Environmental Engineering Reference
In-Depth Information
Thus, we find the integral
I
0
b
A
2
2
d
t
x
+
A
(
t
−
τ
)
1
2
A
t
−
τ
2τ
0
d
e
−
τ
(
t
−
τ
)
2
−
(
x
−
ξ
)
ξ
0
x
−
A
(
t
−
τ
)
(5.54)
t
τ
0
e
−
0
=
τ
0
t
+
τ
(
−
1
)
.
T
−
1
and
T
−
2
The unit of 1 in
u
(
x
,
0
)=
1,
u
t
(
x
,
0
)=
1and
f
(
x
,
t
)=
1is
Θ
,
Θ
Θ
,
respectively.
5.3.4 Physics and Measurement of
τ
0
1. The non-zero
0
u
tt
in the hyperbolic heat-conduction
equations compared with the classical heat-conduction equations. Since
τ
0
yields an additional term
τ
[
u
t
]=
[
τ
0
must be a time constant. While the hyperbolic heat-conduction equa-
tion is a better representation of real heat conduction processes, it has the same
fundamental properties as the classical heat-conduction equation. The solution
of PDS (5.48) is, for example,
u
0
u
tt
]
,
τ
1 which makes sense physically. For the
classical heat-conduction equation, we also have
(
x
,
t
)
≡
d
+
∞
+
∞
e
−
t
2
e
−
(
x
−
ξ
)
2
ξ
2
a
√
π
ξ
2
a
√
π
1
2
a
√
π
2
√
π
u
(
x
,
t
)=
4
a
2
t
d
ξ
=
=
1
.
t
t
−
∞
0
2. By Eq. (5.46), we have
lim
u
=
τ
0
.
(5.55)
t
→
+
∞
Therefore, the
τ
0
is equal to the temperature value of an infinite rod as
t
→
+
∞
and due to
u
t
(
x
,
0
)=
1. We may thus measure
τ
0
by measuring the temperature
t
τ
0
decays very quickly
. The fact that e
−
of the rod due to
u
t
(
x
,
0
)=
1at
t
→
+
∞
as
t
,
t
→
+
∞
necessitates using only relatively large value of
t
.Thevalueof
τ
0
can also be obtained from Eq. (5.46) by measuring the temperature at different
time instant.
3. The
u
t
and
u
tt
can be regarded as the velocity and acceleration of thermal waves,
respectively. By Eq. (5.46), we have
t
τ
0
e
−
u
t
=
(5.56)
and
1
τ
t
τ
0
e
−
u
tt
=
−
.
(5.57)
0
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