MaCKiEŒ2002 - Universiteit Gentdconstal/mackie-workshops/mackie...ii Typeset in LATEX by D. Constales ([email protected]). A special issue of the journal Chemical Engineering Science

MaCKiE–2002

Mathematics in Chemical

Kinetics and Engineering

Book of Abstracts

Ghent, May 5–8, 2002.

ii

Typeset in LATEX by D. Constales ([email protected]).

A special issue of the journal Chemical Engineering Science is in preparation, dedicatedto the full papers corresponding to some of these abstracts.

MaCKiE-2002 is sponsored bythe Fund for Scientific Research Flanders(Belgium) FWO (www.fwo.be)and OMG dmc2 division (www.omgi.com).

Preface

The aim of the Mackie-2002 Workshop is to bring together mathematicians, chemistsand chemical engineers in order to discuss scientific problems that arise in appliedchemistry, and for which advanced mathematical methods are known or are still un-der development; especially the following domains are targeted:

• Multi-scale modeling of transport and complex reactions in porous media

• Models for the nonstationary characterization of catalysts

• Algebraic problems in chemical kinetics

• Analytical and numerical methods in chemical kinetics and, more generally, inapplied chemistry.

Interaction with advanced mathematical research plays an important role in Chem-ical Engineering since the very beginning. For instance, the school of Rutherford Aris(Univ. of Minnesota), a mathematician who even ‘retrained’ to become a chemical engi-neer, has led through Amundson (first Univ. of Minnesota, then Univ. of Houston) andhis student Dan Luss (Univ. of Houston) to the creation of research groups at universi-ties and in the central research and development laboratories of important companiessuch as Du Pont. One of the keynote speakers, V. Balakotaiah, is actually a former stu-dent of Dan Luss. Parallel to this, there is also a Russian school, of which one of theorganizers, G.S. Yablonsky, is an important representative.

At first, the mathematical approach was of necessity strongly analytical, i.e., it aimedat formulating mathematical models that could be solved analytically. This was dueto the limited computational capacity at that time, and did not always allow to pro-duce sufficiently realistic models. In particular, chemical kinetics was often not giventhe share it deserved. As the power of computational units increased, the modelingof chemical kinetics could be refined more and more. Nowadays, chemical kinetics isbased explicitly on reaction mechanisms that involve numerous elementary subreac-tions. The properties of applied catalysts, such as the nonuniformity of active sites andthe texture of the internal pores, can be taken into account with better accuracy. Thecorresponding model equations are solved numerically using adapted computationalalgorithms. These range from procedures for the integration of initial and boundaryvalue problems for systems of nonlinear partial differential equations, to Monte Carlosimulation.

iii

iv

This evolution does present the drawback, however, of increasingly hiding theglobal and general aspects of the problems. One of the major advantages of analyt-ical solutions is the way in which they can provide insight in these aspects, e.g., byanalyzing the limiting behavior of solutions. It is therefore not surprising that we wit-ness today a renewal of interest in analytical methods, which is further supported bythe wide availability of symbolic manipulation and computer algebra.

The purpose of this Workshop is therefore to give a state of the art in this domain,by bringing together a limited number of expert researchers, always bearing in mind tokeep the equilibrium between the two poles, mathematics and chemical engineering, soas to provide a true dialog and cross-fertilization.

The Organizing Committee:

D. ConstalesG.B. MarinG. NicolisR. Van KeerG.S. Yablonsky.

Chapter 1

Averaging Theory and Low dimensional Models for Chemical

Reactors and Reacting Flows

V. Balakotaiah1 S. Chakraborty1

1.1 Introduction

Mathematical models that describe the steady-state and dynamic behavior of chemicalreactors are obtained from the continuity, momentum, species, and energy balances andcombining these with the various constitutive equations for the transport and reactionrate processes. Depending on the level of detail included at the micro, meso and macroscales, these models can vary from one or two ordinary differential equations to severalthousand partial differential equations (in three spatial coordinates and time) containinga very large number of physico-chemical parameters. In addition, due to the strong cou-pling between the transport and reaction rate processes and the nonlinear dependenceof the kinetic and transport coefficients on the state variables (temperature or concen-trations), the model equations are highly nonlinear and are known to exhibit a varietyof complex spatio-temporal patterns. For most cases of practical interest, even with thepresent day computational power, it is impractical to solve such detailed models andexplore the different types of solutions (behaviors) that exist in the multi-dimensionalparameter space. Even in cases where detailed solutions are possible, the numerical re-sults do not provide directly the results an engineer is interested in (such as the averageexit conversion of a reactant or the yield of an intermediate product), unless some av-eraging or coarse-graining is done on the numerical results. Accurate low dimensionalmodels in terms of average (and measurable) variables are desired for the purpose ofdesign, control and optimization of chemical processes.

