-75
studies predict values of about Df = 1.75 and kf = 8.0 [Farias et al. (1995a), Manickavasagam et
al. (1997)]. Observations of the radius of gyration provide predictions of the number of particles
in an agglomerate by using Eq. (D-15).
FIGURE D-7 Comparison of experiment with Mie scattering theory for radiation scattered from
benzene-air flame at = 5461 Å. Theoretical curves based on spheres of diameter 250 A with
0.002% spheres of diameter 1850 Å, all with complex refractive index n - ik = 1.79 – 0.79i
[Erickson et al. (1964)].
Web Appendix-76
The effects of soot shape on radiative energy transfer predictions were examined in
Farias et al. (1998) by using a realistic simulation of the soot aggregation process. It was found
that the effects of soot morphology on spectral and total emissivities were less than 25% and
13% respectively for typical soot volume fractions and flame temperatures. The results suggest
that soot shape effects can be neglected in the emission predictions of soot-laden flames, thereby
simplifying the engineering modeling of radiation in combustion devices.
Thring, Beer, and Foster (1966) put some results of Stull and Plass (1960), along with
their own extensive experimental results, into useful graphs of emittance, extinction coefficient,
and soot concentration for flames applicable in industrial practice. They note, however, that soot
concentration can only be predicted for flames geometrically similar and with the same control
variables as those that have already been studied. Their paper contains a useful review of the
worldwide effort to gather information on, and give methods for, the prediction of radiation from
luminous industrial flames. Other such information is in Viskanta and Menguc (1987), Sarofim
and Hottel (1978), Farias et al. (1998), Thring et al. (1963, 1966), Sato and Matsumoto (1963),
Yagi and Inoue(1962), Bone and Townsend (1927), Leckner (1970), and Viskanta (2005).
In addition to the uncertainties in optical properties and hence in aλ and ε for soot, it is
noted that λ and ε are given in terms of the soot concentration. To use Eq. (D-3), the Ck is
needed. To use Fig. D-6 to determine ε for given flame size, the C in the abscissa must be
known. The C cannot be accurately computed from first principles, knowing the fuel and burner
geometry. Hence some indication of C or Ck must be obtained by examining flames
experimentally. It may be possible to extrapolate performance for a particular application by
Web Appendix-77
examining a similar flame. Examples of mathematical models to predict soot formation are in
Coelho and Carvalho (1995), De Champlain et al. (1997), Rizk and Mongia (1991), and
Magnussen and Hjertager (1977). For gas turbine combustors, because of the complexity in
estimating the luminous emissivity, a luminosity factor has been introduced into the expression
for the emittance of a nonluminous flame [Rizk and Mongia].
EXPERIMENTAL EXAMINATION OF SOOT CONCENTRATION One technique to
obtain information on the soot-concentration quantity Ck2, as in Eq. (D-16) for the visible region,
is to sight through the flame onto a cold black background with a pyrometer and match the
brightness of the pyrometer filament to that of the flame while using a red filter and then a green
filter. With each filter the pyrometer is also sighted on a blackbody source, and the source
temperatures are obtained that produce the same brightness as when the flame was viewed. As a
convenient simplification at the small λT for red and green wavelengths when considering typical
flame temperatures, the blackbody intensity can be approximated very well by Wien’s formula,
Eq. (1-20). Then, if Tr is the blackbody temperature producing the same brightness as the flame
did when the red filter was used, this intensity is
I b ,r 
2C1
 exp(C2 / rTr )
5
r
(D-16)
where λr = 0.665 μm. This intensity can also be written as a spectral emittance of the flame times
the blackbody intensity at the flame temperature,
Web Appendix-78
2C1
2C1
 ε r 5
λ exp  C2 λ rTr 
λ r exp  C2 λ rT f
5
r

This yields
1 1 λr
  ln ε r
T f Tr C2
(D-17)
Similarly, with a green filter (λg = 0.555 μm),
1 1 λg
 
ln ε g
T f Tg C2
(D-18)
Now, as a simple approximation, using (D-6) for λ in the visible region, (D-2) becomes
 Ck2 S 
1.39 
 λ

ε   1  exp 
(D-19)
where S is the path length sighted through the flame. Substitute (D-19) into (D-17) and (D-18) to
obtain
 Ck2 S  
1 1 λr 
 
ln 1  exp   1.39

T f Tr C2 
 λr 
Web Appendix-79
(D-20)
 Ck2 S  
1 1 λg 
 
ln 1  exp   1.39
 λ  
T f Tg C2 
g


(D-21)
These two equations are solved for Tf and Ck2 to yield the needed measure of the soot
concentration as well as the flame temperature.
As an approximation, Ck2 is assumed independent of wavelength and is used in (D-2),
(D-4), and (D-6) to yield, for a path length S,
 Ck2 S 
visible, λ  0.8  m
1.39 
 λ

