NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
Chapter 6 Radiation Heat Transfer
§ 1 The Equation of Radiation Transfer
1,Attenuation by Absorption and Scattering
1) Absorption
? ? dsIdI a b s ??? ??? (6-1)
Where Kη - linear absorption coefficient
Over a geometric path s
? ? ? ? ? ? ?????? ? ????????? ? eIdsIsI s 0e x p0
0
(6-2)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
Where
dss?? 0 ?? ?? Is optical thickness
Absorbtivity:
? ? ? ?
? ?
??
?
??
??
????? e
I
sII 1
0
0
(6-3)
2) Scattering
difference between absorption and scattering
? ? dsIdI ssca ??? ??? (6-4)
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CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
3) Total attenuation
extinction coefficient
??? ??? s??
Optical distance based on extinction
(6-5)
??
s
ds
0 ??
??
(6-6)
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CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
2,Augmentation by Emission and Scattering
1) Emission
? ? dsIdI bem ??? ?? (6-7)
Combined eqs (6-1) and (6-7) for an absorbing -
emitting medium
? ????? ? II
ds
dI
b ??
(6-8)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
For an isothermal gas layer of thickness s is
? ? ? ? ? ??? ????? ?? ??? eIeIsI b 10 (6-9)
If only emission is considered,the emissivity is
defined as
? ? ?????? ???? eIsI b 1/ (6-10)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
2) Scattering
The total spectral radiative heat flux impinging on dA
from solid angle d?i as
? ?? ? ??,??? ddssdAsI iii ??
This flux travels through dV for a distance
ssds i ??/ ?
The total amount of energy scattered away from si is
? ?? ?? ? ? ? dsdd A dsIss dsddssdAsI iis
i
iiis ???? ???? ?????
?
???
?
??? ??????
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CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
The fraction
? ? ?? 4/?,? ?? dss i
is scatteried into the cone
d? around the direction s
?η - scattering phase function
The amount of energy flux from d?i scattered into d?
? ? ? ? ??? dssddsdd A dsI iiis
?
?? ???
4
?,?
?
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
The energy flux scattered into the direction s from all
incoming direction si
? ? ? ? ? ? ? ? ???? ?? ?? ? 4?,??? 4 ????? ? dssdsdd A dsIdd A dsdI iiiss c a
or
? ? ? ? ? ? ? ? iiiss c a dsssIdssdI ??? ?
? ?
?
? ?
?
4
?,??4? (6-10)
Which is valid anywhere inside an arbitrary volume.
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
Omitting the transient term and introducing the extinction
coefficient
? ? ? ?? ????????
? ?
?
?????
?
?
???
4
?,??4? iiisb dsssIIIsdsdI
(6-11)
Define ? ?? ???? s s
s dsds0 0 ???? ???? Optical coordinate
?
?
??
?
? ?
?
??
?? s
s
s ?
?
?
Single scattering adbedo
? ? ? ? ib dssIII
d
dI
?????? ?
? ??
?
???
?
?
?
?
?
? 4
?,?
4
1
(6-12)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
4,Formal Solution to the Equation of Transfer
From eq,(6-14)
? ? ? ?
? ? ? ? ? ? ? ?? ??? ??
?
?
???
??
?
?
??
??
?
???
?
??
?
?
?
???
?
?
0
'
'?,'0
?,
desSeII
esSeI
d
d
Source function
? ? ? ? ? ? ? ?
? ?sSI
d
dI
dsssIIsS iiib
?,
?,??
4
1?,
4
???
?
?
?
??
?
????
?
?
?
?
??
???
????? ?(6-13)
(6-14)
(6-15)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
1) Non-scattering Medium
? ? ? ? ? ? ? ?? ??? ?? ? ??? ? ????????? ??? 0 ' ''0 deIeII b(6-16)
2) The Cold Medium
? ? ? ? ? ? ? ?? ? ??? ??? ? ??? ? ???
? ??
??
??? ???
??
0
'
4 '
?,'40?,dedsIeIsI ii
(6-17)
If ? = 1
? ? ? ? ? ? ? ?? ??? ?? ? ??? ? ?????????? ?????
0
' ''
4
10?,deGeIsI (6-18)
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CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
Where
? ? ? ? ii dsIG ?? ? ? ??? ?? 4 ?,' is known as the incident radiation function
3) Purely Scattering Medium
1???? ? ? ? ? ? ? ? ? ?? ? ??? ???? ? ???
?
?
??
? ???
?
??? ???? 0
'
4 '?,??,'4
10?,dedsssIeIsI
iii
(6-18)
For ? = 1
? ? ? ? ? ? ? ?? ??? ?? ? ??? ? ????????? ???? 0 ' '?,'4 10?,desGeIsI
(6-19)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
4,Boundary Conditions
1) Diffusely Emitting and Reflecting Opaque Surfaces
? ? ? ? ? ? ? ? ? ? ? ? ? ? ???? //?,wwwbwwww rHrrIrrJrIsrI ????
