Biomedical Engineering Reference
In-Depth Information
D =
x
4
D t
D
cos( )
α
D =
y
4
D t
D
sin( )
α
α
=
random
(0,2 )
π
where
D
is the “free” diffusion coefficient, given by Einstein's law:
k T
B
D
=
6
πη
R
H
where
k
B
is the Boltzman constant (1.38 10
-23
J/K), T the temperature (K), h the
dynamic viscosity of the carrier fluid, and
R
H
the hydraulic radius of the particle.
5.4.2.4 Uniformly Narrow ECS
Suppose first that the width of the ECS is approximately constant (as in Figure 5.33).
The cell edges defined in the preceding step are widened to the desired width in or-
der to define a real ECS. Particle location inside the cluster is permanently tracked
and the particles are not allowed to cross the solid (cell) boundary. A random walk
of particles may then be confined inside the ECS as shown in Figure 5.34.
If the time allowed for the calculation is sufficiently large, the ECS is explored
by the diffusing particles as shown in Figure 5.35.
In a porous media, the distance between two points may be defined as the
length of the shortest line in the fluid domain between two points (Figure 5.36),
and tortuosity is then defined as the ratio between the distance in the liquid and the
straight line distance.
It may be theoretically shown [20] that for any 2D regular isotropic lattice of
convex cells, tortuosity has a unique value
τ
= 2
(5.50)
Figure 5.34
Random walk of two particles inside an ECS calculated by a Monte-Carlo method and
constrained by ECS boundaries.
