Environmental Engineering Reference
In-Depth Information
This is a standard form for a continuity equation.
This equation corresponds to a one-component system where internal states of
the particles are not distinguished. In fact, by definition,
R
I
col
(
f
)
d
v
is the varia-
tion of the particle number density due to collision processes, so it is zero for the
total number density summed over internal quantum numbers. If we discriminate
internal states, the right-hand side of the continuity equation is akin to (4.1), so the
continuity equation has the form of the balance equation:
X
X
@
N
i
@
t
C
div(
N
i
w
)
D
k
fi
N
j
k
if
N
j
,
(4.2)
f
f
where indices
i
and
f
refer to internal states of particles and
k
if
is the rate constant
for transition between states
i
and
f
.
Another macroscopic equation that is analogous to the equation of motion for
particles follows from multiplication of the kinetic equation (4.1) by the particle
momentum
m
and integration over particle velocities, where
is a component
v
α
v
α
of the particle velocity, and
x
,
y
,or
z
. The right-hand side of the resulting
equation is the variation of the total momentum of the particles. For a system of
identical particles, collisions do not change the total momentum of the particles,
so the right-hand side of the equation is zero and the macroscopic equation has the
form
α
D
Z
m
Z
m
Z
v
α
@
f
@
f
f
@
v
@
t
d
v
C
d
v
C
F
d
v
D
0.
v
α
v
v
α
@
@
x
Here indices
α
and
both denote vector components (
α
,
D
x
,
y
,
z
)withasum-
mation over
,and
x
is a coordinate. If we change the order of the integration and
summation in the first two terms and integrate the third term by parts, we obtain
Z
Z
f
@
v
@
j
C1
d
v
D
v
α
f
1
δ
α
d
v
D
N
δ
α
,
v
α
where
δ
α
is the Kronecker symbol
δ
α
D
1if
α
D
,and
δ
α
D
0if
α
¤
.
Finally, we obtain
@
@
C
@
@
t
(
mN
v
α
)
(
N
h
m
v
α
v
i
)
NF
α
D
0,
x
where angle brackets denote averaging over the particle distribution function.
We define the pressure tensor as
P
α
Dh
m
(
v
α
w
α
)(
v
w
)
i
.
(4.3)
Inserting this tensor into the above equation, we have
@
@
C
@
P
α
@
x
C
@
t
(
mN
v
α
)
(
mNw
α
w
)
NF
α
D
0.
@
x
