Biomedical Engineering Reference
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given by
T
(
φ
)=
φ
−p
(10.31)
T
(
φ
)=1
−
p
ln
φ
(10.32)
T
(
φ
)=1+
p
(1
−
φ
)
(10.33)
φ
)]
2
T
(
φ
)=[1+
p
(1
−
(10.34)
with
p
as a constant factor and
φ
as porosity. The first, second, and third
model are theoretical models whereas the fourth one is an empirical model. In
sequence, the above equations go back to studies of Archie (1942); Weissberg
(1963); Iversen and Jørgensen (1993); and Boudreau and Meysman (2006),
respectively.
In a recent study, Matyka et al. (2008) developed an LBM (see Section 10.2)
and studied the tortuosity problem from a mathematical perspective. For that
purpose, they considered a rectangular flow domain with randomly distributed
solid squares as solid obstacles with fixed locations (see Figure 10.6a). By cal-
culating the velocity field and the streamlines (Figure 10.6b) the tortuosity
could be calculated, and compared with the models discussed earlier. The com-
parison shows that the hydrodynamic-based tortuosity calculation of Matyka
et al. (2008) matches well with the Weissberg relation (see Figure 10.6c).
For the mathematical modeling presented above, one may ask a question:
what is the minimum size of the model system that is able to predict the
behavior of the particles in the real world? The underlying basic assumption is
that the porous material has to be homogeneous. Large model systems demand
high-computational power. This is the main reason why in simulations, system
sizes are kept as small as possible. To check this, computational analysis of the
path of two particles traveling through a porous medium was performed. Two
different alignments with the gravitational field was depicted (see Figure 10.7).
The model system should be homogeneous and should have similar prop-
erties in all directions (isotropy). Anisotropy is used to describe the variations
of properties depending on the directions. As shown here the model system is
too small. Therefore, the traveling particles do not follow direction determined
by gravity vector. It was shown by Koza et al. (2009) that the model system
has to be at least 100 times larger than the characteristic grain size.
10.3.3 Oscillating Flows over a Permeable Rippled Seabed
The sediment-water interface constitutes a dynamic and significant biolog-
ically active region in marine sediments. Within this region, sediments and
porewater contact with the overlying water, and exchange between these
reservoirs regulates oxygen or nutrient distributions. The importance of the
solute transport across this zone has been long recognized as a key factor for
accurate determination of sediment oxygen demand in marine environments
(Berner 1976).
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