3.

Kinetics of heterogeneously catalysed reactions

The ultimate objective of a catalyst is to accelerate the rate of a reaction by supplying an

alternative reaction pathway. In heterogeneously catalysed reactions the reaction will
take place at the surface of the catalyst. In order to maximise the catalyst surface area,
porous materials are used as a catalyst or as a catalyst support material. This introduces
some complications when considering the kinetics of a heterogeneously catalysed
reaction.

The reactants are being delivered to the particle by fluid flow surrounding the catalyst
particle. A particle in a fluid flowing is typically surrounded by a layer, in which the
velocity of the fluid is reduced. A simplifying assumption is that the layer is stagnant. The
reactant(s) must move through this layer to be able to get to the external surface of the
catalyst particle by diffusion through this layer. The conversion of the reactant(s) takes
place on the surface of the catalyst particle. In porous materials most of the surface is
located inside the pores. Thus, the reactant(s) have to diffuse through the pores to the
active site. The reactant(s) adsorb on the active site and can be converted. The product
molecules desorb, diffuse through pores to the external surface of the catalyst particle,
diffuse through the ‘stagnant’ layer and are swept out of the reaction area by the fluid
flow.

Boundary layer
“Stagnant” layer surrounding
a particle in a flow

Porous particle
Material diffuses through the
void space in the particle

Figure 3.1: A catalyst particle surrounded by a ‘stagnant’ layer (catalyst particle from
[1])

Thus, multiple processes have to be taken into account, when considering

heterogeneously catalysed reactions. Figure 3.2 shows schematically the following
consecutive steps during a heterogeneously catalysed reaction:
1. Bulk fluid flow transporting a reactant to the boundary layer surrounding the
catalyst particle
2. Diffusion of the reactant through the boundary layer to the external surface of the
catalyst particle
3. Diffusion of the reactant through the pore of the catalyst to the active site
4. Adsorption of the reactant on the active site
5. Conversion of the adsorbed reactant yielding the adsorbed product on the active
site
6. Desorption of the adsorbed product from the active site

49
 7. Diffusion of the product through the pores of the catalyst to the external surface
of the catalyst particle
8. Diffusion of the product through the boundary layer surrounding the catalyst
particle
9. Bulk fluid flow transporting the product from the boundary layer out of the
reaction zone

Bulk fluid flow Bulk fluid flow

1 2 P 9
A
A 8
P Boundary
layer

3 7

Pore in catalyst

A P
4 6
5
A P

Figure 3.2: Schematic view of the steps involved in heterogeneously catalysed

reactions (steps numbered according to the text)

In principal the function of a catalyst lies in the functioning of the active site and
catalysts’ activity should be compared based on the activity of the active site. However,
the structure of the heterogeneous catalyst plays an important role, which can obscure
the intrinsic activity of the active site.

3.1 Intrinsic kinetics of heterogeneously catalysed reactions

The intrinsic activity of a heterogeneous catalyst is the activity of the active site. This
incorporates the adsorption properties of the reactant(s) and product compound(s) on
the active site and the conversion of the adsorbed reactant(s) on the active site. The
intrinsic activity can thus only be measured of the sites are exposed to the same
temperature and composition as in the bulk of the fluid, since that is the only position,
where they can be measured.

The intrinsic rate of reaction can be correlated with the partial pressures (or
concentrations) at the active site using a power-law rate expression. For a general
reaction of
aA+bB p P +s S

The rate of consumption of A, the rate of consumption of B, the rate of formation of P,

and the rate of formation of S are linked over the stoichiometric equation, i.e.
 rA  rB rP rS
   .
a b p s

50
The rate of consumption of A can be generally expressed as:
 p s 
     1 pP  p S 
 rA  k  p A  p B  p P  p S  1  
 K p a  p b 
 A B
This formulation of the rate of reaction in this particular way ensures that expression is
thermodynamic consistent, i.e. the rate of reaction becomes zero at equilibrium [2].

The kinetic parameters k, , , , and  can be determined experimentally. The constant

K is the overall thermodynamic equilibrium constant for the overall reaction. The overall
thermodynamic equilibrium constant can be determined from thermodynamic data (see
chapter 1). The term in the brackets represents the so-called thermodynamic driving
force for the reaction to take place and is a measure for the distance of the reaction
mixture from equilibrium.

If the kinetic parameters can be determined far away from equilibrium, the rate
expression simplifies to:

 rA  k  p  
A  pB  pP  p S


and the kinetic parameters k, , , , and  can be determined by multi-linear regression

of the expression:
ln rA   lnk     lnp A     lnpB     lnpP     lnpS 

If the concentration of the product compounds is negligible (or if it can be established

that they do not affect the rate of reaction far away from equilibrium), the expression can
be simplified to:
ln rA   lnk     lnp A     lnpB 

It should however be noted that the obtained rate expression is a purely empirical fit of
the obtained rate data in a given concentration range. Although the temperature
dependency of the rate constant can be fitted with an Arrhenius equation
Ea
k A  e T
R
it should be noted that also the parameters , , , and  will change upon changing
temperature.

Despite the limitations of a purely empirical fit of the data, this type of rate expressions
can be used for up-scaling as long as it takes place within the limits of the obtained
empirical fit. Furthermore, a power-law rate expression often can yield a first insight in
the mechanism of a heterogeneous catalysed reaction. A mechanistic based rate
expression can only be obtained, if the elementary reaction steps are known or can be
postulated.

3.1.1 Langmuir-Hinshelwood approach

In heterogeneously catalysed reactions the catalytic reaction is thought to take place on
an active site. The number of these active sites is limited and thus the rate of reaction is
not only limited by the amount of reactants available for reaction, but also by the amount
of sites on which the reaction can take place. Kinetic expressions describing the rate of
heterogeneously catalysed reaction must take to limited number of active sites on the
catalyst into account. The limited availability of the active site is not taken into account in

51
the general power-law rate, and thus the general power-law rate cannot predict the rate
of reaction at all reaction conditions.

In the Langmuir-Hinshelwood approach, the occupation of the active sites is taken into
account by assuming equilibrium between the species in the bulk of the fluid and the
adsorbed species. For example
A+* A* (adsorption A)
B+* B* (adsorption B)
P+* P* (adsorption P)
S+* S* (adsorption S)
(with * denoting a vacant site)
It is assumed that each of these adsorption steps approach equilibrium. Then, the
fractional coverage of the surface with each species can be determined using the
equilibrium relationship:
 A  K adsorption, A  p A   *
B  K adsorption,B  p B   *
P  K adsorption,P  p P   *
 S  K adsorption,S  p S   *

The total number of sites on the surface is limited, and the sum of all fractional
coverages on the surface equals 1 (site balance; Langmuir-approach):
1    i   A  B  P   S   *
i

Substituting the equilibrium relationships, and solve for * yields:

1
* 
1  K adsorption, A  p A  K adsorption,B  pB  K adsorption,P  pP  K adsorption,S  p S

The rate of reaction can now be modelled for a reaction far away from equilibrium using
a power-law rate taking into account that the reaction required reactants and active sites
to proceed:

 rA  k  p    n
A  pB  pP  p S   *

k  p A  pB  p P  p S

 rA 
 
1  K adsorption, A  p A  K adsorption,B  pB  K adsorption,P  pP  K adsorption,S  p S n

The parameter n can be taken as an indication of the number of active sites required for
the reaction to proceed.

The above equations assume that adsorption follow the Langmuir-adsorption isotherm,
and thus that the sites are energetically homogeneous, no interaction between adsorbed
species, and adsorption/desorption following a single mechanism. Surface heterogeneity
and the adsorbate-adsorbate interaction lead to a decrease in the heat of adsorption
with increasing coverage and would warrant the application of e.g. the Temkin isotherm
rather than the Langmuir isotherm. Nevertheless, it has been argued that the resulting
model is frequently the same [3].

52
3.1.2 Steady-state approach
A more fundamental approach can be taken if the elementary reaction steps are taken
into account. In an elementary reaction, the stoichiometric equation describes the actual
process, and the rate of reaction can immediately deduced from the stoichiometric
equation. If, for example the reaction:
A+B P
is an elementary reaction, it means that upon the collision of a molecule of A with a
molecule of B there is a finite likelihood that product P will be formed. Furthermore, it
indicates that product P may (due to internal vibrations) decompose into molecules A
and B. The rate of the forward reaction describing the collision between A and B is then
given by:
rforward  k forward  p A  pB
(the rate expression can be expressed in terms of partial pressures and concentration)
and the rate for the reverse reaction describing the decomposition of P is given by:
rreverse  k reverse  pP
The net reaction is the difference between the forward and the reverse reaction:
 rA  rB  rP  k forward  p A  pB  k reverse  pP

(At equilibrium the rate of reaction equals zero:

 rA  rB  rP  k forward  p A, eq  p B, eq  k reverse  p P, eq  0

k forward pP, eq
K  )
kreverse p A, eq  pB, eq

A heterogeneously catalysed reaction can be thought of as a sequence of elementary

reaction steps. The intermediates, adsorbed species on the catalyst surface, are
typically not measured or even measurable under reaction conditions. The concentration
of these species must therefore be related to measurable quantities such as
concentration in the fluid phase or partial pressures.

Take for instance a heterogeneously catalysed transformation of reactant A into product

P:
A P
This reaction can be thought to occur with the following elementary reaction steps:
Adsorption of A: A+* A* (1)
Surface reaction : A* P* (2)
Desorption of P: P* P+* (3)
(this is a true catalytic cycle, since the active site – the catalyst – is regenerated in the
cycle; * denotes an active site; A* denotes an active site covered with adsorbed A; P*
denotes an active site covered with adsorbed P). For each of the elementary reaction
steps the rate law can be written down:
r1  k 1, forward  p A   *  k 1,reverse   A *
r2  k 2, forward   A *  k 2,reverse  P *
r3  k 3, forward  P *  k 3,reverse  pP   *

The rate of formation of each of the species involved in the process can be be
formulated:
rA  r1 rA *  r1  r2 rP*  r2  r3 r*  r1  r3 rP  r3

53
In the steady-state approach it is assumed that the net rate of formation of the reactive
intermediates on the surface of the catalyst, in this case A*, P* and * equals zero. This
means that the composition of the pool of adsorbed species on the surface remains
constant and does not change with time. This assumption is certainly true for catalytic
reaction taking place in flow reactors operating at steady-state, but in non-steady-state
operating reactors this approach is only valid, if the surface reactions are sufficiently fast.

The steady-state assumption applied to the example here leads to:

r1  r2  r3
rA *  r1  r2  k 1, forward  p A   *  k 1,reverse   A *  k 2, forward   A *  k 2,reverse  P *  0
rP*  r2  r3  k 2, forward   A *  k 2,reverse  P *  k 3, forward  P *  k 3,reverse  pP   *  0

And thus the steady-state surface coverage of A* and P* can be expressed in terms of
the partial pressure of A, the partial pressure of P and fraction of the surface which is not
covered with adsorbed A or adsorbed P
k1,forward  k 2,reverse  k 3,forward pA  k 2,reverse k 3,reverse pP
A *   *
k1,reverse k 2,reverse  k1,reverse k 3,forward  k 2,forward k 3,forward
k1,forward k 2,forward pA  k 3,reverse  k1,reverse  k 2,forward pP
P *   *
k1,reverse k 2,reverse  k1,reverse k 3,forward  k 2,forward k 3,forward

The rate of consumption of A can now be expressed as a function of the partial

pressures of A and P and the fraction of the surface which is not covered with adsorbed
A or P:
 rA  k 1, forward  p A   *  k 1,reverse   A *
k  k k p k k p
 rA  k 1, forward  p A   *  k 1,reverse  1,forward 2,reverse 3,forward A 2,reverse 3,reverse P   *
k1,reverse k 2,reverse  k1,reverse k 3,forward  k 2,forward k 3,forward
k1,forward k 2,forward k 3,forward pA  k1,reverse k 2,reverse k 3,reverse pP
 rA   *
k1,reverse k 2,reverse  k1,reverse k 3,forward  k 2,forward k 3,forward

k1,forward k 2,forward k 3,forward

Realising that K 1  ;K2  ;K3  and K overall  K1  K 2  K 3
k1,reverse k 2,reverse k 3,reverse
 1 p 
k1,forward k 2,forward k 3,forward pA  1  P 
 K overall pA 
 rA   *
k1,reverse k 2,reverse  k1,reverse k 3,forward  k 2,forward k 3,forward

pA  1 p 
 rA  1 1 1
 1   P    *
   K overall p A 
k1,forward K1 k 2,forward K1 K 2 k 3,forward

The first term in this expression expresses the contribution of the various reaction steps
to the overall rate of consumption of A. Each reaction step can be viewed as a
resistance against the transformation of A into P (and for a serial network the
resistances must be added) [4]. The reciprocal value of the (modified) rate constant is
then a measure for the kinetic resistance against the transformation of A into P. The
second term in the equation describes the driving force of the reaction, i.e. the distance
from equilibrium.