(D-22)
 Ck2 S 
infrared, λ > 0.8  m
0.95 
 λ

(D-23)
ε λ  1  exp  
ε λ  1  exp  
These spectral emittances can be used with the definition in Eq. (D-12) to evaluate the total



  

emittance of the flame as ε Tf , S   Eλb Tf ε λ Tf , S d   Tf4 . The hope is that the Ck2 can
0
be applied to “similar” flames. This is a very rough approximation; so many variables affect the
flow and mixing in the flame that it is difficult to know when the flames will have a similar
character. The detailed nature of flames and their radiating characteristics is a continuing area of
active research [Taylor and Foster (1974), Magnussen and Hjertager (1977), Gibson and
Monahan (1971), Selcuk and Siddall (1976), Kunitomo (1974), Chin and Lefebvre (1990)].
Web Appendix-80
Another way to measure flame emittance is to use a radiometer and a blackbody source
with a blackened chopper in front of it that periodically covers and uncovers the blackbody
aperture (Fig. D-8). With the flame turned off, the voltage signal at the detector from the
uncovered blackbody is Vb, and with the blackbody covered by the chopper it is V0. With the
flame turned on these signals become Vb,f and V0,f. The measurements are made through a filter
so that they are for a small spectral region Δλ centered about λ. The transmittance of a path
through the flame is  λ  1  ε λ . Then Vb,f is the result of transmitted radiation from the
blackbody
and
emission
by
the
flame
so
Vb , f  1  ε λ Vb  ε λ Eλb T f  .
Similarly,
V0, f  1  ε λ V0  ε λ Eλb T f  . The two signals are subtracted, and the spectral emittance obtained
for the path viewed through the flame is
ελ  1
Vb, f  V0, f
Vb  V0

V
b
 Vb , f   V0  V0, f
Vb  V0
FIGURE D-8 Radiometric measurement of flame emittance.
Web Appendix-81

(D-24)
Scattering of laser light can be used to measure local soot concentrations, as discussed in
Santoro et al. (1983, 1987). Although scattering by soot is small, it can be accurately detected
with sensitive instrumentation. Mie scattering theory provides the scattering intensity in a given
direction as a function of soot concentration and optical properties. If values are assumed for the
optical properties, the measured scattering intensities can be used to obtain local soot
concentrations. Usually spherical soot particles are assumed, and the agglomeration of some of
the soot is a source of error. Agglomeration was found in Ku and Shim (1991) to have little
effect on the absorption coefficient, but it did influence scattering. Another difficulty is in
selecting the soot complex refractive index. In Santoro at al. (1983) n  1.57  i0.56 was chosen
for soot in an ethene flame at the argon ion laser wavelength of 0.5145 μm. The phase function
for soot aggregates is in Köylü and Faeth (1996) for observations in the visible region at λ =
0.514 μm. For measuring flame temperature without soot, a two-color emission pyrometry
method is in Bhattcharjee et al. (2000).
UNIFORM SOOT SUSPENSION IN AN ISOTHERMAL RADIATING GAS Usually, for
soot in a flame or in combustion products, there are gaseous constituents that also radiate energy.
To simplify the present discussion, it is assumed that only three radiating constituents are
present—carbon dioxide, water vapor, and soot—but the method can be extended to more
constituents. As spectral radiation passes through the gas-soot suspension, the local attenuation
depends on the sum of the absorption coefficients so, from (9-65), the spectral emittance is
ε λ  λ  1  e
C  H
Web Appendix-82
S  S
(D-25)
where the subscripts C, H, and S refer to CO2, H2O, and soot. Then, from (D-12), the total
emittance is
ε
1
T 4


0
1  e
 λC  λH  λS  S
E
λb
dλ
(D-26)
With ε λJ  1  e  J S for J = C, H, and S, this can be written in the equivalent forms
ε
1
T 4


0
1
=
 T4
1  1  ε λC 1  ε λH 1  ε λS  Eλb dλ

 ε
0
λC
 ε λH  ε λS  ε λCε λH  ε λCε λS  ε λSε λH  ε λCε λH ε λS Eλb dλ
(D-27)
The first three terms in the last integral yield the total emittances of the three constituents, so that
ε  εC  ε H  ε S 
A term such as ε
λC ε λH
1
T 4

