Solid Catalyzed Reactions

GBH Enterprises, Ltd.

Process Engineering Guide:
GBHE-PEG-RXT-808

Solid Catalyzed Reactions

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CONTENTS

Page

0 INTRODUCTION/PURPOSE

3

1 SCOPE

3

2 FIELD OF APPLICATION

3

3 DEFINITIONS

3

4 GENERAL BACKGROUND

4

4.1 General Considerations

4

5 SOLID CATALYZED GAS REACTIONS

7

5.1 Reaction Kinetics

7

5.2 Tests for Transport Limitations

13

5.3 Building a Reaction Kinetic Equation

15


6 INTRAPARTICLE

19

6.1 Types of Pore System

19

6.2 The Catalyst Effectiveness Factor

20

6.3 The Measurement of Effective Diffusivity

25

7 ENHANCEMENT OF INTRAPARTICLE

30

8 NOMENCLATURE

33

8.1 Dimensionless Parameters

34

8.2 Greek Letters

34

8.3 Subscripts

35

9 BIBLIOGRAPHY

35

9.1 Further Reading

36


APPENDICES
A LANGMUIR - HINSHELWOOD KINETICS

37

FIGURES
1

EFFECTIVE RATE CONSTANT

7

2

ITERATIVE APPROACH TO REACTOR MODEL
DEVELOPMENT

8

3

COMMON LABORATORY MICROREACTORS (FLOW TYPE)

9

4

THE BERTY REACTOR

11

5

STEPS IN BUILDING A REACTION RATE EQUATION

15

6

A CENTRAL-COMPOSITE DESIGN FOR TWO FACTORS

17

7

FIRST ORDER ISOTHERMAL IRREVERSIBLE
REACTION WITHIN A CATALYST SPHERE

21

INTEGRAL YIELD vs CONVERSION SHOWING EFFECT OF
PELLET DIFFUSION

24

PREDICTED AND EXPERIMENTAL EFFECTIVENESS
FACTORS

27

STRUCTURAL PERMEABILITY vs PRESSURE PARAMETER Z
FOR BI-MODAL SUPPORTS

29

EFFECTIVENESS FACTOR vs THIELE MODULUS AND
INTRAPARTICLE PECLET NUMBER

31

RELATIVE INCREASE IN CATALYST PERFORMANCE

32

8

9

10

11

12


TABLES
1

EXAMPLES OF REACTOR TYPES

4

2

TYPICAL OPERATING PARAMETERS

5

3

BSERVED REACTION KINETIC PARAMETERS

6

4

REACTOR RATINGS

12

5

EXPERIMENTAL TESTS FOR INTRAPARTICLE AND
INTERPHASE GRADIENTS

14

HEURISTICS FOR BUILDING A HYPERBOLIC-TYPE
RATE EQUATION

19

7

TYPES OF PORE SYSTEM

20

8

COMPARISON OF PORE DIFFUSIVITIES BETWEEN THE
DIFFUSION CELL AND PACKED BED METHODS

26

6


0

INTRODUCTION / PURPOSE

This Guide is one of a series produced under the auspices of the GBH
Enterprises.

1 SCOPE
This Guide covers the reaction kinetics and diffusional aspects of solid catalyzed
reactions. It does not deal with either the fluid dynamics or the mechanical design
of reactors for solid Catalyzed reactions.
2 FIELD OF APPLICATION
This Guide applies to process engineers in GBH Enterprises
world-wide.
3 DEFINITIONS
For the purposes of this Guide, the following definitions apply:
Effectiveness Factor

The ratio of the global rate to the intrinsic rate of
reaction.

Thiele Modulus

A dimensionless group representing the relative
importance of diffusion and reaction taking place in a
pellet.

With the exception of terms used as proper nouns or titles, those terms with initial
capital letters which appear in this document and are not defined above are
defined in the Glossary of Engineering Terms.


4

GENERAL BACKGROUND

4.1
General Considerations
There are numerous possibilities for technical scale realization of solid catalyst
reactors. This is illustrated in the following Table 1 which gives examples.


TABLE 1 EXAMPLES OF REACTOR TYPES

Further information about currently operated reactors may be found by retrieving
information from a data base, see GBHE-PEG-RXT-800, Clause 9.

Although no type is precluded from featuring in a batch process, the boiling bed
is the most usual one to use as a batch reactor.
.
Some general pointers to reactor choice and operating conditions are given in
Table 2.
TABLE 2 TYPICAL OPERATING PARAMETERS

Mixing:

5(i) and 5(ii) demand good premixing. If also 2(ii), might use
back-mixed fluidized bed. 5(iii) and 1(i) call for plug flow.

Temperature:

1(i) and 1(ii) likely to have an optimum temperature.
2(i) with 3(i) needs interbed cooling, cold shot or tubular.
2(i) with 3(iii) needs interbed heating, hot shot, or tubular.

Pressure:

2(i) with 4(i) suggests low pressure.
2(i) with 4(iii) suggests high pressure.

Catalyst:

5(i) surface coating,
5(ii) low porosity particles, surface impregnated,
5(iii) large particles, high internal surface area.