54
The fraction of the surface, which is not covered with adsorbed A or adsorbed P can also
be expressed in terms of the partial pressures of the components A and P using a site-
balance (Langmuir-approach):
 A *  P *  *  1
1
* 
k 2,forward  k 2,reverse  k 3,forward 1 k1,reverse  k 2,forward  k 2,reverse
1  K1 
k k
 pA  
k k
 pP
2,forward 3,forward K3 1,reverse 2,reverse
k 2,reverse  k 3,forward   k1,reverse  k 2,forward
k1,reverse k 3,forward

Thus, the rate of reaction for this simple three-step heterogeneously catalysed reaction
becomes:
pA  1 p 
1 1 1
 1   P 
   K overall pA 
k1,forward K1 k 2,forward K1 K 2 k 3,forward
 rA 
k 2,forward  k 2,reverse  k 3,forward 1 k1,reverse  k 2,forward  k 2,reverse
1  K1 
k
 pA    pP
k K 3 k1,reverse k 2,reverse
k 2,reverse  k 3,forward  2,forward 3,forward  k1,reverse  k 2,forward
k1,reverse k 3,forward

This rather complex looking rate expression in terms of the partial pressures can be
simplified as:
 1 p 
k'p A  1   P 
 K overall p A 
 rA 
1  K 1  p A  K 3  pP
' '

with the parameters to be established by experiment of k’, K’1 and K’3. The parameter k’
is a collective term describing the extent to which each of the three reactions contribute
to the overall rate constant for the reaction. The parameters K’1 and K’3 are not true
equilibrium constants, but express the extent to which reaction 1 and 3 come to
equilibrium.

3.1.3 Principle of the rate determining step

Although the steady-state approach can be used to develop rate expressions for
heterogeneously catalysed reactions in terms of the concentrations/partial pressures in
the fluid phase, the procedure is rather cumbersome. The steady-state approach can be
extended by assuming that one of the reaction steps is much slower than all other
reaction steps and that therefore the other reaction steps can be considered to be at
equilibrium. Figure 3.3 exemplifies the principle of the rate-determining step for a simple
three-step process assuming that the surface reaction is the slowest step, e.g.:
Adsorption of A: A+* A* (1)
Surface reaction : A* P* (2)
Desorption of P: P* P+* (3)

In the derivation it is now assumed that the adsorption of A comes to equilibrium, i.e.
r1  k 1, forward  p A   *  k 1,reverse   A *  0  A *  K 1  p A   *  K adsorption, A  p A   *
Furthermore, it is assumed that the desorption of P is at equilibrium:
1
r3  k 3, forward  P *  k 3,reverse  pP   * P *   pP   *  K adsorption,P  pP   *
K3

55
 net rate
of reaction

Reaction:
10000
9999
A + *  A*
A*  A +*

2
1
A*  P*
P*  A*

1000
999
P* P+*
P + *  P*

Figure 3.3: Principle of the rate determining step for a three-step heterogeneously
catalysed reaction

The rate of reaction is now deduced from the step that is not assumed to be in
equilibrium, i.e. the surface reaction:
r2  k 2, forward   A *  k 2,reverse  P *

Substituting the surface coverage in terms of the partial pressures of A and P:

k 2,reverse
 rA  rP  r2  K 1  k 2, forward  p A   *   pP   *
K3
 1 p 
 K 1  k 2, forward  p A  1 
 rA  rP  r2  P    *
 K overall p A 
Comparing this result to the result obtained with the steady-state approach, the derived
rate expression using the principle of the rate determining step approaches the rate
expression derived using the steady-state approach, if:
1 1 1
  , i.e. when the adsorption of A and the desorption of
k1,forward K1 K 2 k 3,forward K1 k 2,forward
P are fast in comparison to the surface reaction. This was the necessary assumption,
which was made in the derivation of the rate expression using the principle of the rate-
determining step.

The fraction of sites on the surface, which is not covered with adsorbed A or adsorbed P
can be obtained from a site-balance:
 A *  P *  *  1
1
* 
1  K adsorption, A  p A  K adsorption,P  pP

56
Comparing this result to the result obtained with the steady-state approach, the derived
fraction of the surface, which is not covered with either adsorbed A or adsorbed P are in
both approached equal if the surface reactions (both forward and reverse) are much
slower than the other reaction steps. This was the condition on which basis the rate
expression was determined.

Thus, the rate of reaction can be expressed as:

 1 p 
K 1  k 2, forward  p A  1   P 
 rA  rP 
 K overall p A 
1  K adsorption, A  p A  K adsorption,P  pP

The difference between rate expression derived using stead-state approach and the rate
expression using the principle of the rate-determining step lies in the physical meaning
of the denomination. The adsorption constants can be measured independently (using
adsorption measurements) and by fitting to the kinetic data. If the rate-determining
approach is an adequate description of the kinetic process, the independent measured
adsorption constants should be identical. Kabel and Johanson [5] measured the rate of
dehydration of ethanol yielding diethyl ether and water over an ion-exchange resin, and
modeled the rate of reaction using the principle of the rate-determining step. The
equilibrium constants for the adsorption of water and ethanol obtained from the kinetic
modeling were compared with those measured using single component adsorption
measurements (see Figure 3.4). They obtained a reasonable correlation between the
independently determined adsorption constants at the high temperature and therefore
concluded that under these conditions the assumption that one step is much slower than
all other steps, and therefore all other steps can be considered to be at equilibrium, is
justified.

1000

100
K, atm-1

10

1
50 100 150
o
Temperature, C
Figure 3.4: Comparison between the adsorption constants for water and ethanol on
an ion-exchange resin obtained by kinetic modeling of the ethanol
dehydration and by single component adsorption (water:●, ○; ethanol: ■,
□;open symbols: adsorption experiments; solid symbols: kinetic
experiments; drawn from data in [5])

57
If the reaction takes place far from equilibrium and the partial pressure of P is small in
comparison to the partial pressure of A, the rate of reaction can be modelled as:
k'p A
 rA  rP 
1  K adsorption, A  p A
At low partial pressures of A the rate of reaction is proportional to the partial pressure of
A. The observed order of reaction with respect to A in this regime is 1. Increasing the
partial pressure of A further, does not result in a proportional increase in the rate of
reaction due to a saturation of the surface with A. At high partial pressures of A the
reaction order with respect to A becomes equal to zero, i.e. the rate of reaction is
independent of the partial pressures of A.

In a similar manner the rate of reaction for a bimolecular reaction (e.g. the reaction of 2B
 P, in which two adsorbed species of B react to form adsorbed P) can be derived.
Assuming that the reaction takes place with following elementary reaction steps:
Adsorption of B: B+* B* (1)
Surface dimerisation: 2B * P*+ * (2)
Desorption of P: P* P+* (3)

The rate of reaction can derived assuming that one of the reaction steps is the rate-
determining step, i.e.
1. Adsorption of B is rate-determining
2. Surface dimerisation is rate-determining
3. Desorption of P is rate determining

Ad 1: Adsorption of B is rate determining

If the adsorption of B is the rate-limiting step, it means that the surface dimerisation step
and the desorption of adsorbed P on the surface can be considered to be at equilibrium:
2 P *   *
r2  k 2, forward  B *  k 2,reverse  P *   *  0 B * 
K2
p 
r3  k 3, forward  P *  k 3,reverse  pP   * P *  P *
K3

pP
Thus, the surface coverage with adsorbed B is: B *   *
K2  K3

The fraction of the surface that is not occupied by adsorbed B and not with adsorbed P
is given through a site-balance (Langmuir approach):
1
* 
1 1
1  pP   pP
K2 K3 K3

The rate of reaction is now deduced from the step that is not assumed to be in
equilibrium, i.e. the adsorption of B:
 rB  2  rP  r1  k 1, forward  pB   *  k 1,reverse  B *

58
 1
 rB  2  rP  k 1, forward  pB   *  k 1,reverse   pP   *
K2 K3
 1 pP 
k1,forward pB   1 
 K12 K 2 K 3 pB 

 rB  2  rP 
1 1
1 pP  pP
K 2 K 3 K3

If the reaction is taking place far from equilibrium and at low pressure of the product P,
the rate of reaction simplifies to:
 rB  2  rP  k 1, forward  pB
The reaction is then first order with respect to B and the rate of reaction is directly
proportional to the partial pressure of the reactant B.

Ad 2: Surface dimerisation is rate determining

If the surface dimerisation is the rate-limiting step, it means that the adsorption of B and
the desorption of adsorbed P on the surface can be considered to be at equilibrium:
r1  k 1, forward  pB   *  k 1,reverse  B *  0 B *  K1  pB  *
p 
r3  k 3, forward  P *  k 3,reverse  pP   *  0 P *  P *
K3

The fraction of the surface that is not occupied by adsorbed B and not with adsorbed P
is given through a site-balance (Langmuir approach):
1
* 
1
1  K 1  pB   pP
K3

The rate of reaction is now deduced from the step that is not assumed to be in
equilibrium, i.e. the surface dimerisation:
2
 rB  2  rP  r2  k 2, forward  B *  k 2,reverse  P *   *
2 2 2 k 2,reverse
 rB  2  rP  k 2, forward  K 1  pB   *   pP   2*
K3
 p 
2 
 P
1
k 2, forward  K 12  pB  1
 2 
 K 12  K 2  K 3 pB 
 rB  2  rP 
2
 1 
1  K 1  pB   pP 
 K3 

If the reaction is taking place far from equilibrium and at low pressure of the product P,
the rate of reaction simplifies to:
k 2, forward  K 12  pB
2
 rB  2  rP 
1  K1  pB 2

59
At low partial pressure of B the reaction is 2nd order with respect to B, whereas at high
pressure the reaction order changes to 0.

Ad 3: Desorption of product P is rate determining

If the product desorption is the rate-limiting step, it means that the adsorption of B and
the surface dimerisation can be considered to be at equilibrium:
r1  k 1, forward  pB   *  k 1,reverse  B *  0 B *  K1  pB  *
2
B
2
r2  k 2, forward  B *  k 2,reverse  P  *  0 P *  K 2  *
 K 12  K 2  pB
2
 *
*

The fraction of the surface that is not occupied by adsorbed B nor with adsorbed P is
given through a site-balance (Langmuir approach):
1
* 
1  K1  pB  K12  K 2  pB2

The rate of reaction is now deduced from the step that is not assumed to be in
equilibrium, i.e. the desorption of the product P:
 rB  2  rP  r3  k 3, forward   p *  k 3,reverse  p P   *

 rB  2  rP  k 3, forward  K12  K 2  pB
2
 *  k 3,reverse  pP  *
 1 p 
k 3,forward K12 K 2 pB2   1 2  P
 K K K p 2 
 1 2 3 B
 rB  2  rP 
1 K1 pB  K12 K 2 pB2

If the reaction is taking place far from equilibrium and at low pressure of the product P,
the rate of reaction simplifies to:
k 3,forward K12 K 2 pB2
 rB  2  rP 
1 K1 pB  K12 K 2 pB2
At low partial pressure of B the reaction is 2nd order with respect to B.

