Page 153 - Physical Principles of Sedimentary Basin Analysis

6.9 Transient cooling of a sphere ∗ 135
requires the heat flow at the center of the sphere to be zero.) An initial condition is also
needed before the temperature equation can be solved, and it is a constant temperature
T (r, t=0) = T 0 throughout the sphere at t = 0.
r
We introduce the dimensionless radius ˆ = r/r 0 and the dimensionless temperature
ˆ
T = T/T 0 , where both dimensionless variables are numbers between 0 and 1. A step
towards a dimensionless temperature equation (6.128)is

2
r b c b ∂T ˆ 1 ∂ ∂T ˆ
0 2
− ˆ r λ = 0, (6.129)
2
r
λ ∂t ˆ r ∂ ˆr ∂ ˆ
where the coefficient
2
r b c b
0
t 0 = (6.130)
λ
has units of time. We will see that the time t 0 characterizes the temperature transient, and
that it is the natural choice for a quantity to scale time. Dimensionless time is therefore
ˆ t = t/t 0 and the dimensionless temperature equation becomes

∂T ˆ 1 ∂ 2 ∂T ˆ
− ˆ r λ = 0, (6.131)
2
r
∂ ˆ t ˆ r ∂ ˆr ∂ ˆ
which has no (explicit) parameters. The boundary conditions in dimensionless form are
∂T ˆ
ˆ
T (ˆ=1, ˆ t) = 0 and (ˆ=0, ˆ t) = 0 (6.132)
r
r
r
∂ ˆ
and the dimensionless initial condition is
T (ˆ, ˆ t=0) = 1. (6.133)
r
ˆ
The solution of the temperature equation (6.137) is the Fourier series
∞

2
ˆ
r
r
r
T (ˆ, ˆ t) =ˆ a n sin(nπ ˆ) exp −(nπ) ˆ t (6.134)
n 1
as shown in Note 6.3, where the Fourier coefficients are
2 (−1) n+1
a n = . (6.135)
nπ
Figure 6.14 shows the solution (6.134) for the time steps 0.001, 0.02, 0.04, 0.08, 0.16 and
∞, and how the unit sphere cools down from the initial temperature 1 to the stationary
temperature 0.
We see that the terms in the Fourier series decay to zero with a half-life ˆ t 1/2 =
2
ln2/(nπ) . The first term (n = 1) in the series has the longest half-life, and the half-lives
then decrease with decreasing n. It is also seen that the absolute value of the Fourier coef-
ficients decrease as |a n |∼ 1/n with increasing n. The first term, with the largest absolute
value and the longest half-life, can be used to estimate the half-life of the cooling transient