Biomedical Engineering Reference
In-Depth Information
Figure 5.23
Comparison of the concentration profiles between a labeled and a nonlabeled gap.
Axial diffusion is remarkably reduced if the gaps are labeled.
other words, is the apparent weight of the particles negligible? One can conceive
easily that if the particles are small enough they will not sediment and they will
diffuse in the available volume; if they are sufficiently large—like cells or bacte-
riae—they will tend to sediment despite the molecular agitation [3]. We derive here
a criterion to estimate the sedimentation of the microparticles and to decide if the
diffusion equation is valid.
The settling velocity is defined as the uniform vertical velocity of a particle in a
liquid at rest. The settling velocity can be calculated by writing the balance between
gravitational force and hydrodynamic drag. Let
C
D
be the friction factor (hydrody-
namic drag coefficient),
C
D
is defined by
C
= 6
πη
R
(5.43)
D
H
Then, the hydrodynamic drag force on the particle is
F
=
C v
= 6
πη
R v
(5.44)
friction
D
H
And the settling velocity
V
S
is obtained by the force balance
C V
= D
ρ
gVol
(5.45)
D S
P
where D
r
is the buoyancy density (difference between the volumic mass of particle
and liquid),
h
the dynamic viscosity of the fluid, and
g
the gravity acceleration (9.8
m/s
2
). For a spherical particle, the sedimentation velocity is given by
2
2
9
D
ρ
gR
V
(5.46)
=
S
η
Suppose now that a typical dimension of the problem is
d
. For example, the
vertical dimension of a biodiagnostic microchamber is of the order of
d
= 50
m
m.
Lets compute the times
t
1
and
t
2
for the particle to move on the distance
d
by sedi-
mentation and Brownian motion
