DEFINITION OF PROPERTIES FOR OPAQUE SURFACES

Transcription

1 DEFINITION OF PROPERTIES FOR OPAQUE SURFACES Emssvty Absorptvty Rflctvty - ty : ntnsv thortcal - anc : xtnsv xprmntal

2 Emssvty blackbody ral surfac b Black and non-black surfacs

3 Drctonal spctral mssvty ( θ φ ) T = b cosθ cosθ = b = b φ ˆn ˆ Ω = b = b θ da

4 Ex 3-1 ˆn 6 =? (5 μm6 1 K) =.7 = = b π = b b = ε 5 b T T 5 π From Tabl A-5 T = 5 b = T 1.7 ( 1 ) W/(m 2 = = μ m sr) π

5 Drctonal total mssvty ( θ φ ) T = b b = cosθ d cosθ d b = = = b b d d = b φ ˆn θ ˆ Ω ε Snc = b b = b d d π d b = = da d b

6 Ex 3-2 T = 7 K =?.8 5 μm =. > 5 μm cosθ d d π b d = d = b = b cosθd b bd = bd 1 = + b = F + F T 2 T d = F + ( 1 F ) T 1 T From Tabl A-5 1T = 35 F T = ε = ( ) =.553

7 Hmsphrcal spctral mssvty ε ( T ) cosθ dω ε = cosθ dω b = cosθ dω π b = b ε b cosθdω = π b φ ˆn ˆ Ω 1 = ε cosθ dω π θ dω = ε b da

8 Hmsphrcal total mssvty ε ( T ) ε = cosθ ddω cosθ ddω b b = = = = = b cosθdω d 1 π b cosθdω d π ε bd = ε

9 or ε = cosθ ddω cosθ ddω b cosθ b d dω = 1 cosθ επ bd dω = π 1 bd cosθ = dω π 1 = ε cosθ dω π

10 Ex 3-3 ( θ 2 K) =.85cos ε and =? 1 ε = ε cosθ dω π 1 2 π π /2 =.85cosθcosθsnθdθdφ =.567 π θ = ε = 3215 W/m 2

11 Summary Drctonal spctral mssvty ( θ φ T ) = = = ε b b = b b Drctonal total mssvty ( θ φ T ) bd = = = b b Hmsphrcal spctral mssvty ε ( T ) 1 ε = εcosθ dω π = ε b Hmsphrcal total mssvty ε ( T ) ε bd ε = 1 or ε = ε cosθ dω π = ε

12 Ex Fnd: 1) Hmsphrcal total mssvty 2) Total mssv powr 3) Wavlngth at whch spctral mssv powr wll b a max Assumpton: Surfac s a dffus mttr.

13 1) Hmsphrcal total mssvty cosθ = = cosθ ε ε = = b b cosθ dω cosθ dω b 1 π ε b cosθdω = b bcosθdω d = d b ε d b φ θ da ˆn dω Ωˆ bd 2 2 b ε ε d = +

14 2 5 1 E bd 2 2 b ε ε E d ε = + ε = ε F + ε F F 1 2μm 2 5μm 2μm From Tabl A-5 1T = 2μm 16K = 32μm K F 2 μ m= T = 5μm 16K = 8μm K F 5 μ m= ε = ( ) =.558 2) Total mssv powr = ε = ε = = 27 kw/m b 2

15 3) Wavlngth at whch spctral mssv powr wll b a max. ε = = ε b cosθ dω cosθ dω b b Maxmum may occur n < 2 μm or 2 < 5 μm. = b Frst chck whr maxmum b occurs.

