Critical Analysis of Kinetic Modeling Procedures

INTERNATIONAL JOURNAL OF CHEMICAL

REACTOR ENGINEERING

Volume 9 2011 Article A87

Critical Analysis of Kinetic ModelingProcedures

Jose Carlos Pinto∗ Marcos W. Lobao† Andre L. Alberton‡

Marcio Schwaab∗∗ Marcelo Embirucu†† Silvio Vieira de Melo‡‡

∗Universida de Federal do Rio de Janeiro, [email protected]†Universidade Federal da Bahia, [email protected]‡Pontificia Universidade Catolica do Rio de Janeiro, andre [email protected]∗∗Universidade Federal de Santa Maria, [email protected]††Universidade Federal da Bahia, [email protected]‡‡Universidade Federal da Bahia, [email protected]

ISSN 1542-6580Copyright c©2011 Berkeley Electronic Press. All rights reserved.

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Critical Analysis of Kinetic Modeling Procedures∗

Jose Carlos Pinto, Marcos W. Lobao, Andre L. Alberton, Marcio Schwaab,Marcelo Embirucu, and Silvio Vieira de Melo

Abstract

In this work, issues related to the mathematical modeling and statistical anal-yses of kinetic data are discussed. Firstly, problems related to the combinatorialexplosion of the number of plausible kinetic models are analyzed, when com-plex reaction mechanisms are taken into consideration and distinct rate determin-ing steps are assumed. Although modeling procedures based on rate-determiningsteps can lead to oversimplification of kinetic models, these procedures are stillvery popular because the existence of multiple rate-determining steps usually ren-ders the analytical derivation of kinetic rate expressions impossible. However, ifthe derived kinetic models are too simple, one can face serious difficulties to fitthe proposed models to available experimental data. Secondly, problems related tothe statistical analyses of experimental data are discussed. Particularly, very oftenstatistical tools are used even when some of the fundamental assumptions requiredfor their validity are violated. For this reason, the fundamental grounds that sup-port some of the most popular statistical tools are discussed in the framework ofthe kinetic analysis.

KEYWORDS: parameter estimation, mathematical modeling, model discrimina-tion, statistical analysis, kinetics

∗The authors thank CNPq – Conselho Nacional de Desenvolvimento Cientıfico e Tecnologico,CAPES – Coordenacao de Aperfeicoamento de Pessoal de Nıvel Superior, and FAPERJ –Fundacao Carlos Chagas Filho de Apoio a Pesquisa no Estado do Rio de Janeiro, for providingscholarships and for supporting this research.


Introduction

Kinetic modeling of reaction data is of fundamental importance for design of

chemical reaction processes. Actually, reactor design generally constitutes the

heart of chemical process design, since reactor requirements and performance

usually determine how raw materials must be treated before reaching the reactor

and how the product stream must be treated in the subsequent separation steps.

Additionally, around 75% of all chemical reaction processes make use of catalysts

(heterogeneous catalysts, in particular), although this percentage has increased to

about 90% in modern technologies (Hagen, 2006).

Usually, development of catalyst materials and catalyst preparation

methods is based on extensive experimental studies, according to the previous

experience and expertise of catalyst designers. After (or during) catalyst

development, it is also necessary to perform extensive kinetic studies in order to

derive mathematical expressions that can allow for reactor design and provide

reliable simulations of product distributions and of catalyst activity in the

industrial range of interest (Farrusseng, 2008).

As a matter of fact, for most processes, and especially for processes based

on heterogeneous catalytic systems, modeling of kinetic data can constitute a very

complex task, although most times the ‘as simple as possible’ approach is

preferable in order to provide more reliable simulations and simpler reactor

design (Bos et. al., 1997). For this reason, issues related to the mathematical

modeling and statistical analyses of kinetic data are discussed in the following

sections. Firstly, problems related to the combinatorial explosion of the number of

plausible kinetic models are analyzed, when complex reaction mechanisms are

taken into consideration and distinct rate determining steps are assumed (Boudart,

1968; Froment and Bischoff, 1979; Vannice, 2005; Schmal, 2010). Although

modeling procedures based on rate-determining steps can lead to

oversimplification of kinetic models, these procedures are still very popular

because the existence of multiple rate-determining steps renders the analytical

derivation of kinetic rate expressions impossible. However, if the derived kinetic

models are too simple, one can face serious difficulties to fit the proposed models

to available experimental data. As a kinetic model cannot be derived based solely

on theoretical reasoning, subsequent stages of model fitting and estimation of

model parameters are unavoidable (Froment and Bischoff, 1979). For this reason,

problems related to the statistical analyses of experimental data are also discussed

in the following sections. Particularly, very often statistical tools are used even

when some of the fundamental assumptions required for their validity are

violated. Thus, the fundamental grounds that support some of the most popular

statistical tools are discussed in the framework of the kinetic analysis.

1Pinto et al.: Critical Analysis of Kinetic Modeling Procedures

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Derivation of Kinetic Expressions

Let us consider a kinetic model, deduced from a real and known kinetic

mechanism, where it is assumed that different active sites are present and several

surface reactions take place simultaneously. For example, let us assume that the

reaction represented by A → 3B occurs over an ideal catalyst active site and

proceeds through the reaction mechanism defined in Equation (1), where the

elementary reaction rate expressions are also indicated.

[ ][ ]

[ ]

0

0

1

1

2

2

3

3

22* *

1 0 0 0 1

2 3* * * *

1 2 1 1 1 1 2

2 2* * * *

2 3 2 2 2 2 3

*

3 3

2 2

2 3

2 2

k

k

k

k

k

k

k

k

A S I r k A S k I

I S I r k I S k I

I I r k I k I

I B S r

−

−

−

−

−

−

−

→ + = ⋅ − ⋅←

→ + = ⋅ − ⋅←

→ = ⋅ − ⋅←

→ + =← [ ] [ ]*

3 3 3k I k B S− ⋅ − ⋅ ⋅

(1)

Throughout this paper, [ ] represents the molar concentration, A is the

reactant, B is the product, Ii (i=1,..,NI) represent the intermediate reaction species,

NI is the number of intermediate species and S is an empty active site. Thus,

based on Equation (1) the overall reaction rate expressions can be written as:

,

0

* * *

, 1 2 3

0

,

0

Q=A, B

j=I , I , I

NQ

Q i Q i

i

NI

j i j i

i

NS

S i S i

i

r r

r r

r r

υ

υ

υ

=

=

=

= ⋅

= ⋅

= ⋅

∑

∑

∑

(2)

In this case, it is assumed that one is interested in writing the reaction rate

expressions for all chemical species involved in the reaction mechanism.

Assuming that all surface and bulk concentrations of all chemical species can be

measured (which is rare, if not impossible), then it would be possible to monitor

the extent of reaction for each surface reaction occurring at the analyzed active

sites. Therefore, assuming that the elementary rate expressions follow the usual

mass action law and can be represented as the products of the nth

-powers of the

respective reactants concentrations, where the n’s are the stoichiometric

coefficients (Boudart, 1968; Froment and Bischoff, 1979; Vannice, 2005; Schmal,

2 International Journal of Chemical Reactor Engineering Vol. 9 [2011], Article A87


2010), then it would be possible (at least in principle) to estimate the kinetic rate

constants (direct and reverse) for all elementary reaction steps in all analyzed

active sites with the available experimental data. As a consequence, it would be

possible to describe the kinetic behavior of all chemical species simultaneously:

reactants, products and intermediates, including the adsorbed surface species.

Obviously, the proposed scenario is idealized and quite distant from what

researchers can actually measure and perform in a daily laboratory routine. In real

kinetic studies, measuring of all chemical concentrations, identification of all

catalyst sites and definition of all mechanistic steps are practically impossible.

Thus, modeling of kinetic data constitutes an ill posed problem, in the sense that

available information is not sufficient for determination of all unmeasured

variables (or kinetic parameters). As a consequence, some simplifying approaches

must be adopted, such as the Steady State Approximation and the Rate

Determining Step hypotheses.

The Steady State Approximation (SSA)

In short, the Steady State Approximation consists on assuming that the rates of

formation and consumption of intermediate species are equal. According to this,

after a short induction period, the concentration of intermediate species reaches a

stationary equilibrium value (Boudart, 1968; Froment and Bischoff, 1979;

Vannice, 2005; Schmal, 2010). In most cases, the SSA assumes implicitly that the

intermediate concentrations are in equilibrium because of the high reactivity of

the intermediate species. As a consequence, if NI intermediate species are

considered, then NI algebraic mass balance equations can be derived and NI

unknown variables (intermediate species concentrations) can be computed.

