Page 32 - Geothermal Energy Renewable Energy and The Environment

Sources of Geothermal Heat: Earth as a Heat Engine 15

in heat transfer is through infrared emission at the Earth’s surface. It is for that reason that radiation
contributes relatively little in the way of energy or mass transfer within the Earth. That is not the
case, however, for those instances in which it is important to sense or measure heat flow at the Earth’s
surface. Remote sensing techniques rely on heat radiation to characterize surface properties.
Within the last decade infrared sensing aircraft and satellites have been used to map the intensity
of thermal or infrared emissions at the surface as a means of identifying thermal anomalies.
Although complex and still in the formative stages of routine deployment, such efforts have the
potential to significantly impact exploration for and assessment of geothermal resources. We will
leave a detailed examination of thermal radiation for Chapter 6, where the topic of exploration for
geothermal resources is considered in detail.

conducTion
A slab of rock, such as a granite countertop in a kitchen, will achieve a uniform temperature if
left undisturbed for a few hours. By definition, when it reaches that state of uniform temperature
it will be in a state of thermal equilibrium. If a pot of boiling water is placed on the countertop,
then that state of thermal equilibrium will be perturbed. Careful observation and measurement
will document that the granite in contact with the pot will quickly rise in temperature, reaching a
maximum temperature somewhat less than that of the boiling water. Over time, the temperature of
the granite slab will increase at progressively greater distances from the pot, while at the same time
the temperature of the pot, and of the granite immediately in contact with it, will drop. Eventually,
in an ideal case, the granite and the pot will reach a state of thermal equilibrium, at a temperature
just slightly higher than that which the countertop had achieved before the pot was placed on it.
Much of this progressive change in temperature is due to conduction, which is the transfer of heat
through direct physical contact. It is a diffusive process, involving no transfer of mass.
Conduction occurs via transfer of energy between atoms (and electrons) of a material. This
process is often conceptualized as a change in the vibrational frequency of atoms in the material
being thermally perturbed. In the case of the granite counter top and hot pot described above, the
frequency of the vibrating atoms in the minerals making up the granite of the countertop would
increase when they come in contact with the rapidly vibrating atoms of the pot. Since the atoms
in the minerals of the countertop are in physical contact with each other, the increased vibrational
frequency will propagate throughout the granite slab, eventually achieving a state where the tem-
perature is the same everywhere.
The rate at which thermal equilibrium is achieved will depend, primarily, on the thermal con-
ductivity and diffusivity of the material. Thermal conductivity is a measure of the ability of a mate-
rial to conduct heat. Thermal conductivity (k ) has units of W/m-K, and must be measured for each
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material of interest, since it depends upon the microscopic (e.g., atomic structure, bond strength,
chemical composition, etc.) and macroscopic (e.g., porosity, phase state, etc.) properties of a mate-
rial. Since the microscopic and macroscopic properties of a material change with temperature, k
th
will also be a function of temperature, thus requiring that it also be measured at the temperatures of
interest. Table 2.3 provides a list of thermal conductivities for some common materials as a function
of temperature.
The flow of heat (q ) through a material depends directly upon k , as well as on the temperature
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gradient (∇T) over some specified distance (∇x):
q = k × ∇T/∇x. (2.3)
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The resulting heat flow rate is specified for an area (A) by
dq /dt = k × A × dT/dx. (2.4)
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The units for heat flow are J/m – s, which is equivalent to W/m .
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