Biomedical Engineering Reference
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that the mass transport flux
N
A
can be determined by the difference in concentrations
C
A
d
C
AS
. Once the flux relationship is found, mass balance
r
0
A
S
d
¼ N
A
S ¼ k
c
S
d
p
ðC
AS
C
Ab
Þ
(17.100)
can be applied to solve for the concentrations at the intermediate location (
x
d
p
) and thus
relate the reaction rate and/or mass transfer rate to concentrations at the interface and/or
the bulk fluid phase. If there is no porous matrix, the diffusion term (middle of
Eqn
17.100
) is not needed. However, the mass transfer coefficient
k
c
would be changing with
the shrinking core (or a function of
d
).
¼
17.8.1. Time Required to Completely Dissolve a Porous Slab Full
of Fast-Reactive Materials
For linear or rectangular particles,
S
¼
constant and
Eqn (17.99)
is integrated to yield
D
eA
ðC
A
d
C
AS
Þ¼r
0
A
ðd
p
dÞ
(17.101)
which renders the mass transfer flux
N
A
¼ D
eA
C
A
d
C
AS
d
p
d
¼ r
0
A
(17.102)
Assuming that the reaction on the retreating interface is very fast, the concentration of A at
this interface is then the equilibrium concentration of A,
C
Ae
. Combining
Eqn (17.102)
with
Eqn (17.100)
, we obtain
N
A
¼ D
eA
C
Ae
C
AS
d
p
d
¼ k
c
ðC
AS
C
Ab
Þ
(17.103)
Solving for
C
AS
, we obtain from
Eqn (17.103)
D
eA
C
Ae
þ
k
c
ð
d
p
d
Þ
C
Ab
D
eA
þ k
c
ð
C
AS
¼
(17.104)
d
p
d
Þ
Substitute
Eqn (17.104)
into
(17.105)
, we obtain the mass transfer flux or dissolution/reaction
rate at the retreating interface
D
eA
k
c
ðC
Ae
C
Ab
Þ
D
eA
þ k
c
ð
N
A
¼
(17.105)
d
p
d
Þ
Mass balance of the total dissoluble materials in the porous matrix leads to
C
Ae
f
A
S
d
d ¼ N
A
S
d
t
(17.106)
where
C
Ae
is the total concentration of equivalent A (pure), which is the same as the equilib-
rium concentration of A at the retreating surface, and
f
A
is the volume fraction of the dissol-
uble solids in the porous matrix.
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