The classification of chemical reaction rate as more or less fast compares
chemical rate with the rate of diffusive steps. The chemical rate is said to be very
fast relative to the rate at which reacting species can penetrate from the bulk fluid
to the catalyst particle. The rate per unit volume of particle will asymptote to the
first order expression.


where:
k = mass transfer coefficient L/t
A = catalyst particle bulk surface area L2
V = catalyst particle bulk volume L3
C = reactant concentration moles/L3
The chemical rate is described as fast when it is fast relative to the catalyst
particle diffusive processes, but slow relative to external mass transfer. Here, the
rate per unit volume of particle will asymptote to

where:
reaction rate constant t-1 (moles/L3) -(m-1)
order of reaction
effectiveness factor

kR
m
ƞ

=
=
=

for

ɸ 10, isothermal conditions.
ɸ = Thiele Modulus

where:
D = effective diffusivity L2/t
For a fuller discussion of these aspects, see Satterfield [Ref. 2].


A computer program is available for calculating effectiveness factors for complex
reactions in non-isothermal conditions, with a rigorous treatment of multicomponent diffusion. See GBHE-PEG-RXT-819.
If the chemical rate is slow relative to the inner diffusive steps; then the reaction
rate per unit volume of particle will asymptote to :

i.e. ƞ = 1
The observed reaction order (m') and the activation energy (Eapp) depend upon
the particular rate - controlling regime, as indicated in Table 3 below:
TABLE 3 OBSERVED REACTION KINETIC PARAMETERS

Figure 1 illustrates the full range for a spherical particle with first order reaction.
K eff = effective first order reaction rate constant.


FIGURE 1 EFFECTIVE RATE CONSTANT

5

SOLID CATALYZED GAS REACTIONS

This section reviews progress and suggests guidelines in the areas of reaction
kinetics and intrapellet diffusion. The ideal of a mechanistic modeling approach is
unlikely to be reached by the kind of research program feasible in an industrial
laboratory. Limited resources, and particularly time, make it essential to identify a
compromise between this ideal and a completely empirical approach.
However, the precise synthesis of chemical transformations, occurring at the
catalyst surface, and the restricted rates of mass and heat transfer to and from
the surface, into the form of a practical mathematical model, containing the
smallest number of independently measurable parameters, is far from obvious.
A detailed case study is provided in GBHE-PEG-RXT-815 (The Selective
Oxidation of n-Butane to Maleic Anhydride) to illustrate the philosophy.
5.1

Reaction Kinetics

5.1.1 Objectives
A sensible approach would demand the writing down of the bare minimum
number of reactions necessary to describe key features of the observed product
distribution, such as conversion and selectivity to desired product, and key
byproducts which must be removed downstream from the reactor.

Then, classical rate equations of either the power-law type
e.g.

or of the Langmuir-Hinshelwood form (hyperbolic)(see Appendix A and [Refs. 1
and 2]), for example:

are written for each reaction.
The model parameters k1, α and β or k2, KA and KB are estimated by fitting the
proposed model to product distribution data. (see GBHE-PEG-RXT-821)
Preferably, isothermal rate data should be gathered in the chemical ratecontrolling regime. Alternatively, apparent kinetics can be developed on the
actual-sized catalyst particles to be used in the reactor, although this removes
one important variable - particle size – from subsequent optimization.
These data, together with the residence time distribution (RTD) and heat and
mass transfer data (or correlations) can be assembled into a reactor model which
offers the prospect of modest (but certainly not wide-ranging) extrapolations and
practical assessment of alternative reactor designs.
Such an approach can be achieved with a modest outlay of resources and in a
reasonable time scale of months, rather than years.
It will usually be necessary to test the model against other data from which it was
not derived (e.g. semi-technical or pilot-plant) in order to build confidence in its
utility. Iteration back and forth between laboratory and pilot plant is inevitable.
The scheme of model building is portrayed in Figure 2.


FIGURE 2 ITERATIVE APPROACH TO REACTOR MODEL DEVELOPMENT

5.1.2 Laboratory Reactors
In choosing a laboratory reactor for kinetic studies the following points need to
be considered:
(a) ease of accurate chemical analysis.
(b) insurance of isothermality;
(c) an ideal RTD (i.e. plug flow or CSTR);
(d) absence of inter-phase concentration and temperature gradients;
(e) absence of intraparticle concentration and temperature gradients;
(f) ease of data analysis;
(g) cost and ease of construction;
(h) Ability to study deactivating catalysts.

Unfortunately, no single reactor simultaneously meets all these criteria. The
preferred approach, if convenient, is to employ two types of laboratory reactor
which, when combined, span the range of requirements.
Laboratory reactors are classified under three types - differential, integral or
CSTR (recycle), as depicted in Figure 3.
FIGURE 3 COMMON LABORATORY MICROREACTORS (FLOW TYPE)

5.1.2.1 Differential Type
A differential reactor comprises typically a short section of 1/8" or 1/4"
stainless steel tubing containing a small quantity of powdered catalyst (<1
gm) supported on quartz wool and held in a temperature controlled oven
or furnace. The temperature within the catalyst bed is measured
with the aid of a fine thermocouple and sample points are fitted fore and
aft the bed in order to measure the small composition changes across the
reactor (∆CA 0.1Co, typically). From the measured conversion, mass of
catalyst (W) and reactant flow rate (FA), the reaction rate may be
estimated directly:Refinery Process Stream Purification Refinery Process Catalysts Troubleshooting Refinery Process Catalyst Start-Up / Shutdown

Because the conversion is small, isothermality can usually be assured and
the RTD is unimportant, although by-passing is a potential hazard.
The greatest practical limitation of the differential reactor is the intrinsic
difficulty of determining ∆CA. Rather than relying on the accurate
measurement of small changes in reactant composition, it may actually be
preferred to estimate ∆CA from product composition measurements and
stoichiometry.
It is obviously necessary to produce synthetic feeds covering the ranges of
reactant and product compositions anticipated within the commercial
reactor. This aspect, too, may prove troublesome.
5.1.2.2

