Page 230 - Introduction to chemical reaction engineering and kinetics

212 Chapter 8: Catalysis and Catalytic Reactions

The result for ATP,,,, contained in equation 8.540 can be obtained from the follow-
ing energy balance for a control surface or a core of radius Y:

rate of thermal conduction across control surface
= rate of enthalpy consumption/generation within core
= rate of diffusion of A across control surface X (-AH& (8.5-41)

That is, from Fourier’s and Fick’s laws,

(8.542)

Integration of equation 8.5-42 from the center of the particle (r = 0, T = T,, cA =
cAO) to the surface (r = R, T = T,, CA = c&, with k,, D,, and (-AURA) constant, re-
sults in

ATp = T, - To = D,(-AHRA) (cAs - CAo) (8.5-43)
k
e
or, with CA0 + 0,and AT,, -+ AT,,,,,

qvnlzx = D~(-AHRA)CA~
k
as used in equation 8.5-40.
Some of the results of Weisz and Hicks (1962) are shown in Figure 8.12 for y = 20,
with n as a function of 4 and p (as a parameter). Figure 8.12 confirms the conclusions
reached qualitatively above. Thus, vex0 (p 3 0) > qisorh (/3 =’ 0), and vex0 > 1 for rel-
atively high values of p and a sufficiently low value of 4; nendO < visorh < 1. At high
values of /? and low values of 4, there is the unusual phenomenon of three solutions
for n for a given value of p and of 4; of these, the high and low values represent stable
steady-state solutions, and the intermediate value represents an unstable solution. The
region in which this occurs is rarely encountered. Some values of the parameters are
given by HlavaCek and KubiCek (1970).

8.56 Overall Effectiveness Factor q0
The particle effectiveness factor n defined by equation 8.5-5 takes into account con-
centration and temperature gradients within the particle, but neglects any gradients
from bulk fluid to the exterior surface of the particle. The overall effectiveness factor
q0 takes both into account, and is defined by reference to bulk gas conditions (c&, T,)
rather than conditions at the exterior of the particle (c,,+ T,):

q. = t-A(ObSH-Ved)/?-,(C~g, Tg) (8.5-45)

Here, as in Section 8.5.4, we treat the isothermal case for r),, and relate r10 to 7. no may
then be interpreted as the ratio of the (observed) rate of reaction with pore diffusion
and external mass transfer resistance to the rate with neither of these present.
We first relate no to q, kA, and kAs, the last two characterizing surface reaction and
mass transfer, respectively; mass transfer occurs across the gas film indicated in Figure
8.9. Consider a first-order surface reaction. If (-rA) is the observed rate of reaction,