Page 245 - Introduction to chemical reaction engineering and kinetics

9.1 Gas-Solid (Reactant) Systems 227

shrinking-core model (SCM), developed in Section 9.1.2.3, for which explicit solutions
(integrated forms of rate laws) can be obtained for various particle shapes.
In Figure 9.l(c), the opposite extreme case of a very porous solid B is shown. In this
case, there is no internal diffusional resistance, all parts of the interior of B are equally
accessible to A, and reaction occurs uniformly (but not instantaneously) throughout
the particle. The concentration profiles are “flat” with respect to radial position, but
cn decreases with respect to time, as indicated by the arrow. This model may be called
a uniform-reaction model (URM). Its use is equivalent to that of a “homogeneous”
model, in which the rate is a function of the intrinsic reactivity of B (Section 9.3) and
we do not pursue it further here.

9.1.2.2. A General Model

9.1.2.2.1. Isothermal spherical particle. Consider the isothermal spherical particle of
radius R in Figure 9.l(b), with reaction occurring (at the bulk-gas temperature) accord-
ing to 9.1-1. A material balance for reactant A(g) around the thin shell (control volume)
of (inner) radius r and thickness dr, taking both reaction and diffusion into account,
yields the continuity equation for A:

that is,

where Fick’s law, equation 8.5-4, has been used for diffusion, with D, as the effective
diffusivity for A through the pore structure of solid, and (- rA) is the rate of disappear-
ance of A; with (- rA) normalized with respect to volume of particle, each term has units
of mol (A) s-l. If the pore structure is uniform throughout the particle, D, is constant;
otherwise it depends on radial position Y. With D, constant, we simplify equation 9.1-4
to

(9.1-5)

The continuity equation for B, written for the whole particle, is

(-RB) = -2 = -,,$$ (9.1-6)

or

(-rB) = ? = -!A$! (9.1-7)