Biomedical Engineering Reference
In-Depth Information
Mass balance at pseudo-steady state above the retreating fluid
e
solid reactive interface
inside the porous matrix leads to
d
n
A
N
A
Sj
x
N
A
Sj
xþ
d
x
þ r
A
S
d
x ¼
d
t
¼ 0
(17.93)
where
n
A
is the number of moles of A in the differential volume (between
x
and
x
d
x
)of
particle,
S
is the cross-sectional area perpendicular to the path of diffusion/transport
x
.
Since there is no reaction occurring as the reactive interface is not included inside the
volume of mass balance,
Eqn (17.93)
leads to
þ
d
ðN
A
SÞ
d
x
¼ 0
(17.94)
That is,
N
A
S ¼ N
A
Sj
x¼d
(17.95)
where
x
d
is the reactive interface. At the reactive interface, A is being produced for the
fluid phase.
¼
N
A
S
j
x¼d
¼ r
0
A
S
d
(17.96)
where
r
0
A
is the rate of formation of A per unit reactive fluid
e
solid interface area, and
S
d
is the
solid surface area at time
t
.
Since the transport is through diffusion only, we have
d
C
A
d
x
N
A
¼D
eA
(17.97)
where
D
eA
is the effective diffusivity of A. Substituting
Eqns (17.95) and (17.96)
into
Eqn
(17.97)
, we obtain
d
C
A
d
x
¼ r
0
A
S
d
D
eA
S
(17.98)
Noting that reaction rate is evaluated at the retreating interface and thus not a function of the
C
A
on the left-hand side of
Eqn (17.98)
. Separation of variables and integration lead to
d
p
Z
C
AS
D
eA
d
C
A
¼ r
0
A
S
d
Z
d
x
S
(17.99)
C
A
d
d
where
C
A
d
is the concentration of A in the fluid phase right at the retreating reactive fluid
e
solid interface (
x
¼
d
) and
C
AS
is the concentration of A in the fluid phase at the initial solid
surface (
x
d
p
). Integration in
Eqn (17.99)
requires the knowledge of how the cross-sectional
area S changes along the path of diffusion. In practical applications, two cases are common:
diffusion out of spherical particle (catalysts are commonly made spherical) and linear (wood-
chips, for example, may be regarded as slab or linear in each way you observe).
Solution of
(Eqn 17.99)
depends on the geometry as the cross-sectional area perpendicular
to the transport path may change along the transport path. Solution of
(Eqn 17.99)
is to ensure
¼
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