 ε
0
ε λH  ε λCε λS  ε λSε λH  ε λCε λHε λS  Eλb dλ
λC
(D-28)
is nonzero only in spectral regions where both λC and λH are nonzero,
that is, where both constituents are radiating. Thus the terms in the integral represent overlap
regions in the spectrum in which two or three components are radiating. The total emittance ε is
then the sum of the three individual emittances, computed as if the other constituents were absent
minus a correction term for spectral overlap. For the simplified condition of gray constituents,
ε  1  1  εC 1  ε H 1  ε S 
Web Appendix-83
(D-29)
Results for the spectral-overlap terms were calculated in Felske and Tien (1973, 1975) by
using the form of the CO2 and H2O vapor absorption bands and the soot absorption coefficient λ
= Ck/λ discussed earlier. Typical emittances are in Fig. D-9. For low values of CS the soot
concentration is low, especially when S is high. Hence the left sides of the curves are dominated
by the gas emittance, and the vertical displacement of the curves shows the increase in gas
emittance with path length. As CS is increased from 0.001, the curves are somewhat horizontal
(especially at high S, which corresponds to low Q as the soot concentration is not sufficient to
increase ε for the mixture significantly. For larger CS the soot begins to dominate, and for all the
path lengths shown, ε approaches 1 when CS is about 3 × 10−4 cm. This is consistent with the
results in Fig. D-6.
FIGURE D-9 Total emittance of gas-soot suspension as a function of volume fraction of soot
times path length. Gas temperature, 1600 K; total pressure, 1 atm; partial pressures: pH2O = 0.19
atm, pCO2 = 0.09 atm [Felske and Tien (1973)].
Web Appendix-84
Since soot has a rather continuous emission spectrum, it is reasonable [Felske and
Charamapoulos (1982)] to assume gray emission of soot in a nongray gas. This leads to a
particularly simple form relating the emittance of the suspension to the individual emittances of
the soot and gas. If there is no soot, Eq. (D-27) becomes
ε g  1
1
T 4

 1  ε 1  ε  E
λC
0
λH
λb
dλ
Then, from (D-27), if the soot is gray,
1
T 4
 ε s  ε g 1  ε s 
ε  1  1  ε s 

 1  ε 1  ε 1  ε  E
0
λC
λC
λH
λb
dλ  1  1  ε s  1  ε g 
(D-30)
The sum of gray gases model [see Eq. (9-54)] was used in Felske and Charamapoulos to
represent the emittance for a mixture of soot, CO2, and H2O vapor,
J

ε   w j T  1  e
j 1
  j Le

where j is a function of the soot volume fraction and the partial pressures of the CO2 and H2O.
This model was used in Smith et al. (1987) to compute radiative transfer for a gas-soot mixture
between parallel plates. Kunitomo (1974) gives results for the ratio, in a flame, of the soot-cloud
emittance to the emittance of the nonluminous suspending gas; these results are for a liquid fuel.
Web Appendix-85
The ratio increases as the fuel carbon-hydrogen ratio is increased and as the excess air is
decreased. Babikian et al. (1990) made mass absorption coefficient measurements of soot in
spray combustor flames. Values were found to be between those in Dalzell and Sarofim (1969)
and in Lee and Tien (1981).
GASES WITH SUSPENDED PARTICLES OTHER THAN SOOT There are important
applications involving radiation from gases containing suspensions of various types of particles.
An example is ash particles in pulverized coal flames [Viskanta and Mengüc (1987), Im and
Ahluwalia (1993)]. The particles can radiate in the regions between the gas bands, so radiation is
important over a more continuous spectral region than for the radiating gas alone. In Viskanta
and Mengüc it is discussed that the complex refractive index for fly ash ranges from about n =
1.43 – i0.307 to 1.5 – i0.005. When the imaginary part is small the particles are acting as a
nonabsorbing dielectric. Detailed values for the complex refractive index for char and ash
particles are in Im and Ahluwalia in graphical form as a function of wave number. Char and ash
particles also scatter radiation. The scattering albedo of gas-char and gas-ash mixtures exceeded
0.9 in the near-visible spectral region, and was larger than 0.5 at all wave numbers. The radiative
properties of fused and/or sintered slag and ash deposits were measured in Markham et al.
(1992). Fused slag deposits were found to have a high spectral emittance ≥ 0.9 over a wide
temperature range; the measurements included both solid and molten slag. Deposits of sintered
fly ash consisted of finely packed particles. Their spectral emittance varied appreciably with
wave number, and scattering was significant. A curve-fitting method to approximate Mie theory
was used in Caldas and Semião (1999) to predict behavior for soot, carbon particles, and fly ash;
the method reduced computation time considerably.
Web Appendix-86
Another example is the luminosity in the exhaust plume of solid-fueled and some liquidfueled rockets. For a solid fuel the luminosity may be caused by metal particles added to promote
combustion stability. Williams and Dudley (1970) present calculations for a rocket exhaust with
entrained liquid aluminum oxide particles. Edwards et al. (1987, 1990) modeled the radiation
from a solid rocket motor plume with two groups of particles. The very small particles cool
rapidly and are assumed to be cold and scattering. The large particles that remain hot are
emitting and absorbing. The calculations in Tabanfar and Modest (1983) demonstrate the
combination of radiating particles and gases where all constituents have spectrally dependent
properties. Radiation by a cylindrical region at uniform temperature was analyzed in Thynell
(1990) for either CO2 or H2O gas containing soot and gray particles. The absorption of the
nongray gas was obtained from the exponential wide-band model. The absorption coefficient of
the soot was inversely proportional to wavelength, and the scattering from the gray particles had
an anisotropic component. Experiments were performed in Skocypek et al. (1987) with a heated
mixture of carbon dioxide, nitrogen, and particles of BNi-2, a boron nickel alloy. The particulate
scattering increased radiation in the wings of the 4.3-μm CO2 band. Experiments in Walters and
Buckius (1991) show the effect of highly scattering A12O3 particles in CO2 gas. The radiative
behavior of lycopodium particles during combustion in air is considered in Berlad et al. (1991).
The presence of particles in an otherwise weakly absorbing medium can cause the
mixture to be strongly absorbing. Seeding of a gas with particles, such as finely divided carbon,
can increase gas absorption and heating by incident radiation [Lanzo and Ragsdale (1964)], or
the particles can shield a surface from incident radiation [Howell and Renkel (1965), Siegel
Web Appendix-87
(1976), Lee et al. (1984)]. These techniques have possible application in advanced energy
systems and for the collection of solar energy [Abdelrahman et al. (1979)].
Another use for seeding is in the direct determination of flame temperatures for a
nonluminous flame by the line-reversal technique. In this method, a seeding material such as a
sodium or cadmium salt is introduced into an otherwise transparent flame. These materials
produce a strong spectral line in the visible spectrum because of an electronic transition;
cadmium gives a red line and sodium a bright yellow line. A spectrally continuous radiation
source is placed so that it may be viewed through the seeded flame with a spectroscope. The
intensity seen in the spectroscope at the line wavelength is [Eq. (10-15)],
I λ,scope  I λ, cont source e λ   I λb  λ*  e


