Mass Transfer Effects: Immobilized and Heterogeneous Reaction Systems - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 877

Integration can be obtained via either the Trapezoidal or the Simpson's rule. Because central
difference method as well as the Trapezoidal rule or Simpson's rule all have symmetrical
error sequences, an extrapolation technique can be employed to speedup the convergence.
Table 17.3 shows the converged solutions obtained from Excel.
The convergence of the numerical solution is fast. Still, one may not be willing to solve Eqn
(17.8) every time when it is needed. Fig. 17.6 shows the effectiveness factor as a function of
the Thiele modulus and K b . One can observe that the curves in Fig. 17.6 are of the same
shapes as those in Fig. 17.3 . In other words, the effectiveness factor for internal mass transfer
limitation is similar to that of external mass transfer limitation.
Reading numbers from tables or graphs is extremely inconvenient. At this point, we do
know the closed form solutions for the two extreme cases: K b ¼

. It becomes
natural to look for interpolation of the effectiveness factor. For example, Moo-Young M
and Kobayashi T (Effectiveness factors for immobilized enzyme reactions. Can. J. Chem.
Eng . 1972, 50:162 e 167):

0 and K b / N

h 0 0 þ K b h 0 1
1 þ K b

h ¼ (17.57)

where h 0 0 is the effectiveness factor for K b ¼ 0 (or for zeroth-order kinetics) with f evaluated

0, and h 0 1

by setting K b ¼ is the effectiveness factor for K b / N (or for first-order kinetics)

with f evaluated by assuming K b / N . For slab geometry, the maximum error is about

10% and occurs at f 0.2.

An easier way to compute the effectiveness factor is to use the same f values (of finite K b )
with the two extreme cases,

¼ 1 and K b ¼

0 ½0:15 K 1

þ 0:3lnð1 þ K 1
b

h Þ þ h

1

b

h ¼ (17.58)

1 þ 0:15 K 1
b

þ 0:3lnð1 þ K 1
b

Þ

with h 0 the effectiveness factor for K b ¼ 0 and h 1 is the effectiveness factor for K b / N . The

maximum error from Eqn (17.58) is within 3%.

17.3.4.2. Isothermal Effectiveness Factor in a Porous Sphere
For spherical particles ( Fig. 17.4 b), Eqn (17.22) is reduced to

d
d r D eA 4

d C A
d r þ r A 4

pr 2 pr 2 ¼ 0 (17.59)

since the cross-sectional area perpendicular to the mass transport is the surface area of
a sphere with a radius of r . Again, let us consider the kinetics described by Eqn (17.8) , i.e.,

d
d r r 2

d C A
d r r max

C A

K A þ C A r 2 ¼ 0

D eA (17.60)

Let us define the dimensionless radius and concentration by

r
R p ;

C A
C AS

r þ ¼ and C A þ ¼ (17.61)