Civil Engineering Reference
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collisions of gaseous molecules, whereas the characteristic material scale
denotes the size of pores in which the collisions occur.
The gas properties are described by defi nition of the velocity distribution
function
f(r, v)
defi ning the number of gas molecule d
n
having a velocity in
the range [
v, v
+
d
v
] in the volume element
r
+
d
r
. The resulting conductive
heat transfer
φ
cd,g
is expressed as the transport of the particles kinetic energy
(Volz, 2007), as:
1
2
∫
(
3
vv nv
)
ø
cd g
=
m
−
f
⋅
d
with
d
n
=
f
d
rv
d
[9.3]
,
where
m
is the particle mass and
the local average velocity. The change
of this one-particle distribution function
f
is described by the Boltzmann
equation (Boltzmann, 1884) as:
〈
v
〉
∂
∂
f
t
+⋅
∂
∂
f
+⋅
∂
∂
F
f
v
∂
∂
f
t
v
=
[9.4]
r
L
coll
where
F
is an applied external force fi eld. The right-hand side describes the
effect of collisions between particles and the walls of a container deter-
mined by the molecular chaos assumption depicting a binary collision of
particles. The collision term is often approximated as
under the
assumption that collisions in a gas from a non-equilibrium state will tend
to an equilibrium state, with
−
[
f
−
f
0
] /
τ
(
v
) the relaxation time to return to equilib-
rium within d
r
and
f
0
the local equilibrium distribution. The resulting
Boltzmann equation can be expressed by means of dimensionless param-
eters as:
τ
τ
θ
∂
∂ ′
+
f
t
vf
τ
∂
∂ ′
+
vf
τ
∂
∂ ′
[
]
v
′ ⋅
F
′ ⋅
=−
ff
−
0
[9.5]
Λ
r
Λ
v
where the accented parameters are dimensionless. One can notice that the
changes in
f
are driven by
v
τ
/
Λ
, i.e.,
l
/
Λ
in a stationary system, or by
τ
/
θ
in a homogeneous non-stationary system. This ratio expresses the ratio of
the natural scale of the physical problem to that of the material, and is
known as the Knudsen number Kn. Depending on this ratio, we can defi ne
two distinct regimes, i.e., a collisional regime with Kn
1 as considered in
the macroscopic laws with a system state close to
f
0
, and a rarefi ed regime
with Kn
<<
1 where inter-particle collisions are negligible and heat transfer
is ruled by collisions with the walls. Here, the velocity distribution function
evolves according to the Boltzmann equation with the collisional term
equal to zero.
In contrast to the collisional regime, the heat transfer in a rarefi ed gas
proceeds for the major part from the momentum exchange by collision
between the gas molecules and the pore walls instead of by collision between
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