Page 102 - Analysis and Design of Energy Geostructures

74 Analysis and Design of Energy Geostructures

3.4 Conduction
3.4.1 Physical phenomenon and governing equation
Conduction is the mode of heat transfer that occurs at the molecular and atomic levels
between particles of a solid or a fluid that are at different temperatures. This mode of
heat transfer is generally associated with a macroscopically invisible motion of the par-
ticles that constitute the medium and is related to the mechanism of energy diffusion.
The physical phenomenon of conduction can be explained, for example with ref-
erence to the motion of molecules characterising a medium bounded by two surfaces,
which at a meaningful scale can be considered plane walls at different temperatures.
This problem is represented in Fig. 3.2 considering a conduction phenomenon that
can characterise, for example an energy pile. The continuous collision of molecules
involves a transfer of energy from the more energetic regions of the system (at higher
temperature) to the less energetic regions of the system (at lower temperature) accord-
ing to thermodynamics. The motion of molecules through the surfaces allows estab-
lishing a net transfer of energy in the system, which is associated with conduction. In
gases the molecules involved in the molecular interactions are less closely spaced than
in liquids and are characterised by less frequent and lower interactions. In solids atomic
activity in the form of lattice vibrations governs conduction. The previous facts indi-
cate that conduction heat transfer is more pronounced in solids than in fluids.
The rate equation governing conduction is Fourier’s law. According to this law, the
heat flux density (i.e. the rate of heat energy, Q, transferred through a given surface,
, for a medium that possesses
A, per unit time, t) generated by conduction, _q
cond;i
homogeneity and isotropy with respect to the heat conduction phenomenon, is
_
5 Q 5 Q 52 λrT 52 λ @T ^ e x 1 @T ^ e y 1 @T
_ q ^ e z ð3:2Þ
cond;i
At A @x @y @z
where λ is the thermal conductivity of the medium (the parameter, i.e. a positive sca-
lar quantity, that governs heat conduction), r is the vector differential operator (gradi-
ent) and ^e x , ^e y and ^e z are the standard unit vectors in Cartesian (also termed

Figure 3.2 Heat transfer by conduction in an energy pile.