Integral Type

Integral reactors are also of the tubular type and their dimensions may
vary widely from the microreactor scale (0.15m x 0.006m diameter packed
with crushed catalyst) to a single plant scale reactor tube (5m x 0.028m
diameter, say, containing 5 mm catalyst extrudates). High conversions,
typical of commercial practice, are sought over this type of reactor,
perhaps as high as 95% in certain cases. Now, of course, it is much more
difficult to ensure isothermality of the bed, and special cooling (or heating)
arrangements are often necessary, such as immersing the reactor in a
molten salt bath or a fluidized bed. It may even be necessary to dilute the
catalyst with inert packing in order to limit the rate of reaction per unit bed
volume.
Because these reactors tend to be long and narrow, the ideal of plug flow
is closely approached. However, since the reactant conversion changes
considerably over the reactor, Equation (1) must be replaced by the
differential form :


It now becomes necessary to infer a functional form for r (CA), insert it into
Equation (2) and integrate to obtain CA for comparison with observations
of CA at various values of W/FA. This is obviously more complicated than
using Equation (1) to determine the reaction rate directly.
A comparison of reaction kinetics for a complex commercial reaction
system – xylenes isomerization - using both differential and integral
reactors is given by Orr and co-workers [Ref.!3].
High temperature "shift" kinetics employing integral reactors are dealt with
in some depth by Chinchen and Spencer [Ref. 4].
5.1.2.3

CSTR Type

Bearing in mind the limitations of the differential and integral reactors
mentioned above, much effort has gone into developing the so-called
"ideal" gradientless reactor of the CSTR type. The preferred variant seems
to be the internal recirculation or "Berty" reactor, as it has come to
be known. One such type of Berty reactor is shown in Figure 4. The
catalyst is contained within a cruciform basket formed from fine wire mesh.
Recirculation of the gas is achieved by means of a magnetically driven
impeller. Suitable baffling around the periphery of the chamber
containing the basket is designed to promote complete mixing.


FIGURE 4 THE BERTY REACTOR

Since the reactor is supposed to behave like a CSTR, the rate of reaction is
given directly in terms of the integral conversion measured over the reactor, i.e.

In spite of the widespread publicity of the Berty reactor serious shortcomings in
its design were demonstrated by Caldwell [Ref. 5]. These remain to be resolved.
The alternative proposed by Caldwell appears to achieve acceptable rates of
interphase mass transfer, but temperature gradients can exist within the bed
[Ref. 6].
This type of reactor is unsuited to the use of powdered catalyst, so it fails to
address the important question of intraparticle gradients (pore diffusion).
5.1.2.4

Catalyst deactivation

It would be wrong in passing not to mention the problem of catalyst deactivation.
All catalysts deactivate to a greater or lesser extent by one or more of several
mechanisms such as thermal sintering, poisoning by trace impurities (e.g. sulfur)
or fouling due to "coke" formation.
A central feature of the laboratory study may be to isolate the mechanism of
deactivation in a particular case and even to quantify its rate.
If an integral tubular reactor were used, the poison or coke precursors may often
be adsorbed preferentially on the upstream portions of the bed. A band of
deactivated catalyst moves along the bed with increasing time in a
chromatographic effect while the remainder of the bed remains active. The
results may thus be difficult to interpret.
CSTR reactors are better suited for studying catalyst deactivation because of
their gradientless conditions. Ingenious extensions to their design permit the online measurement of coke deposition through thermal gravimetry [Ref. 7].


5.1.2.5

Reactor Ratings

To summarize what has been said the different types of reactor are rated good,
fair or poor according to the requirements stated at the beginning of 5.1.2 in
Table 4 below.
TABLE 4 REACTOR RATINGS

An excellent review of multiphase laboratory reactors and their limitations
is given by Weekman [Ref. 8].
5.2

Tests for Transport Limitations
In order to ensure that kinetic data obtained in a laboratory reactor reflect
truly the chemistry of the reactions concerned, gradients (both thermal and
mass) should ideally be eliminated within the particles (intraparticle) and
between the external surface of the particles and the adjacent fluid
(interphase). Additionally, temperature gradients between the local fluid
regions or catalyst particles (interparticle) must be removed, and
interparticle mass dispersion may need to be measured separately.
Mears [Ref.9] has studied this problem in some depth and draws the
following conclusions:

5.2.1

Heat Transfer
As far as temperature gradients are concerned, the order of importance is


Elimination of interparticle temperature gradients guarantees isothermality
on a local level As well. Tube diameter is the major variable for
consideration in restricting the radial temperature gradient. When the heat
transfer resistance at the wall is significant, Mears shows that for a 5%
deviation in the rate of reaction across the bed radius,

where λr,eff is the effective thermal conductivity of the bed and Biw, the wall Biot
number, is a measure of the ratio of the thermal resistance of the bed to that of
the wall. These two parameters are considered in more detail in GBHE-PEGRXT-810.
Minimizing interparticle temperature gradients is most easily brought about by
limiting dt, rather than the reaction rate per unit bed volume, rb, (by catalyst
dilution, for example).
5.2.2 Mass Transfer
The picture is reversed for concentration gradients, i.e.

