Biomedical Engineering Reference
In-Depth Information
Figure 6.21
At inlet, the uniform distribution is decomposed in spatial Fourier series; after some
translation length
L
, only the first mode remains, and all the others are damped.
2
æ
ö
8
exp
π
Dx
a
=
-
(6.75)
ç
÷
2
2
π
4
w U
è
ø
The number
a
represents the fraction of targets still in suspension in the flow
channel at the length
x
. The fraction of targets transferred to the solvent at the
length
x
is then 1 -
a
. A reduction of 63% (corresponding to 8
e
-1
/
p
2
) of the num-
ber particles continuing to flow in the aqueous channel is reached at the length
L
e
determined by
L
4
Uw
4
=
=
Pe
(6.76)
2
2
w
D
π
π
Equation (6.76) has exactly the same form than (6.69). However, the coefficient
before the Péclet number has been explicitly determined. The ratio
L
e
/
w
is some-
times denoted as the Graetz number (see Chapter 1).
Note that the preceding reasoning uses the assumption that the velocity profile
is flat (velocity
U
). It has been shown numerically that the result is not changed by
considering a quadratic velocity profile—with the same average velocity
U
. After
some traveling distance, the profile of concentration becomes sinusoidal and indis-
cernible from the profile obtained with a uniform velocity.
6.2.10.2 Numerical Approach
Confirmation of the preceding approach can be done by using numerical modeling.
We have two choices to set up a numerical approach. Either a Hagen-Poiseuille flow
