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8.5 Heterogeneous Catalysis: Kinetics in Porous Catalyst Particles 207

Table 8.3 Thiele modulus (4”) normalized with respect to order of
reaction (n) and asymptotic value of 77

Asymptotic value of v
n 4’ 4” 4’ * cc 4” + cc
0 L,(kAlchD,)‘” C$‘/21/2 2”2/@ lh#J”
1 L,(kA/D,)1’2 l/C/+ l/@’
2 Le(b.CAslDe) 112 f$&)l~* (2/3)“*/@ l/#F

8.5.4.4 Effect of Order of Reaction on q
The development of an analytical expression for n in Example 8-4 is for a first-order
reaction and a particular particle shape (flat plate). Other orders of reaction can be pos-
tulated and investigated. For a zero-order reaction, analytical results can be obtained
in a relatively straightforward way for both 7 and I/J (problems 8-14 for a flat plate and
8-15 for a sphere). Corresponding results can be obtained, although not so easily, for an
nth-order reaction in general; an exact result can be obtained for I,!J and an approximate
one for 7. Here, we summarize the results without detailed justification.
For an nth-order reaction, the diffusion equation corresponding to equation 8.5-12 is

d2$ldz2 - I$~+” = 0 (8.5-18)
where the Thiele modulus, 4, is

4 = L(kAc$-j1/D,)1’2 (8.5-19)
The asymptotic solution (4 -+ large) for 77 is [2/(n + 1)lu2/4, of which the result given
by 8.5-14~ is a special case for a first-order reaction. The general result can thus be
used to normalize the Thiele modulus for order so that the results for strong pore-
diffusion resistance all fall on the same limiting straight line of slope - 1 in Figure 8.11.
The normalized Tbiele modulus for this purpose is

(8.5-20)

(8.5-20a)

(8.5-2Ob)

As a result,

n -+ l/+” as 4” + large (8.5-21)

regardless of order n. The results for orders 0, 1, and 2 are summarized in Table 8.3.

8.5.4.5 General Form of Thiele Modulus
The conclusions about asymptotic values of 7 summarized in Tables 8.2 and 8.3, and
the behavior of v in relation to Figure 8.11, require a generalization of the definition
of the Thiele modulus. The result for 4” in equation 8.520 is generalized with respect
to particle geometry through L,, but is restricted to power-law kinetics. However, since