Page 248 - Introduction to chemical reaction engineering and kinetics

230 Chapter 9: Multiphase Reacting Systems

“blurring” is significant, a more general model (Section 9.1.2.2) may have to be used.
The SCM differs from a general model in one important aspect, as a consequence of
the sharp boundary. According to the SCM, the three processes involving mass transfer
of A, diffusion of A, and reaction of A with B at the surface of the core are in series.
In the general case, mass transfer is in series with the other two, but these last two
are not in series with each other-they occur together throughout the particle in some
manner. This, together with a further assumption about the relative rates of diffusion
and movement of the reaction surface, allows considerable simplification of the solution
of equation 9.1-5. This is achieved in analytical form for a spherical particle in Example
9-1. Results for other shapes can also be obtained, and are explored in problems at the
end of this chapter. The results are summarized in Section 9.1.2.3.2.
In Figure 9.2, c.+ is the (gas-phase) concentration of A in the bulk gas surrounding
the particle, cAs that at the exterior surface of the particle, and cAc that at the surface of
the unreacted core of B in the interior of the particle; R is the (constant) radius of the
particle and rC is the (variable) radius of the unreacted core. The concentrations CA8
and cAs are constant, but cAc decreases as t increases, as does r,, with corresponding
consequences for the positions of the profiles for cn, on the left in Figure 9.2(a) and (b),
and cA, on the right.

For a spherical particle of species B of radius R undergoing reaction with gaseous species
A according to 9. l-l, derive a relationship to determine the time t required to reach a
fraction of B converted, fn, according to the SCM. Assume the reaction is a first-order
surface reaction.

SOLUTION

To obtain the desired result, t = t(fB), we could proceed in either of two ways. In one,
since the three rate processes involved are in series, we could treat each separately and add
the results to obtain a total time. In the other, we could solve the simplified form of equation
9.1-5 for all three processes together to give one result, which would also demonstrate the
additivity of the individual three results. In this example, we use the second approach (the
first, which is simpler, is used for various shapes in the next example and in problems at
the end of the chapter).
The basis for the analysis using the SCM is illustrated in Figure 9.3. The gas film,
outer product (ash) layer, and unreacted core of B are three distinct regions. We derive the
continuity equation for A by means of a material balance across a thin spherical shell in
the ash layer at radial position Y and with a thickness dr. The procedure is the same as that
leading up to equation 9.1-5, except that there is no reaction term involving (- r,& since
no reaction occurs in the ash layer. The result corresponding to equation 9.1-5 is

(9.1-17)

Equation 9.1-17 is the continuity equation for unsteady-state diffusion of A through the
ash layer; it iS unsteady-state because CA = cA(r, t). To simplify its treatment further, we
assume that the (changing) concentration gradient for A through the ash layer is established
rapidly relative to movement of the reaction surface (of the core). This means that for an
instantaneous “snapshot,” as depicted in Figure 9.3, we may treat the diffusion as steady-
state diffusion for a fixed value of r,y i.e., CA = CA(r). The partial differential equation,