Page 151 - Physical Principles of Sedimentary Basin Analysis

6.8 Stationary heat flow in a sphere ∗ 133

dr
r
q(r) q(r + dr)

Figure 6.13. Heat flow through a spherical shell.

Energy conservation (6.118) in a spherical geometry can also be written as

1 d
2
r q(r) = S(r) (6.119)
2
r dr
which is verified by carrying out the differentiation. The radial heat flow q is related to the
radial temperature gradient by Fourier’s law
dT
q =−λ(r) , (6.120)
dr
where the heat conductivity is assumed to be a function of r. Inserting Fourier’s law
into expression (6.119) for energy conservation gives the temperature equation for radial
heat flow
1 d 2 dT
r λ(r) =−S(r). (6.121)
2
r dr dr
The temperature equation (6.121) is straightforward to integrate when both the heat con-
ductivity λ and the heat production per unit volume S are constants. Two integrations then
yield

1 S 2 c 1
T (r) =− r − + c 2 (6.122)
6 λ r
where c 1 and c 2 are integration constants that will be found from boundary conditions.
One boundary condition is that dT/dr = 0 at the center of the sphere (r = 0). We see
that the only way to have a finite temperature gradient at the center is to let c 1 = 0.
The other boundary condition is to let T = T 0 be the temperature at the surface, r = r 0 .
The temperature solution is then

1

2 2
S
T (r) = r − r + T 0 (6.123)
0
6 λ