Mass Transfer Effects: Immobilized and Heterogeneous Reaction Systems - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 874

s

2 D eA C AS
r max d p

¼ ð

d x 0 ÞS

dS

¼ f ;

h ¼ for f ¼ 1 (17.38b)

Therefore, the asymptotic behavior is the same as the effectiveness factor when f 1.

17.3.2.2. Effectiveness Factor for a Zeroth-Order Reaction in an Isothermal
Porous Sphere
For a spherical geometry ( Fig. 17.4 b), we can proceed in the same manner as for the slab.
The difference is that the cross-sectional area along the diffusion path is a function of the
radius. Integrating Eqn (17.22) between outer surface and inside the sphere where reactant
just depleted (at r ¼ r 0 ) gives

D eA r 2 d C A
d x

Z Z

r

d C A

d D eA r 2 r max r 2 d r

d r ¼

(17.39)

r 0

0

which yields

d C A

d x ¼ 1

D eA r 2 r max ðr 3 r 0 Þ (17.40)

3

Divide both sides by r 2 and integrate once more,

Z

C A

Z

r

1
3

r max ðr r 0 r 2 Þ D eA d C A ¼ d r (17.41)

r 0

0

Thus, the concentration is given by

r max

6D eA ðr 2 3r 0 þ 2r 0 r 1 Þ

C A ¼ (17.42)

At the outer surface, the concentration is given by

r max

6D eA ðR p 3r 0 þ 2r 0 R 1

C AS ¼ Þ (17.43)

p

which can be solved to obtain the value of r 0 . There are three roots (for method of solution, see
Chapter 18), but only one of them is physical (i.e. greater than 0 and less than R p ):

"
4

!#

r R p ¼ 1

3 þ 1

p 12 D eA C AS

2 þ cos 3 arccos r max R p 1

(17.44)

The other two solutions are such that 4

p
3

in Eqn (17.44) is replaced by either 0 or 2

p
3

.

Clearly, when r 0 ¼ 0, the effectiveness factor is unit (no mass transfer effect). Because