Problems and Solutions on Thermodynamics and Statistical Mechanics

154 Problems €4 Sdutio~ on Thermodynom'ca €4 Statiaticd Mechanics

dz dz dz

dt dt dt

is pL-. Hence we obtain the equation pL- = -j, giving q = - =

-j/pL = n(T - To)/pLz.

(b) The above expression can be written as

dt = pL zdz.

n(T - To)

t = pL(~i - zf)/2n(T - To).

If we take z1 = 1 cm and 22 = 2 cm, then At = 1.2 x lo3 s = 20 min.

1156

Consider a spherical black asteroid (made of rock) which has been

ejected from the solar system, so that the radiation from the sun no longer

has a significant effect on the temperature of the asteroid. Radioactive ele-

ments produce heat uniformly inside the asteroid at a rate of q = 3 x

cal/g.sec. The density of the rock is p = 3.5 g/cm3, and the thermal con-

ductivity is k = 5 x The radius of the asteroid is

R = 100 km. Determine the central temperature T, and the surface tem-

perature T,, of the asteroid assuming that a steady state has been achieved.

(UC, Berkeley)

cal/deg.cmsec.

Solution:

The surface temperature satisfies

4rR3

4rR20T,4 = Q = - 3 PQ 1

80

T, = (z)' = 22.5 K.

The equation of heat conduction inside the asteroid is

V. (-kVT) = Qp.

Using spherical coordinates, we have