Therefore, in principle it becomes possible to infer the concentrations of

intermediate species based on measured products and reactants concentrations and

consequently describe the reaction rates in terms of the measured concentrations

of reactants and main products only.

It must be pointed out that a nonlinear system of algebraic equations can

be generated (and afterwards must be solved) when the stoichiometric coefficients

of intermediates are different from 1 in complex reaction mechanisms. In this

case, explicit derivation of kinetic rate expressions may not be possible even

when the SSA hypothesis is adopted. Therefore, in this case, it seems reasonable

to argue whether the use of the SSA approach is really appropriate and/or useful

to obtain a closed solution for intermediate concentrations, allowing as a

consequence for derivation of explicit expressions for the reaction rates.

In order to illustrate the limitations of the SSA approach, let us consider

that reactions presented in Equation (1) are irreversible, leading to the simple

mechanism presented in Equation (3).




0

1

2

3

*

1

* *

1 2

* *

2 3

*

3

2 2

2 3

2 2

k

k

k

k

A S I

I S I

I I

I B S

+ →

+ →

→

→ +

(3)

Based on the SSA approach, the following expressions can be written for

concentrations of intermediate species:

[ ][ ] [ ] [ ] [ ][ ]20* * *0 01 2 3

1 2 3

3 3

2

k Ak kI A S I S I A S

k k k

⋅ ⋅ ⋅ = = = ⋅

(4)

Considering that the total sites concentration [STot] remains constant, one

can write:

[ ] [ ]

[ ][ ] [ ] [ ] [ ][ ] [ ] [ ]

* * *

1 2 3

200 0

1 2 3

3 3

2

Tot

Tot

I I I S S

k Ak kA S S A S S S

k k k

+ + + =

⋅ ⋅ ⋅+ + + =

⋅

(5)

However, it is not possible to calculate [S] in Equation (5) explicitly,

which means that explicit expressions cannot be derived for the reaction rates.

Similar problems can be observed in most reaction mechanisms that involve

multiple elementary reaction steps. For this reason, very frequently analysts avoid

the proposition of complex mechanisms, as derivation of closed reaction rate

expressions would not be possible. However, one should not expect that simple

reaction mechanisms would necessarily rule the kinetic behavior of real reaction

systems. Simplifying assumptions that cannot be supported by the real

phenomenological behavior of the reaction system will eventually lead to

derivation of poor kinetic rate models. In order to overcome this inherent

limitation of the SSA approach, alternative procedures have also been proposed

and adopted for determination of kinetic models and kinetic rate expressions

(Bengaard et al., 2002; Saeys et al., 2002; Thybaut et al., 2002; Abild-Pedersen et

al., 2005; Boisen et al., 2005; Saeys et al., 2005a; Saeys et al., 2005b; Topsøe et

al., 2005; Saeys et al., 2006; Vang et al., 2006).



Rate Determining Step (RDS)

In order to generate closed reaction rate expressions, the existence of one or more

rate determining steps is often assumed in most reaction mechanisms. According

to the RDS approach, one of the reaction steps is assumed to be much slower than

the others (Boudart, 1968; Froment and Bischoff, 1979; Vannice, 2005; Schmal,

2010). As a consequence, the much faster reaction steps are assumed to reach the

chemical equilibrium and the overall reaction rate is determined by the slowest

reaction step (the rate determining step). This approach allows for derivation of

the well-known Langmuir-Hinshelwood-Hougen-Watson (LHHW) kinetic

models (Boudart, 1968; Froment and Bischoff, 1979; Vannice, 2005; Schmal,

2010).

According to the RDS procedure, all the reaction steps, but the RDS, are

considered to reach the equilibrium. Therefore, for an equilibrium step i, one can

write:

[ ] [ ],, ,*

= free active site

= intermediate species

= reactants/products

i ji S i Q

j i

j Q

S

S I Q K j

Q

υυ υ = ∏ ∏ (6)

where Ki is an equilibrium constant of the elementary step i and υi,S, υi,j and υi,Q

are the stoichiometric coefficients of active sites, intermediate species and

reactants/products Q, respectively, in the elementary step i. For any elementary

step i, the stoichiometric coefficient is negative when the analyzed chemical

species is in the left-hand side of the reaction, positive if the analyzed chemical

species is in the right-hand side of the reaction or zero if the analyzed chemical

species does not take place in the considered elementary step i. Applying the

logarithm on both sides of Equation (6):

[ ] [ ]*

, , ,ln ln ln lni S i j j i Q i

j Q

S I Q Kυ υ υ ⋅ + ⋅ + ⋅ = ∑ ∑ (7)

The main objective pursued by the analyst usually is the determination of

the concentrations of unmeasured species (normally, the intermediate species) as

a function of reactant/product species and free active sites concentrations with the

help of Equation (7). Writing Equation (7) for all elementary steps that are

assumed to be in equilibrium, the following linear system is obtained:




* * *

1 2

*

11,1 1,2 1,

*2,1 2,2 2, 2

,1 2, ,

I I .... I

ln

...

Step 1

...

ln

Step 2

.... .... .... ....

.........

...Step N l

NI

NI

NI

N N N NI

I

I

υ υ υυ υ υ

υ υ υ

[ ][ ]

[ ]

[ ]

*

1, 1, 1,1

2, 2, 2,2

,

n

Eq const Active site Reactants/Products

.... ....

ln

...

ln

ln

....

ln

NI

S A B

S A B

N SN

I

K

KS

K

υ υ υυ υ υ

υ

=

− −

[ ][ ]

, ,

ln

. .... ln

... .... .... .... ...

.... .... .....N A N B

A

B

υ υ

(8)

which can also be written in matrix notation as:

ln[ ]S⋅ = − ⋅ − ⋅I* S Q

ln_I ln_K ln_Qϒϒϒ υυυ ϒϒϒ (9)

where I*

ϒϒϒ is the matrix of stoichiometric coefficients of intermediates, ln_I is the

vector of logarithms of intermediates concentration, ln_K is the vector of the

logarithm of equilibrium constants, υS is the vector of stoichiometric coefficient

of free active sites, Q

ϒϒϒ is the matrix of stoichiometric coefficients of

reactants/products and ln_Q is the vector of the logarithm of concentration of

reactant/products. Equation (9) defines a system of linear equations, which can

usually be solved much more easily than the original set of nonlinear equations

described in Equation (6). Despite that, the RDS approach also leads to some

problems when one is interested in deriving closed reaction rate expressions, as

described below.

The Ambiguous Proposition of RDS Rate Expressions

In order to allow for unambiguous determination of concentrations of unmeasured

chemical species, the linear system of equations presented in Equations (8-9) must

be possible and determined, which means that the number of elementary steps

(lines) must be equal to the number of unknown chemical species (rows), or more

formally that the number of linearly independent linear equations must be equal to

the number of unknown chemical species.

An inconvenient problem may arise when multiple reactions are

considered simultaneously. In order to illustrate this problem, let us assume that a

simple mechanism for two global reactions (A → B and A → C) takes place over



a catalyst and can be described as

*

1

*

1

*

1

A S I

I B S

I C S

+ ←→

←→ +

←→ +

(10)

In this particular case, if the surface reactions are assumed to constitute the

RDS of the investigated mechanism, then the adsorption of A is considered to

reach the equilibrium and must be included in Equation (8). Thus, one equation is

formulated and one unknown variable ([I1*]) can be computed as

( ) [ ]*

1 1ln ln ln[ ] lnI K S A = + + (11)

On the other hand, let us suppose that the adsorption of A (step 1)

constitutes the RDS and controls the reaction rate. As a consequence, the other

two elementary steps are assumed to reach the equilibrium; nonetheless, Equation

(8) can then be formulated as a system of two algebraic equations and a single

unknown variable, as

( ) [ ]( ) [ ]

*

1 2

*

1 3

ln ln ln[ ] ln

ln ln ln[ ] ln

I K S B

I K S C

= − −

= − −

(12)

This apparent inconsistency can only be resolved in the mathematical

model if the two equations presented in Equation (12) are linearly dependent. In

this case, the concentrations of [B] and [C] should necessarily follow

[ ][ ]

3

2

C K

B K= (13)

Equation (13) seems suspicious in real chemical reaction problems, as

products B and C can be manipulated independently and chemical equilibrium

probably cannot be attained instantaneously in most real kinetic systems. As a

consequence, the proposition of a single RDS seems inadequate for practical

purposes in this case. This can be analyzed in a simple manner with the help of

this simple mechanism, but can be very difficult to observe in more complex

mechanisms. For this reason, it is nor surprising to observe that similar analyses

have been systematically ignored in most kinetic studies. In most studies, one of

the equations presented in Equation (12) would be probably selected to describe




the concentration of the intermediate, allowing for definition of at least two

distinct reaction rate expressions.

