Page 256 - Introduction to chemical reaction engineering and kinetics

9.1 Gas-Solid (Reactant) Systems 237
9.1.3 Shrinking Particle

9.1.3.1 General Considerations
When a solid particle of species B reacts with a gaseous species A to form only gaseous
products, the solid can disappear by developing internal porosity, while maintaining
its macroscopic shape. An example is the reaction of carbon with water vapor to pro-
duce activated carbon; the intrinsic rate depends upon the development of sites for
the reaction (see Section 9.3). Alternatively, the solid can disappear only from the
surface so that the particle progressively shrinks as it reacts and eventually disap-
pears on complete reaction (fs = 1). An example is the combustion of carbon in
air or oxygen (reaction (E) in Section 9.1.1). In this section, we consider this case,
and use reaction 9.1-2 to represent the stoichiometry of a general reaction of this
type.
An important difference between a shrinking particle reacting to form only gaseous
product(s) and a constant-size particle reacting so that a product layer surrounds a
shrinking core is that, in the former case, there is no product or “ash” layer, and hence
no ash-layer diffusion resistance for A. Thus, only two rate processes, gas-film mass
transfer of A, and reaction of A and B, need to be taken into account.

9.1.3.2 A Simple Shrinking-Particle Model
We can develop a simple shrinking-particle kinetics model by taking the two rate-
processes involved as steps in series, in a treatment that is simpler than that used for
the SCM, although some of the assumptions are the same:
(1) The reacting particle is isothermal.
(2) The particle is nonporous, so that reaction occurs only on the exterior surface.
(3) The surface reaction between gas A and solid B is first-order.
In the following example, the treatment is illustrated for a spherical particle.

For a reaction represented by A(g) + bB(s) -+ product(g), derive the relation between time
(t) of reaction and fraction of B converted (f,), if the particle is spherical with an initial
radius R,, and the Ranz-Marshall correlation for k,,(R) is valid, where R is the radius at
t. Other assumptions are given above.

SOLUTION

The rate of reaction of A, (- RA), can be expressed independently in terms of the rate of
transport of A by mass transfer and the rate of the surface reaction:

C-Q = k/#W-R2(c+, - c,c,J (9.1-36)

(-RA) = kh4’lrR2ch (9.1-37)

The rate of reaction of B is (cf. equation 9.1-24)

(- RB) = -4~p,,R2(dRldt) (9.1-38)