Historically, chemical engineers have developed low dimensional models of reactorsby making certain a priori assumptions on the length and time scales of reaction, diffu-sion and convection and applying the conservation equations only at the macroscopiclevel. The ideal continuous-flow stirred tank reactor (CSTR) model for describing thebehavior of homogeneous tank reactors is an example illustrating this approach. The as-

1Department of Chemical Engineering, University of Houston, TX 77204-4004, USA.

1

V. Balakotaiah and S. Chakraborty 2

sumptions made in developing such low dimensional models are usually not justifiedsince it involves comparison of the solutions obtained with more detailed (fundamen-tal) models, which are not available. When the predictions of such ad-hoc models didnot match with experimental results, the low dimensional models were modified byexpanding the degrees of freedom using concepts such as residence time distribution,non-ideal flow and mixing, and introducing empirical constants such as dispersion andheat transfer coefficients. The shortcomings of this approach (such as the dependence ofthe effective dispersion coefficients on the kinetic parameters and inconsistencies suchas infinite propagation speed of signals even in convection dominated systems) havebeen recognized recently.

An alternate method of obtaining low dimensional models is the bottom-up ap-proach based on detailed models derived from the fundamental laws. These modelsmay be averaged or simplified to obtain low dimensional or effective models havinglower dimensional state spaces and a smaller number of effective (or lumped) parame-ters. Such averaging may be possible in certain regions in the parameter space in whichthe rates of some transport or reaction processes are much slower or faster compared toothers (i.e., separation of length or time scales exists). This approach is the main focusof this work.

1.2 The Liapunov-Schmidt Method as an averaging technique

Several different empirical as well as rigorous averaging techniques (with different ter-minology such as homogenization, dimension reduction, adiabatic elimination, multi-scale averaging, slaving principle, etc.) are used in different fields for obtaining lowdimensional models. For dynamical systems with scale separation, the Center Man-ifold theorem (Carr, 1981) has been used extensively in recent years to eliminate theslave (or fast decaying) modes and obtain low dimensional models described by a fewordinary differential equations. While this is a powerful technique, a major limitation ofthis technique is that it can only describe the asymptotic behavior of a physical systemclose to a fixed point (such as a trivial solution). Our approach to averaging is basedon the Liapunov-Schmidt (L-S) technique of classical bifurcation theory. This method isbest suited for spatial averaging near one or more zero eigenvalues (corresponding tothe vanishing of a small parameter representing the ratio of length or time scales in thesystem). It can be used to eliminate spatial degrees of freedom and derive accurate lowdimensional models to any order in the small parameter. In addition, unlike most otheraveraging methods, it can be used to determine the region of validity (convergence) ofthe reduced model. The mathematical details of the Liapunov-Schmidt technique arepresented elsewhere (Golubitsky and Schaeffer, 1984; Balakotaiah and Chang, 2002).


1.3 Low dimensional Model for Solutal Dispersion Problem

To illustrate the L-S technique (and our approach), we consider the classical problem ofdispersion of a non-reactive solute in a circular tube of constant cross-section in whichthe flow is laminar. The solute concentration is described by the convective-diffusionequation

∂C

∂t′+ 2〈u〉

(

1− r2

R2

)

∂C

∂z′=D

r

∂

∂r

(

r∂C

∂r

)

; 0 < r < R, z′ > 0, t′ > 0, (1.1)

∂C

∂r

∣

∣

∣

∣

∣

r=0,R

= 0 (1.2)

LC : C(z′, r, 0) = f(z′, r) (1.3)

BC : C(0, r, t′) = g(r, t′). (1.4)

Here, 〈u〉 is the average velocity in the pipe, R is the radius and D is the diffusivity ofthe species. Defining dimensionless variables

z = z′/L, t = 〈u〉t′/L, ζ = r/R, Pe = R2〈u〉/(LD), (1.5)

we can write Eqs. (1.1) and (1.2) as

LC :=1

ζ

∂

∂ζ

(

ζ∂C

∂ζ

)

= Pe

[

∂C

∂t+ 2(1− ζ2)

∂C

∂z

]

;∂C

∂ζ

∣

∣

∣

∣

∣

ζ=0,1

= 0. (1.6)

We note that the transverse operator L is symmetric with respect to the inner product

(v, w) =∫ 1

02ζv(ζ)w(ζ) dζ.