Derivation of Closed RDS Rate Expressions for Complex Mechanisms

Generally, the analyst tries to write the surface concentrations ([I1],[I2],[I3],…) as

functions of the bulk concentrations of the involved chemical species

(f([A],[B],[C],…) ) and the concentration of free active sites [S], in the form:

[ ] [ ] [ ] [ ]( )* , , ...j jI S f A B C = ⋅ (14)

As a consequence, it is possible to write the concentrations of all

intermediate species as a function of reactants/products concentrations and total

active sites concentrations (usually a parameter that must be estimated) as

[ ] [ ]*

1

NI

j Total

j

I S S=

+ = ∑ (15a)

[ ] [ ]

[ ] [ ] [ ]( )1

1 , , ...

Tot

NI

j

i

SS

f A B C=

=+∑

(15b)

In order to write Equation (15), the following condition must be satisfied

(see the Appendix)

, , 0i S i j

j

υ υ+ =∑ (16)

This probably explains why most authors disregard the existence of

multiple active sites in the proposed mechanisms, even when the molecule is quite

complex and adsorption in several sites is likely to occur (Gates et al., 1986;

Saeys et al., 2002; Thybaut et al., 2002; Saeys et al., 2005a; Saeys et al., 2005b;

Saeys et al., 2006). Similarly, this probably explains why catalyst sites are not

assumed to participate in the chemical reactions explicitly and why it is generally

assumed that an adsorbed reactant is transformed into an adsorbed product

without involvement of other active sites (authors write * *A B→ or * * * *A S B C+ → + ; but almost never * * **A S B+ → ). In this sense, proposition of

reaction mechanisms, as presented in most kinetic studies, is influenced very

significantly by the possible derivation of closed reaction rate expressions, and

not necessarily by the real kinetic routes that are believed to occur over the catalyst surface.



For example, let us consider the following mechanism, where step 3

constitutes the RDS

0

0

1

1

2

*

1

* **

1 2

**

2 2

k

k

k

k

k

A S I

I S I

I B S

−

−

→+ ←

→+ ←

→ +

(17)

In this case, Equation (16) is not satisfied and the following expressions

can be obtained when steps 1 and 2 are in equilibrium:

[ ]

[ ]

* 01 1 1

0

** 2 012 2 2

1 0

[ ] ,

[ ] ,

kI K A S K

k

kkI K A S K

k k

−

− −

= ⋅ ⋅ =

= ⋅ ⋅ =

(18)

The balance of sites can be written as:

[ ] * **

1 22 [ ]TotS I I S + + = (19a)

[ ] [ ] [ ] 2

1 2[ ] 2 [ ] [ ]TotS K A S K A S S+ ⋅ ⋅ + ⋅ ⋅ = (19b)

which constitutes a set of non-linear equations and leads to much more complex

expressions for [S]. It is not difficult to conclude that more complex mechanisms

will lead to untreatable analytical solutions. For such reason, researchers rarely

propose a mechanism that does not satisfy Equation (16).

The RDS Depends on the Reaction Conditions

Another limitation is related to possible changes of the controlling RDS as the

experimental conditions change (such as temperature). In this case, the RDS can

change and cause the reaction rate expression to depend on the analyzed

experimental conditions (Choudhary and Doraiswamy; 1975; Blagoeva et al.,

1999; Palo and Erkey, 1999; Redlingshöfer et al., 2002; Redlingshöfer et al.,

2003; Bischof et al., 2010; Lanni and McNeil, 2010). For example, Choudhary

and Doraiswamy (1975) reported that a better fit was obtained for n-butene

isomerization when two kinetic models derived from different RDS assumptions

was used to fit data in different conditions, which was interpreted by the authors




as a change in RDS. As a consequence, one can certainly argue how the kinetic

rate expression should be represented in the experimental region where RDS

transition is observed and more than one RDS can be identified.

The Explosive Number of Candidate Models

When the SSA and RDS approaches are adopted, at least in principle it is possible

to derive the kinetic rate expressions for all chemical components (provided that

closed analytical solutions can be obtained for concentrations of all intermediate

species, as discussed in the previous sections). However, when these approaches

are used and the reaction mechanism presents a complex network of elementary

reaction steps, the number of plausible kinetic models must be equal to the

number of reaction steps in the mechanism, since each step can be regarded as a

possible RDS candidate. If several reaction mechanisms are considered, then the

number of possible kinetic models is even larger and equal to the total number of

reaction steps in all analyzed mechanisms. When the reaction network comprises

more than one reaction, then the number of plausible kinetic models can be

obtained as a combination of the numbers of elementary steps considered for each

reaction, leading to the following expression:

( )

( , )

11

=

kR MecNN

k i

Mod Steps

jk

N N==∑∏ (20)

where NMod, NMec, NR and ( , )k i

StepsN are the total number of plausible models, the total

number of considered mechanisms, the total number of reaction steps in

mechanism k and the total number of steps in mechanism k of reaction i. From

Equation (20) one can see that the number of plausible kinetic models can be very

high, even for relatively simple mechanisms.

For example, if two reactions are involved (NR = 2) and two mechanisms

( (1) (2) 2Mec MecN N= = ) containing five steps ( (1,1) (1,2) (2,1) (2,2) 5Steps Steps Steps StepsN N N N= = = = ) are

used to explain each reaction, then the number of plausible kinetic models equals

100, as illustrated in Figure 1.



Figure 1. Illustration of possible number of kinetic expression derived for 2

reactions and 2 mechanisms that contain 5 steps each.

Therefore, one can certainly conclude that estimation of parameters for all

plausible models is not possible in general and that development and

implementation of model discrimination procedures is a fundamental issue for

those who work in this field (Schwaab et al., 2006; Atkinson et al., 2007;

Schwaab et al., 2008a; Buzzi-Ferraris and Manenti, 2009; Donckels et al., 2009;

Donckels et al., 2010; Alberton et al., 2011a; Alberton et al., 2011b). Similarly,

this example shows how important the development of advanced theoretical and

experimental tools (as described in Section 2.3) can be, when used for reduction

of the total number of model candidates in real and complex kinetic problems and

making the statistical analysis and estimation of model parameters feasible (Gates

et al., 1986; Bengaard et al., 2002; Abild-Pedersen et al., 2005; Boisen et al.,

2005; Saeys et al., 2005a; Topsøe et al., 2005; Vang et al., 2006).

Nevertheless, it is important to emphasize that it is necessary much care

during the analysis of complex reaction mechanisms, when the mechanisms

present some common elementary reaction steps. Let us consider the example of

toluene disproportionation, where toluene and xylene reactions take place

simultaneously at H-mordenite catalyst (Krahl, 1987; Lobão et al., 2011).

Step 01

Step 02

Step 03

Step 04

Step 05

Reaction 01

M

echan

ism

01

Step 01

Step 02

Step 03

Step 04

Step 05

M

echan

ism

02

Step 01

Step 02

Step 03

Step 04

Step 05

Reaction 02

M

echan

ism

01

Step 01

Step 02

Step 03

Step 04

Step 05 M

echan

ism

02

100 possible

kinetic models = 10 possible RDS 10 possible RDS ××××




(toluene) (benzene) (xylene)

(xylene) (toluene) (tri-methyl-benzene)

2

2

T B X

X T TMB

⋅ ←→ +

⋅ ←→ + (21)

The following mechanism has been proposed to describe these reactions as

(Pukanic and Massoth, 1973; Dooely et al., 1990):

*

1

* *

1 2

*

2

Reaction 1

step 1

step 2

step 3

T S I

T I B I

I X S

+ ←→

+ ←→ +

←→ +

*

2

* *

2 1

*

1

Reaction 2

step 1

step 2

step 3

X S I

X I TMB I

I T S

+ ←→

+ ←→ +

←→ +

(22)

An unambiguous way to derive kinetic rate expressions for the two

reactions is assuming that the surface reactions constitute the RDS. The step 1 of

Reaction 1 and step 3 of Reaction 2 are identical; the step 3 of Reaction 1 and step

1 of Reaction 2 are also identical. If one assumes that steps 2 in both reactions do

not constitute the RDS, one might also wonder whether it makes sense to admit

that step 1 of Reaction 1 can be the RDS of Reaction 1 and step 1 of Reaction 2

can be the RDS of Reaction 2, since both steps are common to both mechanisms.

