Geoscience Reference
In-Depth Information
Figure 7.5.
The error
function erf(
x
) and
complementary error
function erfc(
x
).
1.0
0.8
erf(x)
0.6
0.4
erfc(x)
0.2
0.0
3
0
1
2
The differential equation to be solved is Eq. (7.13) with
A
=
0, the
diffusion
equation
:
2
T
∂
T
∂
t
=
κ
∂
(7.33)
∂
z
2
where
c
P
)isthe thermal diffusivity.
Derivation of the solution to this problem is beyond the scope of this topic,
and the interested reader is referred to Carslaw and Jaeger (1959), Chapter 2,or
Turcotte and Schubert (2002), Chapter 4. Here we merely state that the solution
of this equation which satisfies the boundary conditions is given by an
error
function
(Fig. 7.5 and Appendix 5):
κ
=
k
/
(
ρ
T
0
erf
z
2
√
κ
T
=
(7.34)
t
The error function is defined by
x
2
√
π
e
−
y
2
d
y
erf (
x
)
=
(7.35)
0
Yo u can check that Eq. (7.34)isasolution to Eq. (7.33)bydifferentiating with
respect to
t
and
z
. Equation (7.34) shows that the time taken to reach a given
temperature is proportional to
z
2
and inversely proportional to
.
The temperature gradient is given by differentiating Eq. (7.34) with respect
to
z
:
κ
T
0
erf
∂
T
∂
z
=
∂
∂
z
z
2
√
κ
t
2
√
π
1
2
√
κ
t
e
−
z
2
/
(4
κ
t
)
=
T
0
T
0
√
πκ
t
e
−
z
2
/
(4
κ
t
)
=
(7.36)
This error-function solution to the heat-conduction equation can be applied
to many geological situations. For solutions to these problems, and numerous
others, the reader is again referred to Carslaw and Jaeger (1959).