Surface dimerisation or desorption of product P

Rate of consumption of B,

is rate determining
-rB

Adsorption of B is rate determining

Partial pressure for reactant B, pB

Figure 3.5: Influence of the partial pressure of B on the rate of reaction for the
heterogeneously catalysed surface dimerisation in the absence of the
product P far from equilibrium

60
3.1.4 Micro-kinetic modelling
Most rate expressions available in literature are based on a Langmuir-Hinshelwood
approach, although this approach has the same limitations as the use of the Langmuir
isotherm. In particular, the assumptions regarding a homogeneous catalyst surface and
neglect of adsorbate-adsorbate interactions have been shown to be not justified over a
whole range of possible experimental conditions. Hence, the Langmuir-Hinshelwood
approach (and the approaches derived from there) must be viewed as an empirical fit to
experimental data in the measured range of experimental conditions. Extrapolation of
this rate expressions based on this type of approach should be done with care.

An alternative method is to determine the rate of reaction of each elementary reaction

step taking place in the catalytic reaction either experimentally or computationally. The
computational route seems to be preferred due to the speed and the difficulties with the
experimental determination, especially when reactions contain multiple elementary
reaction steps. The set of elementary reactions is then put together to obtain an overall
kinetic description of the system, a micro-kinetic model.

Micro-kinetic modelling is based on theoretical understanding of the elementary

reactions steps and on the activated complex theory or transition state theory. The
reactant is at a minimum point on the potential energy surface (PES) and the
transformation into a product requires the passage through a bottle neck, either an
energetic bottle neck or an entropic bottle neck. The rate constant is then equal to the
probability of attaining the bottle neck and its flux through the bottle neck. Key
assumptions associated with the transition state theory (TST) are related to the life time
of the activated complex/transition state, the uni-directionality of crossing the bottleneck
and the energy distribution of the reactant and activated state complexes. The transition
theory treats the passage through the bottleneck as a classical event, whereas the
energy distribution of the reactant and activated complex is typically determined using
quantum-mechanics. Hence, this theory represents a hybrid formalism.

A simplistic view of the transition theory is the formulation of a quasi-equilibrium between

the reactants and the activated state (or transition state). The activated complex
decomposes towards the products through a vibrational channel with a rate equal to the
vibrational frequency [6-8]. Thus, the rate of reaction for a chemical reaction of A  B
proceeding through a activated state A# is thus given by:

A#
 decomposition

DGA  A# = DHA  A# - T. DSA  A#

A

B
 rA   decomposit ion  C A #   decomposit ion  K A  C A

61
The frequency of passing the barrier can be obtained from the flux of crossing assuming
that the probability of crossing the barrier equals 1, and is then equal to:
kB T
 decomposition 
h
and thus the rate of consumption of the compound A is given by:
k B T
 rA   K A  CA
h
The equilibrium constant for the formation of the activated complex A# from the reactant
A is given by the Gibbs free energy for the formation of the activated complex from the
reactant:
 DGrxn DS rxn DHrxn
A A # A A # AA #
k B T k B T k B T kB k B T
 rA  e  CA  e e  CA
h h

Hence, the classical Arrhenius activation energy is the enthalpy of reaction for the
formation of the activated complex, and the pre-exponential factor is given by the
decomposition frequency of the activated complex and the change in entropy upon
formation of the activated complex from the reactant(s).

The enthalpy of reaction for the formation of the activated complex is obtained from
quantum-chemical calculations [7,8] yielding the difference in the electronic energy of
the activated complex, and the reactant. The change in the translational, rotational and
vibrational states is taken into considerations using partition functions [8]. The latter
typically involves some assumptions regarding the translational and rotational mobility of
the species at the surface. The change in the entropy upon formation of the activated
complex can then also be estimated using partition functions.

It should be realized that the transition theory used for micro-kinetic modeling is in
essence a classical theory, in which the classical partition functions are replaced by
quantum-mechanical ones [8,9]. In a classical theory, the reactant(s) have to cross the
energy barrier, i.e. have sufficient thermal energy to overcome the potential energy
barrier. In a quantum-mechanical treatment reactants(s) may tunnel through the barrier.
Hence, a correction factor needs to be included to account for quantum-mechanical
tunneling for energies below the classical barrier (and also for non-classical reflection
above the barrier). The tunneling effect will result in an enhanced rate of reaction, and is
of particular importance for light elements (such as H) in reactions with a high activation
barrier and a small width of the reaction pathway [10].

The temperature at which the rate of elementary reaction steps are considered in micro-
kinetic modeling is typically taken as the actual reaction temperature. This implies that
the energy dissipation into the bulk of the catalytically active phase must be fast in
comparison to the rate of reaction. This is usually the case (except for very fast
reactions), since the thermal conductivity of many catalytically active materials I high in
comparison to the energy that needs to be dissipated.

3.1.4.1 Modeling an elementary surface reaction step

Modern quantum-chemical methods (such as the Density Functional Theory) can be
used to describe the reaction pathway for the formation of a particular reaction step
occurring on the surface of a catalyst particle. As an example, the hydrogenation of a
surface O on Pt(111) will be described (based on the data from [11]). Typically, the first
step is to estimate the energy of the reactants and the products (see Fig. 3.6). This

62
allows the estimation of the thermodynamics associated with the elementary reaction
step.

O on Pt(111) H on Pt(111) OH on Pt(111)

Eads, eV (incl. ZPE) -1.07 -1.46 -0.22

Normal modes, cm-1 469 3615 1117
385 723 635
383 686 600
388
215
203
Figure 3.6: Adsorption energy of O, H and OH adsorbed on Pt(111) at a coverage of
0.25 ML (incl. ZPE) with respect to gas phase O2, H2 and bare Pt(111) as
calculated using GGA-PBE and the associated vibrational frequencies as
determined using a partial Hessian [11].

Hence, the energy change associated with the reaction is -0.17 eV (or -16 kJ/mol). The
equilibrium constant for this reaction can be written in terms of the partition functions [8]:
 iOH

p.f .OH
e k B T

K eq   i
p.f .H  p.f .O  H i  iO

e
i
k B T
 ei
k B T

and taking into account only the vibrational contributions (i.e. assuming that the motion
of the species on the surface is satisfactorily described by the normal modes):


1
h OH
j
j  DU'0
k B T
K eq 
p.f .OH
 1 e e k B T
p.f .H  p.f .O
 
1 1

hHj h Oj
j j
k B T
1 e 1  e kB T

(with j the number of vibrational modes considered and U’0 the energy difference
between the species including the change in the zero point energy – ZPE).

The first step in the reaction is bringing the reactants (H and O) in proximity (see Fig.
3.7). This step is associated with a change in energy of the system and is repulsive on
Pt(111) (the co-adsorbed state is 0.33 eV less stable than the state in which the
reactants are far from each other) indicating that the probability to find these species in
close proximity is small. This step can be seen as a pre-equilibrium before the actual
reaction takes place and takes into account lateral interactions between the reactants:

63
  
 

 1 
 h Hj O close 
 j
  DU' appraoch
p.f.H and O close  1 e k B T
 H and O close
K eq    e k B T
p.f.H andO far  
 

 1 
 h Hj O far 
 j
k B T 
 1 e  H and O far

(with j the number of vibrational modes considered and U’approach the energy difference
between the species in close proximity and the species far away from each other
(including the change in the zero-point energy – ZPE).

Eco-ads, eV -0.96
Normal modes, cm-1 2280
448
437
433
314
313

Figure 3.7: Co-adsorption of O and H on Pt(111) at a coverage of 0.25 ML

(adsorption energy incl. ZPE with respect to O2 and H2 in the gas phase)
as calculated using GGA-PBE and the associated vibrational frequencies
for the co-adsorbed state as determined using a partial Hessian [11].

The next step in the process is to combine these species yielding a surface hydroxyl-
species (see Fig. 3.8). The rate constant for the formation of the hydroxyl group on the
surface is given by:
 Drxn GTS  Drxn STS  Drxn HTS
k T k T
k OH,f  B  e kB T  B  e kB  e kB T
h h
The change in the Gibbs free energy upon formation of the transition state from H and O
in close proximity is given by:
p.f.TS
D rxn GTS  k B  T  ln
p.f.H andO co-adsorbed
or
 
 

 1 
 j h TSj 
 k B T 
 DU'rxn
kB  T p.f .TS kB  T  1 e  TS
k OH,f      e k B T
h p.f .H and O co-adsorbed h  
 

 1 
 j h Hj O close 
 
 1  e k B T  H and O close

64
 H and O co-adsorbed Transition state OH adsorbed

Eads, eV (incl. ZPE) -0.96 -0.61 -1.46

Normal modes, cm-1 2280 1889 3615
448 592 723
437 513 686
433 351 388
314 244 215
313 570i 203

Figure 3.8: Co-adsorption of O and H, transition state to form OH, and final state on
Pt(111) at a coverage of 0.25 ML (adsorption energy incl. ZPE with
respect to O2 and H2 in the gas phase) as calculated using GGA-PBE and
the associated vibrational frequencies for the co-adsorbed state as
determined using a partial Hessian [11].

The overall rate constant for the formation of surface hydroxyl can now be computed
taking into account the pre-equilibrium to bring O and H in close proximity:
 
 
  1 
 j h TS
j   DU'rxn  DU'appraoch 
 k B T 
kB  T  1 e  TS k B T
k OH,f   e
h  
 
  1 
 j h j 
H O far

 
 1  e k B T  H and O far

This yields an activation energy for the formation of surface hydroxyl groups from O and
H separated on the surface of 0.5 eV (or 48 kJ/mol). The pre-exponential factor
increases slowly with temperature from 0.7.1013 s-1 at 300 K to 1.48.1013 s-1 at 900 K.
Since hydrogen is involved in the reaction, the rate will be enhanced by tunnelling of
hydrogen through the activation barrier. The enhancement factor, , can be
approximated knowing the imaginary frequency of the transition state [10]:
2 4
 h   h 
  7   
k quantummechanical  kB  T   k B  T 
  1  
k chemical 24 5760
(this results in a value for the enhancement factor, , of larger than 1 since the
vibrational frequency is an imaginary number) and decreases with increasing
temperature from 1.37 at 300 K to 1.03 at 900 K resulting in an increase in the pre-
exponential factor from 1.1013 s-1 at 300 K to 1.55.1013 s-1 at 900 K.

65
Knowing the rate constant for the forward reaction for the formation of a surface hydroxyl
species from O and H separated from each other on a Pt(111) surface, the rate for the
reverse reaction is known as well, since the rate constant for the forward reaction and
the rate constant for the reverse reaction are linked to each other through the
thermodynamic constraint:
k HOOH
 K eq
k OHHO

3.1.4.2 Adsorption and desorption processes

Adsorption can be energetically an activated or a non-activated process. For an
energetically non-activated process, the location of the barrier (in this case an entropical
barrier) is often difficult to determine. The rate of adsorption can be estimated by
considering that in the process of adsorption gas phase molecules collide with the
surface. It can be argued that the transition state theory does predict the rate of
adsorption processes with the reactant state being the transition state and the ‘frequency
of decomposition of the activated complex’ viewed as the motion of the adsorbing
molecule in the direction of the reaction coordinate.

Molecules in the gas phase move freely and the average velocity of the gas phase
molecules is given by the Maxwell-Boltzmann distribution. Hence, the probability that a
molecule has a velocity vz perpendicular to the plane of adsorption is given by
mv 2z
1 2k B T
P(v z )  e
kB  T
2 
m
The total number of molecules colliding with the surface per time unit and per unit
surface, with any velocity, is therefore:
mv 2z
ncolliding  
1
 
2k B T
 v z ρ gas  P(v z )  dv z  v z ρ gas  e  dv z
area k T
0 0
2  B
m
ncolliding kB  T
 ρadsorbng gas 
area 2π  m

ncolliding p adsobing gas

or 
area 2π  m  k B  T
(this result is the same as for the effusion of molecules, where molecules pass an
entropical barrier [12]).