As a consequence, Reactions 1 and 2 cannot be analyzed independently. In

accordance to this simple example, the mechanism and rate determining steps

should be proposed carefully when multiple reactions are involved. Theoretical

background and advanced characterization tools can help to verify the consistency

of proposed assumptions in model derivation according to a given mechanism, as

presented in the next section.

Full Set of Unsteady Balance Equations

Based on the previous discussion, one can be tempted to disregard the usual SSA

and RDS approaches and write down the full set of balance equations for all

chemical species involved in the mechanism. Although this can certainly

overcome the many difficulties associated with determination of rate determining

steps and quasi-steady state conditions, the fact is that the proposition of the full

set unsteady mass balance equations does not necessarily leads to well-posed

kinetic problems. First of all, it cannot be guaranteed that all of the proposed

kinetic rate constants can be indeed estimated from the available experimental

data, as discussed in the following section. Second, when some reaction steps are



much faster than the others, the numerical problem cannot be solved efficiently

(stiffness) and numerical model solutions can be unavailable (Hairer and Wanner,

1996). Third, when the model is stiff and the SSA and RDS assumptions are

indeed valid, some model parameters can be lumped, as clearly shown in

Equations (4-5), where k0 cannot be estimated independently from the other

model parameters (Schwaab and Pinto, 2007a). Therefore, there is no guarantee

that the proposition of the full set of unsteady mass balance equations will allow

for better representation of the analyzed kinetic problem. This clearly shows that

modeling of kinetic problems can be regarded as an art, where the expertise of the

analyst should never be underestimated.

Fundamental Proposition of Reaction Mechanisms

As shown in the previous sections, if experimental tools are not available to

monitor bulk and surface concentrations of the reacting chemical species,

proposition of reliable reaction mechanisms is unlikely. For this reason, the use

and development of advanced experimental techniques for in-situ

characterizations, such as infrared spectroscopy (IR), Raman spectroscopy,

nuclear magnetic resonance (NMR), ion scattering spectroscopy (ISS), X-ray

scattering (XPS), scanning tunneling microscopy (STM), among others, can be of

fundamental importance for kinetic studies. As a matter of fact, very significant

advances have been introduced in these fields in the last decades (Dumesic et al.,

1991; Schmal et al., 2001; Granger et al., 2002; Ahola et al., 2003; Topsøe et al.,

2005; Schmal et al., 2005; Meunier and Daturi, 2006; Schmal et al., 2006; Vang

et al., 2006; Bazin et al., 2007; Garland, 2010).

Molecular simulation schemes also constitute powerful tools for

independent identification of RDS and validation of reaction mechanisms. In this

case, the proposition of reaction mechanisms and identification of RDS

candidates are generally based on the independent determination of the energetic

barriers that can be associated with each mechanistic step. For example, if

different mechanisms are proposed and evaluated, the mechanism that presents

the lowest energetic barrier is regarded as the most probable reaction path. Within

a specific mechanism, the reaction step that presents the largest energetic barrier

can be regarded as the most plausible RDS candidate (Bengaard et al., 2002;

Saeys et al., 2002; Thybaut et al., 2002; Saeys et al., 2005a; Saeys et al., 2005b;

Saeys et al., 2006). Based on these very simple principles, molecular simulations

are also being used to guide the preparation of new catalysts and to identify the

active sites in complex structures (Abild-Pedersen et al., 2005; Boisen et al.,

2005; Topsøe et al., 2005; Vang et al., 2006).




Consistency of Kinetic Parameters

Despite the discussion presented in the previous paragraphs, it must be clear that

the analysis of complex networks of sequential and parallel reactions can be very

difficult both experimentally (too many chemical species and intermediates) and

through simulations (too many model equations and too many parameters).

Besides, simulation results may depend on the experimental conditions, so that

reaction mechanisms and RDS can change in the analyzed experimental range, as

already discussed. Therefore, the proposition of kinetic models still depends on

the experience, expertise and inspiration of the analyst. Once mechanisms are

proposed and reaction rate models derived, some guidelines must be used to

perform estimation of model parameters and evaluate the adequacy of the

proposed models. Particularly, attention must be given to the consistency of the

parameter estimates.

Based on the previous discussions, it can be inferred that most kinetic

models described in the literature probably propose very simple explanations for

the real and more complex kinetic behavior of real catalytic reactions. As a

consequence, several phenomenological aspects that are not correctly and

explicitly considered in the proposed kinetic models probably affect the

estimation of the kinetic parameters required for posterior use of these simplified

models. This can possibly explain why some apparent model inconsistencies are

found frequently during building of kinetic models, such as the estimation of

positive entropies and enthalpies of adsorption (Peterson and Lapidus, 1966; Xu

and Froment, 1989; Lobão et al., 2011).

Particularly, tests for thermodynamic consistency of model parameters

have been frequently used to eliminate model candidates from a set of plausible

models (Gut and Jaeger, 1982; Van Trimpont et al., 1986; Lee and Froment,

2008; Specchia et al., 2010). From a pragmatic point of view, as kinetic models

can be regarded as tools for prediction of concentrations and reaction rates and are

necessarily simplifications of reality, the efficiency of model discrimination

procedures based on thermodynamic criteria is doubtful. First, selection of models

that present thermodynamic consistency but poor prediction capacity is useless for

practical purposes. Second, thermodynamic consistency can always be achieved

through proper manipulation of parameter constraints (Gut and Jaeger, 1982; Van

Trimpont et al., 1986; Henriques, 1994; Campos, 1994; Lee and Froment, 2008;

Specchia et al., 2010; Lobão et al., 2011). And finally, it can certainly be argued

whether thermodynamic consistency is a fundamental theoretical issue or a

numerical artifact produced by the simplified description of the real kinetic

problem.

Another important issue regards the comparison of parameter values

described by different authors, mainly the activation energy and pre-exponential



factor of mechanistic steps. Although it seems reasonable to assume that different

authors should obtain similar parameter values, the fact is that such comparisons

should be made carefully, especially if the experiments are different and

performed in distinct ranges of experimental conditions. As already said,

proposed kinetic models are probably very significant simplifications of the real

phenomena. Besides, the pre-exponential factors (A) and activation energies (∆E)

are highly correlated to each other in most kinetic problems (Schwaab and Pinto,

2007b; Schwaab et al., 2008a). In simple terms, this is due to the fact that

different combinations of such parameters can result in similar values for the

kinetic rate constants, as

E

RTk A e

∆ − =

(23a)

S H

R RTK e

∆ ∆ − =

(23b)

Therefore, at least in principle pre-exponential factors (A) and activation

energies (∆E) obtained in a kinetic study should never be compared independently

to other values obtained in a different study; the same is valid for entropy (∆S)

and enthalpy (∆H) of adsorption. Comparative analysis should always take the

intrinsic correlation between A and ∆E values into consideration.

Parameter Estimation

In real problems, kinetic parameters must be estimated in order to make model

predicted data similar to available experimental data. However, it seems

reasonable to assume that model predictions should be more similar to

experimental data obtained with lower uncertainties. Thus, in order to evaluate the

quality of obtained parameter estimates it is necessary to know the behavior of the

experimental errors.

Generally, it is implicitly assumed that experimental errors follow the

normal probability distribution. The validity of this hypothesis is rarely tested,

since a large number of replicates (above 20) is usually required to identify the

statistical nature of random experimental fluctuations (Bard, 1974; Schwaab and

Pinto, 2007a). Additionally, other two hypotheses are frequently assumed: the

model is perfect and the experiments are well done. In this case, model deviations

can be regarded as samples of the experimental errors and the parameter estimates

can be obtained through minimization of the following objective function (Bard,

1974; Fogler, 1999; Schwaab and Pinto, 2007a):




( ) ( )exp 1 expT

calc calc

obj ZF Z Z V Z Z−= − ⋅ ⋅ − (24)

where Fobj is the objective function to be minimized, Zexp

is a vector containing all

available experimental values, Zcalc

is a vector containing all the model calculated

values (which are functions of the parameter values), VZ is the covariance matrix

of experimental uncertainties and the parameter estimates (Θest

) minimize the

objective function as:

( )arg minest

objFΘΘ = (25)

The vector Z contains all variables that are measured experimentally (mass

of catalyst, temperature, volumetric flow, feed concentrations, outlet

concentrations, etc). Some of these variables are associated with the reaction

conditions and are controlled by the analyst during the experiment (independent

design variables, represented by X). The remaining variables are obtained as

experimental responses, such as the outlet compositions (dependent response

variables, represented by Y). The response variables can be calculated with the

model (in our case, the kinetic model) as a function of the reaction conditions X

and kinetic parameters Θ. In other words, the response variables are functions of

the design variables and the kinetic parameters, i.e., Y(X,Θ). The vector of

variables measured experimentally contains both design variables and response

variables, i.e., Z = {X,Y}. The matrix VZ contains information about the

uncertainties of the variables Z. If the uncertainties of design and response

variables are not correlated to each other, then the matrix VZ presents a diagonal

structure and can be represented as

1 1 1 2

1 2 21 2

1 1 1 2

1 2 1 2

X Y

0 0 ....

