Mass Transfer Effects: Immobilized and Heterogeneous Reaction Systems - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 872

Therefore, while Eqn (17.25) is very general, it is of very limited use. At the extreme case of
mass transfer limitation, the diffusion of A into the catalyst occurs only on the thin layer of
the catalyst next to the external surface and thus the cross-sectional area along the diffusion
path can be assumed constant (not changing). Since we are looking for the asymptotic
behavior, this is not a bad assumption to make. Therefore,

"
2

# 2

x¼d p ¼

C AS

Z

d C A
d x

D eA ðr A ÞD eA d C A (17.26)

C Ae ;0

The overall reaction rate is balanced by the diffusion flux into the catalyst particle. Thus, the
effectiveness factor can be computed by

x¼d p
ðr AS ÞV

d C A
d x

D eA S "

2

# 2

Z

C AS

r A ; obs
r AS ¼

a
r AS

h ¼ ¼ ðr A ÞD eA d C A (17.27)

C Ae ;0

where V is the volume of the catalyst particle or a catalyst layer attached on the wall. At this
point, we have determined the asymptotic behavior of the effectiveness factor. One can
observe that the right-hand side of Eqn (17.27) is only a function of concentration at the
outer surface, particle size, and reaction rate and diffusivity parameters. Eqn (17.27) can be
rewritten as

j f / N ¼ f

h (17.28)

where f is called the generalized Thiele modulus. The generalized Thiele modulus is
defined as

" # 2

Z

C AS

¼ ð r AS Þ
a

f 2 ðr A ÞD eA d C A (17.29)

C Ae ;0

From this discussion, one can expect that when mass transfer limitation is extremely strong
or the generalized Thiele modulus f is large, the effectiveness factor can be computed
through Eqns (17.28) and (17.29) for any reaction kinetics and any geometry. When the Thiele
modulus is small, one can expect that the effectiveness factor is a function of catalyst particle
geometry and a weak function of the kinetics. The effect of particle geometry is obvious as
indicated by Eqn (17.24) . The effect of reaction kinetics comes from the fact that the value
of C A at which the flux is zero inside the catalyst is a function of the reaction kinetics.
For the convenience of discussion, we shall define a dimensionless parameter:

K A
C AS

K b ¼ (17.30)

When K b ¼ , Eqn (17.8) reduces

to a first-order kinetics. In most bioreactions, the value of K b is small. We next examine the
effectiveness factor at these two extreme conditions.

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