It has a zero eigenvalue with normalized eigenfunction φ0 = 1. We define the mixing-cup (velocity weighted) and spatial average concentrations by

Cm =∫ 1

04ζ(1− ζ2)C(ζ, z, t) dζ, (1.7)

〈C〉 =∫ 1

02ζC(ζ, z, t) dζ (1.8)

Transverse averaging of Eq. (1.6) gives

∂〈C〉∂t

+∂Cm

∂z= 0. (1.9)

We note that when Pe = 0, 〈C〉 = Cm and substitution of this into Eq. (1.9) gives theleading order evolution equation for the averaged concentration. Writing

C(ζ, z, t) = 〈C〉(z, t) +W (ζ, z, t), W ∈ kerL, (1.10)


we can solve for the slave variable W (ζ, z, t) in terms of 〈C〉(z, t) using a perturbationexpansion in Pe and the L-S technique. To leading order, we have

W (ζ, z, t) = −Pe∂Cm

∂z

[

1

12− 1

4ζ2 +

1

8ζ4]

+O(Pe2). (1.11)

Substitution of this in Eq. (1.10) and transverse averaging (after multiplying by the ve-locity profile) gives the local equation relating Cm and 〈C〉:

Cm − 〈C〉 = − 1

48Pe∂Cm

∂z+O(Pe2). (1.12)

This local equation (when written in dimensional form) defines a characteristic transfertime between the slowly evolving mode Cm (or 〈C〉) and the slave mode Cm − 〈C〉.Eqs. (1.9) and (1.12) complete the reduced model to leading order. We can combinethe two equations to obtain a single equation either for Cm or 〈C〉. Since the cup-mixingconcentration (which is often measured in experiments) is more relevant in applications,the reduced model in terms of Cm in dimensional form is given by

∂Cm

∂t′+ 〈u〉∂Cm

∂z′+ 〈u〉tD

∂2Cm

∂z′∂t′= 0, (1.13)

where the local diffusion or mixing time is defined by

tD =R2

48D.

The corresponding length scale and local diffusivity are given `D = 〈u〉tD, Deff = 〈u〉2tD.To complete the reduced model, we need to specify Cm along the characteristic

curves z′ = 0 and t′ = 0. Thus, to all orders, the initial and boundary conditions areobtained by taking the mixing-cup averages of (1.3) and (1.4):

Cm(z′, t′ = 0) =∫ 1

04ζ(1− ζ2)f(z′, Rζ) dζ =: fm(z′), (1.14)

Cm(z′ = 0, t′) =∫ 1

04ζ(1− ζ2)g(Rζ, t′) =: gm(t′). (1.15)

Eqns. (1.13)–(1.15) complete the hyperbolic model to order Pe. This hyperbolic modelfor solutal dispersion eliminates several inconsistencies of the classical Taylor disper-sion theory (such as infinite speed of propagation of signals, artificial down-streamboundary condition) described by the parabolic model

∂Cm

∂t′+ 〈u〉∂Cm

∂z′= Deff

∂2Cm

∂z′2. (1.16)


1.4 Low dimensional models for homogeneous and catalytic reactors

In order to obtain the low dimensional models for describing the steady-state behaviorof homogeneous reactors, we start with the convection-diffusion-reaction equation incase of laminar flows or the closed Reynolds’ averaged transport equation in case ofturbulent flows. Application of the Liapunov-Schmidt technique reduces the infinitenumber of radial modes in these equations to two radial modes and these equationsare termed as Two-Mode Models (TMMs). As in the solutal dispersion problem, oneof the modes is the cross-sectional (or spatially) averaged concentration 〈C〉, which isrepresentative of the reaction scale of the system, while the other mode is the mixing-cup concentration Cm, which is representative of the convection scale of the system.