Not all collisions will result in an adsorbed molecule, and a sticking coefficient, S, is
introduced. The rate of adsorption is thus given by:
ncolliding p adsobing gas
radsorption  S   S and S
area 2π  m  k B  T k adsorption 
2π  m  k B  T
2
(with p in N/m , m (mass of the adsorbing molecule) in kg, and T in K).

This would yield the rate of adsorption per m2 of surface area. The rate has to be
multiplied by the area of the active site to obtain the rate per active site.

66
The desorption process, associated with an energetically non-activated adsorption
process, is an activated process and may be thought to occur through a vibrational
mode containing sufficient energy. Hence, the rate constant for the desorption process
can then be given by:
E desorption E adsoption

k B T k T k B T
k desorption   desorption  e  B e
h
However this formulation is thermodynamically inconsistent. A thermodynamic
consistent formulation is obtained by considering:
DGads
k adsorption 
k B T
K adsorption  e
k desorption

Hence, the rate constant for the desorption process must be given by:
 DGdes DS des  DHdes DS des  DHdes
k B T k B T S k B T
k desorption  k adsorption  e  k adsorption  e kB
e  e kB
e
2π  m  k B  T
Thus, the rate constant for the desorption process is also dependent on the sticking
coefficient. This sticking coefficient can in terms of the transition state theory be
interpreted in terms of re-crossing of the divide between he reactant and product
(transition state) [13], and hence the sticking coefficient must be valid for both the
adsorption and desorption process. The entropy of the activated complex (transition
state seems to be reduced by one translational degree of freedom in comparison to the
entropy of the reactant.

3.1.4.3 Building a micro-kinetic model

The energetics of elementary reactions can be determined using quantum-chemical
methods such as Density Functional theory (DFT). The adsorption energy of species
and the co-adsorption energy preceding the reaction step are then determined.
However, the surface is covered with a range of different species which may interact
with each other (lateral interactions) resulting in a shift in the adsorption energies. The
lateral interaction can be repulsive or attractive (i.e. destabilising or stabilising a surface
species.

Bonding of species to a surface typically may yield a shift in the electron density
distribution with a net charge on the adsorbing species. This will result in an electro-
static dipole-dipole interaction between the adsorbed species. This interaction is typically
long-range in nature [14-17].

Short-range lateral interactions can be caused by steric hindrance, but also change in
the orbitals (or electron density) on sites adjacent to an occupied site. They can be
alternatively repulsive and attractive [16].

These lateral interactions may modify the strength of adsorption and even the activation
barrier. The presence of lateral interactions may therefore significantly affect the rate of
reaction and must be built into a kinetic model. There are in principle two different
approaches to the construction of a micro-kinetic model. The Monte-Carlo method
determines the energetics of the system in predefined time steps for consecutive
configurations of surface species (structures). This is method depends highly on the size
of the cell considered and requires large computational power for estimating the rate of

67
 reaction. An alternative approach is the mean field approach, which requires an a priori
estimation of the lateral interactions.

The oxidation of hydrogen on Pt(111) will result in the presence of co-adsorbed O-

species on the surface. The adsorption energy decreases strongly with increasing
coverage of O (see Table 3.1) due to lateral interactions. The decrease in the adsorption
energy can be modelled taking into account the position of the adjacent surface oxygen
species. The adsorption energy of O is reduced by ca.0.09 eV, when the O atom shares
one metal atom an adjacent adsorbed O (corner sharing sites), and by 0.49 eV when it
shares two metal atoms with and adjacent adsorbed O (edge sharing sites). Knowing the
interaction energy associated with O located on either an edge sharing or corner sharing
site, the likelihood that one O is located at an edge sharing side is given by:
PO on edge sharing site  1.01  4  0.49  O  4  0.09  O
or in general for having m O on edge sharing and n O atoms on corner sharing sites:

Hence, in a mean field approach, the heat of adsorption can be modelled as:
E ads,O on Pt(111)(eV )  1.01  4  0.49   O  4  0.09   O
(the 2nd and 3rd term are multiplied with 4 each, since there are 4 edge sharing and
4corner sharing sites)

Table 3.1: Adsorption of O on Pt(111) on a p(3x3) cell as determined using GGA-

RPB [11] (adsorption energy with respect to gas phase oxygen)
O 0.11 0.22 0.33 0.44

Eads, eV (incl. ZPE) -1.01 -0.92 -0.82 -0.67

A variety of species are involved in the oxidation of hydrogen on Pt(111), which may
interact in different ways in a variety of elementary reaction steps (see Table 3.2). We
can identify the following reactions:

Adsorption/desorption
 2 
(1) H2+ 2*  2 H* r1  k 1   p H2   *2  H 
 K 1 

 O 
(2) O2+ *  O2* r2  k 2   p O2   *  2 
 K 2 

 H O 
(3) H2O + * H2O* r3  k 3   p H 2 O   *  2 
 K 3 


Formation of peroxide species

   
(4) O2* +H*  OOH* r 4  k 4    O 2   H  OOH * 
 K4 
   
(5) OOH* +H*  HOOH*+* r5  k 5    OOH   H  HOOH * 
K5 
 

68
Cleavage of O-O bond
 2 
(6) O2* +*  2 O* r6  k 6    O 2   *  O 
 K 6 

   
(7) OOH* + *  O* + OH* r7  k 7    OOH   *  O OH 
 K7 
  
2
(8) HOOH* + *  2 OH* r8  k 8    HOOH   *  OH 
 K 8 

Hydrogenation of surface oxygen and hydroxyl species
   
(9) O* + H*  OH* + * r9  k 9    O   H  OH * 
 K9 
 H O   * 
(10) OH* + H*  H2O* + * r10  k 10    OH   H  2 
 K 10 
 
Recombination of surface hydroxyl species
 2 H O   O 
(11) OH* + OH*  H2O* + O* r11  k 11    OH  2 
 K 11 
 
Knowing the equilibrium constants and the rate constants (see Table 3.2) for each of
these reactions, and the site balance
1   *   H   O 2   H2O   OOH   HOOH   O   OH
the surface coverage with the various species can be determined by implementing this
set of rate expressions in an appropriate reactor model.

Table 3.2: Kinetic parameters to model H2 oxidation over Pt(111) at 100 K (data from
[11]; reaction enthalpy and entropy calculated at 100 K; assuming a site
density of 1.5.1019 sites/m2)

Reaction DH DS A(100 K) Ea
kJ/mol J/(mol.K) molecules/s kJ/mol
H2+ 2*  2 H* -97.8 -81.8 1.3.109 1 -
O2+ 2*  O2* -52.4 -160.5 3.2.108 1 -
H2O+ *  H2O* -12.9 -136.8 4.2.108 1 -

O2* +H*  OOH* +* 2.6 -5.1 1.8.1013 49.5

OOH* + H* HOOH* +* 4.0 15.7 4.2.1012 42.7

O2* +*  2 O* -158.1+227.2.O 11.7 2.7.1012 55.6

OOH* + *  O* + OH* -149.8+227.2.O 3.3 4.2.1012 17.2
HOOH* + * 2 OH* -142.8 -25.9 3.0.1012 31.8

O* + H*  OH* + * 11.0-227.2.O -13.5 2.9.1012 62.8-227.2.O

OH* + H*  H2O* + * -42.1 8.0 .
4.6 1012
24.7
OH* + OH*  H2O* + O* -53.0+227.2 O
. 12
.
21.5 2.9 10 -11.02
1
molecules/s/bar
2
activation energy less than zero due to the attraction of surface hydroxyl groups
(adsorption energy with respect to gas phase oxygen)

69
For a reaction taking place in a reactor operating at steady-state, the surface coverage
of the various species does not vary with time. Hence, the surface coverage of the
various species can be estimated from a steady-state balance around the species. The
following balances are obtained for the oxidation of hydrogen over Pt(111):

H*: 0  2  r1  r4  r5  r9  r10
O2*: 0  r2  r4  r6
H2O*: 0  r3  r10  r11
OOH*: 0  r4  r5  r7
HOOH*: 0  r5  r8
O*: 0  2  r6  r7  r9  r11
OH*: 0  r7  2  r8  r9  r10  2  r11

3.2 Internal mass and heat transfer limitations – pore diffusion

The reactants have to be transported from the outside of the catalyst particle to the
active site. Mass is transported either by molecular motion, diffusion, or by convective
flow due to a pressure gradient within a catalyst particle [18].

Molecular motion, diffusion, in the pores of the catalyst can be controlled by molecular
diffusion (ordinary gas or liquid diffusion depending on the phase present in the pores),
Knudsen diffusion or configurational diffusion. Molecular diffusion dominates the
molecular motion if molecules collide more frequently with other molecules than with the
walls of the pores in the catalyst. Molecular diffusion is thus the dominant mechanism for
molecular motion if the concentration of the molecules in the pores is high (e.g. high
pressure). At low pressures molecules are more likely to collide with the walls of the
catalyst pores than with other molecules and Knudsen diffusion becomes the dominant
mechanism.

The ordinary gas diffusion in binary gas mixtures can be estimated using the Fuller,
Schettler and Giddings relation [19]:
 1 1 
10 3  T1.75    
 A
M M B
D AB  2
 1 1 
p     A3    B 3 
 
(with DAB in cm2/s; T in K; p in atm; M the molar mass in g/mol). Table 3.1 shows the
atomic diffusion volumes,  for the estimation of the binary diffusion coefficient. This
estimated diffusion coefficient is typically within 5-10% for pressures to about 10 atm (in
particular for non-polar gases). The diffusion coefficient in multi-component mixtures can
be estimated from binary diffusion coefficients according to Wilke [20]:
1 x A
D A,mix 
xi

j A DA, j

70
Table 3.1: Atomic diffusion volumes, n, for the estimation of the binary gas diffusion
coefficient using the Fuller, Schettler and Giddings relation (values in
bracket are associated with a higher degree of uncertainty)
Atomic and structural diffusion volume increments
C 16.50 (Cl) 19.5
H 1.98 (S) 17.0
O 5.48 Aromatic ring -20.2
(N) 5.69 Heterocyclic ring -20.2
Diffusion volumes for simple molecules
H2 7.07 CO2 26.9
He 2.88 N2O 35.9
N2 17.9 NH3 14.9
O2 16.6 H2O 12.7
Air 20.1 (Cl2) 37.7
Ar 16.1 (Br2) 67.2
Kr 22.8 (SO2) 41.1
CO 18.9 (SF6) 69.7

The estimation of diffusion coefficients in the liquid phase results from the Stokes-
Einstein equation for dilute solutions (e.g. the Wilke-Chang estimation), which can be
used to estimate the diffusion for non-dilute solutions using an activity model [20]. The
estimation of diffusion coefficients in multi-component, liquid mixtures is difficult.

Knudsen diffusion is the dominant mechanism when the mean free path length between
collisions between molecules becomes larger than the pore diameter. The Knudsen
diffusion coefficient is thus a function of the radius of the pores within the catalyst
particle. The Knudsen diffusion coefficient is given by:
2 8 R  T
D A,Knudsen  r 
3 pore   MA

When the pores become in the range of the size of the molecules, the motion of the
molecules is strongly retarded due to the interaction between the wall and the
molecules. This is typically the case in molecular sieves (such as zeolites), in which the
pores have the size of a few Ångstrom (10-10 m), which is in the range of the diameter of
molecules.

Figure 3.6 shows the diffusion of phenol in a dilute phenol in water solution as a function
of the pore size of the catalyst particle at 10 atm and 298K. For a gas-phase mixture of
phenol and water molecular gas diffusion dominates the molecular motion in catalyst
pores larger than ca. 5m. In smaller pores, Knudsen diffusion becomes dominant since
the mean free path length between intermolecular collisions becomes larger than the
pore diameter. In the liquid phase, molecular liquid diffusion dominates the molecular
motion in pores larger than ca. 150 nm (0.15 m). In the liquid phase, molecular diffusion
dominates even in smaller pores due to the higher concentration of molecules in the
liquid phase. In pores with a radius below 1 nm the energetic interaction between the
molecules in the pores and the pore wall become strong resulting in a strong decrease in
the diffusion coefficient (configurational diffusion).

