Mass Transfer Effects: Immobilized and Heterogeneous Reaction Systems - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 902

The external Thiele modulus is a function of the mass transfer coefficient and is a function of
the flow rate and particle size. Noting that r max is proportional to the surface area of the
particle, we obtain

U 0:5931
d 0:4069
p

k c aC Ab
r A ; b f

f 1
e

¼ (17.15)

A condition for negligible external mass transfer effects, i.e. h e > 0.95, is given by

h e K A K A

K A þ C Ab

f 1
e

¼ h e þ

ð1 h e ÞðK A þ C Ab Þ > 0:95 þ 18:05

(17.18)

Therefore, the effect of external mass transfer is influenced by the reaction kinetics. When the
saturation coefficient K A is larger, the external Thiele modulus is larger for the external mass
transfer effects to be negligible.
The internal mass transfer effectiveness factor is defined by

r A ; obs
r AS

h ¼ (17.27)

and the Thiele modulus is defined based on asymptotic behavior of h by Eqn (17.29) ,

" # 2

Z

C AS

¼ r AS
a

f 2 ðr A ÞD eA d C A (17.29)

C Ae ;0

where a is the specific “outer surface” area of the particle, S / V . Here the “outer surface”
area S is the area of the particle including solid surface and “pore” cross-sectional area
at the fluid e particle interface. K b is defined as a dimensionless saturation constant:

K A
C AS

K b ¼ (17.30)

When K b ¼ 0, Eqn (17.8) reduces to a zeroth-order kinetics. When K b / N , Eqn (17.8) reduces

to a first-order kinetics. In most bioreactions, the value of K b is small.
With kinetics given by Eqn (17.8) , the generalized Thiele modulus is given by

r

r max
2D eA C AS

Þ 2 ½1 K b lnð1 þ K 1

f ¼ a

b

(17.31)

1 þ K b

For isothermal systems, the use of the generalized Thiele modulus unifies all the particle
shapes and reaction kinetics when the Thiele modulus is large. When Thiele modulus is
small, both particle shape and reaction kinetics play a role in the effectiveness factor. Table
17.6 shows a summary of the internal mass transfer effects on the reaction kinetics and cata-
lyst particle shape: slab and sphere.
For nonisothermal reaction systems, two new parameters are added even in the case
where the saturation constant K A is assumed not temperature dependent:

¼ ð DH R ; A Þ D eA C AS
k eT T S

b (17.70)