TRANSFER: BASICS OF MASS AND HEAT TRANSFER - Biomass as a Sustainable Energy Source for the Future - page 106

where I λ is the spectral radiation intensity at point r propagating along direction
s ; s is the coordinate along that direction; σ s are the absorption and scat-

tering coefficients of the medium, respectively; and

κ and

( s , s ) is the scattering

Φ

( s , s )/4 phase function. The ratio represents the probability that radiation pro-

pagating in direction s and confined within solid angle d

Φ π

Ω ∗ is scattered through
the angle s s into the direction s confined within solid angle d

Ω . Subscripts b

and κ λ I b λ gives the

amount of energy added to the radiation field per unit distance. I b λ is the black-
body intensity, which is function of the temperature of the medium and accord-
ing to Planck

denote blackbody and wavelength, respectively. The term

λ

' s law is given by

C 1

I b λ = ð Eq : 4 : 13 Þ

C 2
n λ T 5

π n 2 e − 1

λ

Here, n is the index of refraction of the medium (n ≈ 1 for gases), and C 1 = 3.7419 ×

10 − 16 W m 2

and C 2 = 14, 388 μ m K are fundamental constants. The absorption

coefficient κ λ in general depends on both composition and temperature of the

medium. In order to solve the RTE, also boundary conditions have to be provided.
Usually, the boundaries are solid walls, and the material properties of these walls have
to be specified. The RTE may be written in other forms by using the wavenumber or
the frequency as the spectral variable instead of the wavelength. Whatever spectral
variable is chosen, integrating the intensity over this spectral variable, the total
radiation intensity is obtained. If the radiative properties of the medium are independ-
ent of the spectral variable, the medium is called gray. If also the boundary conditions
are gray, the spectral dependence plays no role in the RTE, and only an equation for
total intensity has to be solved.
The radiative heat flux vector, ! [Wm −2 ], is defined as the integral of the spectral
radiative heat flux:

= ð !

λ

I λ ! d

Ω ð Eq : 4 : 14 Þ

4 π

! = ð ∞
0

!

d ð Eq : 4 : 15 Þ

λ

λ

m −2 ] (a scalar), is defined as the integral Similarly, the total incident radiation, G [W

of the spectral incident radiation:

G λ = ð

I λ d Ω ð Eq : 4 : 16 Þ

4 π

G= ð + ∞
0

G λ d ð Eq : 4 : 17 Þ

λ