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)