Mass Transfer Effects: Immobilized and Heterogeneous Reaction Systems - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 865

1
1 þ f e ¼ k c a K A

h e ¼

r max ;

when K A / N (17.17b)

1
1 þ

K A þ C Ab
r max

h e ¼

f e ¼

k c a þ K A þ C Ab ;

when f e / N (17.17c)

which shows a straight line on the log e log plot when mass transfer is limiting. For the case of
K A / N

(i.e. first-order reactions), f e is also known as the Damk¨hler number.
The condition for negligible external mass transfer effects corresponds to low values of f e
as one can observe from Figs 17.2 and 17.3 . One condition normally used is h e > 0.95, i.e. the

external mass transfer contributes to less than 5% reaction rate decrease. Eqn (17.14) can be
used to determine the critical value of the corresponding Thiele modulus. That is,

h e K A K A

K A þ C Ab

f 1
e

¼ h e þ

h e ÞðK A þ C Ab Þ > 0:95 þ 18:05

(17.18)

ð1

Therefore, the effect of external mass transfer is influenced by the kinetics. When the satura-
tion coefficient K A is larger, the external Thiele modulus is larger for the external mass trans-
fer effects to be negligible.
Example 17-1 Glucose oxidase catalyzes the oxidation of glucose. When glucose is present in
excess, oxygen can be considered the limiting substrate. Michaelis e Menten kinetics may be
applied to describe the oxygen-limited reaction rate. The enzyme is immobilized on the
surface of nonporous glass beads. The particles are of d p ¼ 2762 kg/m 3 .

0.001 m, r s ¼

1000 kg $ m 3 , C O 2 b ¼ 0:24 mol $ m 3 .

The medium properties are: m ¼ 0.001 Pa $ s, r L ¼

10 9 m 2 /s, and the catalyst loading

The diffusivity of oxygen in the medium is D AB ¼ 2.3

10 6 g/glass bead. The volume fraction of glass beads in the reactor is 0.4. The
kinetic constants are: r max ¼

is [E] 0 ¼

10 3 mol-O 2 /(s $ g-enzyme) and K m ¼ 0.24 mol $ m 3 . Calcu- 6.5

late the effectiveness factor and the effective rate of reaction assuming

1. no agitation (i.e. diffusion mass transfer);
2. effective medium superficial flow velocity is 1 m/s.

10 3 10 6 /( p 0.001 3 /6) mol-O 2 /m 3 -beads 12.414 mol-O 2 /m 3 -

Solution. r max ¼ 6.5 ¼

beads.
K b b ¼ K m / C Ab ¼ 0.24/0.24 ¼ 1. For spheres, a ¼ S / V ¼ 6/ d p .

1. For pure diffusion to a sphere:

Sh ¼ 2 ¼ k c d p =D AB (E17-1.1)

10 9 /0.001 m/s 10 6 m/s. k c ¼ 2 D AB / d p ¼ 2 2.3 ¼ 4.6

Thus, from Eqn (17.13)

r max
ðk c aÞðK m þ C Ab Þ ¼

12:414
4:6 10 6 ð6=0:001Þð0:24 þ 0:24Þ ¼ 937:05

f e ¼