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COMPARISON OF THE RADIATIVE TWO-FLUX AND DIFFUSION APPROXIMATIONS Charles M. Spuckler NASA Glenn Research Center 21000 Brookpark Rd. Cleveland Ohio 44135 Abstract
Approximate solutions are sometimes used to determine the heat transfer and temperatures in a semitransparent material in which conduction and thermal radiation are acting. A comparison of the Milne-Eddington two-flux approximation and the diffusion approximation for combined conduction and radiation heat transfer in a ceramic material was preformed to determine the accuracy of the diffusion solution. A plane gray semitransparent layer without a substrate and a non-gray semitransparent plane layer on an opaque substrate were considered. For the plane gray layer the material is semitransparent for all wavelengths and the scattering and absorption coefficients do not vary with wavelength. For the non-gray plane layer the material is semitransparent with constant absorption and scattering coefficients up to a specified wavelength. At higher wavelengths the non-gray plane layer is assumed to be opaque. The layers are heated on one side and cooled on the other by diffuse radiation and convection. The scattering and absorption coefficients were varied. The error in the diffusion approximation compared to the Milne-Eddington two flux approximation was obtained as a function of scattering coefficient and absorption coefficient. The percent difference in interface temperatures and heat flux through the layer obtained using the Milne-Eddington two-flux and diffusion approximations are presented as a function of scattering coefficient and absorption coefficient. The largest errors occur for high scattering and low absorption except for the back surface temperature of the plane gray layer where the error is also larger at low scattering and low absorption. It is shown that the accuracy of the diffusion approximation can be improved for some scattering and absorption conditions if a reflectance obtained from a Kubelka-Munk type two flux theory is used instead of a reflection obtained from the Fresnel equation. The Kubelka-Munk reflectance accounts for surface reflection and radiation scattered back by internal scattering sites while the Fresnel reflection only accounts for surface reflections.
https://ntrs.nasa.gov/search.jsp?R=20060051747 2020-07-11T21:56:09+00:00Z
1G
lenn
Res
earc
h C
ente
r at L
ewis
Fie
ld
Com
pari
son
of th
e R
adia
tive
Tw
o-fl
uxan
d D
iffu
sion
App
roxi
mat
ions
Cha
rles
M. S
puck
ler
NA
SA G
lenn
Res
earc
h C
ente
rC
leve
land
Ohi
o
2G
lenn
Res
earc
h C
ente
r at L
ewis
Fie
ld
Obj
ecti
ve
Det
erm
ine
the
erro
r in
usi
ng th
e di
ffus
ion
solu
tion
com
pare
d to
the
Miln
e-E
ddin
gton
two-
flux
sol
utio
n
Det
erm
ine
if u
sing
a r
efle
ctan
ce f
rom
the
Kub
elka
-M
unk
2-fl
ux th
eory
cou
ld r
educ
e th
e di
ffus
ion
solu
tion
erro
r
3G
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earc
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ewis
Fie
ld
App
roac
h
A p
aram
etri
c st
udy
of a
one
dim
ensi
onal
sem
itran
spar
ent
gray
pla
ne la
yer
and
a no
n-gr
ay la
yer
on a
sub
stra
te w
as
perf
orm
ed
Fres
nela
nd K
ubel
ka-M
unk
refl
ecta
nce
wer
e us
ed
Bas
elin
e a
= 0
.13
cm-1
and σ s
= 94
.38
cm-1
Bon
d co
at e
mis
sivi
ty c
hang
ed f
rom
0.7
to 0
.3C
utof
f w
avel
engt
h 5 μm
for
the
laye
r on
a s
ubst
rate
4G
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earc
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ente
r at L
ewis
Fie
ld
App
roxi
mat
ions
Dif
fusi
on a
ppro
xim
atio
n●
sim
ples
t●
radi
atio
n tr
eate
d as
a d
iffu
sion
pro
cess
●ra
diat
ion
is a
bsor
bed
em
itted
and
ref
lect
ed a
t the
su
rfac
e
Tw
o-fl
ux a
ppro
xim
atio
n●
mor
e co
mpl
icat
ed r
equi
res
com
pute
r so
lutio
n●
ther
e is
a r
adia
tive
flux
trav
elin
g in
pos
itive
and
nega
tive
x di
rect
ions
and
act
at d
ista
nce
●ra
diat
ion
is a
bsor
bed
and
emitt
ed in
side
the
mat
eria
l●
surf
ace
refl
ectio
n st
ill o
ccur
5G
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Ref
lect
ions
Fres
nelr
efle
ctio
ns ρ
oan
d ρi
are
surfa
ce re
flect
ions
and
are
func
tions
of r
efra
ctiv
e in
dex.
