(a)
(b)
(c)
dA n = r 2 sin θ dθ dφ
r
n
θ
dθ
dφ
r si n θ
r dθ
r sin θ dφ
r
dα = dl/r
r
dl
dω = dA n /r 2
dA
dA n
�
�
�
The total hemispherical emissive power (integration over wavelength) is
∞
E(T) = E λ (λ, T) dλ
(11.3)
0
For the isotropic surface (independent of circumferential direction angle),
for most of the surfaces,
π/2
E λ = 2π
I λ (θ, λ, T) sin θ cos θ dθ
0
For the isotropic and black surface (independent of vertical direction angle),
π/2
E b,λ = 2πI b,λ (λ, T)
sin θ cos θ dθ = πI b,λ
(11.4)
0
223
Fundamental Radiation
FIGURE 11.3
(a) Conceptual view of hemispheric radiation from a differential element area. (b) Definition of
plane angle. (c) Definition of solid angle.
11.2 Surface Radiation Properties for Blackbody
and Real-Surface Radiation
Total emissive power for black surface (ideal surface)
E b = πI b = σT
4
(11.5)