DEFINITION OF PROPERTIES FOR OPAQUE SURFACES
• Emissivity
• Absorptivity
• Reflectivity
- ity : intensive, theoretical- ance : extensive, experimental
Black and non-black surfaces
e
real surface
blackbody
Emissivity
eb
Directional spectral emissivity ( , , , )e e T
,
,
cos
cose e
b e e
i
i
,
,
e
b e
i
i
b
e
e
, , ,e b ei i be e
e
e
dA
i,e n
Ex 3-1
,
,
e
b ei
i
, ,b eei i be
55be
TT
From Table A-5, 5000T 145
713.97 10be
T
5 2,
140.71000 713.97 10 1519 W/(m m sr)ei
(5 m,60 ,1000 K) 0.7
, ?ei 60
n
Directional total emissivity ( , , )e e T
,0
,0
e
b e
i d
i d
,
e
b e
i
i ,
b
e
e
, ,e b ei i
e
e
dA
i,e n
,0
,0
b e
b e
i d
i d
,0
4
b ei d
T
0
4
be d
T
be e
, ,e b ei i Since
, cosb e ei 0
d
0
d
, cose ei
Ex 3-2T = 700 K
0.8 0 5 m
0.4 5 m
?
,0
,0
cos
cos
e e
b e e
i d
i d
,0
,0
b e
b e
i d
i d
,0
4
b ei d
T
04
be d
T
1
121 0
4 4
bbe de d
T T
1 11 0 2T TF F
0.8 0.38291 0.4 1 0.38291 0.553
From Table A-5, 11 03500 0.38291TT F
1 11 0 2 01T TF F
Hemispherical spectral emissivity ( , )T
,
,
cos e ee
b e
i d
i
,
,
cosb e e e
b e
i d
i
1cos e ed
e
e
dA
i,e n
de
b
e
e
be e
, cosb e ei , cose ei
ed
ed
Hemispherical total emissivity ( )T
,0
,0
cos
cos
e e e
b e e e
i d d
i d d
,04
cosb e e ei d d
T
,0
4
1cosb e e ei d d
T
04
be d
T
b
e
e
4
e
T
4e T
or ,0
,0
cos
cos
e e e
b e e e
i d d
i d d
,0
4
cos e b e ei d d
T
0
4
cos1 e b ei d d
T
04
1cos b
e e
e dd
T
1cos e ed
Ex 3-3 ( , 2000 K) 0.85cos
and e = ?
1cos e ed
2 / 2
0 0
10.85cos cos sin 0.567d d
4 232,150 W/mTe
SummaryDirectional spectral emissivity
Directional total emissivity
Hemispherical spectral emissivity
Hemispherical total emissivity
or
( , , , )e e T
,
,
e
b e
i
i
b
e
e
, , , e b ei i be e
( , , )e e T
1cos e ed
04
be d
T
1cos e ed
04
be d
T
( , )T
( )T
, ,e b ei i be e
be e
4e T
Assumption: Surface is a diffuse emitter.
Ex
Find: 1) Hemispherical total emissivity2) Total emissive power3) Wavelength at which spectral emissive power will be a max
1) Hemispherical total emissivity
e
e
dA
n
de
, ,
, ,
cos
cose e e
b e e b e
i i
i i
,
,
cos
cose e e
b e e e
i d
i d
, cosb e e e
b
i d
e
04
be d
T
0
0
1cosb e e
b
e d d
e d
2 5
1 20 24 4
b be d e d
T T
ie
2) Total emissive power
From Table A-5
2 5
1 20 24 4
b bE d E d
T T
1 0 2 m 2 0 5 m 0 2 mF F F
1 2 m 1600K 3200 m KT 0 2 m 0.31810F
2 5 m 1600K 8000 m KT 0 5 m 0.85625F
0.4 0.31810 0.8 0.85625 0.31810 0.558
4 4 20.558 5.67 16 207 kW/mbe e T
e b
Maximum e may occur
in 0 < ≤ 2 m
or 2 < ≤ 5 m.
3) Wavelength at which spectral emissive power will be a max.