Kub
elka
-Mun
kre
flect
ivity
take
s in
to a
ccou
nt F
resn
elsu
rface
refle
ctio
ns a
nd th
e ra
diat
ive
ener
gy s
catte
red
out o
f the
laye
r. It
is a
func
tion
of ρ
o , ρi
, a, σ
s, an
d D
Fre
snel
Kub
elka
-Mun
k
6G
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r at L
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Fie
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a σ
s
T s1
= 20
00K
q r1o
h 1=
250
w/m
2 K
T g1
= 20
00K
n =
2.1
q r2o
T s2
= 80
0K
h 2=
110
w/m
2 K
T g2
= 80
0K
1 m
m
ρo
ρik =
0.8
w/m
K
Hea
t Tra
nsfe
r M
odel
ρiρo
7G
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Res
earc
h C
ente
r at L
ewis
Fie
ld
Com
paris
on o
f Exa
ct R
adia
tive
Tran
sfer
and
Miln
e-E
ddin
gton
Two-
Flux
App
roxi
mat
ion
Pla
ne L
ayer
Fron
t sur
face
tem
pera
ture
s 0
.075
–0.
28%
Bac
k su
rface
tem
pera
ture
-0
.44
–0.
21%
Hea
t flu
x
0
.14
–3.
6%
100
TT
Tdifference
Percent
solution
exact
solution
exact
solution
flux
two
⋅ ⎟⎟ ⎠⎞⎜⎜ ⎝⎛
−=
−
8G
lenn
Res
earc
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r at L
ewis
Fie
ld
Com
paris
on o
f the
Diff
usio
n an
dM
ilne-
Edd
ingt
onTw
o-Fl
ux A
ppro
xim
atio
n
100
TT
Tdifference
Percent
solution
flux
two
solution
flux
two
diffusion
⋅ ⎟⎟ ⎠⎞⎜⎜ ⎝⎛
−=
−
−
9G
lenn
Res
earc
h C
ente
r at L
ewis
Fie
ld
Per
cent
Dif
fere
nce
betw
een
Dif
fusi
on a
nd T
wo-
flux
So
luti
ons
for
Fro
nt S
urfa
ce o
f Pla
ne L
ayer
Fre
snel
Kub
elka
–M
unk
100
101
102
103
104
23
456
72
34
567
23
456
72
34
567
Scat
terin
g co
effic
ient
cm
-1
-2-1012345678910 Percent difference in temperatureso
lid li
ne F
resn
el re
flect
ivity
brok
en li
ne K
ubel
ka-M
unk
refle
ctiv
ity
a =
0.00
13 c
m-1
a =
0.01
35 c
m-1
a =
0.13
46 c
m-1
a =
1.34
6 cm
-1
a =
13.4
6 cm
-1
10G
lenn
Res
earc
h C
ente
r at L
ewis
Fie
ld
Per
cent
Dif
fere
nce
betw
een
Tw
o-fl
ux a
nd D
iffu
sion
Solu
tion
s fo
r B
ack
Surf
ace
of P
lane
Lay
er
Fre
snel
Kub
elka
–M
unk
100
101
102
103
104
23
456
72
34
567
23
456
72
34
567
Scat
terin
g co
effic
ient
cm
-1
-6-4-202468
Percent difference in temperature
solid
line
Fre
snel
refle
ctiv
itybr
oken
line
Kub
elka
-Mun
k re
flect
ivity
a =
0.00
13 c
m-1
a =
0.01
35 c
m-1
a =
0.13
46 c
m-1
a =
1.34
6 cm
-1
a =
13.4
6 cm
-1
11G
lenn
Res
earc
h C
ente
r at L
ewis
Fie
ld
Per
cent
Dif
fere
nce
in H
eat F
lux
betw
een
the
Dif
fusi
onan
d T
wo-
flux
Sol
utio
ns fo
r a
Pla
ne L
ayer
Kub
elka
–M
unk
