Environmental Engineering Reference
In-Depth Information
where
f
is the velocity distribution function for the particles, and the relation be-
tween the heat flux and the temperature gradient is given by (4.27). For estimation
of the thermal conductivity coefficient, we use the same procedure as in the case of
the diffusion coefficient. Take the heat flux through a given point as the difference
between the fluxes in opposite directions, and express the difference of the heat
fluxes in terms of the difference between temperatures. From (4.39), the heat flux
can be estimated as
q
T
because the energies of particles reaching this
point from opposite sides are different. Because only particles located at a distance
of about
N
v
Δ
T
.Substi-
tuting this into the equation for the heat flux and comparing the result with (4.27),
we find that our estimate for the thermal conductivity coefficient is
λ
reach the given point without collisions, we have
Δ
T
λ
r
p
T
σ
p
m
.
N
v
λ
v
σ
(4.40)
The thermal conductivity coefficient is independent of the particle number density.
An increase in the particle number density leads to an increase in the number of
particles that transfer heat, but this then causes a decrease in the distance for this
transport. These two effects mutually cancel.
Our next step is to derive the heat transport equation for a gas
wh
ere thermal con-
ductivity supplies a mechanism for heat transport. We denote by
the mean energy
of a gas particle, and for simplicity consider a single-component gas. Assuming the
absence of sources and sinks for heat, we obtain the heat balance equation by anal-
ogy with the continuity equation (4.1) for the number density of particles, namely,
ε
@
@
t
(
ε
N
)
C
div
q
D
0.
We take the gas to be contained in a fixed volume. With
@
ε
/
@
T
D
c
V
as the heat
capacity per gas particle, the above equation takes the form
@
T
@
D
t
C
w
r
T
c
V
N
Δ
T
,
(4.41)
where
w
is the mean velocity of the particles, we use the continuity equation (4.1)
for
t
, and employ (4.27) for the heat flux. For a motionless gas this equation
is analogous to the diffusion equation (4.30), and its solution can be obtained by
analogy with (4.34).
@
N
/
@
4.2.5
Thermal Conductivity Due to Internal Degrees of Freedom
An additional channel of heat transport arises from the energy transport associated
with internal degrees of freedom. Excited atoms or molecules that move through a
region with a relatively low temperature can transfer their excitation energy to the
gas, and successive transfers of this nature amount to the transport of energy. The
inverse process can also take place, in which ground-state atoms or molecules pass
