Environmental Engineering Reference
In-Depth Information
or
∂
l
j
=
1
v
j
∂
f
p
υ
l
j
=
1
∂υ
j
∂
f
p
υ
∂
∂
f
p
υ
∂υ
f
p
υ
∂
t
+
r
j
+
j
=
coll
.
(1.49)
∂
∂
t
t
This is the
Boltzmann transport equation
, which can also be written as
∂
l
l
∂
f
j
=
1
v
j
∂
f
j
=
1
∂υ
j
f
∂υ
j
=
∂
f
t
+
r
j
+
coll
,
(1.50)
∂
∂
∂
t
∂
t
where
f
p
υ
(
r
,
v
,
t
)
f
(
r
,
v
,
t
)=
.
N
The collision term
∂
f
in the Boltzmann transport equation is usually writ-
∂
t
coll
ten as (Carey 1999)
∂
f
f
−
f
0
coll
=
−
,
(1.51)
∂
t
τ
0
where
f
0
is the equilibrium distribution for the system, and
τ
0
is the relaxation time.
Suppose that a non-equilibrium distribution of velocities is set up by external forces
which are suddenly removed. Note that
∂
f
0
∂
0 by using the definition of the equi-
librium distribution. The decay of the distribution towards the equilibrium is then
obtained from (1.51) as,
t
=
∂
(
−
)
−
f
f
0
f
f
0
=
−
.
∂
t
τ
0
This equation has the solution
(
f
=
f
0
)
|
t
=(
f
−
f
0
)
|
t
=
0
exp
(
−
t
/
τ
0
)
.
By combining Eqs. (1.50) and (1.51), we obtain the Boltzmann transport equation
with the relaxation time approximation:
l
l
∂
f
j
=
1
v
j
∂
f
j
=
1
∂υ
j
f
∂υ
j
=
−
∂
f
−
f
0
t
+
r
j
+
.
(1.52)
∂
∂
∂
t
τ
0
Dual-Phase-Lagging Constitutive Relation
Consider a three-dimensional heat transfer problem. The position vector
r
has three
components
x
T
.To
study energy transport via particles, we must solve the Boltzmann transport equation
to determine the distribution function
f
,
y
and
z
, and the velocity vector
v
can be expressed as
(
v
x
,
v
y
,
v
z
)
(
r
,
v
,
t
)
. For most cases, however, we can
Search WWH ::
Custom Search