TRANSFER: BASICS OF MASS AND HEAT TRANSFER - Biomass as a Sustainable Energy Source for the Future - page 107

If all the terms of Equation (4.12) are integrated over all directions, spanning a solid
angle of 4 π , the following equation is obtained (see Modest (2003) for details):

ð =

r q λ = κ λ 4 π I b λ − I λ d Ω κ λ 4 ð π I b λ − G λ Þ ð Eq : 4 : 18 Þ

4 π

or, after integration over the full spectrum,

r q = ð ∞

0 κ λ 4

ð π I b λ − G λ Þ d =4 κ P π I b − κ G G ð Eq : 4 : 19 Þ

λ

where I b is the total blackbody intensity

I b = ∞
0

= 1 T 4 = 1

π

I b λ d λ

π σ

E b ð Eq : 4 : 20 Þ

The Planck-mean absorption coefficient κ P and the incident-mean absorption coeffi-

cient κ G are defined as

ð ∞
0 κ λ I b λ d

κ P = 1
I b

ð Eq : 4 : 21 Þ

λ

ð ∞
0 κ λ G λ d

κ G = 1
G

ð Eq : 4 : 22 Þ

λ

Equation (4.19) expresses the conservation of total radiative energy in an elementary
control volume. The divergence of the total radiative heat flux vector at an arbitrary
point in space, given by Equation (4.19), represents the amount of energy per unit time
and per unit volume that is gained or lost at that point as a resultant of the global radiative
heat exchange. It appears as a sink term in the equation for the conservation of energy:

!

Q r = − r q ð Eq : 4 : 23 Þ

Note that scattering does not contribute to Equation (4.19), since it does not cause any
local change of energy, but only a redistribution of radiative energy among the
directions.
The situation with a nonzero radiative source term is the case of a

“ participating

medium. In the case of negligible absorption, emission, and scattering, the intensity

does not change in the medium and travels through space until it reaches a wall.
In the simplest case of a gray medium surrounded by walls at temperature T w and
without taking into account possible reabsorption of emitted radiation, the source term
takes the form

”

!

! = T 4 T w

Q r = − r − 4 κσ − ð Eq : 4 : 24 Þ