Page 128 - Analysis and Design of Energy Geostructures

100 Analysis and Design of Energy Geostructures

its effective volumetric heat capacity. In this context the effective volumetric heat
capacity of porous materials fully saturated with a fluid can be calculated as

ρc p 5 ρ c p;f n 1 ρ c p;s ð1 2 nÞ ð3:23Þ
f s
where ρ c p;f is the volumetric heat capacity of the general fluid filling the pores of the
f
material and ρ c p;s is the volumetric heat capacity of the solid particles. For soils fully
s
saturated with water, ρ c p;f is replaced by the volumetric heat capacity of the water
f
ρ c p;w . The same approach may be applied to calculate the volumetric heat capacity of
w
fully dry soils by using the volumetric heat capacity of the air ρ c p;a . In most cases,
a
however, the contribution of the air volumetric heat capacity is neglected in the calcu-
lation of the effective volumetric heat capacity of the soil because of its small influence
on the result unless for relatively high porosities.

3.8 Initial and boundary conditions for energy conservation
3.8.1 Rationale of initial and boundary conditions
The full mathematical description of the heat transfer problem for any medium needs
initial and boundary conditions to be solved. These conditions describe a state at some
initial time and at the boundaries of the medium over time. They allow obtaining the
temperature distribution in the medium through the solution of the relevant formula-
tion of the energy conservation equation. The unique case in which no initial condi-
tions are needed is the steady-state problem governed by Eq. (3.21), that is a problem
independent of time.
According to Bergman et al. (2011), because the heat equation is second order in
the spatial coordinates, two boundary conditions must be expressed for each coordi-
nate needed to describe the system. However, because the equation is first order in
time, only one initial condition must be specified.
In most problems the typical initial condition employed is to assume a constant ini-
tial temperature for any portion of the bounding surface of a considered medium, that
is T 0 5 const. In contrast, there are five typical conditions that are used (in any combi-
nation) in the mathematical theory of heat transfer as idealisations for any portion of
the bounding surface of a considered medium.

3.8.2 Prescribed surface temperature

The so-called Dirichlet’s boundary condition or boundary condition of the first kind
allows fixing the temperature of any surface as

T H;tÞ 5 f ðH;tÞ ð3:24Þ
ð