Biomedical Engineering Reference
In-Depth Information
solved using a finite element method. For demonstration of concepts, three simple
cases generally encountered in biomedical applications are discussed next.
10.4.1 Case 1: Reaction-Diffusion in Cartesian Coordinates
Consider the situation (Figure 10.10) where there are two baths with different con-
centrations of a chemical
C
. One bath is the chemical source (unlimited supply),
maintained at a concentration
C
0
. The other bath initially contains no chemical and
has a width
L
. At the far edge of the second bath (
x
=
L
), there is an impermeable
boundary. In addition at this boundary the chemical of interest is degraded accord-
ing to a first-order chemical reaction (surface reaction).
The conservation equation in the bulk of the second bath
C
(
x
) reduces to
∂
C
∂
2
C
=
D
+
R
AB
A
2
∂
t
∂
x
At steady-state and for a first-order degradation reaction of the reactant,
∂
2
C
(10.53)
0
=
D
−
kC
AB
2
∂
x
where
k
is the rate constant. Equation (10.53) is analogous to (10.50), and one
could write the analytical solution similar to (10.51)
(10.54)
D
D
AB
AB
−
x
−
x
CAe
=
k
+
Be
k
where
A
and
B
are integration constants. One boundary condition is
C
(0)
=
C
0
at
x
=
0 and the other boundary condition is the flux equation at
x
=
L
, given by:
∂
CL
()
(10.55)
D
=−
kC L
()
AB
∂
x
Figure 10.10
Chemical species diffusing into an “empty” bath with a degradation reaction occur-
ring at
x
=
L
.
