Biomedical Engineering Reference
In-Depth Information
æ
2
ö
é
2
ù
¶
c
r
¶
c
1
¶
c
¶
c
U
D
(6.49)
+
2
1
-
=
+
ê
ú
ç
÷
2
2
¶
t
¶
x
r
¶
r
R
¶
r
è
ø
ë
û
In a Lagrangian system of coordinate moving with the average velocity of the
fluid (6.49) becomes
æ
2
ö
é
2
ù
é
ù
¶
c
¶
c
2
r
¶
c
1
¶
c
¶
c
(6.50)
U
U
D
+
+
1
-
=
+
ê
ú
ê
ú
ç
÷
2
2
¶
t
¶
x
¶
x
r
¶
r
R
¶
r
è
ø
ë
û
ë
û
The boundary condition at the capillary surface is
¶
¶
0
c
r
r R
=
indicating that there is no flow of substance through the wall of the tube. The rea-
soning may be decomposed in two steps: first, assume temporarily that the concen-
tration gradient along the
x
-axis is linear
¶
c
¶
c
(6.51)
=
=
cste
¶
x
¶
x
Here
c
is the average concentration over the cross section of the tube, defined by
R
R
1
2
ò
ò
(6.52)
c
=
c
2
π
r dr
=
c r dr
2
S
R
0
0
Along the axis of the capillary, the concentration must have a finite value. Then,
the solution of (6.50) may be written as
2
æ
2
4
ö
UR
¶
c
r
1
r
c
=
c
+
-
ç
÷
0
2
4
4
D x R
¶
2
R
è
ø
(6.53)
¶
c
t
=
0
¶
where
c
0
is the concentration on the capillary axis (
r
= 0). In the following, remem-
ber that the time derivative is taken in the moving coordinate system. By using
(6.52) and (6.53), we find the following relation between
c
and
c
2
æ
2
4
ö
UR
¶
c
1
r
1
r
c
= +
c
- +
-
ç
÷
2
4
4
D x
¶
3
R
2
R
è
ø
(6.54)
¶
c
t
=
0
¶
The total mass flow of species through any cross section of the capillary is given
by
R
æ
ö
¶
2
2
R U
c
ò
2
Q uc
=
2
π
r dr
= -
π
R
ç
÷
¶
D
x
48
è
ø
0