0 0 ....

.... .... .... .... ....

0

0 .... ...

0 0 .... ...

.... .... .... .... .... ....

x x x x

x x x

Z

y y y y

y y y y

v v

v v

Vv v

v v

=

X

Y

VX

VY

(26)

This hypothesis is seldomly true (particularly in dynamic experiments) but

is often assumed to be correct. Therefore, Equation (24) can be written as:



( ) ( ) ( ) ( )exp 1 exp exp 1 expT T

calc calc calc calc

obj Y XF Y Y V Y Y X X V X X− −= − ⋅ ⋅ − + − ⋅ ⋅ − (27)

If the design variables are controlled with sufficient accuracy, VX = 0 and

the objective function can be written as:

( ) ( )exp 1 expT

calc calc

obj YF Y Y V Y Y−= − ⋅ ⋅ − (28)

If the uncertainties of Y are independent from each other, the matrix VY is

diagonal and the objective function becomes:

( )2expexp

, ,

1 1 ,

calcN Nyk i k i

obj

k i k yi yi

y yF

v= =

−= ∑∑ (29)

where , k yi yiv is the variance of variable yi in experiment k. If the variances

, k yi yiv

are constant for all response variables (which is usually false), then the objective

function becomes:

( )exp

2exp

1 1

N Nycalc

obj i i

i k

F y y= =

= −∑∑ (30)

This is the widely used least-squares function. In most cases, Equation

(30) is used as objective function for the parameter estimation problem. In short,

the least-squares function is obtained when measured variables are subject to

normal fluctuations, errors in the design variables are close to zero (VX = 0), the

errors in the response variables are not correlated (VY is diagonal) and variances of

response variables are constant. All these assumptions are implicitly made when

the analyst selects the least-squares function for formulation of the parameter

estimation problem. Almost never the validity of these many assumptions is

verified, which means that estimation of kinetic parameters is almost always made

on ill posed statistical grounds.

Calculation of Kinetic Rate Values

Determining Kinetic Parameters from Rates Values

Depending on the reactor type and operation, it can be possible to calculate the

reaction rates directly from measured concentrations of reactants and/or products.




However, as a matter of fact, reaction rates are not measured, but inferred from

other measured variables (concentrations). For example, when a continuous

stirred tank reactor (CSTR) is used to perform experiments at steady-state

conditions, the molar balances for species j can be calculated as:

( )exp

in out

j j

j

F Fr

W

−= (31)

where W is the catalyst mass and Fjin

and Fjout

are the inlet and outlet molar flows

of component j. One must observe that calculation of the “experimental” exp

jr

values, which often are used as the response variable Y in the parameter

estimation procedure, the measured values of W and Fjin

, which can be regarded

as the design variables, and of Fjout

, that is the real response variable, must be

known. Considering that the least-squares function can be used, it can be written

in terms of reaction rate values as:

( )( )exp

2exp

1

arg min ,N

est calc

i i

i

r r XΘ=

Θ = − Θ∑ (32)

where Θest

represents the parameter estimates, Nexp is the number of experiments,

riexp

represent the “experimental” kinetic rate data and ricalc

represent the kinetic

rate values calculated with the kinetic model, as a function of the reaction

conditions X and the parameters values Θ, as

( ) ,, ( , , )calc out

i i j ir X f T CΘ = Θ (33)

In this case, it can be observed that the vector of design variables X

includes the reaction temperature Ti and the outlet concentrations of species j

( out

,j iC ) in experiment i. As the outlet concentration is a real response variable and

not a real design variable, the real design variables (the inlet concentrations) are

only considered implicitly in Equation (33). Besides, an interesting point can be

raised regarding the experimental errors, when Equation (33) is considered. As

the outlet concentration clearly is a response variable and is not controlled

independently by the analyst, how can one guarantee that the variances of outlet

concentrations are negligible during parameter estimation? Besides, if all errors

are negligible, how can one formulate the estimation problem on sound statistical

grounds? Finally, if all measurements are free of error and the reaction rates are

calculated with measured values, why would reaction rates contain any significant



amount of uncertainty? Obviously, Equations (31-33) cannot be supported by

sound statistical reasoning, which means that the use of the least-squares function

is meaningless.

It is important to emphasize the importance of computers for formulation

and numerical solution of parameter estimation procedures. In order to calculate

the objective function, it is necessary to inform the values of the unknown

parameters Θ. If the objective function is calculated many times for several

tentative values of Θ, then it is possible to determine the Θ values that allow for

minimization of the objective function, as illustrated in Figure 2. However,

although a numerical solution can be obtained when Equations (31-33) are used

for parameter estimation, reaction rates should not be used for estimation because

they are not measured, but inferred from measured variables, leading to

meaningless statistical results.

Figure 2. Schematic illustration of the parameter estimation procedure for

minimization of Equation (31).

Differential Method and Method of Initial Rates (MIR)

A method used widely for kinetic studies of heterogeneous reaction systems is the

differential method. Basically, the differential method assumes that the dynamic

mass balance equations can be simplified in order to avoid the integration of

reaction rate expressions, by assuming that concentrations remain constant during

the reaction time. Obviously, the validity of the method is restricted to low

conversions, which can be regarded as a major disadvantage. A second

disadvantage is related to the numerical roundoff error introduced by the

discretization of the original differential equation that describes the dynamic

problem. The third disadvantage, as discussed in the previous section, is related to

the fact that reaction rates are not measured variables. For all these reasons, it is

The objective function that

must be minimized is

provided to the computer

software.

Experimental conditions

(X) and experimental rates

(Y) are provided to the

computer software.

The kinetic rate expression

is provided in the form

rcal

= f(X,Θ)

The computer software

provides the parameter

values that minimize the

objective function.




hard to find arguments for recommendation of differential methods nowadays, as

the existence of robust numerical integration procedures can be used for

integration of mass balance equations in reasonable simulation times.

Differential methods are used very often to reduce the number of model

candidates in complex kinetic problems. In this case, differential methods are

usually named as the Method of the Initial Rates (MIR). The MIR involves

“measuring” of the reaction rates at very short reaction times before occurrence of

any significant change of the concentration values (Boudart, 1968; Froment and

Bischoff, 1979; Vannice, 2005; Schmal, 2010). As a matter of fact, the MIR can

be regarded as a numerical procedure used to simplify the parameter estimation

task. According to the MIR, the concentrations are made equal to the initial values

and the mass balance equations are written in a discretized form, such as:

[ ] [ ] [ ] [ ]( )1 2 , ,...,i

i NS

d Qr Q Q Q

dt= (34)

[ ] [ ] [ ][ ] [ ] [ ]( )0

1 2 , ,...,i ii t

i NS

Q Qd Qr Q Q Q

dt tα∆

−≈ = =

∆ (35)

where it is implicitly assumed that the rate values do not change significantly

when ∆t is sufficiently small. As one knows the initial

concentrations [ ] [ ]( )1 , ,...,2Q NQQ Q , and assuming that α can be calculated with

real experimental data, the algebraic equations presented in Equation (35) can be

used for estimation of model parameters.

It is important to emphasize that, although the MRI is very popular and

indeed can be useful for building of kinetic models, it presents many drawbacks.

For example, one can always argue how small ∆t must be for Equation (35) to be

regarded as a fair approximation of Equation (34). That is, as the differential

method, the MRI method is restricted to very low conversion values. Moreover,

when ∆t decreases, the error of α ( 2

ασ ) can increase dramatically in the form

( ) ( )( )2 2

02

2

i iQ t Q

tα

σ σσ

∆ +

=∆

(36)

which has been consistently neglected in the kinetic literature. Therefore, in order

for Equation (35) to represent Equation (34) precisely, the calculation of α can

introduce a significant amount of error, prejudicing the quality of the final

parameter estimates. This poses a very interesting philosophical problem for the

analyst. It is obvious that more involving numerical approaches can be used to



provide better estimates of the initial rates, such as those based on polynomial

approximations of the dynamic trajectories (Froment and Bischoff, 1979; Fogler,

1999). However, as these more involving approaches make use of concentration

measurements when the rates cannot be regarded as constants anymore, one can

always wonder why the analyst should reduce Equation (34) to Equation (35)

before estimation of model parameters, instead of using Equation (34) in its

original form for model building.

