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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