Page 222 - Introduction to chemical reaction engineering and kinetics

204 Chapter 8: Catalysis and Catalytic Reactions

The second boundary condition is not known definitely, but is consistent with reactant A
not penetrating the impermeable face at z = 1. From equations 8.512a to c,

Cl = edl(e+ + e+) (8.5-12d)
C2 = eb/(e+ + e-+) (8.5-12e)

Then equation 8.5-12a becomes, on substitution for C, and C,:
e-4(1-Z) + &l-Z)
e= e4 + e-4 = cosh[4(1 - z)l (8.5-13)
cash 4

where cosh4 = (e+ + e&‘)/2.
Figure 8.10(b) shows a plot of $ = cAIcAS as a function of z, the fractional distance
into the particle, with the Thiele modulus #I as parameter. For 4 = 0, characteristic of a
very porous particle, the concentration of A remains the same throughout the particle. For
4 = 0.5, characteristic of a relatively porous particle with almost negligible pore-diffusion
resistance, cA decreases slightly as z + 1. At the other extreme, for 4 = 10, characteristic
of relatively strong pore-diffusion resistance, CA drops rapidly as z increases, indicating
that reaction takes place mostly in the outer part (on the side of the permeable face) of the
particle, and the inner part is relatively ineffective.
The effectiveness factor 77, defined in equation 8.5-5, is a measure of the effectiveness of
the interior surface of the particle, since it compares the observed rate through the particle
as a whole with the intrinsic rate at the exterior surface conditions; the latter would occur
if there were no diffusional resistance, so that all parts of the interior surface were equally
effective (at cA = c&. To obt ain q, since all A entering the particle reacts (irreversible
reaction), the observed rate is given by the rate of diffusion across the permeable face at
z = 0:

rate with diff. resist. = (-I-~) observed
77= rate with no diff. resist. (-rJ intrinsic

= rate of diffusion of A at z = 0 = (NA at z = O)A,
total rate of reaction at cAs (- RA)int
= -D,A,(dc,/dx),=, _ -D,c,(d$ldz),=,

LAckACAs - L2k,Ck,

That is,

where tanh4 = sinh@cosh$ = (e#’ - e&)l(e’# + e-4).

Note that 7 -+ 1 as r#~ --+ 0 and 77 -+ l/$ as 4 -+ large. (Obtaining the former result
requires an application of L’HBpital’s rule, but the latter follows directly from tanh 4 -+
1 as 4 -+ large.) These limiting results are shown in Figure 8.11, which is a plot of 17 as a
function of 4 according to equation 8.5-14, with both coordinates on logarithmic scales.
The two limiting results and the transition region between may arbitrarily be considered
as three regions punctuated by the points marked by G and H:
!I