It is not Necessary to Determine “Experimental” Rate Values

In more general terms, one does not need to determine the “experimental” rate

values in order to perform the estimation of model parameters. According to this

more effective approach, one can use implicit models to represent the

experimental data, which means that the response variables Y must not be

described explicitly in terms of design variables X and parameters Θ. If X and Θ

are known, it is possible to obtain Y(X,Θ) by solving the mass and energy

balances of the reacting system (eventually, momentum balance can be also

included), leading to concentration values and temperatures, which are real

measured data. For example, if the reactor is operated isothermically mass

balance equations can be written as:

jin out

j j j

W

dNF F r dW

dt− + ⋅ =∫ (37)

where t is the time, Fjin

and Fjout

are the inlet and outlet molar flows, rj is the

reaction rate and Nj is the number of moles of species j in the system. In this case,

the X variables correspond to the reaction conditions, such as temperature T,

catalyst mass W, volumetric flow (vin

) and concentrations (Cjin

) at the reactor inlet

(Fjin

= vin

.Cjin

). The output variables Y can correspond, for instance, to the molar

fractions of chemical components at the reactor outlet, measured with the help of

a chromatograph.

Reaction rate expressions must be inserted into Equation (37) to allow for

integration of the model equations. Very often, Equation (37) cannot be solved

analytically (except for very simple cases) and must be solved numerically. In

order to solve Equation (37), it is necessary to provide the parameter values Θ

(which are unknown and must be estimated). Equation (37) provides the

calculated output molar flows and molar fractions, as these variables are

correlated to each other as:




1

out

ii NS

out

k

k

Fy

F=

=

∑

(38)

where NS represents the number of chemical compounds in the outlet stream.

Then, it is possible to calculate the least-squares function (or any other

statistically sound objective function) as:

( )( )exp

2exp

1 1

,N NS

cal

i i i

i j

F y y X= =

= − Θ∑∑ (39)

Θ can be obtained as described before, through simulation with distinct tentative

Θ values and determination of the minimum value of the objective function

defined in Equation (39)

( )( )exp

2exp

1 1

arg min ,N NS

est cal

i i i

i j

y y XΘ= =

Θ = − Θ∑∑ (40)

Figure 3 illustrates the procedure that should be used for parameters

estimation. In this case one can use the data as obtained experimentally, without

any sort of data manipulation, leading to well posed statistical estimation

procedures. Besides, the application of the proposed procedure is not limited to

specific experimental ranges and can be used simultaneously for low and large

conversion values.

Figure 3. Schematic illustration of the parameter estimation procedure for

minimization of Equation (40).

The objective function that

must be minimized is

provided to the computer

software.

Experimental conditions

(X) and measured response

data (Y) are provided to the

computer software.

The kinetic rate expression

is provided in the form

rcal

= f(X,Θ)

The computer software

provides the parameter

values that minimize the

objective function.

A numerical procedure is

provided to calculate the

output variables Y, as

functions of X and Θ.



Statistical Tests

Statistical tests constitute very useful tools for evaluation of model adequacy. The

statistical analysis of parameter estimation results is of fundamental importance

because measured variables are subject to unavoidable uncertainties. As a

consequence, parameter values and model predictions are also corrupted by

experimental errors to some extent. Despite that, it is important to emphasize that

statistical tests are usually based on the assumption that measured variables are

subject to normal fluctuations, although this hypothesis is rarely verified in real

problems. Therefore, if this hypothesis is incorrect and the number of experiments

is small, the validity of the statistical tests is certainly questionable.

Two classes of statistical tests are commonly used for interpretation of

estimation results: analyses of parameter uncertainties and tests for verification of

model adequacy. These two classes of statistical tests are discussed below.

Uncertainty of Parameter Values

Parameter estimates can be interpreted as sampled parameter values obtained

from sampling of the experimental results. As experimental values are subject to

fluctuations, parameter estimates are also subject to fluctuations. If one assumes

that experimental data are subject to small normal fluctuations, then standard t-

Student tests can be used to evaluate the parameter uncertainties. Particularly, the

confidence interval of parameter estimates with confidence of α% can be

determined as (Bard, 1974; Schwaab and Pinto, 2007a):

( ) ( )1 1, ,

2 2

est true est

NGL NGL

t tθ θα αθ σ θ θ σ− +− ⋅ ≤ ≤ + ⋅ (41)

where ( )1,

2NGL

t α− is the t-Student value (in tabular form or obtained with the help of

a computer program) with NGL degrees of freedom (NGL=number of

experimental points less the number of parameters) and σθ is the standard

deviation of parameters estimates. σθ can be estimated with the help of the

derivatives of the objective function as (Bard, 1974; Schwaab and Pinto, 2007a)

( )1

21

2 2

i j

Obj

Obj

i j

FV Fθ θ θσ

θ θ

−− ∂

= = = ∇ ∂ ∂ (42)




where Vθ is the covariance matrix of parameter uncertainties, whose diagonal

terms contain the variances of the parameter uncertainties.

It must be emphasized that the experimental fluctuations do not necessarily follow

the normal distribution. For example, when conversion values are low and

experimental variances are constant, the normal distribution indicates that

negative conversion values are likely to occur, which is an absurd. In this case,

the t-Student test can provide unrealistic confidence intervals for parameter

estimates.

It is usual to disregard model parameters when the zero belongs to the

confidence interval defined in Equation (41); in this case, the parameter is

considered statistically insignificant, as it cannot be discriminated from zero. In

kinetic studies, many authors also make use of this argument to eliminate models

from the set of model candidates. However, this procedure must be performed

with care, as tests of significance are not intended for model discrimination. When

the parameter is statistically insignificant, this does not necessarily mean that the

parameter should be eliminated, as parameter estimation can be prejudiced by

experimental errors and poor experimental design.

In order to illustrate this point, let us consider a model where the kinetic

rate is given by:

[ ][ ] [ ]

1

2 31

Ar

A B

θθ θ

⋅=

+ ⋅ + ⋅ (43)

where r is the reaction rate, [A] and [B] are the reactant and product

concentrations, respectively, and θ1, θ2, θ3 are the model parameters. Let us

assume that the analyst collected data at low conversions and with no B in feed. In

this case, θ3 will certainly be insignificant, as

[ ][ ]

1

21

Ar

A

θθ⋅

≈+ ⋅

(44)

Both models presented in Equations (43) and (44) will probably fit the

experimental data appropriately, but the analyst will probably prefer the simpler

model presented in Equation (44). However, when using this model in presence of

significant amounts o B, the performance of the model will probably degrade.

This can constitute an important issue when models are discriminated because the

flexibility to fit new experimental data is reduced when parts of the model are

disregarded.



Model Adequacy

Model adequacy is usually evaluated by analyzing the distribution of residuals

between model predictions and measured experimental data. The most popular

adequacy tests are based on confidence tests for variance of model residuals,

assuming that the residuals follow the normal distribution: the chi-square test and

the Fisher test.

The chi-square variable (χ2) is a weighed sum of squared residuals, as

defined in Equations (24, 27-29). The least-squares function defined in Equation

(30) does not define a chi-square variable because the squared residuals are not

weighed by the variances. When the problem presents NGL degrees of freedom

and assuming a confidence level of α%, one can expect the objective function

values to lie in the range (Bard, 1974; Schwaab and Pinto, 2007a):

( ) ( )1 12 2

2 2

, ,objNGL NGLFα αχ χ− +≤ ≤ (45)

The limiting χ2 values can be found in tabular form or computed with the

aid of a software. Models that satisfy (Equation 45) can be regarded as adequate

to represent the experimental data.

Perhaps the main limitation of adequacy tests is the lack of knowledge

about the experimental uncertainties. As described in Equations (25) and (32),

different objective functions can be formulated, depending on the nature of the

experimental errors. However, experimental uncertainties are seldom analyzed,

which means that the statistical tests can provide false results. Despite that, these

statistical tools are used often to evaluate model adequacy and discriminate the

performances of model candidates.

Conclusions

In this work, issues related to the mathematical modeling and statistical analyses

of kinetic data were discussed. Kinetic modeling constitutes an ill-posed problem,

as most variables are not measured and cannot be observed. Steady state

assumptions (SSA) and rate determining step (RDS) approaches can be used for

simplification of mathematical models and derivation of closed kinetic rate

expressions. However, modeling procedures based on SSA and RDS can lead to

oversimplification of kinetic models, while the existence of complex mechanisms

usually renders the analytical derivation of kinetic rate expressions impossible

even when SSA and RDS techniques are used. Particularly, the mathematical

derivation of kinetic rate models based on RDS can be ambiguous and

inconsistent when the mechanism involves multiple elementary reaction steps.