We present the final form of the TMMs for tubular reactors and CSTRs, which isa differential-algebraic system, consisting of two equations. One of them is called theglobal evolution equation, which gives the evolution of the mixing-cup variable, Cm, withtime or residence time, while the other is called the local equation, which describes thedifference between Cm and 〈C〉, in terms of the local mixing time and reaction rates. Theglobal evolution equations for different reactor types are as follows:

Tubular reactor:dCm

dτ= −R(〈C〉), with Cm(τ = 0) = Cin, (1.17)

CSTR:Cm − Cin

τC= −R(〈C〉), (1.18)

where τ is the residence time, τC is the total residence time in the reactor and R(〈C〉) isthe reaction rate. The local equation is the same for both reactor types and is given by

Cm − 〈C〉 = tmixR(〈C〉), (1.19)

where tmix is the local mixing time of the system, which for laminar flows is given bytmix = β1α

2/Dm, where a is the local diffusional length scale (e.g., radius of the tube),Dm is the molecular diffusivity of the species and β1 is a constant (equal to 1/48) whichdepends on the local velocity fluctuations. For the case of turbulent flows, the localmixing time is obtained to be a function of Reynolds number, molecular and turbulentdiffusivities (Chakraborty and Balakotaiah, 2001b). Since diffusion and reaction are thedominant processes at the local scales, the local equation involves only local variables(β1, Dm, a) and no large scale variables like mean velocity or reactor length. As a resultthe local equation retains the same form irrespective of the reactor type. Any alterationin the flow field only alters the local mixing time. Mixing is described in these models asan exchange between the two modes in the local equation, in terms of the local mixingtime.

The Liapunov-Schmidt technique could also be employed to obtain two-mode mod-els for heterogeneous wall-catalyzed reactions, where the two modes of interest are themixing cup concentration Cm and the wall concentration CS , and the local mixing timetmix is replaced by the transfer time between the two phases, tS . However, for the case of


coupled homogeneous-heterogeneous reactions, we require all the three modes Cm, 〈C〉and CS (which are representatives of the convection, homogenous and heterogeneousreaction scales of the system, respectively). These Three-Mode Models are given by

Global equation:dCm

dτ= −RV (〈C〉)−RS(CS), with Cm(τ = 0) = Cm,in,

Local equation 1: Cm − 〈C〉 = tmixRV (〈C〉) + γ1tSRS(CS),

Cm − CS = γ2tmixRV (〈C〉) + tSRS(CS),

where RV (〈C〉) and RS(CS) are the rates of homogeneous and heterogeneous reactions,respectively, and γ1 and γ2 are numerical constants. Here, both the local mixing timetmix and transfer time tS appear in the model, showing that the concept of a single trans-fer time, which has been used in traditional two-phase models, is not valid in case ofcoupled homogeneous-heterogeneous reactions.

1.5 Convergence and Accuracy of low dimensional models

The Liapunov-Schmidt technique not only allows us to obtain the low dimensionalmodels rigorously to all orders of required accuracy, but also specifies the region ofvalidity of the averaged models. For example, for the solutal dispersion problem, thereduced model to all orders may be shown to be

∂〈C〉∂t′

+ 〈u〉∂Cm

∂z′= 0,

〈C〉 − Cm ++∞∑

i=1

βi(tD)∂i〈C〉∂ti

= 0,

where tD is the local time scale and the βi are numerical constants that depend on thevelocity profile and geometry of the channel. Analysis of the radius of convergence ofthe local equation (which is an infinite series in the local time scale tD or, in the caseof homogeneous reactors, the mixing time, tmix) gives us the region of validity of thelow dimensional model in the parameter space. For example, for the case of first orderreactions, the two-mode models are valid if the local Damkohler number Daloc (ratio oflocal mixing time to reaction time) satisfies the following criteria: Daloc < 0.858 for thecase of Pe → +∞ (tube); and Daloc < 0.289 for the case of Pe = 0 (tank). Therefore,low dimensional descriptions are not possible if a very fast reaction is complementedby slow local mixing. Comparison of the solution of the two-mode model with the fullconvective-diffusion equation shows excellent agreement within the region of validity(Chakraborty and Balakotaiah, 2001a). However, these two-mode models, even whentruncated at the lowest order, retain all the parameters present in the convection-dif-fusion-reaction equation and therefore retain most of the qualitative and quantitativefeatures of the full partial differential equation. These low dimensional models, which


are differential-algebraic systems, are particularly advantageous in cases of multiplereactions with non-linear kinetics, where the solution of the full PDEs with laminaror turbulent flows is non-trivial. Unlike other low dimensional reactor models, thesemodels are capable of predicting multiple solutions for the case of autocatalytic kinetics.