Thus, if pore diffusion can be eliminated by restricting the particle diameter dp,
both interphase and interparticle mixing effects are usually negligible, although
the axial dispersion problem becomes more pronounced with increasing
conversion or reaction order, or with decreasing Peclet numbers (i.e. molecular
diffusion).
Direct tests for intraparticle and interphase mass transfer resistances are
summarized in Table 5.


TABLE 5

EXPERIMENTAL TESTS FOR INTRAPARTICLE
AND INTERPHASE GRADIENTS

Note:
The range of dp commonly used in a Berty reactor is limited by
basket dimensions on the one hand (large dp) and pressure drop
across the bed (small dp) on the other.
Consequently, the usual test of reducing dp progressively is not
very satisfactory. Berty suggests the alternative above. It is also
given in Rase's book [Ref. 10].
The intraparticle gradient test for differential and integral plug flow reactors
is well established. Smaller particle sizes are usually produced by
crushing. Since particle size tends to be fixed in the CSTR arrangements,
a rather convoluted test has to be carried out. This involves increasing the
total pressure while keeping the reactant partial pressure fixed. If diffusion
within the catalyst is bulk controlled, the increase in pressure leads to a
decrease in the effective diffusivity, and thus a decrease in observed rate.
However, one should be aware that the changes in partial pressure of
other components could affect the intrinsic reaction rate. Where diffusion
is Knudsen controlled (i.e molecule-pore wall collisions) this test fails
completely, even though the reaction may be diffusion limited.
One would also expect to see the apparent activation energy decrease by
a factor of two in moving from a chemical to a diffusion rate controlled
regime.


It is also expedient to treat the interphase gradient tests with some
caution. At low Reynolds number i.e. Re = G dp /µ <1, as might occur in
micro-reactors containing crushed catalyst, the mass transfer coefficient
kg appears to be independent of FA, and thus no change in the ratio of the
mass transfer rate to the chemical rate is brought about. To reiterate,
however, satisfaction of the intraparticle gradient test should ensure that
interphase mass transfer cannot intrude.
Clearly increasing the stirrer speed in the CSTR should improve
interphase mass transfer, but by-passing can mask this effect.
5.3

Building a Reaction Kinetic Equation
The development of a rate equation that is adequate for design purposes
involves iteration around the 5 steps shown in Figure 5. Each step should
be given careful thought so as to minimize the number of iterations.


FIGURE 5

STEPS IN BUILDING A REACTION RATE EQUATION

5.3.1 Step 1
The first step requires a decision as to which variables are to be studied.


For example, in determining the rate of the single reaction :

A2 + B

C+D

which is known to be essentially irreversible over the range of interest, the
partial pressure of all components can potentially affect the rate. There
would thus be four composition variables and temperature, making 5
variables to consider at the outset.
It may be known independently that the product D does not adsorb on the
catalyst, in which case it can be eliminated from consideration. In cases
where additional components are present, such as a diluent or a selective
poison, which can compete for active sites, additional variables are
created.
In heterogeneous catalysis, in particular, be wary of neglecting the
retarding effects of reaction products, i.e. neglecting terms in the
denominator of Langmuir-Hinshelwood expressions.
5.3.2 Step 2
It is important at an early stage to set realistic ranges of the variables so
as to avoid large extrapolations at the design stage. Close consultation
between chemists and engineering is crucial. Inadmissible combinations
of variables on the grounds of safety and operability need to be identified.
5.3.3 Step 3
Draw up an experimental plan, the basic requirement of which is to
highlight the primary, interactive and non-linear effects of the variables in
the smallest number of experiments.
Some kind of statistical design will be required. Bearing in mind the need
for accurate and direct measurements of the rate of reaction, the choice of
reactor would seem to lie between the differential and CSTR types, with
the integral plug flow reactor forming the vital testing, Step 5.


For the experimental plan a central-composite design is preferred [Ref.
11]. This comprises three parts :
(i) A conventional 2k-1 fractional factorial design for k variables at 2 levels.
(ii) At least three identical experiments at the centre-point conditions to
estimate the percentage of error on the data. These experiments are
spread out through the program in order to check for catalyst deactivation.
(iii) Two out-lier experiments for each variable in order to provide an estimate
of the non-linearity of the reaction rate to each variable.
Such a design for two factors (P1 and P2) is shown in Figure 6. It effectively
saturates the experimental region of interest.


FIGURE 6

A CENTRAL-COMPOSITE DESIGN FOR TWO FACTORS

5.3.4 Step 4
Follow the heuristic ground rules in Table 6 progressively to build a
hyperbolic type rate equation in terms of the dimensionless factors of the
form :


where subscript cp represents the centre-point value. By so doing one
breaks the interdependence of exponential and pre-exponential terms,
thereby removing one of the big problems of non-linear estimation.
For example, if it were suspected in the reaction:
A2 + B C + D
that the rate-controlling step involved reaction between dissociatively
adsorbed A and gas phase B, with D not adsorbed but C adsorbed as a
dimer, the hyperbolic rate model would take the form:

Iterate within Step 4 until an adequate-fitting model containing the smallest
number of adjustable parameters has been found. A brief discussion of
the basis of this type of expression appears in Appendix A.
5.3.5 Step 5
There is no guarantee that the model is adequate for design purposes.
Just fitting the data well is not enough. It is highly desirable, therefore, to
subject it to rigorous testing.
Preferably, the model should be used to predict the performance of a
reactor of different geometry and mixing pattern. For example, if a
laboratory CSTR were used for developing the rate equation, an integral
tubular reactor would be ideal for testing purposes.
As an addition, independent methods, e.g. studies of chemisorption, might
be utilized to cast light on the true nature of the reaction, thereby enabling
some of the assumptions in Step 4 to be questioned.
Further improvements in chemical analysis and laboratory reactor design
bring nearer the day when 10% error in rate data are realizable, as
opposed to 20% in the present-day for complex multi-product reactions
involving difficult analysis.

