Environmental Engineering Reference
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lems of dual-phase-lagging heat-conduction equations (Wang and Zhou 2000, Wang
et al. 2001) by extending those theorems for hyperbolic heat conduction (Wang
2000a). These theorems build relationships between the contributions (to the tem-
perature field) by the initial temperature distribution, the source term and the ini-
tial time-rate of the temperature change, uncovering the structure of the tempera-
ture field and considerably simplifying the development of solutions. Xu and Wang
(2002) addressed thermal features of dual-phase-lagging heat conduction (particu-
larly conditions and features of thermal oscillation and resonance and their contrast
with those of classical and hyperbolic heat conduction).
An experimental procedure for determining the value of
0
has been proposed
by Mengi and Turhan (1978). The general problem of measuring short-time thermal
transport effects has been discussed by Chester (1966).Wang and Zhou (2000, 2001)
developed three methods of measuring
τ
0
. Tzou (1997) and Vadasz (2005a, 2005b,
2006a, 2006b) developed an
approximate
equivalence between Fourier heat conduc-
tion in porous media and dual-phase-lagging heat conduction, and applied the latter
to examine features of the former. Based on that equivalence, Vadasz (2005a, 2005b,
2005c, 2006a, 2006b) showed that
τ
τ
0
in porous-media heat
conduction so that thermal waves cannot occur according to the necessary condi-
tion for thermal waves in dual-phase-lagging heat conduction (Xu and Wang 2002).
However, such waves are observed in casting sand experiments by two independent
groups (Tzou 1997). In an attempt to resolve this difference and to build the intrin-
sic relationship between the two heat-conduction processes, Wang and Wei (2007a,
2007b) developed an
exact
equivalence between dual-phase-lagging heat conduc-
tion and Fourier heat conduction in two-phase systems subject to a lack of local
thermal equilibrium. Based on this new equivalence, Wang and Wei (2007a, 2007b)
also show the possibility of and uncover the mechanism responsible for the thermal
oscillation in two-phase-system heat conduction.
Tzou (1995b, 1997) also generalized Eq. (1.35), for
τ
T
is always larger than
τ
0
>>
τ
T
, by retaining terms
up to the second order in
τ
0
but only the term of the first order in
τ
T
in the Taylor
expansions of Eq. (1.32) to obtain a
τ
0
-second-order dual-phase-lagging model
k
2
q
+
τ
0
∂
q
1
2
τ
0
∂
+
τ
T
∂
∂
2
q
t
+
t
2
=
−
∇
T
t
(
∇
T
)
.
(1.37)
∂
∂
For this case the dual-phase-lagging heat conduction Eq. (1.36) is generalized into
F
t
2
2
T
0
3
T
2
t
3
=
Δ
0
2
F
∂
1
α
∂
T
t
+
τ
0
∂
t
2
+
τ
α
+
τ
T
∂
∂
1
k
+
τ
0
∂
F
t
+
τ
2
∂
T
t
(
Δ
T
)+
,
∂
α
∂
2
α
∂
(1.38)
whis is of hyperbolic type and thus predicts thermal wave propagation with a finite
speed (Tzou 1995b, 1997)
2
k
1
τ
0
τ
T
=
.
V
T
(1.39)
ρ
c
The thermal wave from Eq. (1.26) is obviously different from that in Eq. (1.38).
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