  λ  λ*
0
 d *
λ
(D-31)
If the flame is isothermal, of diameter D, and no attenuation occurs along the remainder of the
path between the continuous source and the spectroscope, (D-31) becomes, as in (10-56),
I λ,scope  I λ, cont source e
 λ,D

 I b, flame 1  e
 λ ,D

(D-32)
where  λ, D      S  dS . In the wavelength region adjacent to the absorbing and emitting
D
0
spectral line, the flame is essentially transparent, so the background radiation observed adjacent
to the line is I λ,
cont source
adjacent background is
Web Appendix-88
. By subtracting I λ,cont source from (D-32), the line intensity relative to the

I λ, scope  I λ,cont source   I λb, flame  I λ, cont source  1  e
  ,D

(D-33)
If the flame is at a higher temperature than the continuous source, (D-33) shows that the
line intensity in the spectroscope will be greater than the continuous background intensity. The
line will appear as a bright line imposed upon a less bright continuous spectrum. Increasing the
temperature of the continuous source causes the source term to override. The line then appears as
a dark line on a brighter continuous spectrum. If the continuous source is a blackbody and its
temperature is made equal to the flame temperature, then I λ,cont source  I λb, flame , and (D-33)
reduces to I λ, scope  I λ,cont source . The line then disappears into the continuum in the spectroscope
because absorption by the flame and flame emission exactly compensate. If the continuous
source is a tungsten lamp, the source temperature measurement is usually made with an optical
pyrometer.
In the derivation of (D-33) it was assumed that the flame is transparent except within the
spectral line produced by the cadmium or sodium seeding. If soot is in the flame, the soot
particles emit the scatter radiation in a continuous spectrum along the path of the incident beam.
The line-reversal technique is of less utility in this instance, as it then depends on the soot
behavior. The effect of soot is analyzed in Thomas (1968). An analysis in Dembele and Wen
(2000) includes nonisothermal constituents, variable concentrations, and soot particles including
scattering. Gaseous combustion products H2O, CO2, and CO are included. Spectral property
variations are incorporated by using the c-k method in Sec. 9-3.3.
Web Appendix-89
Submerged combustion, in which a flame is maintained within a porous solid matrix, is
reviewed in Howell et al. (1996) for premixed gaseous fuels and in Tseng and Howell (1996) for
liquid fuels. These burners combine radiative, conductive, and convective heat transfer, and
combustion chemistry. Recent work is reviewed in Schoegl and Ellzey (2007) and Dixon et al.
(2008).
Web Appendix-90