Biomedical Engineering Reference
In-Depth Information
L
(6.82)
» 32
Pe
R
It is worth comparing the mixing length from relation (6.82) with the entrance
length in a channel (2.25) established in Chapter 2. There is an obvious similarity,
if we write the two relations as
L
»
32
R Pe
»
8(2 )
R Pe
mix
R
D
(6.83)
h
»
0.04 (2 ) Re
R
D
In the case of the hydrodynamic entrance, it is the action of the viscosity that
homogenizes the flow to reach a fully developed flow. In the case of the mixing of
microflows, it is the action of the molecular diffusion that homogenizes concentra-
tion. The physics of the two problems is similar. In both cases the phenomenon
is linked to the growth of a boundary layer. It is no wonder then that the form of
(6.83) is similar. The major difference is that the cinematic viscosity is of the order
of 10
-6
m
2
is whereas the diffusion coefficient is only 10
-10
m
2
is The hydrodynamic
entrance is then very short, whereas the mixing length is very long.
6.2.12.3 Improving the Mixing of Parallel Flows
Usually in bioMEMS, the flow rate is imposed and, from (6.82), the only action
way of reducing the mixing length for parallel flows is to reduce the radius
R
. This
will have a considerable effect since the mixing length
L
mix
varies as the square of
R
. However, if the radius is reduced and if the flow rate is imposed, it is necessary
to divide the flow in multiple branches. A typical design based on the reduction of
the channel cross section is shown in Figure 6.31.
6.2.12.4 Chaoting Mixing
The principle of chaoting mixing is based on successive stretching and bending of
fluid streamlines. In Figure 6.31, we show how a domain of liquid 1 immerged in a
liquid 2 is deformed by chaoting mixing.
If the succession of folding and folding is done rapidly, at a time scale much
smaller than that of diffusion, the time interval for the stretching-folding process
may be neglected and it is possible to compare the corresponding mixing zones in
Figure 6.32. Suppose a time scale
t
; then diffusion length is approximately
λ
» 4
D
τ
(6.84)
If we chose the value of
t
so that the distance
l
is approximately the distance
between the folded regions, the mixing zones at the time
t
corresponding to Figure
6.32 are shown in Figure 6.33. The mixing zone after chaoting mixing has a more
important surface than the original one. This proves the efficiency of chaoting mix-
ing. The important thing here is that the stretching/folding deformations are per-
formed in a short time compared to the diffusion time.
