Environmental Engineering Reference
In-Depth Information
Fig. 5.5
Slender rod and points 0
,
A
1
,
B
Mathematically, we consider
⎧
⎨
u
t
τ
0
+
A
2
u
xx
, −
∞
<
u
tt
=
x
<
+
∞
,
0
<
t
,
K
,
x
∈
[
0
,
x
1
]
,
⎩
u
(
x
,
0
)=
u
t
(
x
,
0
)=
0
.
0
,
x
∈
[
0
,
x
1
]
,
By Eq. (5.49), we know that the temperature
u
(
x
,
t
)
depends only on the
K
value in
the domain of dependence.
Step 3.
Observe the variation of temperature at point
B
and record the time instant
t
0
when there is a reading of temperature at point
B
(
u
(
x
0
,
t
0
)
=
0, no matter how
small).
Step 4.
Calculate
A
)
t
0
using known
x
0
,
x
1
and
t
0
. This comes from the
fact that the characteristic curve
x
=(
−
x
0
x
1
=
x
0
+
A
(
t
−
t
0
)
must cross the
x
-axis at point
A
1
(Fig.5.5), Once
A
is available, we can obtain
τ
0
by Eq. (5.60).
5.3.6 Measuring
τ
0
by a Unit Impulsive Source
δ
(
x
−
x
0
,
t
−
t
0
)
The temperature due to a unit impulsive source
f
has been
found in Section 5.2 as Eq. (5.34). The temperature at
x
0
and
t
0
is, Eq. (5.35) in
Section 5.2,
(
x
,
t
)=
δ
(
x
−
x
0
,
t
−
t
0
)
ρ
1
2
c
u
(
x
0
,
t
0
)=
τ
0
.
k
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