Page 157 - Analysis and Design of Energy Geostructures

Heat and mass transfers in the context of energy geostructures 129

significant groundwater flow, the main heat transfer mode would be
convection.
z. In these conditions, forced convection occurs within the pipes embedded
in the energy wall, conduction characterises the heat exchange between
the pipes and the surrounding reinforced concrete constituting the
energy wall, and natural and/or forced convection can typically govern
the heat exchange between the concrete wall and the adjacent metro sta-
tion environment. If no thermal insulation may be foreseen between the
wall and the soil, conduction or conduction and convection may also
characterise heat transfer between the wall and the soil, depending on the
significance of groundwater flow.

aa. In this form of the Fourier heat conduction equation, λ [W/(m C)] is
2
the thermal conductivity of the medium, r is the Laplace operator of
3

the temperature of the medium T [ C], _q [W/m ] is the internal volu-
v
3
metric heat generation, ρ [kg/m ] is the density of the medium, c p [J/
(kg C)] the specific heat of the medium and t [s] is the time. The term

on the right-hand side of the equation represents the variation of inter-
nal energy in the medium over time.
2
Defining the thermal diffusivity as α d 5 λ=ðρc p Þ [m /s], in case of
no internal heat generation the equation is
@T
2
α d r T 5
@t
while in case of time-independent temperature the equation is
2
λr T 1 _q 5 0
v
and if both conditions occur the equation is

2
r T 5 0
which is also called the Laplace equation.
In general, internal heat generation is applied to a part that will
either act as a heat source or heat sink throughout the analysis, such as
an energy geostructure.
Temperature distribution can be assumed to be independent of
time when the system is at steady-state conditions. Hence, the Laplace
equation often represents the basis for analysis and design considera-
tions of energy geostructures. However, the large heat capacity of soils
delays the effects of external variations in temperature, often requiring
a transient analysis.