Page 229 - Introduction to chemical reaction engineering and kinetics

8.5 Heterogeneous Catalysis: Kinetics in Porous Catalyst Particles 211

For a catalyst particle to be isothermal while reaction is taking place within it, the en-
thalpy generated or consumed by reaction must be balanced by enthalpy (heat) trans-
port (mostly by conduction) through the particle. This is more likely to occur if the
enthalpy of reaction is small and the effective thermal conductivity (k,, analogous to
0,) of the catalyst material is large. However, should this balance not occur, a temper-
ature gradient exists. For an exothermic reaction, T increases with increasing distance
into the particle, so that the average rate of reaction within the particle is greater than
that at T,. This is the opposite of the usual effect of concentration: the average rate is
less than that at ckc. The result is that vex0 > nisoth. Since the effect of increasing T
on rate is an exponential increase, and that of decreasing cA is usually a power-law de-
crease, the former may be much more significant than the latter, and vex0 may be > 1
(even in the presence of a diffusional resistance). For an isothermal particle, nisoth < 1
because of the concentration effect alone. For an endothermic reaction, the effect of
temperature is to reinforce the concentration effect, and r)en&, < r)isarh < 1.
The dependence of q on T has been treated quantitatively by Weisz and Hicks (1962).
We outline the approach and give some of the results for use here, but omit much of
the detailed development.
For a first-order reaction, A + products, and a spherical particle, the material-
balance equation corresponding to equation 8.5-7, and obtained by using a thin-shell
control volume of inside radius r , is

I d2cA 2dcL/CAC -0
dr2 r dr D, A -

(the derivation is the subject of problem 8-13). The analogous energy-balance equation
is
d2T 2 d T (-AHRA)kA _ o (8.535)
dr2+rdr+ ke cA -
Boundary conditions for these equations are:

At particle surface: r = R; T = T,; cA = cAs (8.536)
At particle center: r = 0; dTldr = 0; dcAldr = 0 (8.5-37)

Equations 8.5-34 and -35 are nonlinearly coupled through T, since k, depends expo-
nentially on T. The equations cannot therefore be treated independently, and there is
no exact analytical solution for CA(r) and T(r). A numerical or approximate analytical
solution results in n expressed in terms of three dimensionless parameters:

v(T) = T(&YY,P) (8.5-38)

where C$ (= R(kAID,)l”, Table 8.1) is the Thiele modulus, and y and /3 are defined as
follows:

y = E,IRT, (8.5-39)
p - AT;max - De(-AHRA)CAs (8.5-40)
s k,Ts
where AT,,,,, is the value of ATp when cA(r = 0) = 0. For an exothermic reaction,
/3 > 0; for an endothermic reaction, p < 0; for an isothermal particle, p = 0, since
ATp = 0.