Radial Heat Conduction in a Sphere

Radial Heat Conduction in a Sphere or Spherical Shell We next consider the steady radial conduction of heat in a sphere or spherical shell with volumetric heat production. The temperature distributions in thick planetary lithospheres, such as the lithospheres of the Moon and Mars, are properly described by solutions of the heat conduction equation in spherical geometry. The effects of spherical geometry are not so important for the Earth’s lithosphere, which is quite thin compared with the Earth’s radius. However, on a small body like the Moon, the lithosphere may be a substantial fraction of the planet’s radius. To describe heat conduction in spherical geometry, we must derive an energy balance equation.

Consider a spherical shell of thickness δr and inner radius r, as sketched in Figure 4–13. Assume that the conductive transport of heat occurs in a spherically symmetric manner. The total heat flow out of the shell through its outer surface is 4π(r + δr)2qr(r + δr), and the total heat flow into the shell at its inner surface is 4πr2qr(r).

The subscript r on the heat flux q indicates that the flow of heat is radial. Since δr is infinitesimal, we can expand qr(r + δr) in a Taylor series as qr(r + δr) = qr(r) + δr dqr dr + · · · . (4.34) Thus neglecting powers of δr, the net heat flow out of the spherical shell is 4.9 Radial Heat Conduction in a Sphere or Spherical Shell 261 given by 4π[(r + δr)2qr(r + δr) − r2qr(r)] = 4πr2 ( 2 r qr + dqr dr)

δr. (4.35)

If the net heat flow from the shell is nonzero, then, by conservation of energy, this flow of heat must be supplied by heat generated internally inthe shell (in steady state). With the rate of heat production per unit mass H, the total rate at which heat is produced in the spherical shell is 4πr2ρHδr, 4πr2δr being the approximate expression for the volume of the shell. By equating the rate of heat production to the net heat flow out of the spherical shell, Equation (4–35), we get dqr dr + 2qr r = ρH. (4.36)

The heat balance Equation (4–36) can be converted into an equation for the temperature by relating the radial heat flux qr to the radial temperature gradient dT/dr. Fourier’s law still applies in spherical geometry, qr = −kdT dr .