Page 103 - Analysis and Design of Energy Geostructures

Heat and mass transfers in the context of energy geostructures 75

rectangular) coordinates. The assumption of homogeneity and isotropy indicates that
the thermal conductivity is independent of direction and position and for this reason
appears outside the argument of the gradient.
In Eq. (3.2), the minus sign makes the heat flux density a positive quantity as a
consequence of its direction towards decreasing temperature. The law expressed in
Eq. (3.2) was first stated based on experimental evidence by Fourier (1822) but may
be derived from the principles of irreversible thermodynamics as well (Boley and
Weiner, 1997).
Fourier’s law expresses that thermal equilibrium can only be achieved when no
temperature gradient occurs. In other words, heat flows as far as there is a temperature
gradient.
Fourier’s law can be markedly simplified for problems involving plane geometries
under steady-state conditions. These situations may characterise, for example energy
walls (cf. Fig. 3.3). In this case, the temperature distribution across the wall is linear
and the heat flux in the direction of heat transfer, x, reads

ð
_ q 52 λ dT 52 λ ð T 2 2 T 1 Þ 5 λ T 1 2 T 2 Þ ð3:3Þ
x
dx t w t w
where T 2 and T 1 are the temperatures of the inner and outer surface of the wall and
t w is the wall thickness. It is worth noting that Eq. (3.3) provides the heat flux, that is
the rate of heat transfer per unit area of wall, A. The heat rate (or thermal power)
may be calculated at any time by multiplying _q by the area of wall perpendicular to
_
x
the direction of heat transfer as Q 5 _q A.
x
x
3.4.2 Thermal conductivity values
The thermal conductivity of materials is strongly characterised by the chemical com-
position of the material constituents as well as by physical factors that influence the

Figure 3.3 Conduction heat transfer across a plane energy wall.