The exponential model is used as a stepping stone to the more rigorous
hyperbolic model. A modified exponential rate model of the type :

is developed using linear regression methods. The problem of finding
initial parameter estimates is thereby eliminated. Variables that are not
significant in the exponential model can be omitted from further
consideration - a great help when stepping to a hyperbolic (i.e
mechanistic) model.
At the centre-point of the design, all dimensionless temperature and
concentration terms
become unity. Thus, Equation (6) becomes:

Many of the problems cited above would then disappear. An illustration of
this heuristic approach is given by Cropley [Ref. 12].
The intention here is to provide a rapid and simple approach to the
development of hyperbolic rate models which overcomes most of the
problems that historically have hampered their development. This would
release the experimenter into areas of more critical testing and
modification.


TABLE 6

6

HEURISTICS FOR BUILDING A HYPERBOLIC-TYPE RATE
EQUATION

INTRAPARTICLE DIFFUSION
The need to achieve economic rates of reaction of at least 1 mole/sec m3
catalyst volume necessitates surface areas in the range 0.1 – 10 2 m2/cm3.
Such areas can only be achieved by ensuring sub-division of the catalyst
into the form of a porous particle. An understanding of the pore system,
the types of mechanism and rates of molecular diffusion within each type,
is essential to successful scale-up of particle size. Pore diffusion generally
is an undesirable intrusion, lowering rates of reaction, adversely affecting
selectivity and exacerbating catalyst deactivation.


6.1

Types of Pore System
Pore systems can be divided into three types - lattice, precipitate and
pelleting pores, the characteristics of which are summarized in Table 7.
TABLE 7

TYPES OF PORE SYSTEM

6.1.1 Lattice Pores
Lattice pores are contained in naturally occurring or synthetic zeolites
employed in catalytic cracking, xylenes isomerization and methanol to
gasoline conversion. They form a regular array of cages (15 - 20Å in
diameter) connected by passages 3 - 10Å in diameter, and present a high
surface area. Molecules move through the zeolite structure by
configurational diffusion - a term coined to lump together the importance of
not only translational motion, but also vibrational and rotational modes.
There is much confusion regarding the magnitude of diffusion coefficients
within zeolite crystals, owing to the difficulties of isolation and
measurement of the diffusion mechanism. Reported values of D lie in the
range 10-9 - 10-14 m2/s. Since crystal sizes are in the 1-3µm range, L2/D
may range from 10-2 - 103 seconds.
6.1.2 Precipitate Pores
Precipitate pores form the spaces between particles of a very fine
precipitation of catalytic species and support. At low to moderate
pressures gases diffuse by Knudsen diffusion (i.e. molecule-pore wall
collisions) at rates of the order 1-5 x 10-7 m2/s. For a catalyst pellet of
length 5 mm, L2/D will lie in the range 50 - 250 seconds.


6.1.3 Pelleting Pores
Pelleting pores are introduced In the final stages of catalyst forming, which
may involve tabletting, pressing, or extrusion of a paste followed by
calcination. The extrudate may be in the form of a refractory support, such
as α-Al2O3, which is then impregnated with a solution of a salt of the
catalyst, dried and reduced. Pelleting pores lie in the range 500 – 104Å
and above about 5 bar pressure diffusion is in the bulk regime (i.e.
controlled by molecule-molecule collisions). Diffusion rates are dependent
upon species and composition as well as temperature, pressure and pellet
properties, but tend to lie in the range 4-10 x 10-7 m2/s at moderate
pressure. Again, for a pellet of 5 mm, L2/D is in the range 25 - 60 seconds.
Catalyst pellets may contain two types of pore systems (bi-modal), for
example, types c+b or c+a. When both types of pore system offer
comparable diffusion resistance, scale-up becomes particularly difficult.
6.2

The Catalyst Effectiveness Factor
The whole endeavor of engineering is to reduce dauntingly complex
problems to manageable proportions by judicious choice of assumptions
or simply, inspired guesswork. Diffusion and reaction within a porous
catalyst is a problem of considerable complication falling into this
category. However, in most circumstances, an adequate representation is
possible by the use of effective parameters in a 'smoothed out' continuum
model, in which diffusion coefficients, concentrations and rates of reaction
are continuous functions of position. Occasionally, the continuum model
breaks down and a more detailed approach is needed - for example, in
zeolite catalysts containing a bi-modal distribution of pore sizes, or for
pellets possessing a dense outer skin as a result of the pelleting process.

