A parallel - Consecutive reaction scheme in three-dimensional pore structure catalyst pellet
Elio E. Gonzo
INIQUI (UNSa-CONICET), Universidad Nacional de Salta
Facultad de Ingeniería, Buenos Aires 177, (4400) Salta, Argentina
gonzo@unsa.edu.ar
Abstract The three-dimensional pore model, that characterizes solid materials prepared by aggregating and coalescing spherical microparticles, is used to evaluate the role of structural and transport parameters in a catalytic system of parallel - consecutive first order reactions. The selection of the optimum catalyst structure (porosity), which leads to maximum reaction rates, is illustrated. The effects of intermediate product external concentration and the influence of a non-uniform catalytic activity distribution inside the pellet are also presented.
Keywords Parallel; Consecutive reactions; Diffusion-reaction; Pore structure.
I. INTRODUCTION
A few years ago, Gottifredi et al. (1994) were
able to predict the diffusional effects on
selectivity and the effectiveness factor for a
relatively complex parallel-consecutive reaction
scheme containing a reversible step and in which
all the reactions are of the first order. Also, an
approximate approach that makes possible the
solution for different pellet geometry and catalyst
activity distributions inside the particle, and gives
values extremely accurate with respect to
numerical solutions, was developed.
Today, it is an essential requirement in
heterogeneous catalysis to know structural pore
models through which an optimum balance
between diffusion and reaction can be
performed. Reyes et al. (1990) and Reyes and
Iglesia (1991) have developed very interesting
models for the porous network and transport
properties of a catalyst pellet. Particularly, the
three dimensional pore structure model which
results from the aggregates of spherical particles
partially compressed and sintered, is a very
realistic representation of a class of sol-gel
derived porous materials. The sol-gel technique
is well known (Stober et al., 1968; Xu and
Anderson, 1991). Typically, a solution of a metal
salt is converted by controlled pH precipitation
into a colloidal dispersion of spherical particles of
micrometric or sub-micrometric dimensions.
Finally, this dispersion is compacted and sintered
into pellets or extrudates.
By taking into account this porous structure
model and the parallel-consecutive reaction
scheme previously indicated, an optimal selection
of catalyst structural properties can be
performed, that leads to a maximum rate of
production of the intermediate component B.
In this work it is also shown the effect of pellet
porosity on selectivity and production rates,
when a non-uniform activity distribution is
considered.
II. ANALYSIS
The parallel-consecutive scheme here considered is
in which all the reactions are of the first order.
According to the three-dimensional pore
structure model for the catalyst pellet prepared
from a random aggregate of microspheres, the
diffusivity factor E (Reyes and Iglesia, 1993) is a
unique topological quantity, as function of the
accessible porosity f, that is described by the
equation:
![]() |
(1) |
where E is the relation between the effective diffusivity and the equivalent diffusivity corresponding to the average pore radius (rp), calculated from pellet porosity and surface area per unit volume (av) as
![]() |
(2) |
Equation (1) is valid for porosity higher than
that corresponding to the percolation threshold fc, which is the minimum porosity required to
have macroscopic transport (fc = 0.058).
The surface area av depends on pore size and
exhibits a maximum at intermediate porosity (f » 0.5). The maximum arises from competition
between creation and destruction of pore
surfaces as porosity increase. A good relation
functions between aV and f is (Reyes and Iglesia,
1993)
![]() |
(3) |
The catalyst particle is considered to have either planar, cylindrical or spherical shape (n = 0, 1, 2), respectively; with characteristic length R and with an activity distribution function f(x). This function f(x) is normalized in such a way that (Gottifredi and Gonzo, 1983):
![]() |
(4) |
where x represents the dimensionless spatial
coordinate. The distribution functions, f(x) = 1
(uniform, U) and, f(x) = ax2 (non uniform, NU),
were used.
The dimensionless continuity equations for A
and B in an isothermal pellet can be written as:
![]() |
(5) |
![]() |
(6) |
where CA and CB are the concentrations normalized with respect to the external surface value of reactant A. The groups h2, w, p, q and γ are defined as follows
![]() |
(7) |
where D represents effective diffusivities; k1, k2 and k3 are the reaction rate coefficients and K is
the equilibrium constant for the first step of the
consecutive path. Since our aim is to determine
the effect of pellet porosity on the reacting
system, the Thiele modulus h, for the
consumption of reactant A, is defined at a
reference porosity (f = 0.2) and for a value
equal to one.
Equations (5) and (6) must be solved subject
to the following boundary conditions:
(8) |
Following the work of Gottifredi et al. (1994), where the auxiliary function:
![]() |
(9) |
and the perturbation and matching technique (Gottifredi and Gonzo,1986) were used, the effectiveness factor (η) for the consumption of A is defined as:
![]() |
(10) |
Taking into account the pore structure model previously introduced, η can be obtained from the expression:
![]() |
(11) |
where:
|
(12) |
|
(13) |
|
(14) |
|
(15) |
|
(16) |
(17) |
In Eqn. (14), (Av) represents the dimensionless
surface area per unit volume of the pellet
and E' stands for the diffusivity factor, both
normalized with respect to the corres-ponding
values at an arbitrary reference porosity (f =
0.2, in this case).
