Environmental Engineering Reference
In-Depth Information
Therefore the system is always at overdamped oscillation if
τ T > τ 0 . Consequently,
there are no thermal waves.
ζ m =
1
.
0 separates the underdamped modes from the overdamped modes. Ap-
plying
ζ m <
1 in Eq. (6.220) yields the region of m where the underdamped modes
can occur
1
1
C 1 <
m
<
C 2 ,
if
τ 0 > τ T >
0 ;
(6.238)
π
π
1
π
m
>
C ,
if
τ 0 > τ T =
0
.
(6.239)
Hence C 1 and C 2 are the relaxation distances defined by (Xu and Wang 2002)
C 1 = ατ T τ 0
τ
τ 0
τ
1
T +
T
,
C 2 = ατ T τ 0
τ 0
τ T
1
τ T
,
2 ατ 0 .
C
=
1
1
Therefore, thermal oscillation occurs only for the modes between
and
π
C 1
π
C 2
for the case of
0. This is different from thermal waves in hyperbolic
heat conduction where oscillation always appears for the high order modes (Tzou
1992a).
The behavior of an individual temperature mode discussed above also represents
the entire thermal response if
τ 0 > τ T
>
A sin m
π
x
B sin m
π
x
φ (
x
)=
and
ψ (
x
)=
with A and B
l
l
as constants. In general, a change
ΔΓ m (
t
)
in the m -th mode would lead to a change
sin m π x
l
ΔΓ m (
t
)
in T
(
x , t
)
because
m = 1 Γ m ( t ) sin m π x
T
(
x
,
t
)=
.
l
6.10.2 Resonance
For dual-phase-lagging heat conduction, the amplitude of the thermal wave may
become exaggerated if the oscillating frequency of an externally applied heat source
is at the resonance frequency.
Consider a heat source in the system (6.209) in the form of
e t
F
(
x
,
t
)=
Qg
(
x
)
.
Here Q , independent of x and t , is the strength, g
(
x
)
is the spanwise distribution,
and
Ω
is the oscillating frequency. Expand T
(
x , t
)
and g
(
x
)
by the Fourier sine
 
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