6.2.1 The Single, Irreversible, First Order Reaction
The simplest cases of the single irreversible first order reaction A B and
the consecutive first order reactions A B C provide suitable vehicles for
demonstrating the significant questions. A cottage industry has
mushroomed around this subject and is admirably dealt with by Aris [Ref.
13].
For the single irreversible first order reaction occurring within the spherical
catalyst pellet of radius a, see Figure 7, a steady state material balance
over a differential slice of radius r and thickness dr gives :


FIGURE 7

FIRST ORDER ISOTHERMAL IRREVERSIBLE REACTION
WITHIN A CATALYST SPHERE

The effectiveness factor ƞ is defined by :


where y = r/a and u = C/CS.
Analytical integration of Equations (7), (8) and (9) gives :

in terms of the single dimensionless parameter ɸe, namely the Thiele
modulus :

Figure 7 shows that when ɸe is small the concentration of reactant is
everywhere close to its surface value. Diffusion is very fast by comparison
to chemical reaction and, by definition, ƞ 1. This is usually not of much
practical interest because rates of reaction tend to be too low to be
economically viable. On the other hand, when ɸe is large the concentration
falls exponentially with distance from the outer surface and the reaction is
complete within a thin shell near the surface of the pellet. The process is
said to be diffusion-controlled. A loose interpretation shows that if ɸe < 1,
the process is chemically rate-controlled, whereas if ɸe > 9, it is diffusion
limited. Many industrial catalytic processes operate in the diffusion
controlled regime as a consequence of the high reaction rates required. If
the scale-up problem is simply concerned with reactant conversion, low
catalyst effectiveness factors are not too worrying - they can be offset by
over sizing the reactor. This is not the case, however, when yield of
desired intermediate is the primary concern.


6.2.2 The Consecutive-Parallel First Order Reaction System
A more interesting and practically relevant problem concerns selectivity to
desired product B in the consecutive-parallel scheme :

For the case of a catalyst slab of half-thickness a, the concentrations of A
and B within the catalyst are given by [Ref. 13].

We can take the analysis a stage further by noting that for an isothermal
plug flow reactor, the inter-particle gas concentrations CAS and CBS are
related by the differential equation:

Where ū B and ū A are mean concentrations of B and A within the catalyst
pellet obtained by integrating Equations (13) and (14).

After some algebra, the result of integrating Equations (13) to (15) is given
as a relation between the integral yield to desired product B (Y12) against
the fractional conversion of reactant A (XA):

For slab geometry, the catalyst effectiveness factor for the disappearance
of A by the parallel reactions is given by [Ref. 13].

Figure 8 displays the integral yield as a function of reactant conversion for
a particular case corresponding to a moderate diffusion effect (ƞ = 0.395).
Also shown is the case for no diffusion effect (i.e. ɸ1 0). Diffusion serves
to lower yield by as much as 10% under moderate conditions.


FIGURE 8

INTEGRAL YIELD vs CONVERSION SHOWING EFFECT
OF PELLET DIFFUSION

At a fixed yield (ie raw materials efficiency) of say 0.7, the effect of pellet
diffusion would be to reduce the reactant per pass conversion from 0.88 to
0.1. The consequence of this would be to increase the recycle flow rate by
a factor 9! Since the cost of separating and recycling unreacted material is
proportional to the recycle flow rate, this would likely have serious
implications on production costs and capital cost of separation equipment.
On the other hand, at a fixed conversion, diffusion lowers product yield by
10%, thereby increasing the usage and thus cost of raw materials, as well
as increasing separation costs. In both scenarios, process performance
suffers.


When yield issues are of prime importance, as is often the case, a detailed
understanding of the impact of pellet diffusion on the total process and its
economics is necessary. This information can set targets for process
development and better catalyst systems.
Pellet diffusion can also affect the rate of catalyst deactivation, the scope
of which is beyond this Guide. However, deactivation and diffusion are
treated in detail by Hughes [Ref. 14].
6.2.3 Computer Program – GBHE Proprietary Tools for Reactor Modeling
Consult GBHE-PEG-RXT-819 for information on Proprietary Tools for
Reactor Modeling to calculate the effectiveness factor for multicomponent diffusion, heat transfer and reaction.
6.3

The Measurement of Effective Diffusivity
Table 7 displays ranges of effective diffusivities observed in different types
of pore system which are useful for initial calculations of effectiveness
factors. For more accurate work the effective diffusivity must be measured
experimentally.
Diffusion measurements are usually carried out in the absence of
chemical reaction by using chromatographic techniques applied to single
catalyst particles or packed beds, or by measuring steady counterdiffusion fluxes across individual pellets. Theoretical treatments and
working methods are described elsewhere [Ref. 15].

6.3.1 Simple Uni-modal Pore Size Distribution
If a catalyst has a simple uni-modal pore size distribution, translation of
diffusivity measurements made on non-reacting gases can be securely
directed to the reactive case, since the effective diffusivity De can be
expressed as the product of the structural specific permeability | and a
modified diffusion coefficient D* :

The permeability Ψ is determined from measurements of De by one of the
methods given above, together with D* determined by the relationship:

where D is the bulk diffusion coefficient and DK is the Knudsen diffusivity,
which depends upon the mean pore radius ṝ,

where ṝ is in metres. The mean pore radius can be measured by standard
methods.
Table 8 presents experimental diffusivities for helium in nitrogen diffusion
within the fine pores (ṝ = 3.5 x 10-9 m) of an amorphous silica-alumina
catalyst used in xylenes isomerization [Ref.!3].
TABLE 8

COMPARISON OF PORE DIFFUSIVITIES BETWEEN THE
DIFFUSION CELL AND PACKED BED METHODS [Ref. 15]

At ambient temperature and pressure D = 7 x 10-5 m2/s.
DK = 2.93 x 10-6 m2/s and, from Equation (19),
D* = 2.81 x 10-6 m2/s. It follows from Equation (18) and the packed
bed result in Table 7, that:


Given Ψ, and armed with the knowledge that Knudsen diffusion is
controlling, the effective diffusivity of o-xylene (M = 106) at 723K is
predicted from Equations (18-20) to be 1.33 x 10-7 m2/s. This value was
combined with the intrinsic reaction rate constant k to determine the
effectiveness factor j as a function of pellet sphere diameter from Equation
(11). Predicted and measured effectiveness factors are compared in
Figure 9 . Theory and practice match quite well given the uncertainties in
parameter values. On commercial beads (Dp = 3.5 - 4mm), ŋ ~ 0.5.