As was pointed out in the previous work
(Gottifredi et al. (1994)), the selectivity for the
component B can be predicted from its usual
definition, that is:
![]() ![]() |
(18) |
where RB and RA are the net production rate of
component B and A, respectively.
By combining Eqs. (10), (11) and (18), the
production rate of component B is obtained:
![]() |
(19) |
with
and from Eqns. (10) and (11), the rate of consumption of A may be expressed as
![]() |
(20) |
Finally, the selectivity to component B can be predicted from Eqn. (18).
III. RESULTS AND DISCUSSION
The only structural-dependent parameters of
Eqns. (18), (19) and (20) are E' and AV, both of
which depend on porosity. Consequently, the
value of f that maximizes the selectivity, RA or
RB may be calculated from these equations.
To study the effect of the presence or
absence of the parallel reaction, as well as, the
diffusivities ratio and the equilibrium constant;
parameters q and k3, and p, γ, w and K,
respectively, were varied.
Typical results of the effect of porosity on the
net rate of production of component B are
illustrated in Fig.1 for n = 0 and a Thiele modulus
equal to one. This intermediate value of h is far
from the asymptotic solutions of the system. The
selected parameters for each curve in Fig. 1 and
2 are the following:
U1: CBº = 0.1, w = 0.5, γ = 1, p =1, q = 0.6; U1c: CBº = 0.1, w = 0.5, γ = 1, p =1, q = 1 (k3 = 0); U2: CBº = 0, w = 2, γ = 1, p =10, q = 0.6; U3: CBº = 0.1, w = 2, γ = 2, p =1, q = 0.6; U4: CBº = 0.45, w = 1, γ = 2, p =1, q = 0.6; NU1: CBº = 0.1, w = 0.5, γ = 1, p =1, q = 0.6; NU3: CBº = 0.1, w = 2, γ = 2, p =1, q = 0.6; NU4: CBº = 0.45, w = 1, γ = 2, p =1, q = 0.6.
These simulations clearly show how diffusion
and reaction rates compete as the catalyst
structure is modified. The maximum in RB occurs
at intermediate values of porosity (f = 0.65 to f = 0.85), depending on the values of the different
parameters.
The effect of the parallel reaction can be seen
from a comparison of U1 and U1c series, where
U1c corresponds to a purely consecutive scheme
(q = 1, k3 = 0). Since in U1c A is converted
only in B, the net production rate of B is higher
than in U1 where the parallel reaction exists (q =
0.6, k3/k1 = 2/3). Nevertheless the maximum is
not affected f = 0.75 (η = 0.66 and 0.63 respectively). For the non-uniform activity
distribution of the type considered here
(descending towards the centre of the pellet), RB increases, as may be observed by comparing U1 and NU1. But now the maximum is affected and
is obtained at f = 0.65 with η = 0.75.
Increasing the values of γ (increases the ratio
DA/DB) and increasing the equilibrium constant K
for the first consecutive reaction (p decreases),
the net volumetric rate of B increases
substantially, although the maxi-mum shifts from f = 0.75 (η = 0.51) to f = 0.85 (η = 0.84), as
it is shown in curves U2 and U3.
Furthermore it should be noticed that when a
non-uniform distribution is used (NU3), for the
same conditions than in U3, RB increases
reaching a maximum in f = 0.7 with η = 0.79,
even if the effectiveness factor is lower than in
U3.
The effect of external pellet concentration of
the intermediate product B can be seen in curve
U4 and the role of non uniform distribution of
active site, for the same conditions, in NU4. It is
interesting to note that for uniform distribution
(U4) the rate of production of B is negative for
porosity values lower than 0.63, which means
that B is consumed instead to being produced.
Nevertheless, for an activity distribution
decreasing toward the pellet centre (NU4), RB increases and reaches a maximum at f = 0.8 with η =0.873. Also, the porosity where RB = 0
is smaller than for uniform distribution (f = 0.3
with η = 0.576).
In Fig.2, it is depicted the variation of RA with f for the conditions previously indicated. As
expected, the net rate of consumption of A is
always positive, even for the cases U4 and NU4.
The maxima are reached at lower values of f than those corresponding to RB , under the same
conditions.
The effectiveness factor η and the selectivity S
increase constantly with f for all cases studied.
IV. CONCLUSIONS
A method for evaluating the porosity effect on
the net production rate of intermediate product B
and reactant A is proposed. Furthermore the
importance of selecting the optimum value of the
structural parameter to obtain maximum rates in
complex reaction system is highlighted. The
present approach provides a clear understanding
of the role of diffusion and reaction inside a
catalyst pellet.
The effect of reactant and product
concentrations and the influence of a non-uniform
catalyst distribution on the optimal porosity that
maximizes the rate RA or RB, is clearly shown.
The maximum rates occur at intermediate values
of porosity (0.65 <f< 0.85) and in a diffusionlimited
regime. The effectiveness factor for the
total consumption of A is in the range [0.51 -
0.87].
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Received: April 5, 2000.
Accepted for publication: July 9, 2001.
Recommended by Subject Editor P. Toledo.