Fre
snel
100
101
102
103
104
23
456
72
34
567
23
456
72
34
567
Scat
terin
g co
effic
ient
cm
-1
-50
-250255075100
125
150
175
200
225
Percent difference in heat fluxso
lid li
ne F
resn
el re
flect
ivity
brok
en li
ne K
ubel
ka-M
unk
refle
ctiv
ity
a =
0.00
13 c
m-1
a =
0.01
3 cm
-1
a =
0.13
46 c
m-1
a =1
.346
cm
-1
a =1
3.46
cm
-1
12G
lenn
Res
earc
h C
ente
r at L
ewis
Fie
ld
a σ
s
T s1
= 20
00K
q r1o
h 1=
250
w/m
2 K
T g1
= 20
00K
n =
2.1
q r2o
T s2
= 80
0K
h 2=
110
w/m
2 K
T g2
= 80
0K
1 m
m
ρoρik =
0.8
w/m
K
0.79
4 m
m
ks= 33 w/mKε b
c
Hea
t Tra
nsfe
r M
odel
ε m=0
.6
13G
lenn
Res
earc
h C
ente
r at L
ewis
Fie
ld
Per
cent
Dif
fere
nce
in T
empe
ratu
res
betw
een
Tw
o-fl
ux
and
Dif
fusi
on S
olut
ions
for
Fro
nt S
urfa
ce o
f a L
ayer
λ c=
5.0 μm
and
εbc
= 0.
7
100
101
102
103
23
45
67
23
45
67
23
45
67
Scat
terin
g co
effic
ient
cm
-1
-505101520
Percent difference in temperature
a =
0.00
13 c
m-1
a =
0.01
35 c
m-1
a =
0.13
46 c
m-1
a =
1.34
6 cm
-1
a =
13.4
6 cm
-1
a =
67.2
8 cm
-1
solid
line
Fre
snel
refle
ctiv
itybr
oken
line
Kub
elka
-Mun
k re
flect
ivity
Fre
snel
Kub
elka
–M
unk
14G
lenn
Res
earc
h C
ente
r at L
ewis
Fie
ld
100
101
102
103
23
45
67
23
45
67
23
45
67
Scat
terin
g co
effic
ient
cm
-1
-6-4-2024681012
Percent difference in temperature
a =
0.00
13 c
m-1
a =0
.013
5 cm
-1
a =
0.13
46 c
m-1
a =
1.34
6 cm
-1
a =
13.4
6 cm
-1
a =
67.2
8 cm
-1
solid
line
Fre
snel
refle
ctiv
itybr
oken
line
Kub
elka
-Mun
k re
flect
ivity
Per
cent
Dif
fere
nce
in T
empe
ratu
res
betw
een
Tw
o-fl
ux
and
Dif
fusi
on S
olut
ions
for
Fro
nt o
f Sub
stra
te
λ c=
5.0μ
m a
nd ε
bc=
0.7
Kub
elka
–M
unk
Fre
snel
15G
lenn
Res
earc
h C
ente
r at L
ewis
Fie
ld
Per
cent
Dif
fere
nce
in H
eat F
lux
betw
een
the
Dif
fusi
onan
d T
wo-
flux
Sol
utio
ns fo
r a
Lay
er w
ith
a Su
bstr
ate
λ c=
5.0μ
m a
nd ε
bc=
0.7
100
101
102
103
23
45
67
23
45
67
23
45
67
Scat
terin
g co
effic
ient
cm
-1
a =
0.00
13 c
m-1
a =
0.01
35 c
m-1
a =
0.13
46 c
m-1
a =
1.34
6 cm
-1
a =
13.4
6 cm
-1
a =
67.2
8 cm
-1
-20
-1001020304050
Percent difference in heat flux
solid
line
Fre
snel
refle
ctiv
itybr
oken
line
Kub
elka
Mun
k re
flect
ivity
16G
lenn
Res
earc
h C
ente
r at L
ewis
Fie
ld
Eff
ect o
f Bon
d C
oat E
mis
sivi
ty o
n th
e P
erce
nt D
iffe
renc
e in
T
empe
ratu
re b
etw
een
Tw
o-F
lux
and
Dif
fusi
on S
olut
ions
F
ront
Sur
face
of L
ayer
F
resn
elR
efle
ctiv
ity
λ c
= 5.
