Biomedical Engineering Reference
In-Depth Information
where
D
is the effective diffusivity evaluated as the product between molecular
diffusivity and a constant depending on the matrix microstructure. in a more
complex example with convection,
J
s
can be described as:
-
VD
d
d
J VD
c
z
[4.32]
s
JJc
=
J VD
s
Jc
where
z
is the distance from the membrane surface (longitudinal coordination
axis),
c
=
c
(
z
) is the local solute concentration,
V
is the average velocity of
the solvent fl uid and is proportional to D
P
. it must be noted that
D
presented
here is different from the effective diffusivity introduced above, owing to
dependence on the convective motion. When considered at the microscopic
level, convection ignores the diffusion phenomenon, therefore the microscopic
Peclet number (
Pe
d
=
V
d/
D
m
>> 1) is very high and the effective diffusivity
can be expressed by:
D
=
a
-1
D
m
Pe
d
=
a
-1
Vd
[4.33]
where
a
represents the tortuosity which is highly dependent on the
microstructure of the matrix. From this, it is evident that
D
could be
very much higher than
D
m
. Under constant conditions, solving d
J
s
/d
z
= 0 with boundary conditions
c
(0) =
c
s
and
c
(
L
) = 0, it is evident that
Sh
=
Pe
/(1 -
e
-
Pe
), where
Sh
=
J
s
/(
Dc
s
/
L
) and
Pe
=
VL
/
D
=
L
/
ad
. Here
Sh
indicates the Sherwood number, the ratio between the solute mass fl ow and
its microscopic diffusion component; whereas
Pe
is the macroscopic Peclet
number as the ratio between the convective macroscopic mass fl ow and the
diffusive fl ow. Peclet represents the ratio between the macroscopic and the
microscopic scales. Because here
Pe
>> 1:
kc
L
[4.34]
s
lim
Sh
Pe
Æ
lim
lim
J
J
kc
kc
L
s
D
P
D
P
P
=
=
·
s
L
J
s
Pe
>>1
Pe
>
>
>>1
m
m
L
Pe
It is necessary to consider enzymatic erosion in any detailed discussion
of this topic. This process takes place in two steps: fi rstly, amide penetration
through the membrane (transported by water) and pore walls; secondly, the
enzymatic reaction occurs and the amide reacts with the enzyme trapped
within the matrix, producing maltose. The characteristic duration of this
process is generally the sum of the penetration time (
t
p
) and the reaction
time (
t
r
). Therefore, because the reaction velocity (
R
s
) is proportional to the
amide concentration
c
s
, the following is obtained:
1
=
1
+
[4.35]
Rrc
he
s
Rr
w
c
w
he
re
r
Rr
=
RRr
r
=
s s
R
s s
Rr
s
Rr
=
Rr
c
c
s
w
r
r
s
s
he
re
tt
s
t
tt
s
s
p
r
The slowest process determines the hydrolysis kinetic. Here t
r
<<
t
p
and
therefore:
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