Page 221 - Geothermal Energy Renewable Energy and The Environment

Direct Use of Geothermal Resources 209

materials emit radiation in more complex ways that depend upon both the surface properties of an
object and the physical characteristics of the material of which the object is composed.
In addition, when considering heat transfer from one object to another via radiation, the viewing
aspect of the heat source, as seen by the object receiving radiation, must also be taken into account.
A spherical object of a given surface area, for example, will impart via radiation a relatively small
amount of heat to an object some distance away, compared to a flat plate with the same surface
area and radiation state if the flat plate directly faced the receiving object. Taking these effects into
account Equation 11.3 is generally written as:

Q = ε × ζ × σ × A × (T − T ), (11.4)
4
4
1
2
rd
where ε is the emissivity of the material (ε equals 1.0 for a perfect black body), ζ is a function
that accounts for the geometrical effects influencing heat transfer, and T and T are the respective
2
1
temperatures of the bodies. For most considerations involving radiative heat transfer in direct use
applications, interfaces are commonly flat plates or enclosed bodies in a fluid, thus rendering the
geometrical factor of minimal importance, and Equation 11.4 simplifies to
4
Q = ε × σ × A × (T − T ), (11.5)
4
2
1
rd
where ε is the emissivity of the radiating body at temperature T , and A is its effective surface area.
1
heaT Transfer by evaporaTion
Heat transfer via evaporation can be an efficient energy transport mechanism. The factors that influ-
ence evaporation rate are temperature and pressure of the vapor that is overlying the evaporating fluid,
the exposed area, the temperature of the fluid, the equilibrium vapor pressure, and the velocity of the
vapor. Although these properties are relatively simple to formulate individually, the process is affected
by factors similar to those that influence convective heat transfer and thus become quite complex.
One complicating factor is that the boundary layer behavior with respect to the partial pressure
varies both vertically away from the interface as well as along the interface due to mixing via turbu-
lent flow. As a result, the ambient vapor pressure is not easily represented rigorously.
In addition, the rate of evaporation is affected by the temperature gradient above the interface,
which is affected by the properties of the boundary layer, and which, in turn, influence the equi-
librium vapor pressure. Since the gradient is the driving force for diffusional processes, the rate at
which diffusion will transport water vapor from the surface will be affected by the local tempera-
ture conditions as well as the fluid velocity.
These complications have lead to an empirical approach for establishing evaporation rates in
which various functional forms are fit to data sets that span a range of conditions. A summary of
various results using this method are presented by Al-Shammiri (2002). The form below is from
Pauken (1999):
E = a × (P − P ) , (11.6)
b
a
w
ev
2
where E is the evaporation rate, in g/m -hr, P and P are the water vapor saturation pressures
w
a
ev
(kPa) at the temperature of the water and air, respectively, and
a = 74.0 + (97.97 × v) + (24.91 × v )
2
and
b = 1.22 − (0.19 × v) + (0.038 × v ),
2
where v is the velocity of the fluid moving over the interface (m/s).