Finally, the use of SSA and/or RDS approaches can lead to the combinatorial

explosion of the number of plausible kinetic models when complex reaction

mechanisms are taken into consideration.

The quantitative interpretation of experimental kinetic data requires the

estimation of model parameters. As discussed, the objective function used for

parameter estimation can assume several forms, depending on the nature of the

experimental errors. Despite that, the nature of the experimental fluctuations is

seldom analyzed, making the statistical analyses of the experimental data

questionable. Particularly, statistical tests find widespread use for analysis of

model adequacy and parameter significance. It was shown that model

discrimination based on parameter significance can lead to formulation of limited

models, with poor predictive capability.

Given the importance of the measured data in the field of kinetics, it was

shown that the proper statistical characterization of the experimental

measurements should be strongly encouraged. Besides, given the availability of

high speed computers, the use of rigorous modeling and numerical procedures

should also be encouraged, in order to avoid the oversimplification of the studied

problem.

As shown in this text, modeling of kinetic problems can be regarded as an

art, where the expertise of the analyst should never be underestimated. As stated

by Albert Einstein: “All our science, measured against reality, is primitive and

childlike – and yet it is the most precious thing we have.”

Appendix

In kinetic studies, the intermediate surface concentrations are generally written as:

[ ] [ ] [ ] [ ]( )* , , ...j jI S f A B C = ⋅ (14)

According to Equation (9), it is possible to obtain the logarithm of the

intermediate concentrations as:

1 1 1ln[ ]S− − −= ⋅ − ⋅ − ⋅

I* I* S I* Qln_I ln_K ln_Qϒϒϒ ϒϒϒ υυυ ϒϒϒ ϒϒϒ (A.1)

where 1−I* S

ϒϒϒ υυυ is the exponent of [S]. Let us assume that all exponents are equal to

1, as shown in Equation (15). Then



1

1

1

...

1

−

= −

I* Sϒ υϒ υϒ υϒ υ (A.2)

Thus:

1

1

...

1

= − ⋅

S I*υ ϒυ ϒυ ϒυ ϒ (A.3)

1, 1,1 1,2 1,

2, 2,1 2,2 2,

, ,1 2, ,

.... 1

.... 1

.... .... .... .... .... ....

.... 1

S NI

S NI

N S N N N NI

υ υ υ υυ υ υ υ

υ υ υ υ

= −

(A.4)

which can be written for each step i as:

, ,i S i j

j

υ υ= −∑ (A.5)

, , 0i S i j

j

υ υ+ =∑ (16)

References Abild-Pedersen F., Lytken O., Engbæk J., Nielsen G., Chorkendorff I. and

Nørskov J.K., “Methane Activation on Ni(111): Effects of Poisons and Step

Defects”, Surface Science, 2005, 590, 127–137.

Ahola J., Huuhtanen M. and Keiski R.L., “Integration of in Situ FTIR Studies and

Catalyst Activity Measurements in Reaction Kinetic Analysis”, Industrial &

Engineering Chemistry Research, 2003, 42, 2756-2766.

Alberton A.L., Schwaab M., Lobão M.W.N. and Pinto J.C., “Experimental

Design for the Joint Model Discrimination and Precise Parameter Estimation

through Information Measures”, Chemical Engineering Science, 2011a, 66, 1940-

1952.




Alberton A.L., Schwaab M., Lobão M.W.N. and Pinto J.C., “Design of

Experiments for Discrimination of Rival Models Based on Multiobjective

Optimization Procedures”, Chemical Engineering Science, 2011b, submitted.

Atkinson A.C., Donev A.N. and Tobias R.D., “Optimum Experimental Designs,

with SAS”, 2007, Clarendon Press, Oxford.

Bard Y., “Nonlinear Parameter Estimation”, 1974, Academic Press, New York.

Bazin P., Marie O. and Daturi M., “General Features of in situ and Operando

Spectroscopic Investigation in the Particular Case of DeNox Reactions”, Studies

on Surface Sciences and Catalysis, 2007, 171, 97-143.

Bengaard H.S., Nørskov J.K., Sehested J., Clausen B.S., Nielsen L.P.,

Molenbroek A.M. and Rostrup-Nielsen J.R., “Steam Reforming and Graphite

Formation on Ni Catalysts”, Journal of Catalysis, 2002, 209, 365-384.

Bischof S.M., Ess D.H., Meier S.K., Oxgaard J., Nielsen R.J., Bhalla G., Goddard

W.A. and Periana R.A., “Benzene C-H Bond Activation in Carboxylic Acids

Catalyzed by O-Donor Iridium(III) Complexes: An Experimental and Density

Functional Study”, Organometallics, 2010, 29, 742–756.

Blagoeva I.B., Kirby A.J., Koedjikov A.H. and Pojarlieff I.G., “Multiple Changes

in Rate-Determining Step in the Acid and Base Catalyzed Cyclizations of Ethyl

N-(p-nitrophenyl)hydantoates Caused by Methyl Substitution”, Canadian Journal

of Chemistry, 1999, 77, 849-859.

Boisen A., Dahl S., Nørskov J.K. and Christensen C.H., “Why the Optimal

Ammonia Synthesis Catalyst is not the Optimal Ammonia Decomposition

Catalyst”, Journal of Catalysis, 2005, 230, 309–312.

Bos A.N.R., Lefferts L., Marin G.B. and Steijns M.H.G.M., “Kinetic Research on

Heterogeneously Catalysed Processes: A Questionnaire on the State-of-the-art in

Industry”, Applied Catalysis A: General, 1997, 160, 185-190.

Boudart M., “Kinetics of Chemical Processes”, 1968, Prentice-Hall, New Jersey.

Buzzi-Ferraris G. and Manenti F., “Kinetic Models Analysis”, Chemical

Engineering Science, 2009, 64, 1061-1074.



Campos R.S., “Estimação de Parâmetros com Restrições de Desigualdade

Aplicada a Modelos Cinéticos”, MSc. Thesis, 1994, Universidade Federal do Rio

de Janeiro, Rio de Janeiro. (in Portuguese)

Choudhary V.R. and Doraiswamy L.K., “A Kinetic Model for the Isomerization

of n-Butene to lsobutene”, Industrial & Engineering Chemistry Process Design

and Development, 1975, 14, 227-235.

Donckels B.M.R, De Pauw D.J.W., Baets B.D., Maertens J. and Vanrolleghem

P.A., “An Anticipatory Approach to Optimal Experimental Design for Model

Discrimination”, Chemometrics and Intelligent Laboratory Systems, 2009, 95, 53-

63.

Donckels B.M.R., De Pauw D.J.W., Vanrolleghem P.A. and Baets B.D., “An

Ideal Point Method for the Design of Compromise Experiments to Simultaneously

Estimate the Parameters of Rival Mathematical Models”, Chemical Engineering

Science, 2010, 65, 1705-1719.

Dooely K.M., Brignac S.D. and Price G.L., “Kinetics of Zeolite-Catalyzed

Toluene Disproportionation”, Industrial & Engineering Chemistry Research,

1990, 29, 789-795.

Dumesic J.A., Rudd D.F., Aparicio L.M. and Rekoske J.E., “The Microkinetics of

Heterogeneous Catalysis”, 1991, American Chemical Society, Washington DC.

Farrusseng D., “High-Throughput Heterogeneous Catalysis”, Surface Science

Reports, 2008, 63, 487-513.

Fogler H.S., “Elements of Chemical Reaction Engineering”, 1999, Prentice-Hall,

New Jersey.

Froment G.F. and Bischoff K.B., “Chemical Reactor Analysis and Design”, 1979,

Willey, New York.

Garland M., “Combining Operando Spectroscopy with Experimental Design,

Signal Processing and Advanced Chemometrics: State-of-the-art and a Glimpse of

the Future”, Catalysis Today, 2010, 155, 266–270.

Gates S.M., Russel Jr. J.N. and Yates Jr. J.T., “Bond Activation Sequence

Observed in the Chemisorption and Surface Reaction of Ethanol on Ni(111)”,

Surface Science, 1986, 171, 111-134.




Granger P., Delannoy L., Lecomte J.J., Dathy C., Praliaud H., Leclercq L. and

Leclercq G., “Kinetics of the CO + NO Reaction over Bimetallic Platinum-

Rhodium on Alumina: Effect of Ceria Incorporation into Noble Metals”, Journal

of Catalysis, 2002, 207, 202-212.

