chap 12 (2) [compatibility mode]

8/6/2019 Chap 12 (2) [Compatibility Mode]

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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

chap 12 (2) [compatibility mode]

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