We also show that when low dimensional models derived by the L-S method donot converge, the governing partial differential equations have continuous spectrum(i.e., no scale separation exists). In such cases, averaging is not possible since three-dimensional solutions with all length scales leading to ‘chemical turbulence’ may exist.Examples will be provided to illustrate this.

1.6 Bibliography

[1] V. Balakotaiah and H-C. Chang, Hyperbolic homogenized models for thermal andsolutal dispersion, SIAM J. Appl. Math., in review (2001).

[2] V. Balakotaiah and H-C. Chang, Applied Non-Linear Methods for Engineers, CambridgeUniversity Press, to be published (2002).

[3] J. Carr, Applications of Center Manifold Theory, Springer (1981).

[4] S. Chakraborty and V. Balakotaiah, Low dimensional models for describing mixingeffects in laminar flow tubular reactors, Chem. Engng Sci., in review (2001a).

[5] S. Chakraborty and V. Balakotaiah, Two-mode models for describing mixing effectsin homogeneous reactors. AIChE J., in review (2001b).

[6] M. Golubitsky and D.G. Schaeffer, Singularities and Groups in Bifurcation Theory, Vo-lume 1, Springer, (1984).

Chapter 2

Mathematical modelling of endothermic reactions in the catalyst unit

with ordering elements

V.K. Belnov1 N.M. Voskresenskii1 S.I. Serdyukov1 I.I. Karpov1 V.V. Barelko2

2.1 Abstract

The search for problem solving of heat removal from overheated surfaces is an issue ofthe day. The endothermic process of cyclohexane dehydrogenation

C6H12 → C6H6 + 3H2 −Q1, (2.1)

steam conversion of methane

CH4 + H2O ↔ CO + 3H2 −Q2; CO + H2O ↔ CO2 + H2 +Q3, (2.2)

and decomposition of ammonia

2NH3 → N2 + 3H2 −Q4, (2.3)

can be considered as alternate versions of a thermocatalytic method of cooling.An urgent problem is the creation of unified catalyst modules (units), effectively

working in extreme conditions. The development and analysis of a non-steady statemathematical model of heat exchange is a key stage for the optimization of catalystunits

At present, catalysts implemented as coverings applied to regular constructional el-ements of a module (planar catalysts) are more promising. We consider such a modulewhich is a set of plane-parallel plates made of heat-conducting material with the cata-lyst divided by transport channels through which the reaction mixture flows (see Fig.2.1).

When the carrier represents a combined unit and porous or heterogeneous catalystmaterial (for example, porous iron), chemical reactions, heat and mass transfer in the

1Department of Chemistry, Moscow State University, 119 899 Moscow, Russia.2Institute of Problems of Chemical Physics RAS, Chernogolovka, Russia.

8

V.K. Belnov, N.M. Voskresenskii, S.I. Serdyukov, I.I. Karpov and V.V. Barelko 9

Figure 2.1: Fragment of the catalytic block


block can be described by the following system of differential equations where the heatconduction in a carrier is subject to the Maxwell-Cattaneo heat flux model:

ε∂ρ

∂t+∂J

∂z= 0,

ερ∂x1

∂t+ J

∂x1

∂z= −AW,

εcpρ∂T

∂t+ cpJ

∂T

∂z= αA(Tk − T ),

(1− ε)ρkcpk∂Tk

∂t= −(1− ε)

∂q

∂z− αA(Tk − T )− AW∆H,

τq∂q

∂t+ q = −λk

∂Tk

∂z.

where q is heat flux, tqrelaxation time, A the solid material surface area in one unitof package volume, b the channel half-width, cp the specific heat capacity of gas, cpk

the specific heat capacity of the carrier, ∆H the increase of gas mixture enthalpy dueto chemical reaction per one kilogram of converted methane or ammonia, l the half-thickness of the carrier with catalytic covering, J the gas mass rate per one unit of reac-tor cross section, t time, T gas mixture temperature, Tk temperature of the carrier andcatalyst, x1 the mass fraction of methane or ammonia or cyclohexane in the mixture;z axial distance along the package in gas flow direction, W rate of methane or ammo-nia transformation per one unit of contact surface area, α the heat transfer coefficient,ε = b/(b + l) the fraction of void section in package; λk the heat conductivity coefficientof the carrier, ρ the gas mixture density

