Environmental Engineering Reference
In-Depth Information
4
πΔ
r
, and letting
Δ
r
→
0, we can obtain
d
d
r
(r
2
N
A
(r))
r
2
r
σ
A
s
=
−
0.
(6.63)
The expression for
N
A
(r)
is given by Fick's law of diffusion,
N
A
(r)
=
D
ef
A
(
d
C
A
/
d
r)
. The surface reaction rate
r
σ
=
−
k
σ
C
A
(r)
.
Hence,
r
2
d
C
A
d
r
1
r
2
d
d
r
k
σ
A
s
D
eff
A
−
C
A
=
0.
(6.64)
The above differential equation has to be solved with appropriate boundary conditions
to obtain
C
A
(r)
.
Let the outer surface of the sphere be at a constant concentration
C
A
σ
. This implies
that external film diffusion is not important. Hence the first boundary condition is
C
A
=
R
. A second boundary condition is that the concentration at the
center of the spherical particle is finite, that is, d
C
A
/
d
r
C
A
σ
at
r
=
0. At this point, we
can introduce a set of dimensionless variables that will simplify the form of the
differential equation. These are
|
r
=
0
=
Ψ =
C
A
(r)/C
A
σ
and
Λ =
r/R
. This gives
Λ
Ψ
1
Λ
d
d
2
d
2
− Φ
Ψ =
0,
(6.65)
2
Λ
d
Λ
Φ =
k
A
s
/D
ef
A
1
/
2
is called the
Thielemodulus
. Upon inspection of the equa-
where
σ
, we note that it is the ratio of the surface reaction rate to the rate of diffusion.
Thus the Thiele modulus gives the relative importance of reaction and diffusion rates.
For large
tion for
Φ
Φ
, the surface reaction rate is large and diffusion is rate limiting, whereas
for small
Φ
, surface reaction is rate limiting. The transformed boundary conditions
are
Φ =
1at
Λ =
1 and d
Ψ
/
d
Λ|
Λ
=
0
=
0. The solution to the above differential
equation is (Fogler, 2006; Smith, 1970)
C
A
C
A
σ
=
1
Λ
sinh
(
ΦΛ
)
)
.
(6.66)
sinh
(
Φ
Figure 6.13a is a plot of
C
A
/C
As
versus
Λ
(i.e.,
r
) for three values of the Thiele
modulus,
.
In the field of catalysis it is conventional to define a related term called the
overall
effectivenessfactor
,
Φ
, which is an indication of how far a molecule can diffuse within
a solid before it disappears via reaction. It is defined as the ratio of the actual overall
reaction rate to the rate that would result if the entire interior surface were exposed
to the surface concentration
C
As
. It is given by (Fogler, 2006)
ξ
3
Φ
ξ =
2
(
Φ
coth
Φ −
1
)
.
(6.67)
The above function is plotted in Figure 6.13b, which shows that with increasing
,
the effectiveness factor decreases. With increasing Thiele modulus, the accessibility
Φ
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