(6-21)
H ? ?
wr - hemispherical irradiation
? ? ? ?? ?? ??? 0'?? ''??'?,sn ww dsnsrIrH
If ? = 0
? ? ? ?wbw rIsrI ??,(6-22)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
2) diffusely Emitting,Specularly Reflecting,Opaque
Surfaces
? ? ? ? ? ? ? ? ? ? ? ? ? ?? ?? ????? 0?? ?,''??'?,?,sn swwswwdwbww srIrdsnsrIrrIrsrI ????
(6-23)
ss is the specular direction
5,Radiative Heat Flux
? ?? ? ?? ? ??? 0 0 4 ??? ?? ?? ddssIdqq? (6-24)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
From eq,(6-13)
? ? ? ????? ???? ???? GIdIIq bb ??????? ? 44 4
? ? ? ??? ??????? ? ??? ????? ???? 44 ??4 iisb dsIdsIIq (6-25)
Using ?? ?????s?
The divergency of the total heat flux
? ? ? ? ??????? ???? ???? dGIddIIdqq bb ?? ? ? ?? ? ?????????? 00 0 4 44(6-26)
When ????= const.
? ? ? ?GTIdTq ??????? ? 444 44 ???? ?
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
Overall Energy Conservation
For a moving compressible fluid
'''QvpquvtuDtDu ?? ???????????????? ?????? ??? (6-28)
Where u is internal energy,v velocity vector,
?dissipation function,Q’’’ heat source
assume du=cvdT
? ? rvv
rrc
qQvpTTv
t
Tc
Dt
DTc
qTqqq
??????????????
?
??
?
? ???
?
??
??????
'''??? ????
?
(6-29)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
7,Solution Methods for the Equation of Transfer
1) Two direction of research
(i) Exact solution -highly idealized situation
(ii) Approximate solution - generalized situation
2) Phenomena,Geometry
Temperature field
Scattering
Properties
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
3) Advanced methods
(i) spherical harmonics method
(ii) discrete ordinate method
(iii) zonal method
(iv) Monte-Carlo method
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
§ 2 Radiative Properties of Particulate Medium
1,Introduction
An electro-magnetic wave passing through a particle’s
vicinity
(i) the shape of the particle
(ii) the material
m=n-i k complex index of refraction
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CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
(iii) its relative size x=2?a/?
(iv) the clearance between particles c/?
Scattering,diffraction
reflection
refraction
independent scattering
dependent scattering
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
2,Absorption and Scattering for a Single Sphere
The amount of scattering and absorption by a particle
Csca - scattering cross section
Cabs - absorption cross section
Cext - Cabs+Csca extinction cross surface area
absorption efficiency factor,Qabs= Cabs/?a2
scattering efficiency factor,Qsca= Csca/?a2
extinction efficiency factor,Qext= Cext/?a2
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
3,Radiative Properties of a Particle Cloud
1) Clouds of uniform size particles
? ? ? ??????
????
??
??
?
???
?
?
????
??
??
T
extTextTs
absTabsT
s c aTs c aTs
QNaCN
QNaCN
QNaCN
2
2
2(6-32)
(6-33)
(6-34)
(6-35)
It means the scattering phase function in a cloud of
uniform particles is the same for each particle.
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
If total properties are desired,
Planck-mean coefficient
?? ? ?? ?? ? 04 dyITy bp (6-34)
Rosselland-mean coefficeint
? ?? 03 141 ??? ?
?
d
dT
dI
yTy
b
p
(6-35)
Y = ?,?s or ?
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
2) Clouds of Non-uniform Size Particles
Modified Gamma distribution
? ? ? ??BaAaan r ?? ex p 0≤a<∞ (3-36)
Which vanishes at a=0 and a →∞
?=1,→gamma distribution
? ? ? ?
??
?
/1
1
0 ?
?
?
??
?
? ??
?? ?
?
rB
rA
daanN T
(6-37)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
? - Gamma function? ?
? ?
? ?
? ? ? ???
?
?
??
?
?
?
??
??
?
?
?
?
?
? ?
?
??
??
0
2
0
/)4
3
0
0
1
3
4
4
3
4
daanQadaanCy
B
r
A
daanaf
dttez
x
rv
zt
?
?
?
?
?
?
?
(6-38)
(6-39)
(6-40)
y?? ? s ?,??,??,x=sca,abs,ext
? ? ? ? ? ? ? ?
? ? ? ?
? ? ? ??
?
? ?
?
?
???
??
???
0
0
0,1
,
daanaC
daanaC
daanaaC
s c a
ss c a
s c a
T
?
? ?
(6-41)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
4,Radiation Properties of Small Particles
(Reyleigh Scattering)
? ? ? ??????
?
?
?
?
?
?
?
?
?
???
?
?
?
2
2
2
4
2
2
2
c o s1
4
3
2
1
4
2
1
3
8
a b se x t
s c a
Qx
m
m
Q
x
m
m
Q
(6-42)
(6-43)
(6-44)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
5,Radiation Properties of large Spheres
(geometric optics)
For large diffusely reflecting spheres
? ? ? ????????
?
?
c o ss i n
3
8
?
?
?
s c a
abs
Q
Q (6-45)
(6-46)
(6-47)