At low speed of chemical reaction and low convective transfer between plates (J ≈ 0,W ≈ 0), the equation set (2.4) reduces to the equation set

εcpρ∂T

∂t= αA(Tk − T ),

(1− ε)ρkcpk∂Tk

∂t= −(1− ε)

∂q

∂z− αA(Tk − T ),

τq∂q

∂t+ q = −λk

∂Tk

∂z,

which can be reduced to the dimensionless equation containing parameters A and B:

A∂3θk

∂ω3+∂2θk

∂ω2+ 2

∂θk

∂ω=∂2θk

∂Y 2+N

∂

∂ω

(

∂2θk

∂Y 2

)

, (2.4)

where qk, Y and ω are the dimensionless temperature of catalyst, unit length and time,respectively.

The heat transfer according to equation (2.4) represents a dual-phase-lag model. Inthis work such conditions, at which the heat transfer represents heat waves, were ana-lyzed. The wave conditions of model (2.4) corresponds to a value of coefficient β < 1/2[1].


Figure 2.2: Heat waves propagation in the catalyst unit.

We used a numerical simulation of catalyst unit based on the equations set (2.4) and(2.4). During computational modeling we used some calculus of approximations suchas method of difference scheme, method of chaser and method of simple iteration.

The calculation results for model (2.4) and conditions, at which one the non-stationary process of heat transmission represents dissipating heat waves, are presentedin Fig. 2.2.

Then we modeled a catalyst block in which endothermic the chemical reactions (2.1–2.3) proceed. As catalysts we used earlier investigated plasma-deposited and gauzecatalysts on metal carrier [2]. For the calculations we used actual kinetic functions ofreactions (2.1–2.3).

Our results demonstrate a significant advantage of ammonia decomposition overmethane conversation and cyclohexane dehydrogenation in the chemothermal coolingof overheated surfaces.

2.2 Acknowledgement

The work is supported by the International Scientific Foundation ‘Science for Peace’,Project No SfP 971897 and the Russian Foundation for Basic Researches, Project No 01-03-36662.


2.3 Bibliography

[1] K.K. Tamma and X. Zhou, Macroscale and Microscale Thermal Transport andThermo-Mechanical Interactions: Some Noteworthy Perspectives, J. Thermal Stresses,21:405, 1998

[2] S.I. Serdyukov, M.S. Safonov, A.A. Fomin and I.S. Nasonovskii, RF Inventor’s Cer-tificate no. 2040330, Byull. Izobret., 21, 1995.

Chapter 3

From Single Events Theory to Molecular Kinetics – Application to

industrial process modeling

D. Guillaume1 K. Surla1 P. Galtier1

3.1 Introduction

The single events theory, first developed at Ghent University [1], is based on a detailedand exhaustive network of reactions and species. It is generated by computer and basedon some assumptions that reduce the number of parameters involved in the descriptionof the mechanism.

The single events theory has been proved to be efficient in the modelling of acid-catalysed petrochemical processes. The lumping approach was intensively developedfor heavy feed processes containing species at thermodynamical equilibrium. The mainaim was the reduction of the kinetics and computational burden. The same concept can,however, be applied to processes with light feed whose models need fewer parameter.The advantages of this interpretation are meaningful: verification of the thermodynam-ical coherence, comparisons with apparent kinetics, links to the traditional point of viewon kinetic modelling, simplification of the overall procedures. . .

This paper reports the way to reformulate the kinetics according to some chemicalassumptions and the use of this methodology to actual industrial cases.

3.2 Modelling by single events

For detailed information on single events modelling, the reader may refer to [1].A complete reaction network is generated by means of a computer algorithm which

takes into account the following elementary steps:

1. dehydrogenation of paraffin

2. protonation of olefin

1IFP CEDI ‘Rene Navarre’ BP3, 69390 Vernaison France.

13

D. Guillaume, K. Surla and P. Galtier 14

3. isomerization by hydrogen shift

4. isomerization by methyl shift

5. isomerization by PCP branching

6. b-cracking

7. deprotonation of carbenium ion

8. hydrogenation of olefin

The main advantage of the single events theory is that it allows to reduce the numberof parameters. Indeed, the rate coefficient of an elementary step, k, can be written as the

product of a single-event rate coefficient, k, and the number of single events, ne.This number of single-events, ne, can be calculated from the symmetry number of

the reactant(s), sr, and the activated complex, s′.

