Page 37 - Geothermal Energy Renewable Energy and The Environment

20 Geothermal Energy: Renewable Energy and the Environment

Table 2.4
dynamic Viscosities of Geological and common materials
o
material Temperature ( c) Viscosity (pa-s)
Water 20 .001
Honey 20 10.0
Tar 20 30,000
Molten rhyolite a ~1400°C ~3.55 × 10 11
Upper mantle b ~1000°C ~1 × 10 19
Lower mantle c ~3500°C ~1 × 10 to ~3 × 10 22
21

a Webb, S. L and Dingwell, D. B., Journal of Geophysical Research, 95, 15695–701 1990.
b Hirth, G. and Kohlstedt, D., Geophysical Monograph, Washington, DC: American
Geophysical Union, 138:83–105, 2003.
c Yamazaki, D. and Karato, S.-I., American Mineralogist, 86:385–91, 2001.

Qualitatively, the physical conditions that favor the onset of convection are low viscosity, significant
thermal expansion, the presence of gravity (necessary for buoyancy effects), and low thermal con-
ductivity (or large distance between heat source and heat sink). In addition, the temperature increase
between heat source and sink must be greater than the temperature increase that would normally occur
simply as a result of increasing pressure with depth (the “adiabatic gradient”; see Chapter 3).
The conditions that favor convective flow can be quantitatively represented by the ratio

3
R = (g × α × ∇T × d )/(ν × κ). (2.9)
a
This ratio evaluates the relative magnitudes of buoyancy forces and viscous forces. In this equa-
tion, g is the acceleration of gravity (9.8 m × s ), ∇T is the vertical temperature gradient (units of
–2
K), α is the coefficient of thermal expansion (units are 1/K), d is the depth interval (in meters) over
which the temperature difference occurs, ν is the kinematic viscosity (m × s ) and κ is the thermal
–1
2
diffusivity (m × s ). The R is a dimensionless number; that is, it has no units. This can be demon-
–1
2
a
strated by substituting into the equation the units for each parameter and carrying out the appropri-
ate reconciliation of units. The R is called the Rayleigh number and provides an indication of the
a
relative contribution of conduction and convection to heat transfer. For values of R greater than
a
about 1000, convection is the dominant heat transfer mechanism, while lower values of R indicate
a
that conduction is the dominant heat transfer mechanism.
For the mantle, estimated values of R range between 10 and 10 , depending upon the model used
5
7
a
for the mineralogy of the mantle, the temperature distribution with depth and the scale of possible
convective overturn (Anderson 1989). Clearly, heat transfer from the lower mantle to the Earth’s sur-
face is strongly dominated by convective processes. It is this fundamental attribute of the Earth that
drives plate tectonics and accounts for the distribution of geothermal resources around the globe.
plaTe TecTonIcs and The dIsTrIbUTIon oF GeoThermal resoUrces
In 1912, Alfred Wegner published his landmark paper on what became known as continental drift.
That paper was followed in 1915 by his book Die entstehung der kontinente und ozeane (The Origin
of Continents and Oceans). Over the next 14 years the book went through three revisions, culmi-
nating with his 1929 fourth edition, which provided the most complete discussion of his argument
for movement of the continents. The hypothesis proved to be extremely controversial. It was not
until the 1960s that the geological community overwhelmingly came to accept the view that the
continents and ocean basins are, in fact, mobile.