The transport through diffusion occurs whenever a concentration gradient of a particular

compound exists in a pore. The origin of the gradient is the chemical reaction, which

71
takes place within the pores of the catalyst. Matter can also be transported by flow
through the pores in catalyst particles. Flow can occur if a pressure gradient exists within
the pores of the catalysts and the dominant mechanism for transport is not Knudsen
diffusion. This will thus only be important for gas phase reactions in catalyst pores with a
large pore radius. Liquids can be viewed as incompressible and thus the pressure
gradient within a liquid in a pore is minimal. For large pore radii the transport through the
flow caused by the pressure gradient can be much faster than transport through the
molecular motion. The flow in the pores can be viewed as a Poiseuille flow [18].

molecular diffusion Knudsen diffusion configurational diffusion

Figure 3.6: Influence of the pore radius on the diffusion coefficient

Top: Diffusion coefficient of phenol in dilute water mixtures at 10 bar
and 298 K (data points for phenol in water in TS-1 and Ti-Beta
obtained from [21])
Bottom: Schematic representation of molecular diffusion, Knudsen
diffusion and configurational diffusion

The catalyst particle can be thought of as a loose pile of bricks. The interstitial space
between the solid making up the catalyst particle is the pore space (see Figures 2.1, 2.2
and 3.1). Pores in catalyst particles do not possess a cylindrical shape (as schematically
shown in Figure 3.6). The average pathway a molecule follows is thus not straight. To
account for this the diffusion coefficient is divided by the tortuosity factor, , which
represents the actual distance travelled relative to the shortest distance between two
points. Furthermore, catalyst particles consist of solids and pore space. Molecules can

72
only move through the pore space, but not through the solid phase. Hence, the diffusivity
is corrected with the porosity. The effective diffusivity through a catalyst particle is thus
given by:
D
D effective 


3.2.1 Concentration gradient in catalyst pores

Catalysts are typically porous materials. The catalytic reaction is typically taking place
within the porous structure of the catalyst particle, since the surface area of the catalyst
is mainly associated with the pore structure and the external surface area typically
contributes very little to the overall surface area. The chemical reaction in the pore will
lead to a concentration gradient within the pore structure. Thus, the concentration of the
reactant(s) will vary depending on the position within the pore. The rate of reaction is
dependent on the concentration of the reactant(s) at the active site and will thus also
vary with the position within the pore. The occurrence of a chemical reaction induced
concentration profile leads to diffusion of the reactant(s) in the pore.

To illustrate the concentration profile within a catalyst particle, we will assume that an
isothermal, catalytic reaction A  P takes place within a cylindrical pore of of length Lpore
(see Figure 3.7). The reaction takes place without a change in the number of moles and
thus transport of the reactant(s) takes place through diffusion. The chemical reaction is
assumed to be 1st order with respect to A.
 mol 
 rA    k  CA with the units for k: m3/(gcat.s)
 gcat  s 
The rate of reaction per unit surface area is thus given by:
'  mol  k  CA
 rA  2   with SA the surface area in m2/g
 m s  SA

CA,s

CA

JA,z -rA JA,z+Dz 2.rpore

z z+Dz

Lpore
Figure 3.7: Schematic model for a reaction in a pore

73
The rate constant for the reaction is normalised with respect to the surface area of the
catalyst and has therefore the units Following the approach described by Thiele [22], a
steady-state material balance for the reactant A over a volume element between the
pore length z and z+Dz within the catalyst pore is given by:
2 2 k
  r  J A,z    r  J A,z  Dz   C A  2    r  Dz  0
SA
(with JA the molar flux of A in and out of the volume element in the catalyst pore)
Rearranging and taking the limit of Dz0
dJA 2 k  C A

dz r  SA

According to Fick’s law the molar flux of A for equimolar counter-diffusion is given by:
dC A
J A  D A
dz
d2 C A k
  CA  0
dz 2 r  D A  SA
This 2nd order differential equation can be solved with the boundary conditions, that at
the pore entrance the concentration of the reactant A is CA,s and that at the centre of the
pore, z=Lpore/2, the concentration profile is symmetrical:
z=0 CA=CA,s
z=Lpore/2 dCA/dz=0
With these boundary conditions the concentration of the reactant A is given by:
  2  z  
cosh   1 
  L pore   L pore 2k
CA
   
with  
C A, s cosh 2 r  D A  SA

The term  is called the Thiele module. This constant can be interpreted as the ratio of
L2pore
the time constant for transport (diffusion), , and the time constant for reaction
DA
r  SA
. The Thiele module is small, if the time constant for transport (diffusion) is small in
k
comparison to the time constant for the reaction, i.e. when the rate of the chemical
reaction is slow in comparison to the rate of transport. The Thiele module is large, if the
time constant for diffusion is very large in comparison to the time constant for reaction,
i.e. when diffusion is slow in comparison to the chemical reaction.

Figure 3.8 shows the concentration profile in a pore for various values of the Thiele
module. The concentration of the reactant A in the pore is always smaller than the
concentration of A at the pore mouth. The concentration of the reactant A is expected to
decrease with increasing value for the Thiele module. With a Thiele module of larger
than 5, the concentration of A in the centre of the pore becomes very small.

Similar profiles can be derived for other type of reactions. Table 3.2 show the
concentration profile as a function of the pore in an open, cylindrical pore for an
isothermal chemical reaction for reactions with a different reaction order. The intrinsic
kinetics of catalytic reactions is often described using Langmuir-Hinshelwood type of
expressions (see chapter 3.1). However, catalytic reactions can often be approximated

74
 by more simple forms of a kinetic expression in a limited concentration range, such as 0
order, 1st order or 2nd order type of reaction.

centre of the pore

1
=0.5

0.8
=1
CA/CA,s

0.6

=2
0.4

0.2
=5

0
0 0.2 0.4 0.6 0.8 1
pore pore
mouth z/Lpore mouth

Figure 3.8: Concentration profile for a reactant A undergoing a first order, irreversible
reaction under isothermal conditions in an open, cylindrical pore

Table 3.2: Concentration of a reactant A in an open cylindrical pore as a function of

 rA
the Thiele module (with  rA'  )
SA  r
Reaction 0 order 1st order 2nd order
Irreversible Irreversible Irreversible
Rate ' k ' k CA k  C2A
 rA   rA  '
 rA 
SA SA SA
Thiele L pore k L pore k L pore k  CA
module      
2 D A  r  SA  C A, s 2 D A  r  SA 2 D A  r  SA
L pore  rA' L pore  rA' L pore  rA'
     
2 D A  C A, s 2 D A  C A, s 2 D A  C A, s
  2z 
 2 cosh   1  
CA 2  2z 1  2z 
    L pore 
CA,pore=f(z)  1     CA    Complex see [23]
 L pore 2  L pore   
cosh
C A, s   
 C A, s

The Thiele module for the various reaction orders is seemingly defined differently. The
rate constant in the Thiele module can be viewed as the ratio of a pseudo first-order rate
constant (normalised per unit surface area) relative to the effective diffusion coefficient.
The solution for the concentration profile for a 0 order reaction given in Table 3.2 differs
from the solution given by Wheeler [18]. Wheeler argues that at the point the
concentration of A becomes zero, the gradient (dCA/dz) will become zero. This is not

75
correct for a reaction with a reaction order of 0 at all concentrations of the reactant A and
an abrupt change in the concentration profile is expected when the concentration of A
becomes zero (in reality the reaction order will change at very low concentrations).

3.2.1.1 Concentration gradient in a catalyst particle

The catalytic activity is typically not measured in a single pore, but rather in a catalyst
particle of a certain shape. The concentration gradient in a catalyst particle can be
analysed in a similar manner by considering a steady-state material balance over a shell
in a catalyst particle (see Figure 3.9).

Rparticle

Dr
r+
r

0.8
second order
CA/CA,s

0.6
first order
0.4
zero order
0.2

0
-1 -0.6 -0.2 0.2 0.6 1
Position in catalyst pellet, r/Rparticle

Figure 3.9: Concentration profile for a reactant A undergoing an isothermal

irreversible reaction in a spherical catalyst particle with Thiele module, ,
of 2.

4    r  Dr  2  J A,r  Dr  4    r
2 n
 J A,r  k    C A 
4
3

   r  Dr 3  r 3  0 
(with JA the molar flux of A in and out of the shell in the catalyst particle. Substituting
Fick’s 1st law for the molar flux of A and taking the limit of Dr0:
d  2 dC A  k   n
 r  D A,effective    CA
3  dr  3
dr
In this expression, the effective diffusivity appears rather than the diffusion coefficient,
since a catalyst particle consists of void space and solid material. The reactants can only

76
diffuse through the void space and not through the solid phase within the catalyst
particle.

If the effective diffusivity is not a function of the position within the catalyst particle:
d  2 dC A  k 
r    CnA
2
r  dr  dr  D A, effective

The differential equation can be made dimensionless by introducing the following

dimensionless parameters:
r C
 A  A
R particle C A,s
d 2 dA 
      2  An
2
  d  d 

with , the Thiele module defined as:

k  CnA,s1    rA  
  R particle   R particle 
D A,effective D A,effective  C A,s

This second order differential equation can be solved numerically for various reaction
orders (see e.g. [23]) with the following boundary conditions:
 1 A  1
dA
0 0
d
For a 1st order reaction the concentration of the reactant A relative to the concentration
of the reactant A at the outer surface of the catalyst particle is given by:
 r 
sinh   
 R particle 
CA
  
 sinh
C A, s r
R particle

The concentration profile originates from a reaction taking place within the pore system
of the catalyst particle. The rate of reaction is dependent on the local concentration of
the reactant. Thus, a decrease in the concentration will lead to a reduction in the rate of
consumption of the reactant for a reaction with a reaction order larger than 0. The
decrease in the rate of reaction is more pronounced for reactions with a high reaction
order. The decrease in the rate of reaction is related to the concentration gradient of the
reactant in the pore. A lower rate of reaction results in a smaller gradient. This will lead
to a higher concentration of the reactant in the catalyst particle in reactions with a higher
reaction order.

The concentration profile of the reactant A shows a minimum at the centre of the catalyst
particle. For a reaction of AP, the concentration profile of the product P is the mirror
image of the concentration profile of the reaction A. The total concentration remains
constant. (In the case of a temperature profile within the catalyst particle (see 3.2.2) the

77
change in the total concentration due to a variation in the local temperature is minimal in
comparison to the change in the concentration due to mass transfer limitations and can
thus be considered to be approximately constant.) The concentration of product P is thus
maximum in the centre of the catalyst particle.

3.2.2 Pressure gradient in a catalyst particle

Reactions, in which the number of moles of product compounds differs from the number
of moles of reactant(s), will generate a pressure gradient within the catalyst particle
[18,24-27]. Thus, the gas phase reaction of A s P with s1, will result in a pressure
difference between the outside of the catalyst particle and the centre of the catalyst
particle. The pressure in the catalyst particle is expected to be higher than the pressure
on the outside, when the number of moles increases due to reaction (s>1). The pressure
in the catalyst particle is expected to be lower than the pressure on the outside, when
the number of moles increases due to reaction (s<1).

A pressure gradient over the catalyst particle will also result in a pressure gradient within
the catalyst particle. The pressure drop over a single catalyst particle is typically very
small and can be neglected [18].

The occurrence of a pressure gradient within a catalyst particle will lead to a forced flow
in addition to the normal diffusion process taking place within the catalyst particle. The
forced flow will be Knudsen flow when the pore radius in the catalyst particle is smaller
than the mean free path length of the molecules between collisions (and the diffusion
process is dominated by Knudsen diffusion). Knudsen flow is indistinguishable from
Knudsen diffusion [9], and thus the mass transfer of material in the catalyst particle is not
affected. The forced flow will result in Poiseuille flow, if the mean free path length of the
molecules is smaller than the pore radius.