16 From Wn s dsplacmnt law 2898μm K max = = 1.81 μ m < 2 μ m 16K ( ) b Thus maxmum occurs at = 1.81 μm or = 2 μm

17 b 5 = ε b = ε T 5 T at = 1.81 μm From Tabl A-5 T = 2898 μm. b 13 K = T (1.81 μm) = = 5. kw/m μm at = 2 μm From Tabl A-5 T = 32 μm. b K = T (2 μm) = = 15.5 kw/m μm Maxmum spctral mssv powr occurs at = 2 μm. 13

18 b b Pak msson

19 Absorptvty dpndnc on th drctonal and spctral dstrbutons of th ncdnt radaton thus not a matral proprty xcpt α Drctonal spctral absorptvty α ( θ φ ) α ( θ φ) = absorbd nrgy at and cosθ Ωˆ φ θ da T ˆn

20 Krchhoff s law b blackbody at T θ dω da at T absorbd nrgy mttd nrgy = dacosθdωd n qulbrum α ( θφ T) = ( θφ T) = α dacosθdωd b b : no rstrcton

21 Drctonal total absorptvty absorbd nrgy at α ( ) T ( T)cosθd α ( T )cosθ d = = ε ( T ) ( Td ) ( T ) d ε α ( θ φ ) T and ( θ φ ) = α cosθ = d α ( T ) ( Td ) b = ( T ) d = d b d ) whn ( θ φ T) = C( θ φ) b ( T) α = ) whn not functon of = α = drctonal-gray surfac b

22 Hmsphrcal spctral absorptvty α ( T ) α cosθdω cosθdω α cosθ dω cosθ dω = = ε cosθdω = G 1 ε = εcosθ dω π ) whn ( θ φ) ( ) only: α = ε dffus rradaton ) whn ndpndnt of drcton = ε α = ε dffus-spctral surfac =

23 α α Hmsphrcal total absorptvty = = α cosθ dω d cosθ dω d cosθdωd ε cosθ dω d α( T ) α Gd α cosθ dω = α = G cosθ dω ) whn = ε α = ε : dffus-gray surfac ) whn ( θ φ) = Cb ( T) ) whn = and ( θ φ) = ( ) v) whn = ε and ( θ φ ) = C( θ φ ) ( T) α = ε b = cosθ dω d b cosθ dω d b

24 Ex μm ε (3 K) =.2 > 3 μm Fnd α 1) for dffus ncdnt radaton from a black sourc at T = 1 K and 2) for dffus ncdnt solar radaton 1) T = 1 K ncdnt radaton: (1 K)cos θ d ω d absorbd nrgy: α (3 K) (1 K)cos θ dω d α(3 K) = α (3 K) (1 K)cosθ dω d (1 K)cosθ dω d

25 α(3 K) = (3 K) (1 K)cosθ dω d b (1 K)cosθ dω d b (1 K) (3 K)cos b = = ε θ dω d 1 π b(1 K) (3 K)cosθdω d π = ε (3 K) (1 K) d b =.8F +.2F = ε (3 K).8 3 μm =.2 > 3 μm

26 2) T = 578 K α(3 K) = ε (3 K) (578 K)cosθ dω d b (578 K)cosθ dω d b = ε (3 K) (578 K) d b =.8F +.2F = Rmark: ε (3 K) = ε (3 K) (3 K) d 5 5.8(8.7 1 ).2( ).2 b = + =

27 Rflctvty Spctral rflctvty bdrctonal spctral rflctvty drctonal spctral rflctvty drctonal-hmsphrcal spctral rflctvty hmsphrcal-drctonal spctral rflctvty hmsphrcal spctral rflctvty dω d r dω r da

28 Bdrctonal spctral rflctvty (spctral rflcton dstrbuton functon) ρ ( θ r φr θ φ) dω ˆ ˆ ˆ ˆ d r( Ωr Ω) ρ ( Ωr Ω) ( Ωˆ )cosθ dω ( Ωˆ ) = d ( Ωˆ Ωˆ ) r r r r d r : contrbuton of from Ωˆ drcton to r n Ωˆ r drcton ˆ ˆ ˆ ˆ r( Ω r) = ρ ( Ωr Ω) ( Ω)cosθdω Rcprocty: ρ ( Ωˆ Ωˆ ) = ρ ( Ωˆ Ωˆ ) r r da d r