Gut G. and Jaeger R., “Kinetics of the Catalytic Dehydrogenation of

Cyclohexanol to Cyclohexanone on a Zinc Oxide Catalyst in a Gradientless

Reactor”, Chemical Engineering Science, 1982, 37, 319-326.

Hagen J., “Industrial Catalysts: A Practical Approach”, 2nd

Edition, 2006, Wiley-

VCH, Weinheim.

Hairer E. and Wanner G., “Solving Ordinary Differential Equations II: Stiff and

Differential-Algebraic Problems”, 2nd

Edition, 1996, Springer, Berlin.

Henriques C.A., “Isomerização de Xilenos sobre Mordenitas: Estudo Cinético da

Reação e Caracterização do Coque Formado”, DSc Thesis, 1994, Universidade

Federal do Rio de Janeiro, Rio de Janeiro. (in Portuguese)

Krahl C.A., “Cinética de Desproporcionamento de Tolueno sobre Mordenita”,

MSc. Thesis, 1987, Universidade Federal do Rio de Janeiro, Rio de Janeiro. (in

Portuguese)

Lanni E.L. and McNeil A.J., “Evidence for Ligand-Dependent Mechanistic

Changes in Nickel-Catalyzed Chain-Growth Polymerizations”, Macromolecules,

2010, 43, 8039–8044.

Lee W.J. and Froment G.F., “Ethylbenzene Dehydrogenation into Styrene:

Kinetic Modeling and Reactor Simulation”, Industrial & Engineering Chemistry

Research, 2008, 47, 9183-9194.

Lobão M.W.N., Alberton A.L., Melo S.A.B.V., Embiruçu M., Monteiro J.F.L.

and Pinto J.C., “Kinetics of Toluene Disproportionation: Modeling and

Experiments”, Industrial & Engineering Chemistry Reserach, 2011, submitted.

Meunier F. and Daturi M., “Preface”, Catalysis Today, 2006, 113, 1-2.

Palo D.R. and Erkey C., “Kinetics of the Homogeneous Catalytic

Hydroformylation of 1-Octene in Supercritical Carbon Dioxide with



HRh(CO)[P(p-CF3C6H4)3]3”, Industrial & Engineering Chemistry Research,

1999, 38, 3786-3792.

Peterson T.I. and Lapidus L., “Nonlinear Estimation Analysis of the Kinetics of

Catalytic Ethanol Dehydrogenation”, Chemical Engineering Science, 1966, 21,

655-664.

Pukanic G.W. and Massoth F.E., “Kinetics of Mesitylene Isomerization and

Disproportionation over Silica-Alumina Catalyst”, Journal of Catalysis, 1973, 28,

304-315.

Redlingshöfer H., Kröcher O., Böck W., Huthmacher K. and Emig G., “Catalytic

Wall Reactor as a Tool for Isothermal Investigations in the Heterogeneously

Catalyzed Oxidation of Propene to Acrolein”, Industrial & Engineering Chemistry

Research, 2002, 41, 1445-1453.

Redlingshöfer H., Fischer A., Weckbecker C., Huthmacher K. and Emig G.,

“Kinetic Modeling of the Heterogeneously Catalyzed Oxidation of Propene to

Acrolein in a Catalytic Wall Reactor”, Industrial & Engineering Chemistry

Research, 2003, 42, 5482-5488.

Saeys M., Reyniers M.F., Marin G.B. and Neurock M., “Density Functional Study

of the Adsorption of 1,4-Cyclohexadiene on Pt(1 1 1): Origin of the C–H Stretch

Red Shift”, Surface Science, 2002, 513, 315–327.

Saeys M., Reyniers M.F., Neurock M. and Marin G.B., “Ab Initio Reaction Path

Analysis of Benzene Hydrogenation to Cyclohexane on Pt(111)”, Journal of

Physical Chemistry B, 2005a, 109, 2064-2073.

Saeys M., Reyniers M.F., Thybaut J.W., Neurock M. and Marin G.B., “First-

Principles Based Kinetic Model for the Hydrogenation of Toluene”, Journal of

Catalysis, 2005b, 236, 129–138.

Saeys M., Reyniers M.F., Thybaut J.W., Neurock M. and Marin G.B.,

“Adsorption of Cyclohexadiene, Cyclohexene and Cyclohexane on Pt(111)”,

Surface Science, 2006, 600, 3121–3134.

Schmal M., Souza M.M.V.M., Aranda D.A.G. and Perez C.A.C., “Promoting

Effect of Zirconia Coated on Alumina on the Formation of Platinum

Nanoparticles - Application on CO2 Reforming of Methane”, Studies on Surface

Sciences and Catalysis, 2001, 132, 695-700.




Schmal M., Souza M.M.V.M., Resende N.S., Guimaraes A.L., Perez C.A.C., Eon

J.G., Aranda D.A.G. and Dieguez L.C., “Interpretation of Kinetic Data with

Selected Characterizations of Active Sites”, Catalysis Today, 2005, 100, 145–150.

Schmal M., Souza M.M.V.M., Alegre V.V., Silva M.A.P., César D.V. and Perez

C.A.C., “Methane Oxidation – Effect of Support, Precursor and Pretreatment

Conditions – in situ Reaction XPS and DRIFT”, Catalysis Today, 2006, 118, 392-

401.

Schmal M., “Cinética e Reatores: Aplicação na Engenharia Química”, 2010,

Synergia, Rio de Janeiro. (in Portuguese)

Schwaab M., Silva F.M., Queipo C.A., Barreto Jr. A.G., Nele M. and Pinto J.C.,

“A New Approach for Sequential Experimental Design for Model

Discrimination”, Chemical Engineering Science, 2006, 61, 5791-5806.

Schwaab M. and Pinto J.C., “Análise de Dados Experimentais I: Fundamentos de

Estatística e Estimação de Parâmetros”, 2007a, E-Papers, Rio de Janeiro. (in

Portuguese)

Schwaab M. and Pinto J.C., “Optimum Reference Temperature for

Reparameterization of the Arrhenius Equation. Part 1: Problems Involving one

Kinetic Constant”, Chemical Engineering Science, 2007b, 62, 2750-2764.

Schwaab M., Lemos L.P. and Pinto J.C., “Optimum Reference Temperature for

Reparameterization of the Arrhenius Equation. Part 2: Problems Involving

Multiple Reparameterizations”, Chemical Engineering Science, 2008a, 63, 2895-

2906.

Schwaab M., Monteiro J.L. and Pinto J.C., “Sequential Experimental Design for

Model Discrimination Taking into Account the Posterior Covariance Matrix of

Differences Between Model Predictions”, Chemical Engineering Science, 2008b,

63, 2408-2419.

Specchia S., Conti F. and Specchia V., “Kinetic Studies on Pd/CexZr1-xO2

Catalyst for Methane Combustion”, Industrial & Engineering Chemistry

Research, 2010, 49, 11101–11111.

Thybaut J.W., Saeys M. and Marin G.B., “Hydrogenation Kinetics of Toluene on

Pt/ZSM-22”, Chemical Engineering Journal, 2002, 90, 117–129.



Topsøe H., Hinnemann B., Nørskov J.K., Lauritsen J.V., Besenbacher F., Hansen

P.L., Hytoft G., Egeberg R.G. and Knudsen K.G., “The Role of Reaction

Pathways and Support Interactions in the Development of High Activity

Hydrotreating Catalysts”, Catalysis Today, 2005, 107–108, 12–22.

Vang R.T., Honkala K., Dahl S., Vestergaard E.K., Schnadt J., Lægsgaard E.,

Clausen B.S., Nørskov J.K. and Besenbacher F., “Ethylene Dissociation on Flat

and Stepped Ni(111): A Combined STM and DFT Study”, Surface Science, 2006,

600, 66–77.

Vannice M.A., “Kinetics of Catalytic Reactions”, 2005, Springer Science, New

York.

Van Trimpont P.A., Marin G.B. and Froment G.F., “Kinetics of

Methylcyclohexane Dehydrogenation on Sulfided Commercial Platinum/Alumina

and Plantinum-Rhenium/Alumina Catalysts”, Industrial & Engineering Chemistry

Fundamentals, 1986, 25, 544-553.

Xu J. and Froment G.F., “Methane Steam Reforming, Methanation and Water-

Gas Shift: I. Intrinsic Kinetics”, AIChE Journal, 1989, 35, 88-96.




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Critical Analysis of Kinetic Modeling Procedures

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