Environmental Engineering Reference
In-Depth Information
only obtain an approximate distribution function. By using
f
(
r
,
v
,
t
)
, the rate of
energy flow per unit area (the energy flux) can be expressed as
(
,
)=
(
,
)
(
,
,
)
ε
(
)
.
q
r
t
v
r
t
f
r
v
t
v
d
v
(1.53)
all
v
Here
q
(
r
,
t
)
is the energy flux vector,
v
(
r
,
t
)
is the velocity vector, and
ε
(
v
)
is the
kinetic energy of the particle as a function of particle velocity. Note that
f
)
is the fraction of system particles in the ensemble per unit volume per unit velocity.
Therefore
f
(
r
,
v
,
t
d
r
d
v
is the fraction of system particles in the ensemble that have
phase points in d
r
d
v
around
r
and
v
.
In terms of an integral over momentum, Eq. (1.53) reads (Tien et al. 1998)
(
r
,
v
,
t
)
q
(
r
,
t
)=
v
(
r
,
t
)
f
(
r
,
p
,
t
)
ε
(
p
)
d
p
,
(1.54)
all
p
where the vector
p
is the momentum, and the distribution
f
(
r
,
p
,
t
)
is the fraction of
system particles per unit volume per unit momentum.
f
d
r
d
p
is the fraction
of system particles in the ensemble that have phase points in d
r
d
p
around
r
and
p
.
Because
p
(
r
,
p
,
t
)
=
m
v
,wehave
f
(
r
,
p
,
t
)
d
r
d
p
=
mf
(
r
,
p
,
t
)
d
r
d
v
.
(1.55)
This implies that the fraction of system particles per unit volume and per unit veloc-
ity can also be expressed as
mf
(
r
,
p
,
t
)
. Therefore we have
f
(
r
,
v
,
t
)=
mf
(
r
,
p
,
t
)
.
(1.56)
This equation enables us to rewrite Eq. (1.53) into Eq. (1.54).
By introducing the spherical coordinates for the integral in Eq. (1.54),
=
ϕ
,
=
ϕ
,
=
θ
,
p
x
p
sin
θ
cos
p
y
p
sin
θ
sin
p
z
p
cos
p
x
+
where
p
=
p
y
+
p
z
,wehave
∞
π
2π
p
2
sin
q
(
r
,
t
)=
v
(
r
,
t
)
f
(
r
,
p
,
t
)
ε
(
p
)
θ
d
p
d
θ
d
ϕ
.
(1.57)
0
0
0
Applying the relation between
p
and the kinetic energy
ε
yields
∞
π
2π
m
√
2
m
q
(
r
,
t
)=
v
(
r
,
t
)
f
(
r
,
p
,
t
)
ε
ε
sin
θ
d
ε
d
θ
d
ϕ
.
(1.58)
0
0
0
For the case of no external forces acting on the heat transfer medium, the particle
randomly accesses every direction with the same probability so that the distribution
function
f
(
r
,
p
,
t
)
only depends on
r
,
ε
and
t
. The density of states
D
(
ε
)
can then be
defined as
π
2π
m
√
2
m
m
√
2
m
D
(
ε
)=
ε
sin
θ
d
θ
d
ϕ
=
4
π
ε
.
(1.59)
0
0
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