First check where maximum eb occurs.
b
e
e
be e
,
,
cos
cose e e
b e e e
i d
i d
e b
Thus, maximum occurs at = 1.81 m or = 2 m
max
2898 m K1.81 m < 2 m
1600K
From Wien’s displacement law
at = 1.81 m
From Table A-5, T = 2898 m.K
at = 2 m
From Table A-5, T = 3200 m.K
Maximum spectral emissive power occurs at = 2 m.
be e 55be
TT
135
128.67 10be
T
13 5 2(1.81 m) 0.4 128.67 10 1600 54.0 kW/m me
135
125.76 10be
T
13 5 2(2 m) 0.8 125.76 10 1600 105.5 kW/m me
Peak emission
e b,
e
eb
e
Directional spectral absorptivity
dependence on the directional and spectral distributions of the incident radiation, thus not a material property except
Absorptivity
( , , , )i i T
,
( , )
cosi i
i ii
absorbed energy at and
i
i
dA
n
i,i
absorbed energy
emitted energy
in equilibrium
dA at T
d
ib
i
blackbody at T
: no restriction
cosbi dA d d cosbi dA d d
( , , , ) ( , , , )T T
Kirchhoff’s law
Directional total absorptivity
directional-gray surface
( , , )i i T
, ( , ) cosi i i ii absorbed energy at and
,0
,0
( ) ( )cos
( )cos
i i i
i i i
T i T d
i T d
,0
,0
( ) ( )
( )
i i
i i
T i T d
i T d
,0
,0
( ) ( )
( )
i i
i i
T i T d
i T d
0
4
be d
T
0
0
b
b
i d
i d
i) when , ,( , , , ) ( , ) ( , )i i i i i i b ii T C i T
ii) when not function of
Hemispherical spectral absorptivity
diffuse-spectral surface
( , )T
,
,
cos
cosi i i
i i i
i d
i d
,
,
cos
cosi i i
i i i
i d
i d
, cosi i ii d
G
1cos e ed
i)when only:
, ,( , , ) ( )i i i ii i
ii) when independent of direction
diffuse irradiation
Hemispherical total absorptivity
: diffuse-gray surface
iv) when and
( )T
,0
,0
cos
cos
i i i
i i i
i d d
i d d
0
G d
G
,
,
cos
cosi i i
i i i
i d
i d
,0
,0
cos,
cos
i i i
i i i
i d d
i d d
0
0
cos
cos
b e e
b e e
i d d
i d d
i) when ii) when , ,( , , ) ( , )i i i b ii Ci T
iii) when and , ,( , , ) ( )i i i ii i
, ,( , , ) ( , ) ( , )i i i i i b ii C i T
Ex 3-6 0.8 0 3 m(300 K)
0.2 3 m
Find 1) for diffuse incident radiation from a
black source at Ti = 1000 K and 2) for diffuse incident solar radiation
,0
,0
(300 K) (1000 K)cos(300 K)
(1000 K)cos
i i i
i i i
i d d
i d d
1) Ti = 1000 K
, (1000 K)cosi ii , (1000 K)cosi i ii d ,0
(1000 K)cosi i ii d d
incident radiation:
absorbed energy:
, (1000 K)cosii ,(300 K) (1000 K)cosi ii ,0
(300 K) (1000 K)cosi i ii d d
0
0
(300 K) (1000 K)cos(300 K)
(1000 K)cos
b i i
b i i
i d d
i d d
04
(1000 K) (300 K)cosb i i
i
i d d
T
0
4
1(1000 K) (300 K)cosb i
i
i d d
T
04
(300 K) (1000 K)b
i
e d
T
0 3000 30000.8 0.2 0.364F F
0.8 0 3 m(300 K)
0.2 3 m
2) Ti = 5780 K
0
0
(300 K) (5780 K)cos(300 K)
(5780 K)cos
b i i
b i i
i d d
i d d
04
(300 K) (5780 K)b
i
e d
T
0 17340 173400.8 0.2 0.787F F
04
(300 K) (300 K)(300 K) be d
T
5 50.8(8.70 10 ) 0.2(1 8.70 10 ) 0.200
Remark:
30,000 people gathered in Yoido Square to get express bus ticket bounded for Kwangju (noon on Sep. 19, 1982)
The road to home town in Thanks Giving Days is far and hard. 100,000 people gathered at Seoul Station (Sep. 29,1985)
Heavily crowded people at Seoul Station (early in the morning on Sep. 24)
