Heat - The Solid Earth - page 322

18. Assume that the Earth is solid and that all heat transfer is by conduction. What value
of internal heat generation distributed uniformly throughout the Earth is necessary to
account for the Earth's mean surface heat flow of 87 × 10 − 3 Wm − 2 ?How does this
value compare with the actual estimated values for the crust and mantle?
19. Calculate the rate at which heat is produced in (a) the crust and (b) the mantle. Assume
that the crust is 10 km thick and that the volumetric heat-generation rates are 1.5 ×
10 − 6 Wm − 3 in the crust and 1.5 × 10 − 8 Wm − 3 in the mantle.
20. Calculate the steady-state surface heat flow for a model solid Earth with the following
constant thermal properties: k ,4Wm − 1 ◦ C − 1 ; and A ,2 × 10 − 8 Wm − 3 .
21. It takes about 4 min to boil a hen's egg of mass 60 g to make it edible for most people.
Forhow long would it be advisable to boil an ostrich egg weighing about 1.4 kg?
(From Thompson (1987).)
22. (a) Calculate the conductive characteristic time for the whole Earth.
(b) Calculate the thickness of the layer that has a characteristic time of 4500 Ma.
(c) Comment on your answers to (a) and (b).
23. (a) A sphere has radius r and uniform density ρ . What is the gravitational energy
released by bringing material from infinitely far away and adding a spherical
shell, of density r ,tothe original shell?

(b) By integrating the expression for the gravitational energy over r from 0 to R ,
calculate the gravitational energy released in assembling a sphere of density

ρ and thickness

ρ

and radius R .
(c) Use the result of (b) to estimate the gravitational energy released as a result of the
accretion of the Earth.
(d) Assume that all the energy calculated in (c) became heat and estimate the rise in
temperature of the primaeval Earth. Comment on your answer.

References and bibliography
Alfe, D., Gillan, J. and Price, G. D. 1999. The melting curve of iron at the pressures of the
Earth's core from ab initio calculations. Nature , 401 , 462-4.
Batt, G. E. and Brandon, M. T. 2002. Lateral thinking: 2-D interpretation of
thermochronology in convergent orogenic settings. Tectonophysics , 349 , 185-201.
Bloxham, J. and Gubbins, D. 1987. Thermal core-mantle interactions. Nature , 325 , 511-13.
Bott, M. H. P. 1982. The Interior of the Earth: Its Structure, Composition and Evolution .
Amsterdam: Elsevier.
Bukowinski, M. S. T. 1999. Taking the core temperature. Nature , 401 , 432-3.
Carlson, R. L. and Johnson, H. P. 1994. On modeling the thermal evolution of the oceanic
upper mantle: an assessment of the cooling plate model. J. Geophys. Res. , 99 , 3201-14.
Carslaw, H. S. and Jaeger, J. C. 1959. Conduction of Heat in Solids , 2nd edn. New York:
Oxford University Press.
Celerier, B. 1988. Paleobathymetry and geodynamic models for subsidence. Palaios , 3 ,
454-63.
Clark, S. P. 1966. Thermal conductivity. In S. P. Clark, ed., Handbook of Physical Constants .
Vol. 97 of Memoirs of the Geological Society of America. Boulder, Colorado:
Geological Society of America, pp. 459-82.