Environmental Engineering Reference
In-Depth Information
The former shows that velocity decays exponentially with increasing
t
, with a
decaying constant
1
τ
0
)
(
−
. The latter yields
1
τ
0
lim
t
0
u
tt
=
−
.
(5.58)
→
Therefore, the decaying constant of the velocity is equal, in its value, to the ini-
tial value of the acceleration of thermal waves. This leads to another method of
measuring
τ
0
simply by measuring the initial value of
u
tt
. Under the CDS in
PDS (5.40), we have, by Eqs. (5.55) -(5.58),
u
|
+
∞
·
u
tt
|
t
=
0
=
−
1and
u
t
=
−
τ
0
u
tt
.
4. By Eq. (5.53), we obtain
0
e
−
1
t
τ
0
2
lim
u
=
lim
τ
0
t
+
τ
−
=+
∞
.
t
→
+
∞
t
→
+
∞
Therefore, the temperature at all points increases as
t
tends to infinity. This is
similar to mechanics, where the PDS governing the displacement
s
(
t
)
s
(
t
)=
1
,
(5.59)
s
(
s
(
0
)=
0
)=
0
is similar to PDS (5.51) in hyperbolic heat conduction. The solution of PDS (5.59)
is
s
1
2
t
2
. It also increases as
t
tends to infinity, but with a different speed.
=
5.3.5 Measuring
τ
0
by Characteristic Curves
a
√
τ
0
, we obtain
By the definition of thermal wave speed
A
=
a
2
A
2
=
1
A
2
τ
0
=
k
c
,
(5.60)
ρ
where
k
,
ρ
and
c
are thermal conductivity, density and specific heat, respectively.
and
c
are normally taken as material constants for most materials, we can
obtain values of
Since
k
,
ρ
τ
0
of different materials by measuring
A
. The steps of measuring
A
can be summarized as follows.
Step 1.
Mark three points
O
,
A
1
and
B
on a slender rod (Fig. 5.5). The distances
of
A
1
and
B
from
O
are denoted by
x
1
and
x
0
, respectively. Here point
B
is the test
point.
Step 2.
Apply an initial temperature
u
(
x
,
0
)=
K
to the part of bar between
O
and
A
1
,i.e.in
.Here
K
is a positive constant. The larger the
K
, the higher the
measuring accuracy.
[
0
,
x
1
]
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