Chemistry Reference
In-Depth Information
single gases, the Bhatnagar-Groos-Krook operator is the most known collision
term [6]. For gaseous mixtures, numerous operators, like the Sirovich model
[7], the Hamel-Morse model [8, 9], the model of Andries et al. [10] and the
model of Kosuge [11] have been developed. The most used linearized kinetic
model for mixtures is the McCormack operator [12].
If the gas is slightly perturbed, the distribution function can be linearized
such that
f
α
(
v
,
r
) =
f
α
(
v
,
r
)[1 +
h
α
(
v
,
r
)]
,
(6)
where
h
α
(
v
,
r
)
is the perturbation function and
f
α
(
v
,
r
) =
n
α
(
r
)
π
−3/2
v
−3
exp(
−
v
2
/v
α
)
(7)
α
Here,
n
α
(
r
)
is the equilibrium distribution function of component
α
.
is the
p
equilibrium value of the density, and
v
α
=
2
R
g
T /m
α
is the mean velocity of
the molecules for species
α
.
The variation of the relative density and the macroscopic velocity around the
equilibrium values can be obtained as the moments of the perturbation function
Z
n
α
−
n
α
n
α
=
π
−3/2
v
−3
d
v
h
α
exp(
−
v
2
/v
α
)
,
(8)
α
Z
u
α
=
π
−3/2
v
−3
d
vv
h
α
exp(
−
v
2
/v
α
)
.
(9)
α
The perturbation function is governed by the linearized Boltzmann equation
X
L
αβ
(
h
α
, h
β
)
−
∂f
α
∂
r
∂h
α
∂
r
1
f
α
,
v
(10)
=
β
where
L
αβ
(
h
α
, h
β
)
is the linearized collision term, and the last term on the
right hand side represents a driving force arising from the spatial variation of
the equilibrium properties.
The kinetic equations, Eqs. (5) and (10), are supplemented by boundary
conditions at the solid-gas interfaces. There are numerous models to describe
the gas-surface interaction. In typical applications under isothermal conditions,
however, the diffuse or diffuse-specular type boundary conditions can provide
physically accurate results. In the diffuse boundary condition, the molecules
hitting the solid surface are reflected in accordance with the local equilibrium
