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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