Environmental Engineering Reference
In-Depth Information
and emitted by the walls, and these processes establish the equilibrium between
the radiation and the walls of the vessel. This radiation is called blackbody radia-
tion.
To calculate the average number of photons in a particular state, we use the fact
that photons obey Bose-Einstein statistics. Therefore, the presence of a photon in a
given state does not depend on whether other photons with this energy are also in
this state. Then according to the Boltzmann formula (1.43), the relative probability
that
n
photons with energy
„
ω
are found in a given state is exp(
„
ω
n
/
T
). Thus,
the mean number of photons in this state is
P
n
n
exp(
„
ω
n
/
T
)
1
n
ω
D
P
n
exp(
D
exp(
1
.
(1.56)
„
ω
n
/
T
)
„
ω
/
T
)
This formula is called the Planck distribution [55].
We introduce the spectral radiation density
U
ω
as the radiation energy per unit
time and volume in a unit frequency range. We shall show below how this quantity
can be determined. The radiation energy in the frequency interval from
ω
to
ω
C
d
is the volume of a region where the radiation is located.
Alternatively, this quantity can be expressed as 2
ω
is
Ω
U
ω
d
ω
,where
Ω
)
3
, where the factor
2 accounts for the two polarizations of an electromagnetic wave,
k
is the photon
wave number,
n
ω
„
ω
n
ω
Ω
d
k
/(2
π
)
3
is the
number of states in an element of the phase space. Using the dispersion relation
ω
D
is the number of photons in a single state, and
Ω
d
k
/(2
π
and wave vector
k
of the photon (
c
is light
velocity), the equivalence of these two aspects of the same quantity yields
ck
between the frequency
ω
„
ω
3
U
ω
D
2
c
3
n
ω
.
(1.57)
π
When the Planck distribution (1.56) is inserted into (1.57), we obtain the Planck
radiation formula:
3
„
ω
U
ω
D
2
c
3
exp(
1
.
(1.58)
π
„
ω
/
T
)
In the limiting case
„
ω
T
, this result transforms to the Rayleigh-Jeans formu-
la [56, 57]:
ω
2
T
U
ω
D
,
„
ω
T
.
(1.59)
2
c
3
π
This expression corresponds to the classical limit, and hence does not contain the
Planck constant. The opposite limit yields the Wien formula [58]:
2
c
3
exp
,
3
„
ω
„
T
U
ω
D
„
ω
T
.
(1.60)
π
We shall now apply (1.58) to find the radiative flux emitted by a blackbody sur-
face. It may be defined as the flux of radiation coming from a hole in a cavity with
