Heat - The Solid Earth - page 320

promotes metamorphic reactions and has a major impact on the thermodynamics
of re-equilibration.

Problems

1. Estimate Table 7.2 for 4000 and 4500 Ma. Discuss the implications of the rela-
tive heat-generation values in this table, particularly with respect to the Archaean
Earth.
2. Calculate the phase difference between the daily and annual surface temperature vari-
ations that would be measured at depths of 2 and 5 m in a sandstone.
3. Taking T 0 as 40 ◦ C for the annual variation in surface temperature, calculate the depth
at which the variation is 5 ◦ C. What is the phase difference (in weeks) between the
surface and this depth?
4. Calculate an equilibrium geotherm for 0 ≤ z ≤ d from the one-dimensional heat-flow
equation, given the following boundary conditions:
(i) T = 0at z = 0 and
(ii) T d .

Assume that there is no internal heat generation.
5. Calculate an equilibrium geotherm from the one-dimensional heat-flow equation given
the following boundary conditions:
(i) ∂ T /∂ z = 30 ◦ Ckm − 1 at z = 0km and
(ii) T = 700 ◦ Cat z = 35 km.
Assume that the internal heat generation is 1 Wm − 3 and the thermal conductivity is
3Wm − 1 ◦ C − 1 .
6. On missions to Venus the surface temperature was measured to be 740 K, and at
three sites heat-producing elements were measured (in percentage of total volume)
as follows (ppm, parts per million).

= T d at z =

Venera 8 Venera 9 Venera 10

K (%) 0.47 ± 0.08 0.30 ± 0.16 4 ± 1.2

U (ppm) 0.60 ± 0.16 0.46 ± 0.26 2.2 ± 0.2

Th (ppm) 3.65 ± 0.42 0.70 ± 0.34 6.5 ± 2

The density of the Venusian crust can be taken, from a measurement by Venera
9, to be 2.8 × 10 3 kg m − 3 . Calculate the heat generation in Wm − 3 at each site.

(From Nisbet and Fowler (1982).)
7. Using the one-dimensional equilibrium heat-conduction equation, calculate and plot
the Venus geotherms (Aphroditotherms) of Problem 6 down to 50 km depth at each
site. Assume that the conductivity is 2.5 W m − 1 ◦ C − 1 (a typical value for silicates) and
that, at a depth of 50 km, the heat flow from the mantle and deep lithosphere of Venus
is 21 × 10 − 3 Wm − 2 . What have you assumed in making this calculation? What do
these Aphroditotherms suggest about the internal structure of the planet?