Page 218 - Introduction to chemical reaction engineering and kinetics

200 Chapter 8: Catalysis and Catalytic Reactions

The variation of chemical potential may arise as a result of variation of concentration or
temperature or by other means, but we consider only the effect of concentration here.
From a molecular point of view inside a catalyst particle, diffusion may be consid-
ered to occur by three different modes: molecular, Knudsen, and surface. Molecular
diffusion is the result of molecular encounters (collisions) in the void space (pores) of
the particle. Knudsen diffusion is the result of molecular collisions with the walls of
the pores. Molecular diffusion tends to dominate in relatively large pores at high P,
and Knudsen diffusion tends to dominate in small pores at low P. Surface diffusion re-
sults from the migration of adsorbed species along the surface of the pore because of a
gradient in surface concentration.
Since we don’t usually know enough about pore structure and other matters to assess
the relative importance of these modes, we fall back on the phenomenological descrip-
tion of the rate of diffusion in terms of Fick’s (first) law. According to this, for steady-
state diffusion in one dimension (coordinate x) of species A, the molar flux, NA, in, say,
mol rnp2 (cross-sectional area of diffusion medium) s-r, through a particle is

NA = -D,dc/Jdn (8.54)

where D, is the effective diffusivity for A.
The effective diffusivity D, is a characteristic of the particle that must be measured for
greatest accuracy. However, in the absence of experimental data, D, may be estimated
in terms of molecular diffusivity, DAB (for diffusion of A in the binary system A + B),
Knudsen diffusivity, D,, particle voidage, l p, and a measure of the pore structure called
the particle tortuosity, rp.
An estimate for D,, is (Reid et al., 1987, p. 582):

0.00188T3”[(M* + Mn)/M*Mn]1’2
D A B = (8.54a)
PMDdh

where D, is in cm2 s-l, T is in K, MA and MB are the molar masses of A and B,
respectively, in g mol-l, P is pressure in kPa, dAB is the collision diameter, (dA + dB)/2,
in nm, and fi, is the so-called collision integral.
The Knudsen diffusivity may be estimated (Satterfield, 1991, p. 502) from

D, = 9700re(TIM)1’2
where D, is in cm2 s-r , re is the average pore radius in cm, and M is molar mass. Equa-
tion 8.54b applies rigorously to straight, cylindrical pores, and is an approximation for
other geometries.
The overall diffusivity, D*, is obtained from DAB and D, by means of the conventional
expression for resistances in series:

1 1
-=- +’ (8.54~)
D* DAB DK

The effective diffusivity is obtained from D*, but must also take into account the two
features that (1) only a portion of the catalyst particle is permeable, and (2) the diffusion
path through the particle is random and tortuous. These are allowed for by the particle
voidage or porosity, l p, and the tortuosity, rp, respectively. The former must also be
measured, and is usually provided by the manufacturer for a commercial catalyst. For
a straight cylinder, TV = 1, but for most catalysts, the value lies between 3 and 7; typical
values are given by Satterfield.