Page 160 - Analysis and Design of Energy Geostructures

132 Analysis and Design of Energy Geostructures

ff. The energy conservation equation reads in this case:
_ q cond 2 _q conv 2 _q rad 5 0

2
where _q cond 52 λ ΔT is the conductive heat flux [W/m ], with λ the
t slab
thermal conductivity of the medium [W/(m C)], ΔT the temperature

difference through the slab [ C] and t slab is the slab thickness [m];
2
_ q conv 5 h c ðT s 2 T N Þ is the convective heat flux [W/m ], with h c the
convective heat transfer coefficient [W/(m 2 C)], T s the temperature of
the surface exposed to the fluid and T N the bulk temperature, both
expressed in [ C]; and

4
_ q rad 5 Eσ SB ðT 2 T Þ is the radiative heat flux [W/m ], with E the
4
2
sur
s
emissivity of the body [ ], σ SB is the Stefan Boltzmann constant [W/
4
(m 2 C )] and T sur is the surrounding environment temperature [ C].

gg. In heat transfer analysis the ratio of the thermal conductivity λ [W/

(m C)] to the volumetric heat capacity ρc p [J/(m 3 C)] is termed the
2
thermal diffusivity α d [m /s] and reads
λ
α d 5
ρc p
hh. Soils of large thermal diffusivity respond:
i. Quickly to variations in their thermal environment
ii. Slowly to variations in their thermal environment
iii. Irrespectively of the variations of their thermal environment
iv. Depending on the thermal conductivity
2
ii. The thermal diffusivity is defined as α d 5 λ=ðρc p Þ [m /s].
Material Thermal conductivity, Volumetric heat Thermal diffusivity, α d
2
λ [W/(m C)] capacity, ρc p [m /s]

[MJ/(m 3 C)]
Dry Saturated Dry Saturated Dry Saturated
Clay 0.2 0.3 1.1 1.6 0.3 0.6 2.1 3.2 0.5 0.67 0.5 0.52
Silt 0.2 0.3 1.2 2.5 0.6 1.0 2.1 2.4 0.3 0.33 0.57 1.04
Sand 0.3 0.4 1.7 3.2 1.0 1.3 2.2 2.4 0.3 0.31 0.77 1.33
Gravel 0.3 0.4 1.8 3.3 1.2 1.6 2.2 2.4 0.25 0.82 1.375
Concrete 0.9 2.0 1.8 2.0 0.5 1
Water 0.57 4.186 0.14
Air 0.025 0.0012 20.83

jj. In general, the different values of thermal diffusivity for different materials
cannot be neglected: differences in thermal conductivity, density and spe-
cific heat are observed between the different materials and affect the heat
transfer.