Page 153 - Analysis and Design of Energy Geostructures

Heat and mass transfers in the context of energy geostructures 125

Solutions
a. Isotropic means having identical values of a certain property in all direc-
tions in space.
b. Conduction, convection and radiation.
c. Heat transfer generally occurs in the form of general thermal energy
propagating because of a spatial temperature difference. When a temper-
ature gradient exists in a medium, which can be a solid or a fluid, the
term conduction is used to refer to the heat transfer that occurs across
the medium. In contrast, the term convection refers to the heat transfer
that occurs between a surface and a moving fluid when they are at differ-
ent temperatures. Finally, the term radiation is used to refer to the heat
transfer characterising all surfaces of finite temperature that emit energy
in the form of electromagnetic waves, even in the absence of an interven-
ing medium.
d. The driving cause of heat transfer by conduction is generally considered to be:
i. The motion of a fluid
ii. An invisible motion of the particles that constitute a medium
iii. Waves of the electromagnetic field propagating at the speed of light
e. Fourier’s law is an empirical relationship between the conduction rate in
a material and the temperature gradient in the direction of energy flow,
first formulated by Fourier in 1822 who concluded that ‘the heat flux
resulting from thermal conduction is proportional to the magnitude of
the temperature gradient and opposite to it in sign’.
f. The general formulation of Fourier’s law is
_ q 52 λrT
i
2
where _q [W/m ] is the heat flux (i.e. the flow of thermal energy per unit
i

of area per unit of time), λ [W/(m C)] is the thermal conductivity of
the medium, r represents the gradient and T [ C] is the temperature of

the medium.
λ
g. A temperature gradient is a physical quantity that describes in which
direction and at what rate the temperature variations around a particular
location. The temperature gradient is a dimensional quantity expressed
in units of degrees (on a particular temperature scale) per unit length.

The metric SI unit is K/m, or equivalently C/m.
h. For the Cartesian coordinates x;y;z, the general form of Fourier’s law is

_ q 52 λ @T ^ e x 1 @T ^ e y 1 @T ^ e z , where _q [W/m ] is the heat transfer rate
2
@z
@y
@x
in space per unit area perpendicular to the direction of transfer,