The Knudsen diffusion coefficient depends on the molecular mass of the diffusing
species. Compounds with a different molar mass will thus diffuse at a different rate, but
at steady-state the net molar flux of the reactants and the products in a shell with a
thickness of Dr is in a spherical catalyst particle will be zero. This yields a relation
between the flux of the reactant and the flux of the products, from which a relationship
can be developed between the pressure in the catalyst particle relative to the pressure
on the outside of the catalyst particle as a function of the stoichiometric coefficient of the
reaction, s, the mole fraction of the reactant on the outside of the catalyst particle, and
the ratio of the concentration of the reactant A relative to the concentration of A at the
outside of the catalyst particle (the latter term can be related to the Thiele module) [27]:
p
ps
   C 
 1  1  s 0.5  y A, s   A  1
C 
 A, s 
and the maximum pressure difference can be estimated using CA=0:
 p 
 
p
 
 1  s 0.5  1  y A, s
 s  max

In a similar manner, the pressure in the catalyst particle relative to the pressure on the
outside of the catalyst particle can be related to the mole fraction of the reactant A at a
certain position in the catalyst particle and the mole fraction of A on the outside of the
catalyst particle, if molecular diffusion is the dominant diffusion mechanism. Only the
case of a gas phase reaction has to be considered, since liquids can be considered as

78
incompressible fluids and thus the pressure gradient will be minimal. If molecular gas
diffusion is the dominant diffusion mode, the flow in the particle due to the pressure
difference has to be taken into account. For gas phase systems this can be related to
the ratio of the concentration of A relative to the concentration of A at the outer surface
using an iterative process.

p 2    D AB, effective  
 1  1  s 0.5  y
 ln
1  s  1  y A, s 
 
A
 1 
ps   ps  1  s  1  y A 1  1  s 0.5  y 
 A, s 
The relative pressure in the particle is now a function of the viscosity of the mixture, ,
the effective diffusivity, DAB,effective, and the permeability through the porous particle, .
The permeability through a catalyst particle can be estimated to be between 10 -16 and
10-13 m2 [25]. Estimating the gas phase molecular diffusivity as 10-5 m2/s and the gas
phase viscosity as 10-5 Pa.s yields for a pressure of 1 atm:
2    D AB, effective 2  10 5  10 5
  ps

 
10 16  10 13  10 5
 0.2  200

The maximum pressure drop over a catalyst particle, when the molecular
 p



0.2  200   ln 1  s  1  y A,s 
 (with p in bar)
 ps  max
 1
ps
  
 1  1  s 0 .5  y
A, s


s

Figure 3.10 shows schematically the pressure gradient within a catalyst particle. The
pressure gradient exists independent of the reaction kinetics and only dependent on the
reaction stoichiometry. The pressure gradient, if molecular diffusion is the controlling
diffusion mechanism, is strongly dependent on the permeability. For materials with a low
permeability, the pressure gradient can be substantially higher than the pressure
gradient, if Knudsen diffusion is the limiting case. It should however be realised that both
Knudsen diffusivity and permeability are related to the average pore diameter in the
catalyst particle. In catalyst with a low permeability, Knudsen diffusion is much more
likely to become the dominant diffusion mechanism [26]. For catalyst particles with a
high permeability the pressure gradient in the catalyst particle is less if molecular
diffusion controls the diffusion mechanism than when Knudsen diffusion is the controlling
the diffusion mechanism. This can be rationalised, since if molecular diffusion controls
the diffusion mechanism, flow within the catalyst particle will reduce the pressure
gradient.

The maximum pressure difference between the pressure on the outside of the catalyst
particle and inside the catalyst particle can be significant, if the reaction is carried out
under severe mass transport limitations (i.e. if the concentration of the reactant A in the
centre of the catalyst particle tends to zero). Depending on the controlling diffusion
mechanism and the reaction stoichiometry, the maximum pressure difference can be up
to 50% of the total pressure, if the gas phase on the outside of the catalyst particle
consists of pure reactant. In industrial operations, it is thus expected that the highest
pressure gradients within catalyst particles occurs at the bed entrance. Furthermore, the
catalyst must be able to withstand quite high pressure differences, if the reaction is
carried out under mass transfer limitations. For example, in a reaction of 2AB
operating at 100 bar, the maximum pressure difference is ca. 30 bar. This large pressure
difference may cause disintegration of the catalyst particle.

79
 1.5

s=2 (Knudsen)

s=2 (molec.)

p/ps 1
equimolar reaction, s=1
s=0.5 (molec.)

s=0.5 (Knudsen)

0.5
0 0.2 0.4 0.6 0.8 1
CA/CA,s
Maximum relative pressure,

1.5
s=2 (Knudsen)

s=2 (molec.)
equimolar reaction, s=1
p/ps

1
s=0.5 (molec.)

s=0.5 (Knudsen)

0.5
0 0.2 0.4 0.6 0.8 1
Mole fraction of reaction A on the outside of the
particle, yA,s

Figure 3.10: Pressure gradient within catalyst pellets if Knudsen diffusion is the
dominant diffusion mechanism (dotted line) and if molecular gas diffusion
is the dominant mechanism (solid line). (Gas diffusion estimated using
2    D AB, effective
 1)
  ps
Top: Pressure in the catalyst particle as a function of the local
concentration relative to concentration in the gas phase for
yA,s=1
Bottom: Maximum pressure difference in catalyst particle relative to
pressure on the outside of the catalyst particle as a function of
the mole fraction of the reactant A on the outside of the catalyst
particle

3.2.3 Temperature profile in a catalyst particle

Every chemical reaction is associated with a heat of reaction. In some cases the
temperature effect can be marginal (e.g. in isomerization reactions), but in other cases

80
(such as oxidation reactions) the heat of reaction can be significant. The amount of heat
released by the chemical reaction is dependent on the rate of reaction and the heat of
reaction. The rate of reaction is a function of the concentration of the compounds
involved and the temperature at a certain position within the catalyst particle. Heat is
being transported in the catalyst particle by thermal conductivity.

The thermal conductivity of porous materials has been investigated [28-31]. A large part
of the heat conduction takes place through the solid. The contribution of the void space
within the catalyst particle and in particular of the gas within the void space has a relative
small influence on the overall conductivity, but the overall thermal conductivity is a
function of the thermal conductivity of the solid and the thermal conductivity of the gas
within the pore system. The thermal conductivity of a porous solid in vacuum is only a
weak function of temperature [29]. The thermal conductivity of bulk solids passes
through a maximum as a function of temperature [29], whereas the thermal conductivity
of gases and liquids increase with increasing temperature.

Table 3.3: Thermal conductivity,  (W/(m.K) of some selected materials at 300 K

(data from [29])
Al2O3 Graphite Fe MgO SiO2 H2O(l) H2 (g) O2 (g)
, W/(m K)
. a
36 10 (2000) 80 48 1.34 0.62 0.182 0.027
a
: Thermal conductivity of graphite is dependent on its orientation

1
Ag-He (1 atm)
Thermal conductivity, ,

Ag-Air (1 atm)
0.8
Ag-vacuum
Al2O3-He (1atm)
W/(m K)

0.6 Al2O3-Air (1 atm)

.

Al2O3-vacuum
0.4

0.2

0
0 0.5 1 1.5 2 2.5 3 3.5
3
Density, g/cm
Figure 3.11: Thermal conductivity of silver and alumina particles in various gases as a
function of the pellet density (particle density is related to the void fraction
in the catalyst pellet) (Figure redrawn from [28,29])

The temperature profile in a spherical catalyst particle is related to the concentration

profile within the catalyst particle. The relationship between the temperature profile and
the concentration profile can be obtained by considering the material balance and the
energy balance at steady-state over a shell in a spherical catalyst particle (see Figure
3.9).

Material balance:

81
   ( rA )       r  Dr 3  r 3   0
 dC A   dC A  4
 4    r  Dr 2  D A, effective    4    r 2  D A, effective  
 dr  r  Dr  dr r 3

Energy balance:
  ( rA )  DHrxn        r  Dr 3  r 3   0
 dT   dT  4
 4    r  Dr  2       4   r2   
 dr  r  Dr  dr  r 3

Taking the limit of Dr0 :

Material balance: Energy balance:
d  2
 r  D A,effective 
r  dr 
2
dC A 
dr 
  rA  
2
d  2
r   
r  dr 
dT 
dr
  rA    DH

rxn
 
Elimination of the rate of reaction results in and realising that the temperature gradient
equals zero, when the concentration gradient equals zero:
dT D A, effective  DH

rxn

dC A 
dr  dr

Substituting the boundary condition that at r=Rparticle the concentration of the reactant A is
CA,s and the temperature is Ts:

T  Ts 
D A, effective   DHrxn    C  CA 
A, s


Thus, a fundamental relationship exists between the temperature profile in the catalyst
particle and the concentration profile. This relationship is independent on the rate
expression describing the intrinsic kinetics of the chemical transformation. The
temperature profile will thus show a maximum at the centre of the catalyst particle for
exothermic reactions and a minimum for endothermic reactions. From this relationship
the maximum temperature difference between the temperature on the outside of the
catalyst particle and the temperature in the centre of the catalyst particle can be
estimated. This maximum temperature difference will occur when the concentration of
the reactant A in the centre of the catalyst particle approaches zero:
 T  Ts


 

D A, effective   DHrxn  C A, s 
   Pr ater number
 Ts  max   Ts

82
 2

Relative temperature, T/T s

=0.8
1.6 Exothermic reactions
=0.6
=0.4
1.2 =0.2
=0
0.8 =-0.2
=-0.4
Thermo-neutral reactions
=-0.6
0.4
=-0.8 Endothermic reactions
0
0 0.2 0.4 0.6 0.8 1
Relative concentration, CA/CA,s

Figure 3.12: Relationship between the local temperature in a catalyst particle and the
local concentration in a catalyst particle relative to the temperature and
concentration on the external surface of the catalyst particle as a function
of the Prater number

3.2.4 Effect of mass and heat transfer limitations on activity

The rate of reaction is governed by the concentration of the compounds and the
temperature at the position within the catalyst particle, at which the reaction takes place.
The reaction rate may differ significantly from the rate of reaction occurring at the
external surface of the catalyst particle, since the concentration and temperature within
the catalyst particle may differ due to mass and heat transfer limitations. The difference
in the rate of reaction can be taken into account by defining an effectiveness factor, .
The effectiveness factor is defined as the rate of reaction in the catalyst particle relative
to the rate of reaction if all catalytic sites are exposed to the concentration and
temperature prevalent at the external surface.

 r Ci, V , TV   dV
r Vparticle
  observed 
r C i, s , Ts  r C i, s , Ts   Vparticle

The observed rate is obtained by integrating over the whole catalyst particle taking into
account the special variation in the concentration and temperature. It should be realised
that the observed rate can also be obtained by realising that the observed rate
corresponds to the rate at which the reactants diffuse into the catalyst particle. Thus, for
a spherical particle the observed rate per unit mass of catalyst is given by:
2
4    R particle  D A, effective  dC 
A
robserved   
4 3  dr  r  Rparticle
     R particle
3

The effectiveness factor for a spherical particle is thus given by:

r 3  D A, effective  dC A 
  observed   
r Ci, s , Ts    R particle  r Ci, s , Ts   dr  r  R
particle

Introducing the following dimensionless parameters:

83
 r CA  rA  
 A    R particle 
R particle C A,s D A, effective  C A, s

3  d A 
   
  d
2
  1

3.2.4.1 Isothermal reaction AP in a spherical catalyst pellet

The concentration profile for the isothermal reaction A P is given by the dimensionless
differential equation:
d  2 dA 
      2  An  1 A  1
2
  d  d 
dA
0 0
d

r CA  rA  
with  A    R particle 
R particle C A,s D A,effective  C A,s

The effectiveness factor for a 1st order reaction of AP (n=1) can be determined
analytically. The relative concentration of the reactant A, A=CA/CA,s is given by:
sinh     dA  1
A      1
  sinh  d   1 tanh 

Thus, the effectiveness factor is given by:

3  1 1
     
  tanh   

For other reaction orders (n≠1), a numerical approach is required to solve the differential
equation. Figure 3.13 shows the effectiveness factor as a function of the Thiele module.
The effectiveness factor is essentially equal to unity for a Thiele module of less than 0.5
for a 1st or a 2nd order reaction. The behaviour of a zero order reaction differs
significantly from the other reaction orders, since the rate of reaction is not a function of
the local concentration of the reactant A: the rate of reaction has a constant value
unequal zero when the reagent is present is equals zero if no reagent is present any
more. The effectiveness factor for zero order reactions is a measure for the fraction in
the catalyst pellet where no reactant is present, and equals one in an isothermal,
constant volume reaction system in a spherical catalyst pellet, if the Thiele module is
less than 6 . It should however be kept in mind that a true zero order reaction does not
exist, but that reactions showing with a Langmuir-Hinshelwood kinetics can be
approximated by a zero order reaction, if the adsorption of the reactants is strong. At low
concentration of the reactant the reaction order changes from zero order to 1st order.

