Page 250 - Introduction to chemical reaction engineering and kinetics

232 Chapter 9: Multiphase Reacting Systems

where the total rate (for the particle) (-RA) is in mol s-l, kh is in m s-l, and the spe-
cific rate (-Ye), normalized with respect to the area of the core, is in mol rnd2 s-l. The
solution of equation 9.1-18, to obtain an expression for cAc for use in equation 9.1-22, is
straightforward (but tedious), if we use the substitution y = dc*/dr. We integrate twice,
first to obtain dc*ldr and second to obtain CA, using the boundary conditions to evaluate
the integration constants, and eliminate C~ to obtain c.&. The first integration, together
with equation 9. l- 11, results in

dc, _ kA,R2 (4
- D, (CAg - %);
dr
Integration of (A), together with equation 9.1-21, gives

kA,R2

CA = CA, + ~ c - c.&(; - ;)
D, ( Ag c
Applying equation (A) at the surface of the core (r = r,), together with equation 9.1-20,
we obtain one expression for ckr in terms of cAc:

cAs = cAg - (kA,r,2~kA,R2>CA, (Cl

Similarly, from equation (B), together with equation 9.1-19, at the particle surface, we
obtain another expression:

kA,R2
ch = cAc +
D,(%
On elimination of cAs from equations (C) and (D), and substitution of the resulting expres-
sion for cAc in equation 9.1-22, we obtain, with rearrangement,

4TCAg
(-RA) = 1 (9.1-23)
R-r 1
+ ti ’ k&r:
kA,R2
This is the end of step (l), resulting in an expression for (-RA) in terms of (fixed) r,.
In step (2) of the solution of equation 9.1-17, we allow the core surface, fixed in step
(l), to move, and integrate the continuity equation for B, using the first part of equation
9.1-6. For this purpose, we substitute both equations 9.1-23 and 9.1-6, the latter written in
the form

(-RB) = -2 = -$(pBmimf) = -4n-pB,rT% (9.1-24)

into the stoichiometric relation 9.1~8a, resulting in

(9.1-25)

Integration of equation 9.1-25, from t = 0, r, = R to t, rc, results in

(9.1-26)