Environmental Engineering Reference
In-Depth Information
The counterpart of PDS (5.99) in classical heat-conduction equations is
u t
a 2
=
Δ
u
(
r
,
t
) ,
0
<
r
< + ,
0
<
t
,
(5.102)
u r
(
0
,
t
)=
0
,
u
(
r
,
0
)=
1
.
After an even continuation and a function transformation v
(
r
,
t
)=
ru
(
r
,
t
)
,it
reduces to
v t =
a 2
Δ
v
(
r
,
t
) , − <
r
< + ,
0
<
t
,
(5.103)
v
(
r
,
0
)=
r
.
Its solution (also the solution of PDS (5.102)) is available in Chapter 3 as
+
+
2
2
4 a 2 t
e ( r ρ )
e ξ
1
2 a π
1
2 a π
u
(
r
,
t
)=
4 a 2 t
ρ
d
ρ =
( ξ
r
)
d
ξ
tr
tr
+
+
2
4 a 2 t d
e ξ
1
2 a π
1
π
2
e η
=
ξ =
d
η =
1
.
t
Therefore, this property is also preserved by the hyperbolic heat-conduction
equation.
2. By Eq. (5.98), the solution of
u t τ 0 +
) τ 0
A 2
u tt =
Δ
u
(
r
,
t
)+
f
(
r
,
t
,
0
<
r
< + ,
0
<
t
,
u r (
0
,
t
)=
0
,
(5.104)
u
(
r
,
0
)=
0
,
u t (
r
,
0
)=
0
is
t
u
=
W f τ (
r
,
t
τ )
d
τ
0
I 0 b
2
t
r + A ( t τ )
t
τ
2
1
2 A
e
2
=
τ 0 d
τ
(
At
)
(
r
ρ )
ρ
f
( ρ , τ )
d
ρ .
τ 0 r
0
r
A
(
t
τ )
(5.105)
When f
(
r
,
t
)= δ (
r
r 0 ,
t
t 0 )
,wehave
τ 0 r I 0 b A 2
2 e
r 0
2 A
t t 0
2
2
u
(
r
,
t
)=
(
t
t 0 )
(
r
r 0 )
τ 0
,
(5.106)
t 0 ) τ 0 ,thetem-
which shows that under the effect of a point source
δ (
r
r 0 ,
t
perature is higher inside the spherical surface r
=
r 0 than that outside the surface,
and is relatively lower on the surface r
=
r 0 .Furthermore,
ρ
1
2
c
lim
r 0 u
(
r
,
t
)= ,
lim
r
u
(
r
,
t
)=
0
,
lim
r
u
(
r
,
t
)=
0 .
(5.107)
k
τ
r 0
t
t 0
 
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