Page 123 - Analysis and Design of Energy Geostructures

Heat and mass transfers in the context of energy geostructures 95

fine-grained soils (Rees et al., 2000). In view of this latter evidence, radiation is
usually neglected in the analysis of heat transfer characterising geomaterials.
• The contribution of radiation can be considered negligible with respect of that of
convection for the heat carrier fluid flowing in the pipes.

3.7 Energy conservation equation
3.7.1 General

The energy conservation equation expresses the principle of conservation of energy.
This governing equation is also often termed the energy equation.

3.7.2 Fourier heat conduction equation

The equation that governs the conservation of energy in the context of the analysis of
heat transfer only characterised by conduction is typically termed Fourier heat conduc-
tion equation. This equation can be derived by considering Fourier’s law of heat con-
duction expressed in Eq. (3.2) for a representative volume subjected to arbitrary
thermal conditions on its surfaces with internal volumetric heat generation _q per unit
v
time and neglecting any conversion of mechanical energy into heat (cf. Fig. 3.11).
The energy balance for the elementary volume reads

Rate of heat entering through Rate of heat Rate of energy
1 5
the bounding surfaces of a volume generation in a volume storage in a volume
Accordingly, the generalised Fourier heat conduction equation reads

@T
ð ð3:14Þ
r λrTÞ 1 _q 5 ρc p
v
@t
where ρ and c p are the density and the specific heat of the considered medium, respec-
tively, and t is the time. The term on the right-hand side of the equation represents
the variation of internal energy in the medium over time.

dx
dz
y
dy
– ∂
–λ ∂T λ ∂T + λ ∂T dx dydz
dydz
∂x ∂x ∂x ∂x
x
z
Figure 3.11 Balance of variables over the representative volume.