84
 1

Effectiveness factor, 
0.8

n=0
0.6 n=1
n=2

0.4

0.2

0
0.1 1 10 100
Thiele module, 
Figure 3.13: Effectiveness factor, , as a function of the Thiele module for an
isothermal reaction, without change in volume for different reaction orders
taking place in a spherical catalyst particle

Under severe mass transfer limitations (Thiele module,  > 5), the effectiveness factor
shows an approximate reciprocal dependency on the Thiele module, i.e.
cons tan t


(the constant a depends on the reaction order; it approaches a value of 3 for 1st order
reaction kinetics).

The observed rate of reaction and the intrinsic rate of reaction (i.e. the rate of reaction in
the absence of mass and heat transfer limitations) are linked over the effectiveness
factor:
Ea
robserved    r Ci,s , Ts     k  CnA    A  e RT  CnA

The activation energy of a reaction is typically obtained from the slope in a semi-
logarithmic plot of the observed rate versus the reciprocal value of the absolute
temperature. Two limiting cases can be considered, i.e. no mass transfer limitations and
severe mass transfer limitations. The effectiveness factor approaches unity if mass
transfer limitations are small, i.e.

lnrobserved   ln A  CnA 
Ea
RT

The true activation energy of the intrinsic chemical reaction is thus obtained from the
slope of the observed rate versus the reciprocal value of the absolute temperature in a
semi-logarithmic plot.

In the case of severe mass transfer limitations in an isothermal catalyst pellet, the
effectiveness factor is reciprocal dependent on the Thiele module, i.e.

85
robserved    r Ci,s , Ts     k  CnA 
a cons tan t
 k  CnA   k  CnA
 k  CnA,s1  
R particle 
D A,effective
cons tan t D A,effective  k C nA
robserved   
R particle  C nA,s1
Under severe mass transfer limitations the observed activation energy is reduced. The
effect of temperature on the diffusion is typically small in comparison to the effect of
temperature on the rate constant and can thus be neglected. A semi-logarithmic plot of
the observed rate of reaction versus the reciprocal value of the absolute temperature will
yield:
 
 cons tan t D A,effective
lnrobserved   ln 


CnA 
  ln k  
 R particle C n1 
 A,s 

   Ea 
 cons tan t D A,effective CnA   
lnrobserved   ln     ln A  e RT

 R particle  C n1   
 A ,s   
 
 cons tan t D A,effective
lnrobserved   ln 


CnA 
 
Ea
  ln A  2  R  T
 particle
R C A,s 
n  1

Thus, under severe mass transfer limitations over an isothermal catalyst pellet, the
observed activation energy is only half of the true activation energy for the chemical
reaction.

3.2.4.2 Isothermal reaction AsP in a porous catalyst pellet

Most chemical reactions, with the exception of isomerisation reactions, are associated
with a change in the total number of moles in the reaction mixture. This may lead to a
pressure gradient in the catalyst pellet (see 3.2.2) for reactions taking place in the gas
phase (reactions taking place in the liquid phase do not show this behaviour due to the
incompressible nature of liquids). The pressure gradient may lead to a forced flow in the
catalyst particle.

In gas phase reactions, the diffusion mechanism can either be dominated by Knudsen
diffusion or by bulk gas diffusion. In the former case, the effectiveness factor is not
affected by the pressure gradient present in the catalyst particle, since the Knudsen
diffusion is independent of pressure. Furthermore, the forced flow due to the pressure
gradient (Knudsen flow) is indistinguishable from Knudsen diffusion and will not
contribute either.

It can be shown [18], that forced flow due to a pressure gradient if bulk gas diffusion is
the dominant diffusion mechanism becomes only significant for catalyst particles with
very large pores (>1 m) or for catalysts operating at high pressure (ca. 100 atm). In all
other cases, the contribution of the forced flow to the mass transfer within the catalyst
pellet is rather small and can thus be neglected.

Bulk gas diffusion is reciprocal dependent on the pressure. The bulk gas diffusion
coefficient decreases with increasing pressure. Hence, it can be expected that for

86
reactions with an increase in the number of moles (s>1), the diffusion coefficient
decreases with the extent of reaction and for reactions with a decrease in the number of
moles (s<1) the diffusion coefficient will increase with the extent of reaction. This will
affect the effectiveness factor. It has however been shown [32], that the effectiveness
factor for a reaction with s=2 is only ca. 10% smaller than for reactions with s=1. This
means that for these kind of reactions the effectiveness factor can be roughly estimated
neglecting the change in the number of moles in the reaction (only if s becomes very
large, >10, or very small the effect of the change in the number of moles should be taken
into account).

3.2.4.3 Non-isothermal reactions in a spherical catalyst pellet

Chemical reactions are usually associated with a change in the enthalpy. The
temperature in a catalyst pellet will therefore differ from the temperature outside the
catalyst pellet, if mass transfer to the catalytically active sites is limited. For exothermic
reactions the temperature inside the catalyst pellet is higher than the temperature
outside the catalyst pellet and for endothermic reactions the temperature inside the
catalyst pellet is lower than the temperature outside the catalyst pellet.

The rate of a chemical reaction increases strongly with temperature (Arrhenius’

dependency). Thus, the rate of reaction at an active site in a catalyst pellet depends on
the local temperature and local concentration of the kinetically important compounds.

The rate of reaction in a catalyst pellet will be lower than the rate of reaction at the outer
surface of the catalyst pellet for endothermic reactions, since the concentration of the
reactants will be lower in a catalyst pellet and the temperature will be lower. Both factors
point to a lower rate of reaction. Thus the observed rate of reaction over a catalyst pellet
will be lower than the intrinsic rate of reaction. The non-isothermal conditions in a
catalyst pellet in the case of an endothermic reaction will thus lead to a decrease in the
effectiveness factor compared to the isothermal case.

Exothermic reactions lead to a temperature increase in the catalyst pellet, but

simultaneously the concentration of the reactant(s) is decreasing. Hence, the rate of
reaction may first increase due to the temperature increase, but under more severe
mass transfer limitations decrease due to depletion of the reactant(s).

Non-isothermal reactions can be modelled in a similar manner as isothermal reactions

taking into account the temperature gradient. Considering the non-isothermal, 1st order
reaction of AP, the mass balance yields:
d  2 d A 
      2  A
  d 
2 d 
r CA k 
with  A    R particle 
R particle C A,s D A,effective

The temperature dependency of the diffusivity can be neglected in relation to the

temperature dependency of the rate constant:
Ea
d 2 d A  k  A e RT 
     R particle
2
  A  R particle
2
  A
  d 
2 d  D A,effective D A,effective

87
 Ea Ea  Ts 
 1
RTs RTs  T 
d  2 dA  Ae e 
     R particle
2
  A
  d 
2 d  D A,effective
Ea  Ts 
 1
d  2 dA 
      2  e RTs  T   A
  d 
2 d 
with Ts: temperature on the outside of the catalyst pellet
k Ts   
  R particle  Thiele module at temperature on the outside of the
D A,effective
catalyst pellet
E 
 A Arrhenius’ number
R  Ts

The temperature is also a function of the position in the catalyst pellet, but can be
expressed as a function of the relative concentration A:
T
 1
 
D A,effective   DHrxn  C A,s
 1  A   1    1  A 
Ts   Ts

Thus, the differential equation to be solved is:

 1 

  1 
d  2 dA  11A  
      2  e   A
 2  d  d 
The boundary conditions for solving the differential equation are:
 1 A  1
dA
0 0
d

3 d A 
and the effectiveness factor can be found using:    
  d
2
  1

Figure 3.14 shows the dependency of the effectiveness factor on the Thiele module
(based on the temperature on the outside of the catalyst particle) for a 1 st order reaction.
The effectiveness factor is also a function of the Prater number, , and the Arrhenius
number, . The effectiveness factor for endothermic reactions (<0) is lower than the
effectiveness factor for isothermal reactions, because the temperature in the catalyst
pellet is lower than the temperature on the outside of the catalyst pellet reducing the
local rate of reaction further.

88
Figure 3.14: Effectiveness factor, , as a function of the Thiele module, , for a non-
isothermal, 1st order reaction, without change in volume for different
values of the Prater number, , and an Arrhenius’ number, , of 25
(graphical depiction of the dependency of the effectiveness factor on
other values of  and  can be found in [33])

The effectiveness factor for exothermic reactions (>0) can be larger than unity, i.e. the
observed rate is larger than expected for the intrinsic rate of reaction based on the
concentration and the temperature on the outside of the catalyst pellet. This is caused by
the higher temperature inside the catalyst pellet, which accelerates the chemical
reaction. At high values for the Thiele module, the effectiveness factor becomes less
than unity due to the strong depletion of the reactant.

Multiple solutions to the differential equations (up to 3) are obtained at values of the
Thiele module, , between 0.2 and 0.6 for reactions with =25 and =0.5. Only two of the
solutions represent a stable situation, which can be achieved in praxis (the middle
solution is unstable, since a small variation in the concentration/temperature on the
outside of the catalyst particle will result in the other states being achieved). The ultimate
effectiveness factor that will be obtained, depends on how the reaction is initiated.

3.2.5 Effect of internal mass and heat transfer limitations on selectivity

Selectivity is often of more importance than activity in catalytic reactions. The selectivity
of a reaction is affected by the concentration of the reactants and the reaction
temperature. Mass and heat transfer limitations in heterogeneously catalysed reactions
lead to a decrease in the concentration of the reactant(s) and lead to an increase in the
local reaction temperature at the active site for exothermic reactions and to a decrease
in the reaction temperature at the active site for endothermic reactions.

3.2.5.1 Parallel reactions of a common reactant

A reactant A might be converted in two independent reactions to two types of products.
Consider the conversion of a reactant A to a product P in a 1st order reaction and the
conversion of the reactant A to a product S in a second order reaction:

89
AP rP  k P  C A
AS rS  k S  C 2A
(the rate of consumption of the reactant A is now of course given by
 rA  k P  C A  k S  C 2A ). The selectivity for the formation of P, SAP, can now be
defined as the rate of formation of S relative to the rate of consumption of A:
kP  CA 1
S A P  
kP  CA  k S  CA 1 k S  C
2
A
kP
It can be easily seen that the selectivity for the product P will increase with decreasing
concentration of the reactant A. Mass transfer limitations present in catalyst pellet may
thus result in an increase in the selectivity for the product P, if the activation energy for
both reactions are identical. In general, it may be stated that at low concentration
reactions with a low reaction order are favoured over reactions with a high reaction
order.

