Page 221 - Introduction to chemical reaction engineering and kinetics

8.5 Heterogeneous Catalysis: Kinetics in Porous Catalyst Particles 203

That is, on applying equation 8.5-4 to both faces of the strip, we have

-D,Ac% = -D,A, [$$ + & &)dx] + (-rA)A,dx (8.5-6)

for any surface kinetics, where A, is the cross-sectional area perpendicular to the direction
of diffusion of A (A, is constant here and cancels). The rate law for (- rA) is not specified,
but the units of (-I*) are mol A rnd3 (particle) s-l. If we introduce first-order kinetics
((-rA) = kAcA), equation 8.5-6 becomes

d2c,ldx2 - k,c,lD, = 0 (8.5-7)

To obtain a nondimensional form of this equation, we define dimensionless concentration,
I/J, and length, z, respectively, as
t+b = c,/c, (8.5-8)
z = XlL (8.5-9)

Equation 8.5-7 in nondimensional form is then

d2$ldz2 - (kAL2/D,>+ = 0 (8.5-10)

The coefficient of $ in equation 8.5-10 is used to define a dimensionless group called the
Thiele modulus,2 4:

C#I = L(k,lD,)“2 (n = 1) (8.5-11) 1

so that equation 8.5-10 becomes

d2+ldz2 - 42$ = 0 (8.512)

The importance of 4 is that its magnitude is a measure of the ratio of intrinsic reaction
rate (through kA) to diffusion rate (through 0,). Thus, for a given value of kA, a large value
of 4 corresponds to a relatively low value of D,, and hence to relatively high diffusional
resistance (referred to as “strong pore-diffusion” resistance). Conversely, a small value of
$J corresponds to “negligible pore-diffusion” resistance.
The solution of equation 8.5-12 provides the concentration profile for I,!J as a function of
z, +(z). On integrating the equation twice, we obtain

t/t = Cle4z + C2e-4z (8.5-12a)

where Ct and C2 are integration constants to be obtained from the boundary conditions:

at z = 0, $=l (8.512b)
atz = 1, dr+Wdz = 0 (SS-lk)

*Equation 8.5-11 applies to a first-order surface reaction for a particle of flat-plate geometry with one face
permeable. In the next two sections, the effects of shape and reaction order on I$ are described. A general form
independent of kinetics and of shape is given in Section 8.5.4.5. The units of kA are such that 4 is dimensionless.
For catalytic reactions, the rate constant may be expressed per unit mass of catalyst (k&,,. To convert to kA for
use in equation 8.5-11 or other equations for C#J, (k,&, is multiplied by pP, the particle density.