Environmental Engineering Reference
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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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