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Assumption:
Surface is a diffuse e mitter.
Exam ple 12.5
Find:
1) Hemispherica l total emissivity, 2) Total e missive po wer, E
3) Wavelength a t which spec tral emissive pow er willbe a max
1) Hemispherica l total e missivity
e
e
e
dA
n
de
, ,
, ,
cos
cos
e e e
b e e b e
I I
I I
,
,
cos
cos
e e e
b e e e
I d
I d
,
cos b e e e
b
I d
E
0
0
1cos b e e
b
E d d
E d
0
4
bE d
T
2 5
1 , 2 ,0 2
4 4
b bE d E d
T T
2) Tota l emissive p owe r
From Tab le 12.1
2 5
1 , 2 ,0 2
4 4
b bE d E d
T T
1 (0 2 m) 2 (0 5 m) (0 2 m) F F F
1 2 m 1600K 3200 m KT ( 0 2 m ) 0.318102F 2 5 m 1600K 8000 m KT ( 0 5 m) 0.856288F
0.4 0.318102 0.8 0.856282 0.318102 0.4 0.318102 0.8 0.53818 0.558
bE E 4T 4 20.558 5.67 16 207 kW/m
Maximum E may occurin 0 < 2 m or 2 < 5 m.
3) Wavelength a t which spectral
emissive powe r will be a max
Thus, maximum may oc cur at = 1.81 m or = 2 m
From Wiens law
First c hec k w here ma ximum Eboccurs.
,
,
cos
cos
e e e
b e e e
I d
I d
bEE
bE E
max 2898 m K 1.81 m < 2 m1600K
= 1.81 m
at = 1.81 mFrom Tab le 12.1, T= 2898 m.K
at = 2 mFrom Tab le 12.1, T= 3200 m.K
Maximum spec tral emissive pow er oc curs at = 2 m.
bE E bI 55bI TT 4
50.722 10b
I
T
4 8 5(1.81 m,1600 K) 0.4 0.722 10 5.67 10 1600E
254 kW/m m4
50.706 10b
I
T
4 8 5(2 m,1600 K) 0.8 0.706 10 5.67 10 1600E
2105.5 kW/m mPea k em ission
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dA
Absorptivity
Direc tional spec tral ab sorptivity
depe ndence on the direc tional and spectraldistributions of the inc ident rad iation, thusnot a m aterial prope rty except
( , , , )i i T
ii
, i iI n
,
( , )
cos
i i
i iI
ab sorbed energy at and
Kirchhoffs law
absorbed energy
emitted energy
in eq uilibrium
dA at T
dIb
I
blackbodyat T
: no restriction
( , , , ) ( , , , )T T cos
b I dA d d
cosb I dA d d
Direc tional tota l absorptivity
directional-gray surface
( , , )i i
T ,
cosi iI absorbed energy at and (i, i)
,0,0
cos
cos
i i
i i
I d
I d
,0
,0
i
i
I d
I d
,0
,0
i
i
I d
I d
0
4
bE d
T
0
0
b
b
I d
I d
i) when , ,( , , ) ( , ) ( , )i i i i i b i I C I T
ii) when not function of
Hemispherical spectral absorptivity
diffuse-spec tral surfac e
( , )T
,
,
cos
cos
i i i
i i i
I d
I d
,,cos
cos
i i i
i i i
I d
I d
,
cosi i i
I d
G
1
cose ed
i) when only:, ,( , , ) ( )
i i i iI I
ii) when independent of direction
diffuse irradia tion
Hem isphe rical tota l absorptivity
: diffuse-gray surfac e
iii) when ( , , , ) ( , , )e e e eT T , ,( , , ) ( )i i i iI I andiv) when ( , , , ) ( , )e e T T , ,( , , ) ( , ) ( , )i i i i i b i I C I T andii) when
, ,( , , ) ( , )i i i b i I CI T
( )T
,0,0
cos
cos
i i i
i i i
I d d
I d d
0
G d
G
,
,
cos
cos
i 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 0cos
cos
b e e
b e e
I d d
I d d
i) when ( , , , ) ( )
e eT T
Rela tions amo ngreflectivity, absorptivity, and emissivity
Kirchhoffs law
for a d irec tional-gray surface,
a) ( , , , ) ( , , , ) 1T T ( , , , ) ( , , , )T T
( , , , ) ( , , , ) 1T T b) ( , , ) ( , , ) 1T T
( , , ) ( , , )T T ( , , ) ( , , ) 1T T
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for a d iffuse-spec tral surfac e,
for a diffuse-gray surfac e,
c ) ( , ) ( , ) 1T T ( , ) ( , )T T ( , ) ( , ) 1T T
d) ( ) ( ) 1T T
( ) ( ) 1T T ( ) ( )T T
Exam ple 12.7
Find:
1) Spe c tral distribution of reflec tivity
2) Hemispherical total a bsorptivity
3) Nature of surfac e temp erature cha nge
bE E G500 K
0.8
sT
1) Spec tral reflectivity
1 2) Hemispherica l total a bsorptivity
, cosi iI , cosi iI
incident radiation in at :absorbed energy:
Why is the va lue of (= 0.76) closer to unity ?
