Environmental Engineering Reference
In-Depth Information
In Eq. (5.78),
2
2
1
(
x
−
x
0
)
+(
y
−
y
0
)
c
τ
0
,
lim
t
→
+
∞
=
0
,
2
A
2
(
t
−
t
0
)
2
A
2
t
−
t
0
2τ
0
ch
c
A
e
−
lim
t
→
+
∞
(
t
−
t
0
)
2
−
(
x
−
x
0
)
2
−
(
y
−
y
0
)
2
O
e
−
2τ
0
1
2
+
t
−
t
0
=
(
t
→
+
∞
)
.
Thus
O
e
−
τ
0
t
−
t
0
2
1
2
+
1
lim
t
→
+
∞
u
2
(
x
,
y
,
t
)=
lim
t
→
+
∞
A
2
2
=
0
.
2
π
A
τ
0
(
−
)
2
−
(
−
)
2
−
(
−
)
t
t
0
x
x
0
y
y
0
This is also physically correct because the temperature at any point due to
f
(
x
,
y
,
t
)=
δ
(
x
−
x
0
,
y
−
y
0
,
t
−
t
0
)
should decay as
t
approaches
∞
.
3. When
x
0
=
y
0
=
0, both
u
1
(
x
,
y
,
t
)
and
u
2
(
x
,
y
,
t
)
are even functions of
x
and
y
so that they are symmetric around the origin
. This is again physically
grounded because both
u
1
and
u
2
come exclusively from
f
(
0
,
0
)
(
x
,
y
,
t
)=
δ
(
x
−
0
,
y
−
0
,
t
−
t
0
)
and because diffusion is direction-independent.
5.6.3 Different Properties
1. Solution (5.77) indicates that
u
1
(
x
,
y
,
t
)
=
0 no matter how far point
(
x
,
y
)
is from
)
can thus propagate over a very long distance in a very short time period, thus the
speed of temperature propagation is infinite. This is clearly physically impossi-
ble.
(
,
)
(
−
)
(
,
the source
x
0
y
0
andnomatterhowshort
t
t
0
is. The disturbance at
x
0
y
0
However, this is not the case for solution (5.78). When point
(
x
,
y
)
is suffi-
ci
ently far away from
source
(
x
0
,
y
0
)
and
(
t
−
t
0
)
is sufficiently short such that
(
)=
0. This implies a finite speed of temperature propagation. Thus the hyperbolic
heat-conduction equation is a better representation of a real heat conduction pro-
cess.
x
−
x
0
)
2
+(
y
−
y
0
)
2
>
A
(
t
−
t
0
)
for
M
(
x
,
y
,
t
)
outside of
Ω
P
0
,then
u
2
(
x
,
y
,
t
While the classical heat-conduction equation assumes an infinite speed,
u
1
(
x
,
y
,
t
)
2
2
and a small
will quickly decay to zero for a large
(
x
−
x
0
)
+(
y
−
y
0
)
(
t
−
t
0
)
.
This is the rationale behind its wide applications.
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