Environmental Engineering Reference
In-Depth Information
through a region with a relatively high temperature, are excited in this region, and
then transport this excitation energy to other regions in the gas. This mechanism of
heat transport is significant if a typical distance over which excited and nonexcited
atoms reach equilibrium is small compared with the length of this system.
The heat flux can be represented as a sum of two terms:
q
D
T
T
,
r
r
tr
i
where
tr
is the thermal conductivity coefficient due to the transport of translation-
al energy, whereas the second term is due to transport of energy in the internal
degrees of freedom. The thermal conductivity coefficient is then just the sum of
these two terms, or
D
C
i
.
(4.42)
tr
We shall now analyze the second term.
The internal state of the gas particle is denoted by subscript
i
.Becauseofthe
presence of a temperature gradient, the number density of particles in this state is
not constant in space, and the diffusion flux is given by
D
i
@
N
i
@
j
i
D
D
i
r
N
i
D
T
.
T
r
This leads to the heat flux
X
X
i
D
i
@
N
i
@
q
D
ε
j
i
D
ε
T
,
T
r
i
i
i
where
i
is the excitation energy of the
i
th state. If the diffusion coefficient is the
same for excited and nonexcited particles, the thermal conductivity coefficient due
to internal degrees of freedom is
ε
X
T
X
i
i
D
i
@
N
i
@
D
@
@
D
@
@
D
ε
T
D
ε
i
N
i
D
T
(
N
ε
N
)
D
Dc
V
,
(4.43)
i
i
ε
D
P
i
D
P
i
N
i
is
where
ε
i
N
i
/
N
is the mean ex
ci
tation energy of the particles,
N
the total number density, and
c
V
T
is the heat capacity per particle. Using
estimates (4.29) and (4.40) for the diffusion coefficient and the thermal conductivity
coefficient, one can conclude that
D @
ε
/
@
tr
if the excitation energy of internal states
is of the order of a thermal energy of the particles.
Another example of this mechanism for thermal conductivity occurs when a dis-
sociated gas undergoes recombination upon entering a cold region. We examine
the example of a gas that consists of atoms coexisting in thermodynamic equilibri-
um with a small admixture of diatomic molecules representing the bound system
of these atoms. The dissociation and recombination proceed in accordance with
thesimplescheme
i
A
C
A
!
A
2
.
