k eff : effective coefficient of thermal conductivity (WÁm
À1
ÁK
À1 )
T: temperature (K)
The bulk thermal conductivity for a soil (k bulk ) results not only from the thermal
conductivity of the soil particles (k s ) but from that of the fluids contained in the pores
(k f ) (Eq. 3.25):
k eff ¼ k s 1 À n
ð
Þþk f n
ð Þ
ð3:25Þ
where,
k eff : effective thermal conductivity of the bulk (WÁm
À1Á
K
À1 )
k s : thermal conductivity of the soil particle (WÁm
À1
ÁK
À1 )
k f : thermal conductivity of the fluid in the bulk (W.m
À1
ÁK
À1 )
n: total soil porosity (À)
3.3.1.3 Heat Radiation
Heat radiation is the energy emitted by electromagnetic waves. All bodies with
temperatures above 0
C emit energy according to the Stefan–Boltzmann law. The
general equation for radiation with component i is defined as (Eq. 3.26) (Howell
et al. 2011):
J ri ¼ ε i σT
4
i þ 1 À ε i
ð
Þ
X N
j¼1
F ij J j
ð3:26Þ
where,
J ri : molecular radiation flux of component i (molÁm
À2
Ás
À1 )
ε i : surface emissivity of component i
σ: Stefan–Boltzmann constant (5.67 Â 10
À8 WÁK
À4
Ám
À2 )
T: temperature of component i (K)
F ij : view factor between element i and j
J j : molecular flux of component j (molÁm
À2
Ás
À1 )
3.3.2 Modeling Heat Transfer in Porous Media
Energy balance equations in fluid ( f ) and in solid (s) are respectively defined by
Eqs. (3.27) and (3.28) (Hsu 2005; Kaviany 1999):
174
S. Colombano et al.