Geology Reference
In-Depth Information
The boundary conditions for ™ arise immedi-
ately from those for
T
.Wehavethat™(
z
,
t
)
D
0for
t
0and™(0,
t
)
D
1for
t
> 0. Furthermore:
d
2
™
dǜ
2
C
2ǜ
d™
dǜ
D
0
(12.34)
With respect to ǜ, the boundary conditions
assume the form:
lim
z
!1
™.
z
;t/
D
0 for any t>0
(12.29)
lim
ǜ!1
™.ǜ/
D
0
I
™.0/
D
1
(12.35)
A solution by similarity is based on the idea
that two solutions of (
12.28
) for different times
should have a “similar” spatial distribution of
temperatures. Therefore,
z
and
t
must appear in
a particular combination, just like solutions of
the plane wave equation require that distance
x
and time
t
always appear in the combination
t
-
x
/
v
,
v
being the wave velocity. In the case
of
th
e diffusion equation, the quantity
L
(
t
)
D
p
›t represents a characteristic
thermal diffusion
distance
, thereby it is reasonable to assume that if
™
D
™(
z
,
t
) is a solution to (
12.28
), then it
wi
ll be a
Now let us set:
d™
dǜ
¥
(12.36)
Substituting into (
12.34
) reduces this equation
to a first-order differential equation:
d¥
dǜ
C
2ǜ¥
D
0
(12.37)
function of the dimensionless ratio
z
=
p
›t rather
than of an arbitrary combination of the variables
z
and
t
. To simplify the results, it is convenient
to define a dimensionless
similarity variable
ǜ as
follows:
The solution is:
¥.ǜ/
D
ae
ǜ
2
(12.38)
Therefore, the general solution for ™ is:
z
2
p
›t
Z
ǜ.
z
;t/
(12.30)
e
2
d
C
b
™.ǜ/
D
a
(12.39)
0
Formally, the similarity implies that tempera-
ture distribution at time
t
can be obtained from the
distribution at time
t
0
by stretching the distance
z
by the square root
of
t
0
/
t
. In fact, the transfo
rma
-
Finally, substituting the boundary conditions
allows to express the solution to (
12.34
)interms
of error function:
tion
z
!
z
0
D
z
p
t
0
=t gives ǜ
0
D
z
0
=
2
p
›t
0
D
z
=
p
t
0
=t=
2
p
›t
0
Z
ǜ
D
z
=
2
p
›t
D
ǜ.Now
let us rewrite the diffusion equation in terms of ǜ.
By the chain rule, we have:
2
p
e
2
d
1
erf
.ǜ/
D
erfc
.ǜ/
™.ǜ/
D
1
0
(12.40)
This solution can be easily converted into
@
™
@t
D
d™
dǜ
@ǜ
@t
D
ǜ
2t
d™
dǜ
(12.31)
a
temperature
distribution
using
(
12.27
)and
(
12.30
). We have:
T.
z
;t/
D
.T
0
T
i
/
erfc
z
@
™
@
z
D
d™
dǜ
@ǜ
@
z
D
1
2
p
›t
d™
dǜ
(12.32)
2
p
›t
C
T
i
@
2
™
@
z
2
D
d
2
™
dǜ
2
d
2
™
dǜ
2
1
2
p
›t
@ǜ
@
z
D
1
4›t
(12.41)
(12.33)
Aplotof
T
vs
z
isshowninFig.
12.7
for
a sudden increase of the surface temperature
(
T
0
>
T
i
). The near-surface region where the
Substituting into (
12.28
) gives the following
ordinary linear differential equation:
