Environmental Engineering Reference
In-Depth Information
where
is the chemical potential, k B is the Boltzmann constant and T is the tem-
perature. For the phonon, f 0 is the Bose-Einstein equilibrium distribution
μ
1
f 0 ( ε )=
exp
1 ,
(1.67)
ε
k B T
where
is the angular frequency of the quantum harmonic oscillator.
Obviously, for both cases, f 0 is an even function of the velocity v . Therefore (Xu
and Wang 2005)
ε =
h
ω
,and
ω
f 0
ε
D
( ε )
v d
ε =
0
.
(1.68)
ε
Multiplying Eq. (1.65) by
ε
D
( ε )
v and integrating over all possible energies
yields
ε τ
v
·
f
(
r
, ε (
v
) ,
t
+ τ T )
v
ε
D
( ε )
d
ε +
q
(
r
,
v
,
t
+ τ )=
0
,
(1.69)
in which we have used Eq. (1.68).
Under the assumption that the relaxation times
τ T are independent of
the system energy and the system has achieved a quasi-equilibrium state, we have
τ 0 and
f
=(
d f 0 /
d T
)
T . Eq. (1.69) becomes
q
(
r
,
t
+ τ 0 )=
k
·
T
(
r
,
t
+ τ T ) ,
(1.70)
where k is the thermal conductivity tensor
vv d f 0
k
=
τ
d T ε
D
( ε )
d D
( ε ) .
For isotropic materials, k
k I with k and I being a constant and the unit tensor,
respectively. Eq. (1.70) reduces to
=
(
,
+ τ
)=
(
,
+ τ
) ,
q
r
t
k
T
r
t
(1.71)
0
T
which is the dual-phase-lagging constitutive relation [Eq. (1.32)].
1.3.3 Three Types of Heat-Conduction Equations
Heat-conduction equations come from the application of the first law of thermo-
dynamics (also called the conservation of energy) to heat conduction. By the ap-
proaches for developing equations of mathematical physics outlined in Sect. 1.2.2
 
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