Reactions performed under mass transfer limitations will result in a change in the
temperature at the active site. At a high reaction temperature, reactions with a high
activation energy are more favoured than reactions with a low activation energy.
Consider the conversion of a reactant A to a product P and to a product S both in a 1 st
order reaction with different activation energies, EaP and EaS respectively:
Ea P
AP rP  k P  C A  A P  e RT  CA
Ea S
AS rS  k S  C A  A S  e RT
 CA
The selectivity for the formation of P, SAP, can now be evaluated as:
Ea P
kP  CA AP  e RT  CA 1
S A P   Ea P Ea S
 Ea S Ea P 
kP  CA  k S  CA A
AP  e RT
 CA  AS  e RT
 CA 1 S  e RT
AP
The selectivity for P increase with increasing temperature, if EaP>EaS.

3.2.5.2 Parallel reactions of two reactants

The feed gas may contain more than one reactant, e.g. A and B, which are converted
independently in a first order reaction to the products P and S respectively:
AP  rA  rP  k A  C A
BS  rB  rS  k B  CB

The selectivity ratio for the conversion of A yielding P, SRAP, can be expressed as the
ratio of the rate of consumption of A relative to the rate of consumption of B:
r k  CA
SR A P  A  A
 rB k B  CB
If the reaction is taking place over a heterogeneous catalyst, the rates of reaction must
be replaced by the observed rate of reaction:
 rA,observed  A  k A  C A,s
SR A P,observed  
 rB,observed B  k B  CB,s

90
When mass transfer limitations can be neglected, the selectivity ratio is determined by
the ratio of the rate constants and of the relative concentrations.
k C A,s
A , B  1 SR A P,observed  A 
k B C B,s
Under severe mass transfer limitations, the effectiveness factor becomes proportional to
the reciprocal value of the Thiele module
a a  k C A,s k A D A C A,s
A  B  SR A P,observed  B A    
A B  A  k B CB,s k B DB CB,s
Thus, under severe mass transport limitations the selectivity for the transformation of A
into P relative to the transformation of B into S is altered. The selectivity for the faster of
the two reactions is reduced. Thus, mass transfer limitations must be reduced to
maintain the selectivity for the faster reaction (if the activation energies for both reactions
are approximately the same or if the temperature profile in the catalyst pellet can be
neglected).

3.2.5.3 Consecutive reactions: A  P  S

Consecutive reactions, in which the reactant A is converted into the product P, which is
subsequently converted in the product S are quite common in catalysis (often the
product P is even the desired product!). Consider an isothermal consecutive type of
reaction of A  P  S, with both reactions being 1st order. Thus, the rate of
consumption of A and the rate of formation of P are given by:
 rA  k A  C A rP  k A  C A  k P  CP

In the absence of mass transfer limitations the selectivity for the formation of P can be
found from:
r k C
SP  P  1  P  P
 rA k A CA
The selectivity for the intermediate product P is thus a function of the ratio of the rate
constants and of the ratio of the concentrations. The selectivity can be expressed as a
function of the amount of A converted to eliminate the influence of the concentration ratio
[18]. Figure 3.15 shows the yield and the selectivity of P as a function of the conversion
of A. A 100% selectivity can only be obtained with this type of reactions at zero
conversion of the reactant A. The selectivity for the product P drops with increasing
conversion. The drop is more severe, if the rate constant for the conversion of P
becomes larger relative to the rate constant for the consumption of A. The maximum
yield that can be obtained for product P is obtained at a much higher conversion level,
but this is at the cost of selectivity.

91
 kP/kA =
1 1
kP/kA = 0.1
0.2
0.8 0.8 0.5
0.1

Selectivity of P
1
Yield of P
0.2
0.6 0.6 2
0.5
0.4 0.4
1
0.2 0.2
2

0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
Conversion of A Conversion of A

Figure 3.15: Yield (left) and selectivity (right) of the product P in a consecutive reaction
of A  P  S as a function of the conversion of A for various ratios of the
rate constant for the conversion of P relative to the rate constant for the
conversion of A in the absence of mass transfer limitations

Under severe mass transfer limitations, the concentration of the reactant A in the
catalyst pellet will be low, and the concentration of the products will be high. Thus, mass
transfer limitations will favour the consecutive reaction (i.e. the consumption of P yielding
S) and thus will reduce the selectivity for P. Figure 3.16 shows the yield and selectivity
for the product P in a consecutive reaction of A  P  S under severe mass transport
limitation (i.e. the effectiveness factor is proportional to the reciprocal value of the Thiele
module). The maximum selectivity for the product P that can be obtained under severe
mass transport limitations is significantly lowered. The selectivity drops with increasing
conversion. The drop is however less than in the absence of mass transfer limitations
(see Figure 3.15), because the concentration of the reactant A in the catalyst pellet is
already very low. The maximum yield of the product P that can be obtained under severe
mass transport limitations is reduced in comparison to the case when mass transfer
limitations could be neglected.

Mass transfer limitations must be eliminated in order to maximize the selectivity for an
intermediate product P. Furthermore, a catalyst must be developed with a high rate
constant for the conversion of the reactant A with at the same time a low rate constant
for the conversion of P.

1 1

0.8 0.8 kP/kA =

Selectivity of P

0.1
Yield of P

0.6 kP/kA = 0.6 0.2

0.1 0.5
0.4 0.4 1
0.2
2
0.5
0.2 1 0.2
2
0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
Conversion of A Conversion of A

Figure 3.16: Yield (left) and selectivity (right) of the product P in a consecutive reaction
of A  P  S as a function of the conversion of A for various ratios of the
rate constant for the conversion of P relative to the rate constant for the
conversion of A under severe mass transfer limitations

92
3.2.6 Elimination of internal mass and heat transfer limitations
Catalysts should be evaluated for their intrinsic kinetic properties. In catalyst pellets
mass and heat transfer limitations may exist resulting in an observed rate of reaction,
which differs from the intrinsic rate of reaction at the measured temperature and
concentration on the outside of the catalyst pellet.

Internal mass transfer limitations are governed by the Thiele module. The effectiveness
factor becomes unity for small Thiele modules. Hence, it is important to be able to vary
the Thiele module. At a given set of reaction conditions, the only variable in the Thiele
module that can be varied is the radius of the catalyst particles. For an isothermal first
order reaction, the effectiveness factor is larger than 0.95, when the Thiele module is
less than unity. Thus, the absence of mass transfer limitations can be assessed using:
 rA,observed  
Weisz-Prater criterion:  2  R particle
2
 1 [34]
D A,effective  C A,s
Hence, measuring the observed over a given catalyst particle at a given reaction
condition (CA,s) and estimating the effective diffusivity can be used to verify the absence
of mass transfer limitations. This criterion can be made more general for non-isothermal
conditions and reactions orders different from one by changing the maximum value of
the Thiele module in the Weisz-Prater criterion (e.g.  2  0.01).

It is however advisable to verify the absence of mass and heat transfer limitations by
performing experiments with different particle sizes, if the estimated Thiele module is
larger than 0.1. In the absence of mass and heat transfer limitations, the observed rate
of reaction is independent of the catalyst particle size. Hence, a series of experiments, in
which the catalyst particle size is varied and all other experimental conditions are kept
constant, shows the region, in which experiments are performed in the absence of mass
and heat transfer limitations. Figure 3.17 shows the possible effect of particle size on the
observed conversion in a fixed bed reactor for a 1st order reaction and the range of
particle sizes in which mass and heat transfer limitations can be neglected.

1
Conversion in a fixed bed

0.8 exothermic reaction

>0
internal
0.6 mass and heat
reactor

transfer limitations
negigible
0.4
isothermal reaction
=0
0.2
endothermic reaction
<0
0
Catalyst particle size
Figure 3.17: Expected conversion in a fixed bed reactor for a 1st order irreversible
reaction as a function of the catalyst particle size keeping all other
experimental conditions constant ( k  '  0.33 )

93
3.3 External mass and heat transfer limitations

3.3.1 Mass transfer through the boundary layer

A catalyst particle in a flowing fluid is thought to be surrounded by a thin boundary layer,
in which there is no flow. Reactants have to be transported from the bulk of the fluid to
the outer surface of the catalyst particle through this boundary layer. Product
compounds have to be transported from the outer surface of the catalyst pellet to the
bulk of the fluid through the boundary layer as well. Heat will be transported through this
boundary layer as well.

The rate of transport of a reactant A through the boundary layer can be modelled as:
 
rA,transport  k L  a  C A,bulk fluid - C A,outer surface catalyst particle
with kL: mass transfer coefficient
a: external surface area per unit mass
At steady-state the rate of transport through the boundary layer equals the observed rate
of consumption of the reactant A. The extent of decrease of the concentration of the
reactant A from the bulk of the fluid to the outer surface of the catalyst particle can thus
be estimated by:
C A,outer surface catalyst particle  rA,observed
 1
C A,bulk fluid k L  a  C A,bulk fluid
The decrease in the concentration of the reactant A relative to the concentration of the
reactant A in the bulk of the fluid can thus be neglected, if the mass transfer coefficient is
large in comparison to the observed rate of reaction. The concentration of the reactant A
on the outer surface becomes almost zero, if the mass transfer coefficient is very small.
The observed rate of reaction is then dominated by the rate of transport through the
boundary layer and no longer by the chemical reaction. The observed reaction order is
then one:
severe mass transport limitations:  rA,observed  k L  a  C A,bulk fluid

The mass transfer coefficient is related to the flow around a catalyst particle. The mass
transfer coefficient for spherical catalyst particles can be estimated using the Frössling
correlation [35]:
 1 1
D A,boundary layer    fluid  u  dparticle  2   fluid  3 
kL    2  0.6      
   
dparticle
   fluid   fluid  D A,boundary layer  
 
with DA,boundary layer: diffusion coefficient of A through the boundary layer
dparticle: diameter of catalyst particle
u: linear velocity of the bulk fluid
fluid: density of the fluid in the boundary layer
fluid: viscosity of the fluid in the boundary layer

The external mass transfer coefficient can thus be enhanced by changing the linear
velocity of the bulk fluid. It should however be noted that there is not a strong
dependency of the mass transfer coefficient on the linear velocity.

94
The mass transport coefficient is hardly dependent on temperature ( k L  T 0.52.25
depending on the flow regime and the type of diffusion). Hence, ta semi-logarithmic plot
of the observed rate of reaction versus the reciprocal value of the absolute temperature
can be used to distinguish the various regimes of operation (see Figure 3.18), viz.
1. Kinetic regime: Eaobserved  Eachemicalreaction
Ea chemical reaction
2. Internal mass transport limitations: Ea observed 
2
3. External mass transport limitations: Eaobserved  0

Figure 3.18: Expected temperature dependency of the observed rate of reaction over
an isothermal catalyst pellet (=0)

Figure 3.18 shows an idealised situation of the observed rate of reaction as a function of
the reciprocal value of the absolute temperature. It should however be noticed that the
transition between chemical reaction control and external mass transfer control does not
necessarily show internal mass transfer control.

3.3.2 Heat transfer through the boundary layer

Heat transport through the boundary layer can be modelled in a similar manner to mass
transport. The rate of heat transport through the boundary layer depends on the heat
transfer coefficient and the temperature gradient over the boundary layer:
Q  
through boundary layer  h  a  Tbulk fluid - Touter surface catalyst particle
with h: heat transfer coefficient
a: external surface area per unit mass
At steady-state the rate of heat transfer through the boundary layer equals the heat
generated by the chemical reaction in the catalyst particle. The ratio of the temperature
at the outer catalyst surface to the temperature in the bulk of the fluid can thus be
expressed as::
Touter surface catalyst particle
 1
 
 rA,observed  DH
rxn

Tbulk fluid h  a  Tbulk fluid

rxn
For exothermic reaction (DH <0), the temperature at the outside of the catalyst particle
is larger than the temperature in the bulk of the fluid.

95
The temperature difference over the boundary layer and the concentration difference are
linked (similar to the case of internal mass and heat transfer limitations).

k  DHrxn
Tbulk fluid - Touter surface catalyst particle  L
 
 C A,bulk fluid - C A,outer surface catalyst particle
h
Hence, elimination of external mass transport limitations will result in the elimination of
external heat transfer limitations.

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