Moreover, the parameter k is supposed to be identically the same for the same reac-tion type involving the same reactive and produced ions nature (secondary or tertiary)and it is not dependant of the ions carbon number.

These assumptions drastically reduce the number of parameters involved for de-scribing a network of hundred thousands of reactions to a finite fixed number whateverthe maximum carbon number is.

3.3 Rate equations at a molecular level

Consider here the isomerization of a Pi paraffin into a Pj paraffin by acid catalysis, withthe following assumptions :

• hydrogenation/dehydrogenation at equilibrium

• protonation/deprotonation at equilibrium

• reaction in gas phase

• rate determining step on the acid phase.

Note :

• R+li

ions linked to paraffin Pi;

• R+mj

ions linked to paraffin Pj.


With the assumptions of the rate determining step on the acid phase, the equations ofthe kinetics of the isomerization of the paraffin Pi into the paraffin Pj can be written asthe sum of the elementary steps that link ions issued of paraffins Pi and Pj :

R(Pi ↔ Pj) =∑

li,mj

(kisom(mj, li)R+mj − kisom(li, mj)R

+li).

By taking into account the assumptions and the free sites balance, the kinetics can bewritten as a well known Langmuir type kinetics :

R(Pi ↔ Pj) =kjiPj − kijPi

PH2+∑

h bhPh,

where the parameters kij and bh are calculated from the intrinsic single events parame-ters.

The parameters are temperature dependant only and equal multiple sums on thewhole reaction network. At isothermal conditions, they can be computed once outsidethe solver and considerably reduce the simulation time.

3.4 Use to some theoretical consideration

The modelling of a process can highly benefit of this new reformulation. The realisa-tion of the model on computer is technically complex because of the numerous sumsappearing.

This new formulation allows :

• to check the coherence of the sums,

• to check the coherence of the parameters value by comparison with literature data.

3.4.1 Thermodynamical coherence

One key point of the modelling is the thermodynamical coherence between direct andreverse kinetics parameters.

The ratio of the direct and reverse parameters can be written as :

K =

∑

(li,mj) kisom(li, mj)Kpro/dep(Oki, li)Khyd/deh(Pi, Oki

)

kisom(mj, li)Kpro/dep(Onj, mj)Khyd/deh(Pj, Onj

)

The sums are computed on the same reaction pathways because of the reversibility ofall the reactions (in the isomerization case).

Because of the equilibrium, the equilibrium constant between Pi and Pj can be writ-ten on a particular pathway :

kisom(mj, li)Kpro/dep(Onj, mj)Khyd/deh(Pj, Onj

)K(Pi ↔ Pj)

= Khyd/deh(Pi, Oki)Kpro/dep(Oki

, li)kisom(li, mj),


and finally :K = K(Pi ↔ Pj).

The ratio of the direct and reverse kinetics parameters is so formally identical to theequilibrium constant.

Comparison of the ratio and actual equilibrium constant constitutes a checkpoint forthe computer model (before any estimation of the kinetics parameters).

3.4.2 Comparison to molecular models

Literature reviews present some Langmuir-Hinshelwood models for isomerization andprovide data for molecular kinetics. This new point of view make possible the compar-ison between literature data and estimation results.

It is also easier to compare the activity of several catalysts by mean of the apparentkinetics parameters that is not directly possible with the single events parameters.

3.5 Application to an industrial case : isomerization of n-butane

In order to simulate an industrial IFP process of butane isomerization, a single eventskinetics model has been developed.

Various research groups suggest that, unlike the isomerization of paraffins with 5or more hydrocarbons, the isomerization of butane proceeds by a bimolecular reactionpathway:

2nC4k1→ iC8 + H2

k2→ 2iC4.

It was observed from the parameter estimations and the experimental results that thefirst step was rate determining.

The final model can be reduced to the following molecular form :

r =k1PnC4

1 + b1PnC4+ b2PiC4

+ b3(P2iC4/PH2

).

3.6 Conclusion

This new way of writing the rate equation makes it easier to apply the single eventsmodelling. Indeed, the equations look closer to the usual Langmuir equation type.Thus, you can directly compare apparent kinetic parameters of different model or ob-tained through estimation on experimental data. It also allows to find realistic initialvalues for kinetics parameters estimation.


