Biomedical Engineering Reference
In-Depth Information
The corresponding flux is
æ
2
2
ö
¶
Q
R U
c
j
=
= -
ç
(6.55)
÷
¶
2
D
x
π
R
è
48
ø
Relation (6.55) shows that the mass flux of the concentration
c
has the form of
Fick's law with the effective diffusion coefficient
2
2
R U
(6.56)
D
=
eff
D
48
Now, if we assume that the axial concentration gradient is no more constant
¶
c
, we are entitled to write
¹
cste
¶
x
¶
c
¶
j
= -
(6.57)
¶
t
¶
x
and we obtain
2
¶
c
¶
c
(6.58)
=
D
eff
2
t
¶
¶
x
This last equation shows that the average concentration is governed—in the
moving coordinate system—by the usual diffusion equation for a stationary me-
dium with the effective diffusion coefficient
D
eff
defined by (6.58).
A numerical example of the Taylor model is shown in Figure 6.16. The diffu-
sion front progresses with the flow and, at the same time, smears out due to mo-
lecular diffusion.
One can make a simplified picture of the situation (Figure 6.17). In the case of
the Taylor-Aris method, the concentration front has a parabolic shape; thus, the dif-
fusional surface is much larger than if the front were flat. For this reason, the same
situation for a flow induced by electro-osmosis presents less diffusion because the
diffusional front is nearly flat (Figures 6.17 and 6.18).
6.2.9.2 Conditions of Applicability of the Method
One question remains: What are the conditions for the Taylor-Aris approach to be
valid?
First, we have neglected the axial diffusional
¶
¶
2
c
x
in (6.47) in front of the radial
2
2
1
c c
r r
r
. This radial term is important if the radial velocity gra-
dient is large (the velocity profile varies from 2
U
to 0 along the radius) (i.e., if the
average velocity is sufficiently important). In such a case, we have
¶
¶
diffusional term
+
2
¶
¶
D D
<<
eff
which is equivalent to
UR
D
>> 7
