Biomedical Engineering Reference
In-Depth Information
Figure 6.12
Schematic view of the flow and concentration boundary layer in a tube.
é
2
2
ù
c
c
c
c
¶
¶
¶
¶
u
+
v
=
D
+
(6.33)
ê
ú
2
2
¶
x
¶
y
¶
x
¶
y
ë
û
Boundary layer hypothesis assume that the vertical convection term is negligible
as well as the axial diffusion term, so we are left with
2
¶
c
¶
c
u
=
D
(6.34)
2
¶
x
¶
y
After substitution of the Hagen-Poiseuille flow profile in (6.34), we obtain
é
2
ù
¶
2
y
c
c
3
¶
U
1
-
=
D
(6.35)
ê
ú
¶
2
2
2
x
d
¶
y
ë
û
where
U
is the average velocity and
d
is the half vertical distance between the plates
(Figure 6.13). Now, we introduce the following scaling
Figure 6.13
Typical mass transfer boundary layer on a partially labeled solid surface in an
analysis chamber. Results obtained by solving the advection-diffusion equation with a finite difference
algorithm (average velocity 1 mm/s, labeled distance 6 mm).