0μm
100
101
102
103
23
45
67
23
45
67
23
45
67
Scat
terin
g co
effic
ient
cm
-1
-505101520 Percent difference in temperature
solid
line
ε
bc =
0.7
brok
en li
ne ε
bc =
0.3
a =
0.00
13 c
m-1
a =
0.01
35 c
m-1
a =
0.13
46 c
m-1
a =
1.34
6 cm
-1
a =
13.4
6 cm
-1
a =
67.2
8 cm
-1
17G
lenn
Res
earc
h C
ente
r at L
ewis
Fie
ld
Eff
ect o
f Bon
d C
oatE
mis
sivi
ty o
n th
e P
erce
nt D
iffe
renc
e in
T
empe
ratu
re b
etw
een
Tw
o-F
lux
and
Dif
fusi
on S
olut
ions
F
ront
Sur
face
of L
ayer
K-M
Ref
lect
ivit
y
λ
c=
5.0μ
m
100
101
102
103
23
45
67
23
45
67
23
45
67
Scat
terin
g co
effic
ient
cm
-1
-5-4-3-2-10
Percent difference in temperature
solid
line
ε
bc =
0.7
brok
en li
ne ε
bc =
0.3
a =
0.00
13 c
m-1
a =
0.01
35 c
m-1
a =
0.13
46 c
m-1
a =
1.34
6 cm
-1
a =
13.4
6 cm
-1
a =
67.2
8 cm
-1
Kub
elka
–M
unk
18G
lenn
Res
earc
h C
ente
r at L
ewis
Fie
ld
Max
imum
Cha
nge
in P
erce
nt D
iffer
ence
fo
r a C
hang
e in
Bon
d C
oat E
mis
sivi
ty
from
0.7
to 0
.3
-4.0
4-1
.90
-1.1
10.
86K
-Mre
flect
ion
-11.
33-2
.94
-2.9
8-1
.96
Fres
nel
refle
ctio
n
Hea
t flu
xB
ack
ofsu
bstra
teFr
ont o
fsu
bstra
teFr
ont o
fla
yer layer
substrate
layer
substrate
layer
substrate
19G
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Res
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ewis
Fie
ld
Con
clus
ions
The
per
cent
err
or u
sing
the
diff
usio
n so
lutio
n is
low
est
for
all s
catte
ring
con
side
red
whe
n th
e ab
sorp
tion
is h
igh
The
larg
est e
rror
s oc
cur
mos
tlyfo
r hi
gh s
catte
ring
and
lo
w a
bsor
ptio
n w
hen
the
Fres
nels
urfa
ce r
efle
ctio
n is
use
d
The
err
or a
t hig
h sc
atte
ring
can
be
redu
ced
by u
sing
a K
ulbe
lka-
Mun
kre
flec
tanc
e
For
a la
yer
with
a s
ubst
rate
the
Kub
elka
-Mun
kre
flec
tanc
e is
abl
e to
han
dle
a ch
ange
in b
ond
coat
emis
sivi
ty b
ette
r
20G
lenn
Res
earc
h C
ente
r at L
ewis
Fie
ld
Gen
eral
ized
Con
clus
ions
For
the
heat
ing
cond
ition
s pr
esen
ted
here
the
abso
rptio
n th
ickn
ess
(a∙D
) no
t the
opt
ical
thic
knes
s [(
a + σ s
)∙D
] is
the
impo
rtan
t fac
tor
in d
eter
min
ing
the
use
of th
e di
ffus
ion
appr
oxim
atio
n. N
eed
a la
rge
abso
rptio
n th
ickn
ess.
The
wor
st c
ondi
tions
to u
se th
e di
ffus
ion
appr
oxim
atio
nar
e fo
r hi
gh s
catte
ring
thic
knes
ses
(sσ∙
D)
with
no
abso
rptio
n or
low
abs
orpt
ion
thic
knes
s an
d on
ly s
urfa
cere
flec
tions
acc
ount
ed f
or.
Kub
elka
-Mun
kre
flec
tivity
can
red
uce
the
diff
usio
n so
lutio
nap
prox
imat
ion
erro
r fo
r la
rge
scat
teri
ng th
ickn
esse
s
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