Page 154 - Analysis and Design of Energy Geostructures

126 Analysis and Design of Energy Geostructures

λ [W/(m C)] is the thermal conductivity, T [ C] is the temperature and

^ e x , ^e y and ^e z are the standard unit vectors in Cartesian coordinates.
For the cylindrical coordinates r;θ;z, the general form of Fourier’s law
is _q 52 λ @T ^ e r 1 1 @T ^ e θ 1 @T ^ e z , where ^e r , ^e θ and ^e z are the standard unit

r @θ
@z
@r
vectors in cylindrical coordinates.
For the spherical coordinate system r;θ;φ, the general form of

Fourier’s law is _q 52 λ @T ^ e r 1 1 @T ^ e θ 1 1 @T ^ e φ , where ^e r , ^e θ and ^e φ
@r r @θ r sinθ @φ
are the standard unit vectors in spherical coordinates.
i. _q 52 λ dT 52 λ ð T 2 2 T 1 Þ 5 λ T 1 2 T 2 Þ where _q [W/m ] is the heat flux, λ
2
ð
x
x
dx
t w
t w
[W/(m C)] is the thermal conductivity of the medium, T [ C] is the

temperature of the medium and t w [m] is the wall thickness.
j. The thermal conductivity is a property that provides an indication of the
rate at which energy is transferred by the diffusion process characteristic of
conduction and depends on the physical structure of matter and its state.
BasedonFourier’s law, the thermal conductivity associated with conduc-
tion in the x-direction of a Cartesian coordinate system is defined as
_ q
x

λ x 2 ½W=ðm Cފ
@T
@x
2
where _q [W/m ] is the heat flux (i.e. the flow of thermal energy per unit
x
of area per unit of time), T [ C] is the temperature of the medium and x

is the Cartesian coordinate representing the direction of the heat flux
vector. Heat transfer occurs at a lower rate across materials of low ther-
mal conductivity than across materials of high thermal conductivity.
k. As the intermolecular spacing is much larger and the motion of the
molecules is more random for the fluid state than for the solid state, ther-
mal energy transport is less effective. The thermal conductivity of gases
and liquids is therefore generally smaller than that of solids and, for the
same reason, in the specific the thermal conductivity of gases is smaller
than liquids.
l. The thermal conductivity of a saturated geomaterial is greater than that
of the same dry geomaterial because of the higher magnitude of the ther-
mal conductivity of water with respect to that the air filling the geoma-
terial pores in the saturated and dry case, respectively (λ w 5 0.57 W/

(m C) while λ a 5 0.025 W/(m C) at a temperature of T 5 15 C).

m. Thermal conductivity of air at ambient temperature is λ a 5 0:025 W/(m C).

Therefore the effective thermal conductivity calculated as a weighted arith-
metic mean of the thermal conductivities of its components reads