29 Drctonal-hmsphrcal spctral rflctvty ρ ( θ φ) dω d r dω r ρ ( Ωˆ ) = ρ ( Ωˆ ) = = r r r d ( Ωˆ Ωˆ ) cosθ dω r r r r ( Ωˆ )cosθ dω ρ ˆ ˆ ˆ ( Ωr Ω) ( Ω)cosθdω cosθrdωr ˆ ( Ω)cosθdω ρ ( Ωˆ Ωˆ )cosθ dω da r r r

30 Hmsphrcal-drctonal spctral rflctvty ρ ( θ φ) r dω r d r r da da ˆ ˆ ˆ r( Ωr) r( Ωr) ρ ( Ωr ) = = 1 a ( Ωˆ )cosθ dω π ( Ωˆ ) = ( ˆ ˆ ) ( ˆ ρ Ω Ω Ω )cosθ dω r r r

31 avrag ncdnt ntnsty cosθ dω = cosθ dω a 1 π a = cos θdω a = cosθdω π ρ ( Ωˆ ) r = ρ ( Ωˆ Ωˆ ) ( Ωˆ )cosθ dω r 1 ˆ ( Ω)cosθdω π rcprocty: whn ( θ φ) s unform ovr all ncdnt drctons ρ ( θ φ) = ρ ( θ φ) r r

32 Hmsphrcal spctral rflctvty ρ ( ) dω ˆn r dω r ρ ( ) = r r = ( Ωˆ )cosθ dω r r r r da ( Ωˆ )cosθ dω ρ ( ˆ ˆ ) ˆ Ωr Ω ( Ω)cosθdω cosθrdωr ( Ωˆ )cosθ dω

33 r = ˆ ˆ ˆ ( Ω ) ρ ( Ωr Ω)cosθrdω r cosθdω r ( Ωˆ )cosθ dω = ρ ( ˆ ˆ ) ˆ Ωr Ω ( Ω)cosθdω cosθrdωr ( Ωˆ )cosθ dω = ρ ( Ωˆ ) ( Ωˆ )cosθ dω G ρ ( ˆ ) ( ˆ ˆ Ω = ρ Ωr Ω)cosθrdωr r

34 Hmsphrcal total rflctvty ρ ρ ( ) = ρ ( Ωˆ ) ( Ωˆ )cosθ dω G ρ ρ ( Ωˆ ) ( Ωˆ )cosθ dω d = Gd = ρ Gd G

35 Rlatons among Rflctvty Absorptvty and Emssvty a) α ( θφ T) + ρ ( θφ T) = 1 Krchhoff s law ( θφ T) + ρ ( θφ T) = 1 α ( θφ T) = ( θφ T) b) α ( θφ T) + ρ ( θφ T) = 1 for a drctonal-gray surfac α ( θφ T) = ( θφ T) ( θφ T) + ρ ( θφ T) = 1

36 c) α ( T) + ρ ( T) = 1 for a dffus-spctral surfac α ( T) = ε ( T) ε ( T) + ρ ( T) = 1 d) α( T) + ρ( T) = 1 for a dffus-gray surfac α( T) = ε ( T) ε ( T) + ρ( T) = 1

37 Ex 3-9 rn =? black hmsphr at T = 15 K da at 5 K.3 < 2μm n(5 K) =.8 2 < 5μm.5 5μm Assumpton: Th lmnt has a spcularly rflctng surfac.

38 rn =? black hmsphr at T = 15 K da at 5 K In th normal drcton ncdnt nrgy = absorbd nrgy + rflctd nrgy b ω = ε nb dadωd + rn dad d d = d d rn b n b ( ) = d = ρ d 1 n b n b dadωd

39 d = ρ d rn n b 1 rn = rn d = ρ nb d = ρ nb d π b = ρ n d π 1 b 2 b b = ρ 1 n d + ρ 2 n d + ρ 3 n d 1 π 2.7 < 2μm ρ n(5 K) =.2 2 < 5μm.5 5μm ( ) rn =.7F 3 +.2F F75 π 2 = 35.3 kw/(m sr)