Biomedical Engineering Reference
In-Depth Information
(b) Plot the changes in concentration of oxygen (
C
/
C
0
) as a function of the
percentage of maximum thickness (
R
2
−
R
1
) for a series of values of
R
2
=
4 mm, 5 mm, 8 mm, and 13 mm. For plotting graphs, you can use either
the Taylor series or any mathematical software such as Mathematica or
MathCAD [2].
k
(
R
2
-
R
1
)
2
/
D
, then
what are the Damkohler number values for each of the above cases?
(d) Explain the relationship between the oxygen concentration plots and the
Damkohler number values.
(e) What design modifications would you consider using to improve the
oxygen availability in cases where oxygen is the limiting factor?
10.32 Consider a long hollow fiber bioreactor where an enzyme is immobilized
within a bundle of
n
hollow fibers. Each hollow fiber has an inner diameter
ID and an outer diameter OD. The wall of the hollow fiber is permeable to the
substrate and the product, but not the enzyme. The substrate at concentration
S
i
is fed at a flow rate
F
. The feed passes through the shell side of the hollow
fiber reactor and exits at the other end with a bulk substrate concentration of
S
o
. The inside of the hollow fiber is stagnant.
(c) If the derived Damkohler number expression is
Da
=
(a) Given the diffusivity of the substrate through the hollow fiber wall
D
w
and
that through the medium
D
m
and given the intrinsic reaction rate
expression
v
(
s
), set up the material balance equation(s) that needs to be
solved to find the substrate concentration profile within the fiber wall and
within the inner space of the hollow fiber.
(b) Provide applicable boundary conditions.
(c) Describe briefly how you would solve the above ODE to arrive at a
solution
s
(
r
), where
r
is the radius from the center of the hollow fiber.
(d) Assume that you have solved the equation from the last part, and you now
have the function
s
(
r
) for
r
= 0 to
r
=
R
o
, where
R
o
is the outer radius of the
fiber. Find the observed rate of reaction per unit length of the fiber per
fiber.
(e) Provide the definition of the effectiveness factor, and find the effective-
ness factor as a function the substrate concentration at
r
=
R
o
(i.e., at the
shell side).
(f) List the equation(s) that must be solved to find the bulk substrate concen-
tration at the exit,
S
o
.
References
[1]
Limbut, W., et al., “Capacitive Biosensor for Detection of Endotorin,”
Analytical and Bio-
analytical Chemistry,
Vol. 389, No. 2, 2007, pp. 517-525.
[2]
Deen, W. M.,
Analysis of Transport Phenomena,
Oxford, U.K.: Oxford University Press,
1998.