,
,
cos
cos
i i i
i i i
I d
I d
,cosi i iI d
G
,0
,0
cos
cos
i i i
i i i
I d d
I d d
0G d
G
6 8 16
2 6 8
6 12 16
2 6 12
0.2 500 1.0
0.76
G d d G d
G d G d G d
= 5000 W/m2
3) Nature of surface temperature change
Does the surface tempe rature inc rease o r dec rease?
bE E G500 K
0.8
sT
st in out g E E E E 4
b sG E G T 8 40.76 5000 0.8 5.67 10 500
23800 2835 965 W/m Since , the surface tem pe rature will increa se
with time.st 0E
Exam ple 12.9
Coals
Brickwall
(diffuse)
Find:
1) Hemispherica l total emissivity of the fire brick w all 2) Total em issive po wer of the brick w all 3) Absorptivity of the w all to irradia tion from the c oals
Assumptions:
Spectral distribution of irradiation at the brick wall
ap proximates that due to e mission from a b lack bod yat 2000 K
spectral emissivity of brick wall
500 KsT 2000 KcT
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Spec tral distribution of sola r radia tion Direc tional distribution of sola r radia tionat earths surface
ac tual diffuse
Scattering of solar radiationin the earths atmosphere
Assumptions:
steady-state
ab sorbe r surface diffuse
Example 12.11
: useful heat removal
flat-plate solar
collector
sky 10 CT skyG 1/ 330 C, 0.22 sT h T T
120 CsT 2750W/mSG
0.1
0.95
uq
Find:
1) Useful heat removal rate p er unit area,
2) Efficiency of the collector.u
2[W/m ]q
sky skyGconv
q
uq
1) Useful heat removal rate
S SG E
skyG
S0.95 0.1
uq : useful heat remova l
1/ 3, 0.22 sT h T T 120 CsT
sky10 CT 2
S750W/mG
in outE E
S S sky s c uky onvqG G q E
S S sky sky co vu nG G qq E
4
sky sky,G T sky 0.1
4/ 3conv 0.22s sq h T T T T 4
sE T 4/ 34 4S su S ky 0.22 s sG T Tq T T
2712.5 27.1 88.7 135.2 516 W/m
2) The collector efficiency
2
u
2
S
516W/m0.69
750W/m
q
G
skyG
S0.95 0.1
uq : useful heat remova l
120 CsT sky
10 CT 2S
750W/mG
Comments:
1) Since the spec tral range of Gsky is entirely differentfrom that of GS, it would be incorrect to a ssume thatsky = S .
2) With a co nvection co efficient , the
useful heat flux and the efficienc y are reduc ed to
and . A cover plate can
co ntribute significantly to reducing c onvec tion (and
radiation) heat loss from the absorber plate.
2
u161W/mq 0.21
25W/m Kh
1/ 3 1/ 3 20.22 0.22 120 30 0.986 W/m K s h T T
1/ 3, 0.22 sT h T T
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ReflectivityDirectional-hemispherical spectral reflectivity
dA
I,i didI,r
dr
( , , )i i
,
,
( , , , , )cos( , , )
( , , )cos
r r r i i r r
i i
i i i i i
dI d
I d
,( , , , , )cos
r r r i i r r dI d ,
( , , ) ( , , )cosi i i i i i i
I d
,( , , )cos
r r r r rI d ,
( , , ) ( , , )cosi i i i i i iI d d
A
d
A
I,r
I,i
d
A
Hemispherical-directional spectral reflectivity
average incident intensity
d
A
I,r
I,i
( , , )r r
,
,
( , , )( , , )
1( , , )cos
r r r r
r r
i i i i i i
I d
I d d
,
cos cos a i i i i i I d I d
, ,
1cos , cos
a i i i a i i i I I d I I d
Rec iproc ity:when is uniform ove r allincident directions
,( , , )
i i iI
( , , ) ( , , ) r r i i
Hemispherical spectral reflectivity
dA
I,i diI,rdr
( )
,
,
( , , )cos
( , , )cos
r r r r r
i i i i i
I d
I d
,( , , ) ( , , )cos
i i i i i i iI d
G
, ( , , )cos r r r r rI d ,( , , ) ( , , )cosi i i i i i iI d
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Hemisphe rical total reflec tivity
,0
0
( , , ) ( , , )cosi i i i i i i
I d d
G d
0G d
G
,
( , , ) ( , , )cosi i i i i i iI d
G
A transmitting Layer with Thickness L>
Reflectance
2 22 2 2 2 2 2111 1 11 1
Transmittan c e
22 2 2 4 4 2 211 11
T
1-
(1-)
(1-)2
(1-)
(1-)2
(1-)22
2(1-)2
(1-)3
2 (1-)23
3 (1-)3
3 (1-)4
3 (1-)24
(1-)(1-) (1-)(1-)
(1-)2(1-)3 (1-)3(1-)
4 (1-)4
2 2 2 2 4 41 1 1R
2 2 3 3 1 11 1 11
A
spectral transmittanc e 2
2 2
1
1T
22 20
1
1G d
G
,trG T G
total transmittance
Absorptance
1-
(1-)(1-)(1-)
(1-)2
(1-)
(1-)(1-) (1-)2
(1-)22
2(1-)2
(1-)2(1-) (1-)3
2 (1-)23
3 (1-)3
3 (1-)3(1-)3 (1-)4
3 (1-)24
4 (1-)4
,tr,
G
G
0 T G d
G
,tr0
0
G dT
G d
R+ T+ A =1
the amount of radiation energy streaming out through aunit area perpendic ular to the direction of propag ation ,
per unit solid angle around the d irec tion , per unitwavelength around , and pe r unit time about t.
I
2 2cosdA
1 1cosdA
the amount of radia tion energy intercepted b ydA2 per unit wavelength, pe r unit time [W/m]
d 41 1cos
d QI
dA d d dt 4
2
1 1cos
d Q d q I dA d
d dt
2 2
2
cosdAd
r
1dA
2dA
1n
2n
1
2
r
22 21 1 2
coscos
dA d q I dA r