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Heterogeneous Catalysis and Solid Catalysts
reactor temperature in this regime has little
effect on the reaction rate, and the apparent
activation energy drops.
Effectiveness Factor [89,455,456]
The
effectiveness factor h is the ratio of the actual
reaction rate observed on a porous catalyst
particle to the rate that would be obtained if the
inside of the particle were exposed to the temperature and reactant concentrations of the fluid. Mathematical analysis [89,457–466] of mass
transfer in porous particles of different shapes
has shown that the effectiveness factor is a
function of a dimensionless quantity, called the
Thiele modulus w [457]: for a sphere of radius R
and for a plate sealed on one side and on the
edges the thickness of which is L, w is defined by
the following equations:
m1 1/2
kv cs
sphere : ws ¼ R
ð31Þ
Deff
m1 1/2
kv cs
plate : wL ¼ L
Deff
where kv is the reaction rate constant per unit of
gross catalyst volume, cs is the concentration on
the surface, m is the reaction order, and Deff the
effective diffusion coefficient, given by
Deff ¼ D
Q
t
ð32Þ
where D is the diffusion coefficient for a pair of
fluids taking into account binary and Knudsen
diffusion, Q the void fraction of the porous
mass, and t a factor allowing for tortuosity and
varying cross sections of the pores.
Equation (31) for the plate can also be used
for arbitrary catalyst geometry if L is interpreted
as characteristic diffusion length, i.e. the ratio of
catalyst particle volume and its external surface
area.
For first-order reactions (m ¼ 1), the effectiveness factors are as follows (tan is hyperbolic
tangent):
3
1
1
Sphere : h ¼
ð33Þ
ws tanh ws ws
Plate : h ¼
tanh wL
wL
Figure 19. Effectiveness factor w as a function of the Thiele
modulus ws or sL
Flat plate sealed on one side and on edges, first-order
reaction; D Same, second-order reaction; þ Spherical particle, first-order reaction
* Reproduced with permission [462]
Correlation between the effectiveness factor and
the Thiele modulus for nonexothermic reactions
is shown in Figure 19 [462]. The effectiveness
factor is about unity for w < 1 and inversely
proportional to w for w > 3.
If the intrinsic velocity rate constant kv
(Eq. 32) cannot be determined directly, another
dimensionless modulus Q has been derived [460,462]. For first-order reactions occurring in a sphere, it is defined by
Q w2 h ¼
R2
Deff Vc Cs
dn
dt
ð34Þ
where dn/dt is the conversion rate in moles per
second of the reactant in the catalyst volume Vc.
The effectiveness factor as a function of Q is
shown in Figure 20 for a moderate energy of
activation (E ¼ 10 RT; first-order reaction in a
spherical particle) and variable enthalpy change
DH (l is the thermal conductivity of the catalyst). For exothermic reactions (b > 0), the
effectiveness factor goes through a maximum
value exceeding unity because of the interaction
of two opposing effects. Poor mass transfer
lowers the efficiency of the catalyst, whereas
insufficient heat transfer raises catalyst temperature and reaction rate.
Effects on Selectivity [89,455,456] The
effect of mass- and heat-transport processes on
the selectivity of reactions yielding more than
one product depends on the selectivity type. In a
type I reaction at low effectiveness factors, the
observed selectivity factor changes from k1/k2 to