Page 252 - Introduction to chemical reaction engineering and kinetics

234 Chapter 9: Multiphase Reacting Systems

same as the corresponding part of equation 9.1-28. (This can be repeated for each of the
two other cases of single-process control, gas-film control and ash-layer control (the latter
requires use of the QSSA introduced in Example 9-1); see problem 9-1 for these and also
the comment about other cases involving two of the three processes as “resistances.“)

SOLUTION

Referring to the concentration profiles for A in Figure 9.2, we realize that if there is no
resistance to the transport of A in either the gas film or the ash layer, c, remains constant
from the bulk gas to the surface of the unreacted core. That is,

cAg = cAs = cAc

and, as a result, from equation 9.1-22,

Combining this with equation 9.1-24 for (-Rn) and equation 9.1-8a for the stoichiometry,
we obtain

o r

dt = -(pBm/bkhcAg)drC
Integration from rc = R at t = 0 to rc at t results in

t = t,(l - r-,/R)

= t,[l - (1 - fn)‘“] (9.1-31)
from equation 9.1-27, where the kinetics parameter tl, the time required for complete
reaction, is given by

tl = bmR/bkhcAg (9.1-31a)

These results are, of course, the same as those obtained from equations 9.1-28 and -29 for
the special case of reaction-rate control.
9.1.2.3.2. Summary of t(&) for various shapes. The methods used in Examples 9-1
and 9-2 may be applied to other shapes of isothermal particles (see problems 9-1 to 9-3).
The results for spherical, cylindrical, and flat-plate geometries are summarized in Table
9.1. The flat plate has one face permeable (to A) as in Figure &lo(a), and the variable
1, corresponding to r,, is the length of the unreacted zone (away from the permeable
face), the total path length for diffusion of A and reaction being L. For the cylinder, the
symbols r, and R have the same significance as for the sphere; the ends of the cylinder
are assumed to be impermeable, and hence the length of the particle is not involved in
the result (alternatively, we may assume the length to be > r,.).
In Table 9.1, in the third column, the relation between fn and the particle size pa-
rameters (second column), corresponding to, and including, equation 9.1-27, is given
for each shape. Similarly, in the fourth column, the relation between t and fu, corre-
sponding to, and including, equation 9.1-28 is given.