Transport - Environmental Modeling - page 68

which can be compared with the diffusivity in Fick's Law (Eq. 3.5 ). In the literature
one can also find the term conductive heat flux . The physical unit of l T is [energy/

temperature)] (mostly: [Watt ( Km) 1 ]). Some values of thermal
conductivities can be found in Table 3.1 .
Like Fick's Law, Fourier's Law was at first stated for a single phase situation,
but the formulation ( 3.26 ) does not have to be changed for multi-phases. The
proportionality factor usually is not the same: in the saturated porous medium,

(length time

l T

is a weighted mean of the phases involved, i.e. the fluid and the solid phase:

l T ¼ 1 y ð Þl s þ yl f (3.27)

where the subscripts denote the thermal conductivities for the pure phases.
Table 3.1 provides a list of 'material properties,' which are relevant for heat
transport. Some are related to pure phases (water and calcite), some to mixed phases
(gravel, sand).
A parameter with the unit [area/time] results when thermal conductivity is
divided by the specific heat capacity. In analogy to mass diffusion, this parameter
is termed thermal diffusivity, and Fourier's Law can be understood as the law of
heat diffusion. Note that the transformation from single phase to multi-phase is not
the same for heat and mass diffusion. The important difference is that heat diffusion
takes place in all phases, whereas mass diffusion is relevant only in the fluid phase.
Replacing the energy flux vector in ( 3.21 ) by the terms for advective and
diffusive fluxes, as formulated in ( 3.25 ) and ( 3.26 ), one obtains the heat transport
equation:

ðrCÞ @

@t T ¼r

lrT yðrCÞ f v T þ q e (3.28)

which for a divergence-free flow field simplifies to:

ðrCÞ @

@t T ¼r lrT yðrCÞ f v rT þ q e

(3.29)

Division by rC delivers:

@
@t T ¼r D T rT yk

q e
rC

v rT þ (3.30)

ðrCÞ f

with thermal diffusivity D T ¼ rC

rC .Some
values of thermal diffusivities are listed in Table 3.1 . Note that thermal diffusivities
are more than two orders of magnitude higher than molecular diffusivities for
species. In systems, in which heat and mass diffusion act simultaneously, heat
diffuses much faster than any species.
Aside from the

and the ratio of heat capacities k ¼

term, the advection term ( 3.30 ) and the different representa-
tion of the sources, the heat transport formulation is identical to the mass transport

yk