Page 125 - Analysis and Design of Energy Geostructures

Heat and mass transfers in the context of energy geostructures 97

where α d is the thermal diffusivity of the medium given by
λ
α d 5 ρc p ð3:20Þ

where ρc p is the volumetric heat capacity. The thermal diffusivity measures the ability
of a material to conduct thermal energy relative to its ability to store thermal energy
(Hermansson et al., 2009). Materials of large α d will respond quickly to variations in
their thermal environment, while materials of small α d will respond more slowly. The
thermal diffusivity of a medium is indicative of the propagation speed of the heat into
the body during temperature variations. The higher the value of α d is, the faster prop-
agation of heat within the medium is.

3.7.4 Laplace’s equation
The particular case in which the temperature distribution is independent of time and
no heat sources are present can be of interest, and involves the Laplace’s equation (e.g.
for the temperature field)
2
r T 5 0 ð3:21Þ
Eq. (3.21) is associated with steady-state conditions and often represents the basis
for analysis and design considerations.

3.7.5 Energy conservation equation
When convection and conduction characterise the heat transfer, the energy conserva-
tion equation reads

2 @T
ð3:22Þ
1 ρ c p;f v rf ;i rT
λr T 1 _q 5 ρc p
f
v
@t
where v rf ;i is the fluid velocity vector.
The expression of the energy conservation reported in Eq. (3.22) presents two
unknowns: the temperature field, T, and the displacement field of the moving fluid,
u f ;i (included in the term v rf ;i 5 @u f ;i =@t). Therefore the only Eq. (3.22) makes the
solution of conduction convection related problems undetermined.
Conduction convection heat transfer problems can be addressed by solving Eq. (3.22)
and the mass conservation equation, under the assumption of negligible influences of
the phenomena involved in the equilibrium of the moving fluid (i.e. incompressible
inviscid flow). The previous aspect explicates the essence of the thermohydraulic cou-
pling between heat transfer and mass transfer that takes place as soon as convection is
considered.