Biomedical Engineering Reference
In-Depth Information
c
t
*
*
c
=
,
t
=
(5.40)
RL
c
0
D
It is straightforward to see that the nondimensional diffusion equation is
æ
ö
¶
c
*
L
¶
c
*
¶
2 *
c
R c
¶
2 *
1
(5.41)
=
+
+
ç
÷
t
R
r
r
r
L
z
¶
*
è
*
¶
*
¶
*2
ø
¶
*2
This equation is an axisymmetrical diffusion equation with the anisotropic dif-
fusion coeficients
R
L
*
*
(5.42)
D
=
,
D
=
z
r
L
R
In order to compensate for the change in geometry, the diffusion coefficients
are now strongly anisotropic; the equivalent diffusion coefficient in the direction
r
is large whereas that in the direction
z
is small. The ratio between the
r
and
z
dif-
fusion coefficient is
D
*
L
2
r
z
=
>>
1
*
2
D
R
Remark that (5.41) verifies Buckingham's Pi theorem [8]. There are four in-
dependent parameters in (5.38):
c
0
,
L
,
R
, and
D
. These parameters are measured
with three different units: kilos or moles (if we count the concentration in kilos or
moles), meters, and seconds. According to Buckingham's theorem, there should be a
4 - 3 = 1 dimensionless parameter in the nondimensional equation. This parameter
is evidently
L/R
.
5.3.8.4 Numerical Solution
Equation (5.41) may be solved by using a standard finite element method. The
computational domain is defined by
r
*
Î {0,1},
z
*
Î {0,1}. At the wall, the condition
of impermeability yields
*
¶
c
¶
c
r
. An initial condition
c
*
0
= 1 is im-
posed in a very small volume at the periphery of the tube. Concentration contours
obtained at two different times are shown in Figure 5.21.
Note that the nondimensional calculations results have to be transformed back
into dimensional values. It is a straightforward process if we use (5.39) and (5.40),
and if we remark that the initial concentration
c
0
is obtained by converting the sur-
face concentration in immobilized primers into a volume concentration.
=
=
0
r
*
=
r R
=
*
1
r
¶
¶
5.3.8.5 Diffusion Barriers
It can be checked that the analytical and numerical results agree (Figure 5.22).
Most of the time analytical results when sufficiently accurate should be preferred.
However, in the present case, the numerical approach—although more complex to
set up—is more powerful. For example, it is possible to investigate whether axial
