Biomedical Engineering Reference
In-Depth Information
10.4.3 Case 3: One-Dimensional Radial Diffusion in Spherical Coordinates
Consider the case of unsteady one-dimensional radial diffusion through a sphere
of radius
R
(Figure 10.13). Assume the concentration of the diffusing chemical
(could be oxygen, nutrient, enzyme, or a drug encapsulated in a polymeric capsule)
is independent of the angle of orientation and a function of radius only, that is,
C
=
C
(
r
). An important factor affecting concentration is the absorption or metabolism
of the component that can be considered as a reaction
R
i
(units are mol/m
3
.s). It is
conceivable that the reaction will decrease the concentration of the component and
affect the diffusion. A material balance on a thin spherical element with an inner
surface area of 4
π
r
2
and an outer surface area of 4
π
(
r
+
Δ
r)
2
yields
C
rr
∂
(
)
(
)
2
(10.62)
4
π
2
Δ
i
=
Nr N
2
−
r r
+ Δ
4
π
+
4
π
rrR
2
Δ
ir
ir
+Δ
r
i
∂
t
Note the 4
r
is the volume of the shell. Divide both sides of the equation by the
volume element 4
π
r
2
Δ
π
r
2
Δ
r
, taking the limit as
Δ
r
→
0, and using the definition of the
derivative results in
2
∂
C
1
∂
r
(10.63)
i
=
ir
+
R
2
i
∂
t
r
∂
r
∂
C
Substituting Fick's law for diffusion
i
assuming
D
AB
to be con-
(
ND
=
=
)
ir
AB
∂
t
stant with respect to the radius gives
∂
C
D
∂
r
⎛
∂
C
⎞
i
=
AB
r
2
i
+
R
(10.64)
⎜
⎟
2
⎝
⎠
i
∂
t
r
∂
r
∂
t
Equation (10.64) can be solved for various applications such as change in con-
centration of a drug entrapped in a spherical particle or oxygen profile within the
aggregate of cells with appropriate boundary conditions.
Figure 10.13
One-dimensional radial diffusion in spherical coordinates.
