Biomedical Engineering Reference
In-Depth Information
where
H
is the half-distance between the two plates. The general form for the equiv-
alent diffusion coefficient using the Péclet number is
D
1
eff
2
Pe
(6.65)
=
D
β
where
b
is a geometric coefficient depending on the shape of the cross section. Note
that the lower limit for the Peclet number (6.59) is not universal and depends on the
cross section of the capillary tube. The general formulation would be
Pe
>2
β
(6.66)
6.2.9.3 Applicability to Microflows
The conditions defined in the preceding section are very often satisfied by micro-
flows in microsystems. Generally, tube diameters are in the range of 100
m
m to
1 mm, and velocities vary from a few microns per second to a few millimeters per
second. Thus, for a water-based flow, the Reynolds number is smaller than 10, and
the flow is strongly laminar. Because diffusion coefficients are very small (seldom
larger than 10
-10
m
2
is the Peclet number is at least of the order of 10 and most of
the time larger than that. Finally, the condition
L
/
R
>> 7 requires that the micro-
channel be sufficiently long, which is usual.
When applicable, the Taylor-Aris approach is very simple and useful. It has
many applications in chemistry [6, 7] and for immunoassays, as we will see in
Chapter 8.
6.2.10 Distance of Capture in a Capillary
In biotechnology, the capture of particles advected by a carrier fluid flowing inside a
capillary tube is a fundamental question. For example, we may want to dimension
an annular surface to capture a certain type of particles in the carrier fluid. In this
section we do not deal with the capture itself (this will be done in Chapter 8), but
with the contact of the particles with the solid wall.
6.2.10.1 Analytical Approach
Scaling Analysis
A very simple approximation may be done by comparing an axial convection to a
radial diffusion. Particles near the wall are not going to have a very long axial dis-
placement before impacting the wall, whereas the particles initially located at the
center of the capillary will follow the longest trajectory before impacting the wall
(Figure 6.19).
The average maximum time necessary for a particle to diffuse radially to the
wall is
2
R
D
τ
»
(6.67)
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