Environmental Engineering Reference
In-Depth Information
properties are assumed to be constant (see Section 1.3.3 for the derivation),
F
2 T
1
α
T
t + τ 0
+ τ T
1
k
+ τ 0
F
t 2 = Δ
T
t ( Δ
T
)+
(1.36)
α
t
This equation is parabolic when
τ 0
< τ
T (Wang and Zhou2000, 2001). Al-
2 T
2 t exists in the equation, the mixed derivative
( τ
/ α )
/
though a wave term
0
( Δ
)
τ
t completely destroys the wave structure. The equation, in this case,
therefore predicts a nonwave-like heat conduction that differs from the usual dif-
fusion predicted by the classical parabolic heat conduction (1.24). When
T
/
T
τ 0 > τ T ,
however, Eq. (1.36) can be approximated by Eq. (1.26) and then predominantly pre-
dicts wave-like thermal signals.
The dual-phase-lagging heat-conduction equation (1.36) forms a generalized,
unified equation that reduces to the classical parabolic heat-conduction equation
when
0,the
energy equation in the phonon scattering model (Joseph and Preziosi 1989, Guyer
τ T = τ 0 , the hyperbolic heat-conduction equation when
τ T =
0and
τ 0 >
α = τ R c 2
3
9
5 τ
R , and the energy equa-
tion in the phonon-electron interaction model (Kaganov et al. 1957, Anisimòv et al.
and Krumhansi 1966) when
,
τ
=
N and
τ 0 = τ
T
1
c e +
1
k
c e +
c l
G and
1
G
1
c l
1974, Qiu and Tien 1993) when
.In
the phonon scattering model, c is the average speed of phonons (sound speed),
α =
,
τ T =
τ 0 =
c l
τ R
is the relaxation time for the umklapp process in which momentum is lost from the
phonon system, and
τ N isthe relaxation time for normal processes in which momen-
tum is conserved in the phonon system. In the phonon-electron interaction model,
k is the thermal conductivity of the electron gas, G is the phonon-electron coupling
factor, and c e and c l are theheat capacity of the electron gas and the metal lattice,
respectively. This, together with its success in describing and predicting phenomena
such as ultra-fast pulse-laser heating, propagation of temperature pulses in super-
fluid liquid helium, nonhomogeneous lagging response in porous media, thermal
lagging in amorphous materials, and effects of material defects and thermomechan-
ical coupling, heat conduction in nanofluids, bi-composite media and two-phase
systems (Tzou 1997, Tzou and Zhang 1995, Vadasz 2005a, 2005b, 2005c, 2006a,
2006b, Wang et al. 2007a, Wang and Wei 2007a, 2007b), has given rise to the re-
search effort on various aspects of dual-phase-lagging heat conduction (Tzou 1997,
Wang and Zhou 2000, 2001).
The dual-phase-lagging heat-conduction model that is based on Eqs. (1.36) has
been shown to be well-posed in a finite region of n -dimensions ( n
1) under any
linear boundary conditions including Dirichlet, Neumann and Robin types (Wang
and Xu 2002, Wang et al. 2001). Solutions of one-dimensional (1D) heat conduc-
tion has been obtained for some specific initial and boundary conditions by Antaki
(1998), Dai and Nassar (1999), Lin et al. (1997), Tang and Araki (1999), Tzou
(1995a, 1995b, 1997), Tzou and Zhang (1995), Tzou and Chiu (2001). Wang and
Zhou (2000, 2001) obtained analytical solutions for regular 1D, 2D and 3D heat-
conduction domains under essentially arbitrary initial and boundary conditions. The
solution structure theorems were also developed for both mixed and Cauchy prob-
 
Search WWH ::




Custom Search