FIGURE 9

PREDICTED AND EXPERIMENTAL EFFECTIVENESS FACTORS

6.3.2 Bi-modal Pore Structures
Industrial catalysts sometimes possess a bi-modal distribution of pores for example, if a fine porous powder of nearly uniform particle size is
pelletized and calcined. At moderate pressures, diffusion within the
micropores is of the Knudsen type whereas that within the macropores is
dominated by bulk diffusion. This presents a number of problems to the
reaction engineer.

Firstly, can the simple continuum model containing the single effective
diffusivity De, outlined in Clause 6.2, be used to calculate the
effectiveness factor? Secondly, if so, how may De be evaluated?
Extrapolation of a measurement of De for a non-reacting gas at
atmospheric pressure to that for a reacting gas at, say, 50 bars, is fraught
with uncertainty - the controlling diffusion regime may, in fact, change!
A rather elegant solution was proposed by Hugo and Koch [Ref. 16].
The ratio of modified diffusion coefficients for the micropores (D*m) and
macropores (D*M) may be written as :

From equation (19):
At very high pressures, D/DK

0 and Z 1 with:

The structural specific permeability ΨG in Equation (22) contains the
contributions of all pores, irrespective of size, since bulk diffusion controls
in every pore.
On the other hand, at very low pressures D/DK 0 and z dm/d M, the ratio
of micropore to macropore diameters. Since this ratio is usually very small,
it is permissible to write :

These two limiting forms of equation for De are similar to the trivial unimodal case. Neither requires assumptions about the arrangement of
micropores and macropores.

Hugo and Koch suggest an interpolating form of relation for De to span the
whole pressure range in the form :

where Ψ(z) depends upon pressure, and is such that Ψ(o) = ΨM and Ψ(1)
= ΨG. These authors found a linear correlation between Ψ and z to be of
general validity; i.e. for all porous materials with typical bi-modal pore size
distributions. Some of their data are shown in Figure10. They show that Ψ
may vary by as much as a factor 3 and always increases with pressure.
Their results suggest that two measurements of De are sufficient to locate
Ψ at any intermediate pressure - one at normal pressure using a nonreactive gas and, preferably, a second using a liquid to simulate and
simplify measurement employing a high pressure gas.
Park and Kim's study [Ref. 17] of reaction within biporous catalysts
suggests a note of caution in the use of effective diffusivities. They claim
that diffusivities measured under reaction conditions can be an order of
magnitude lower than those found for non-reactive gases.


FIGURE 10 STRUCTURAL PERMEABILITY vs PRESSURE
PARAMETER Z FOR BI-MODAL SUPPORTS

7

ENHANCEMENT OF INTRAPARTICLE DIFFUSIVITY
The flow of material through a catalyst packed bed reactor will require a
pressure drop in the direction of flow. If the catalyst particles have macropores which penetrate through the particles, this pressure drop will cause
forced convection of fluid through the pores which are aligned in the
direction of flow.


Nir and Pismen [Ref. 18] have analyzed the case of an irreversible first
order liquid phase reaction accompanied by pressure driven flow and
diffusion in a porous catalyst slab. Their main findings are shown in
Figures 11 and 12. These give the usual effectiveness factor vs Thiele
modulus diagram with an additional parameter n where :

an intraparticle Peclet number, where :
Vo = linear velocity in pore
L = half the pore length
D = diffusivity of limiting reactant within the pore.
The maximum effect of convection (λ) is seen to occur when moderate diffusional
effects (1 < Ø < 10) are apparent.
The pore velocity Vo is related to the pressure gradient across the pore ∆P/2L
by:

In the case of a single tube of circular cross-section :


Where:
dt

=

diameter of the tube

In a porous medium we would also expect K to be proportional to the square of
the pore diameter.
It is best to measure K, but an approximation may be made using pore diameter
distribution data by calculating an average value for d from:

Where ni is the number of pores of diameter di.


FIGURE 11 EFFECTIVENESS FACTOR vs THIELE MODULUS AND
INTRAPARTICLE PECLET NUMBER


FIGURE 12 RELATIVE INCREASE IN CATALYST PERFORMANCE


8

NOMENCLATURE

All effective transport parameters are defined in terms of total areas (void + nonvoid) normal to the direction of transport


9

BIBLIOGRAPHY
1

Smith, J.M., "Chemical Engineering Kinetics", McGraw-Hill, Tokyo,
2nd Edition (1970).

2

Satterfield C.N. "Mass Transfer in Heterogeneous Catalysis" MIT
Press (1970).

3

Orr, N.H., Cresswell, D.L. and Edwards D.E. "Model Building in
Complex Catalytic Reaction Systems. A Case Study: p-Xylene
Manufacture", I&EC Proc. Design and Development, (1983) 22,
135.

4

Chinchen C.C., Logan, R.H. and Spencer, M.S. "Water-Gas Shift
Reaction over an Iron oxide/Chromium oxide catalyst. I: Mass
Transport Effect". Applied Catalysis (1984) 12,69.

5

Caldwell, L. "An Improved Internal Gas Recirculation Reactor for
Catalytic Studies", Applied Catalysis (1983) 8, 199.

