Environmental Engineering Reference
In-Depth Information
Chapter 6
Dual-Phase-Lagging Heat-Conduction
Equations
In this chapter we first develop the solution structure theorem for mixed problems of
dual-phase-lagging heat-conduction equations. We then apply it to solve the prob-
lems under some boundary conditions. We also develop the solution structure the-
orem for Cauchy problems and discuss the methods of solving Cauchy problems.
Finally we examine thermal waves and resonance and develop equivalence between
dual-phase-lagging heat conduction and heat conduction in two-phase systems.
6.1 Solution Structure Theorem for Mixed Problems
In this section we develop the solution structure theorem for mixed problems of
dual-phase-lagging heat-conduction equations
2
T
t
+
τ
q
1
α
∂
T
∂
+
τ
T
∂
∂
t
2
=
Δ
T
t
Δ
T
+
F
(
M
,
t
)
.
(6.1)
∂
α
∂
Here,
α
,
τ
q
and
τ
T
are all positive constants.
T
=
T
(
M
,
t
)
stands for the temperature
at spatial point
M
and time instant
t
.
6.1.1 Notes on Dual-Phase-Lagging Heat-Conduction Equations
For the benefit of developing the solution structure theorem and solutions and for
comparing with the results in Chapters 3- 5, we rewrite the dual-phase-lagging heat-
conduction equations (6.1) into
u
t
τ
B
2
∂
∂
A
2
0
+
u
tt
=
Δ
u
+
t
Δ
u
+
f
(
M
,
t
)
,
(6.2)
=
α
τ
=
ατ
T
τ
)=
α
τ
,
A
2
,
B
2
where
u
(
M
,
t
)
≡
T
(
M
,
t
)
,
f
(
M
,
t
F
(
M
,
t
)
and
τ
0
=
τ
q
.
q
q
q
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