Page 246 - Introduction to chemical reaction engineering and kinetics

228 Chapter 9: Multiphase Reacting Systems

From the stoichiometry of reaction 9.1-1, (--TV) and ( -rB) are related by

(-r~) = b(-rd (9.1-8)
or

(-43) = b(-K4) (9.1-8a)

where (- RA) is the extensive rate of reaction of A for the whole particle corresponding
to (-RJj).
Equations 9.1-5 and -7 are two coupled partial differential equations with initial and
boundary conditions as follows:

att = 0, cB = cBo = PBm (9.1-9)

(9.1-10)
cA = cAg

(9.1-11)
= kAg(cAg - ch,>

which takes the external-film mass transfer into account; kAg is a mass transfer coef-
ficient (equation 9.2-3); the boundary condition states that the rate of diffusion of A
across the exterior surface of the particle is equal to the rate of transport of A from
bulk gas to the solid surface by mass transfer;

at r = 0, (dcA/dr),,o = 0 (9.1-12)
corresponding to no mass transfer through the center of the particle, from consideration
of symmetry.
In general, there is no analytical solution for the partial differential equations above,
and numerical methods must be used. However, we can obtain analytical solutions for
the simplified case represented by the shrinking-core model, Figure 9.l(a), as shown in
Section 9.1.2.3.

9.1.2.2.2. Nonisothermal spherical particle. The energy equation describing the pro-
file for T through the particle, equivalent to the continuity equation 9.1-5 describing the
profile for CA, may be derived in a similar manner from an energy (enthalpy) balance
around the thin shell in Figure 9.l(b). The result is

where k, is an effective thermal conductivity for heat transfer through the particle (in
the Fourier equation), analogous to D, for diffusion, AHRA is the enthalpy of reaction
with respect to A, and CPn is the molar heat capacity for solid B (each term has units of
J mP3 s-l, say). The initial and boundary conditions for the solution of equation 9.1-13
correspond to those for the continuity equations:

at t = 0, T = Tg (9.1-14)

at r = R, k,(dTldr),,, ‘= h(T, - Tg) (9.1-15)