Page 255 - Introduction to chemical reaction engineering and kinetics

236 Chapter 9: Multiphase Reacting Systems

For each shape and each rate-process-control special case in Table 9.1, the result for
ti may be obtained by setting fn = 1. For the cylinder, in the term for ash-layer diffusion,
it may be shown that (1 - fn) In (1 - fn) + 0 as fn + 1 (by use of L’Hopital’s rule).

9.1.2.3.3. Rate-process parameters; estimation of kAg for spherical particle. The
three rate-process parameters in the expressions for t(fn) (kAg, D,, and kh), may each
require experimental measurement for a particular situation. However, we consider
one correlation for estimating kAg for spherical particles given by Ranz and Marshall
(1952).
For a free-falling spherical particle of radius R, moving with velocity u relative to a
fluid of density p and viscosity p, and in which the molecular diffusion coefficient (for
species A) is D,, the Ranz-Marshall correlation relates the Sherwood number (Sh),
which incorporates kAg, to the Schmidt number (SC) and the Reynolds number (Re):

Sh = 2 + 0.6Sc1”Reu2 (9.1-32)

That is,

2RkA,lDA = 2 + 0.6(plpD,)1”(2Ruplp)1” (9.1-33)

This correlation may be used to estimate kAg given sufficient information about the
other quantities.
For a given fluid and relative velocity, we may write equation 9.1-33 so as to focus on
the dependence of kAg on R as a parameter:

Kl K2 (9.1-34)
kAg = y + R1/2
where Kl and K2 are constants. There are two limiting cases of 9.1-34, in which the
first or the second term dominates (referred to as “small” and “large” particle cases,
respectively). One consequence of this, and of the correlation in general, stems from
reexamining the reaction time t(fn), as follows.
In Table 9.1, or equations 9.1-28 and -29 for a sphere, kAs appears to be a constant,
independent of R. This is valid for a particular value of R. However, if R changes from
one particle size to another as a parameter, we can compare the effect on t(fn) of such
a change.
Suppose, for simplicity, that gas-film mass transfer is rate controlling. From Table 9.1,
in this case, for a sphere,

PB,$~B (9.1-35)
t = 3bCAgkAg

= PB~R~.~B (“small” particle) (9.1-35a)
3bCAgKl

= ~~~~3’~ (“large” particle) (9.1-35b)
g

from equation 9.1-34. Thus, depending on the particle-region of change, the dependence
of t on R from the gas-film contribution may be R2 or R312.
The significance of this result and of other factors in identifying the existence of a
rate-controlling process is explored in problem 9-4.