6

Gut, G., Jaeger, R., "Kinetics of the Catalytic Dehydrogenation of
Cyclohexanol to Cyclohexanone on a Zinc Oxide catalyst in a
Gradientless Reactor", Chem. Eng. Sci. (1982), 37, 319.

7

Sundaram, K.M., Froment, G.F., "Kinetics of Coke Deposition in the
Thermal Cracking of Propane", Chem. Eng. Sci. (1979) 34, 635.

8

Weekman, V.W. Jr., "Laboratory Reactors and their Limitations".
A.I.Ch.E.J. (1974) 20, 833.


9

Mears, D.E., "Tests for Transport Limitations in Experimental
Catalytic Reactors", I&EC Proc. Design and Development (1971)
10.

10

Rase, H.F. "Chemical Reactor Design for Process Plants" John
Wiley (1977).

11

Davies, O.L., "Design and Analysis of Industrial Experiments", 5345, Hafner Publishing Co. New York. (1960).

12

Cropley, J.B. "Heuristic Approach to Complex Kinetics", A.C.S.
Symposium Series No.65, Chemical Reaction Engineering Houston p292 (1978).

13

Aris, R. "The Mathematical Theory of Diffusion and Reaction in
Permeable Catalysts", vol 1: The Theory of the Steady State",
Oxford University Press (1975).
Hughes, R. "Deactivation of Catalysts", Academic Press (1984).

14
15

16

Hugo, P. and Koch, E.N. "Diffusion within Bi Porous Catalysts",
German. Chem. Eng. (1985) 8, 234.

17

Park, S.H. and Kim, Y.G. "The Effect of Chemical Reaction on
Effective Diffusivity within Bi!Porous Catalysts - II Experimental
Study". Chem. Eng. Sci. (1984) 39, 533.

18

9.1

Cresswell, D.L. and Orr, N.H. "Measurement of Binary Gaseous
Diffusion Coefficients within Porous Catalysts" from Residence
Time Distribution Theory in Chemical Engineering, edited by A.
Petho and R.D. Noble, Verlag Chemie, Weinheim, p.41 (1982). IC
07039/C Cresswell, D.L. Simultaneous sorption and diffusion within
adsorbent granules using pulse chromatography. (Nov 1986).

A. Nir and L.M. Pismen "Simultaneous intraparticle forced
convection, diffusion and reaction in a porous catalyst." Chem. Eng.
Sci. (1977) 32, 35.

Further Reading
Berty, J.M., "Reactor for Vapour phase Catalytic Studies", Chem.
Eng. Prog. (1974) 70,578.


Jackson R, "Transport Processes within Porous Catalysts", Elsevier
Amsterdam (1977).
Satterfield C.N. "Heterogeneous Catalysis in Practice" McGraw-Hill
(1980).
Sharma, R.K., Cresswell, D.L. and Newson, E.J. Selective
Oxidation of Benzene to Maleic Anhydride at Commercially
Relevant Conditions", I.S.C.R.E. 8 p353, Edinburgh, September
(1984); Inst. Chem. Eng. Symp. Series No.87.


APPENDIX A

LANGMUIR - HINSHELWOOD KINETICS

The Langmuir-Hinshelwood form of rate expression; e.g. see 5.1.1, arises
out of the mechanism of solid catalyzed reactions. This mechanism is
simplified as :
(a) adsorption of reactants onto the catalyst surface;
(b) reaction of (adsorbed) species;
(c) desorption of products.
A simple example will illustrate. Consider the solid catalysed reaction
between A and B to produce C and D.

A+BC+D

Steps (a) would be written :
A + s
B + s

As
Bs

Here s is a vacant catalyst active site
As and Bs are adsorbed species
Step (b) reaction between two adsorbed species :
As + Bs

Cs + Ds

Steps (c)
Cs
Ds

C+s
D+s

If we now consider concentrations X, we could write physical equilibrium
equations:


If we let XS be the total "concentration" of available catalyst active sites, at
physical equilibrium we can write:

Now if step (b) were rate controlling and the adsorption steps were at
equilibrium, the rate expression might be :


If adsorption of A were rate controlling, the rate expression might be :

Notice that the KA X A term has disappeared from the denominator
because the reaction consumes A as fast as the adsorption step can
supply it, i.e. XAs is near zero, it is not in equilibrium. If a desorption step
were rate controlling, the catalyst would be flooded with the limiting
product and the reaction rate could be apparently zero order.

A variety of different rate expressions results from different concepts of the
reaction mechanism.

DOCUMENTS REFERRED TO IN THIS PROCESS ENGINEERING GUIDE
This Process Engineering Guide makes reference to the following
documents:
GBHE ENGINEERING GUIDES
Glossary of Engineering Terms (Referred to in Clause 3).
GBHE-PEG-RXT-800

Guide on How to use the Reactor Technology
Guides (Referred to in 4.1)

GBHE-PEG-RXT-802

Residence Time Distribution Data
(Referred to in Clause 4 and 4.1)

GBHE-PEG-RXT-809

Homogeneous Reaction - Gas Solid Systems
(Referred to in 5.2.1)

GBHE-PEG-RXT-812

Case Studies in Reactor Technology
(Referred to in Clause 5)

GBHE-PEG-RXT-817

Tools for Reactor Modeling
Part C (referred to in 4.1 and 6.2.3)
Part F (referred to in 5.1.1).


Solid Catalyzed Reactions

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