Environmental Engineering Reference
In-Depth Information
does not change the total momentum of the system, the characteristic time for the
momentum transfer satisfies the relation
m
(
q
)
N
(
q
)
τ
m
(
s
)
N
(
s
)
τ
D
.
(4.8)
qs
sq
In particular, the Euler equation for a multicomponent system in vector form is
given by
X
@
w
q
@
C
r
p
F
q
m
q
D
(
w
s
w
q
)
τ
t
C
(
w
q
r
)
w
q
q
.
(4.9)
qs
s
4.1.2
Equation of State for a Gas
The relation between the bulk parameters of a gas (pressure
p
, temperature
T
,and
particle number density
N
) is given by the equation of state. Below we derive this
equation for a homogeneous gas. For this purpose we employ a frame of reference
where the gas (or a given volume of the gas) is at rest. The pressure is the force
in this frame of reference that acts on a unit area of an imaginary surface in the
system. To evaluate this force we take an element of this surface to be perpendicular
to the
x
-axis, so the flux of particles with velocities in an interval from
v
x
to
v
x
C
d
v
x
is given by
dj
x
v
x
,where
f
is the distribution function. Elastic reflection
of a particle from this surface leads to inversion of
v
x
,thatis,
v
x
D
v
x
fd
!
v
x
as a result
of the reflection. Therefore, a reflected particle of mass
m
transfers to the area the
momentum 2
m
v
x
. The force acting on this area is the momentum change per unit
time. Hence, the gas pressure, that is, the force acting per unit area, is
Z
m
Z
mN
˝
v
x
˛
.
2
x
fd
2
p
D
2
m
v
x
fd
D
D
v
x
v
v
x
v
x
>
0
We account for the fact that the pressure on both sides of the area is the same.
In the above expression
v
x
is the velocity component in the frame of reference
where the gas as a whole is at rest. Transforming back to the original axes, we have
mN
˝
(
v
w
x
)
2
˛
,
p
D
(4.10)
x
where
w
x
is the
x
component of the mean gas velocity. Because the distribution
function is isotropic in the reference frame where the gas as a whole is at rest, the
gas pressure is identical in all directions and is given by
mN
˝
(
v
w
x
)
2
˛
D
mN
˝
(
v
w
y
)
2
˛
D
mN
˝
(
v
w
z
)
2
˛
.
p
D
(4.11)
x
y
z
Let us use the definition of the gas temperature through the mean particle veloc-
ity in the frame of reference where the mean velocity is zero, that is,
2
˝
(
v
w
)
2
˛
.
3
T
2
D
m