Struggling to get tickets to home town at Seoul Station (Aug. 6, 1992)
dA
i,i di di,r
dr
ReflectivitySpectral reflectivity• bidirectional spectral reflectivity• directional spectral reflectivity directional-hemispherical spectral reflectivity hemispherical-directional spectral reflectivity• hemispherical spectral reflectivity
Bidirectional spectral reflectivity
( , , , , )r r i i
,
,
ˆ ˆ( , )ˆ ˆ( , ) ˆ( )cosr r i
r i
i i i i
di
i d
ˆ ˆ ˆ ˆ( , ) ( , )r i i r Reciprocity:
(spectral reflection distribution function)
, ,ˆ ˆ ˆ( ) ( , )
ir r r r ii di
, ,ˆ ˆ ˆ ˆ( ) ( , ) ( )cos
ir r r i i i i ii i d
dA
di,r
i,idi
di,r: contribution of i,i from direction to i,
r in direction
ˆi
ˆr
Directional-hemispherical spectral reflectivity
( , , )i i
,
,
cosˆ
ˆ ˆ( , )( ) ˆ( )cos
rr r i r r
i
i i i i
di
d
d
i
ˆ ˆ( , )cosr
r i r rd
,
, coˆ ˆ ˆ( , sˆ( ) ˆ( )cos
) ( )cosr
r i i i i r r
i
i i i i
i d
d
i d
i
dA
i,i di di,r
dr
Hemispherical-directional spectral reflectivity
dA
i,r
dA
( , , )r r di,r
i,idi
, ,
, ,,
ˆ ˆ( ) ( )ˆ( )1 ˆ( )cos
i
r r r rr
i ai i i i
i i
i i d
, ,ˆ ˆ ˆ ˆ( ) ( , ) ( )cos
ir r r i i i i ii i d
average incident intensity
, , ,cos cosi a i i i i ii d i d
, , , , , ,
1cos , cosi a i i i i a i i ii i d i i d
reciprocity: when is uniform over all incident directions
, ( , , )i i ii
,
,
ˆ ˆ ˆ( , ) ( )cosˆ( )
1 ˆ( )cos
i
i
r i i i i i
r
i i i i
i d
i d
( , , ) ( , , )r r i i
Hemispherical spectral reflectivity ( )
,
,
ˆ( )cos( ) ˆ( )cos
r
i
r r r r
i i i i
i d
i d
dA
i,i
di
i,r
dr
n
,
,
ˆ ˆ ˆ( , ) ( )cos cos
ˆ( )cosr i
i
r i i i i i r r
i i i i
i d d
i d
,
,
ˆ ˆ ˆ( ) ( , )cos cos
ˆ( )cosi r
i
i i r i r r i i
i i i i
i d d
i d
,ˆ ˆ( ) ( )cos
ii i i i ii d
G
,
,
ˆ ˆ ˆ( , ) ( )cos cos
ˆ( )cosr i
i
r i i i i i r r
i i i i
i d d
i d
ˆ ˆ ˆ( ) ( , )cosr
i r i r rd
Hemispherical total reflectivity
,0
0
ˆ ˆ( ) ( )cosi i i i ii d d
G d
0G d
G
,ˆ ˆ( ) ( )cos
( ) ii i i i ii d
G
Relations among Reflectivity, Absorptivity, and Emissivity
Kirchhoff’s law
for a directional-gray surface,
a) ( , , , ) ( , , , ) 1T T
( , , , ) ( , , , )T T
( , , , ) ( , , , ) 1T T
b) ( , , ) ( , , ) 1T T
( , , ) ( , , )T T ( , , ) ( , , ) 1T T
for a diffuse-spectral surface,
for a diffuse-gray surface,
c) ( , ) ( , ) 1T T
( , ) ( , )T T
( , ) ( , ) 1T T
d) ( ) ( ) 1T T
( ) ( )T T
( ) ( ) 1T T
Ex 3-9
,
0.3 0 2 m
(500 K) 0.8 2 5 m
0.5 5 mn
dA at 500 K
ir,n = ?
black hemisphereat Ti = 1500 K
Assumption: The element has a specularly reflecting surface.
In the normal direction,
incident energy
= absorbed energy + reflected energy
dA at 500 K
ir,n = ?black hemisphereat Ti = 1500 K
,b ii dAd d , ,n b ii dAd d ,r ni dAd d
, , , ,r n b i n b ii d i d i d
, , , ,1 n b i n b ii d i d
,0, r nr n i di , ,0 n b ii d , ,0
1n b ie d
4
0 3000 3000 7500 750, 00.7 0.2 0.5r niT
F F Fi
4,
, 40
b iin
i
eTd
T
, , ,r n n b ii d i d
,
0.7 0 2 m
(500 K) 0.2 2 5 m
0.5 5 mn
1 2
1 2 31 2
4, , ,
, , ,4 4 40
b i b i b iin n n
i i i
e e eTd d d
T T T
235.3 kW/(m sr)