3.7 Bibliography

[1] M.A. Baltanas and G.F. Froment, Computer generation of reaction networks andcalculation of product distributions in the hydroisomerization and hydrocracking ofparaffins on Pt-containing bifunctional catalysts, Comp. Chem. Engin. 9 (1985) 71–81.

Chapter 4

Calculation of Three-Dimensional Flow Fields in Cracking Furnaces

Arno J.M. Oprins1 Geraldine J. Heynderickx1

4.1 Introduction

Thermal cracking of hydrocarbons is an endothermal process that takes place in tubularreactor coils suspended in large gas-fired furnaces. Heat transfer to the reactor tubesis mainly due to radiation from the furnace refractory walls with temperatures up to1500K and to radiation from the flue gas with temperatures up to 2100K.

For the simulation of the thermal cracking process inside the reactor tubes a detailedkinetic radical reaction scheme was developed at the Laboratorium voor Petrochemis-che Techniek, containing over 1,000 reactions between 128 species (Heynderickx andFroment, 1996). In addition, coke formation models were derived for both relativelylight and heavy hydrocarbon feedstocks (Plehiers et al., 1990; Reyniers et al., 1994).

For reactor tubes with smooth internal surfaces under typical operating conditions,it is satisfactory to combine the above kinetics with a one-dimensional plug-flow reactormodel to obtain a high degree of accuracy. This is related to the fact that radial gradientsare suppressed by a high degree of turbulence corresponding to Reynolds numbersof well over 250,000. More detailed information on the reactor simulation is given inWillems and Froment (1988a,b).

With respect to the furnace, simulations of the combined furnace-reactor system forthe calculation of the heat fluxes to the reactor tubes are based on the zone method ofHottel and Sarofim (1967). The furnace is divided into a number of surface and vo-lume zones that are considered to be isothermal. For these zones the energy balances,containing radiative, convective and conductive contributions, are constructed. Theradiative contributions are obtained through Monte Carlo simulations, calculating theview factors between the different zones in the furnace. Rao et al. (1988) and Plehiersand Froment (1989) give detailed information on the coupled simulation of furnace andreactor.

These calculations require the assumption of a flow and heat release field in the fur-nace. With respect to the first assumption, calculations have shown that the choice of an

1Laboratorium voor Petrochemische Techniek, Krijgslaan 281 (S5), B9000 Ghent, Belgium.

18

A.J.M. Oprins and G.J. Heynderickx 19

idealized flow field of the flue gas (such as plug-flow), compared to simulations witha realistic flow field inside the furnace, leads to significant differences in simulationresults for the reactor coils. These differences are directly associated with the second as-sumption of the localization of the heat release. Especially when combustion of the fuelgas takes place inside the furnace (so called long-flame burners as opposed to radiationburners) the choice for the location of the heat release becomes more or less arbitraryand simulation of the flow field and heat release (combustion) field is necessary (Sun-daram and Albano, 1997).

The calculation of accurate flow fields in the furnace requires the three-dimensionalNavier-Stokes equations to be solved (De Saegher et al., 1996; Detemmerman and Fro-ment, 1998). Since radiative heat transfer accounts for most of the heat flux towards thereactor, there is also need for an appropriate radiation model. These models finally haveto be combined with a combustion model (including combustion kinetics, NOx forma-tion and turbulent mixing). In this work the model equations and boundary conditionswill be discussed, followed by an outline of the numerical methods used.

4.2 Model equations

The simulation software is constructed using a modular approach. This gives the usermaximum flexibility in generating initial values and omitting not-used or non-relevantmodules (e.g., the combustion model when simulating a furnace with radiation burn-ers). In the following section the model equations containing flow, radiation and com-bustion are summarized.

4.2.1 Flow model equations

The calculation of the steady-state flow field is based on the compressible RANS orReynolds-Averaged Navier-Stokes equations. Because of the time averaging a closuremodel is required to account for turbulence in addition to the continuity, momentumand energy equations. In this work the standard k-e turbulence model is applied. Theresulting equations are:

3∑

i=1

∂

∂xi(ρgUi) = 0 (4.1)

3∑

j=1

∂

∂xj(ρgUiUj) = −∂P

∂xi+

3∑

j=1

∂

∂xj

(

µtot

(

∂Ui

∂xj+∂Uj

∂xi− δij

3∑

k=1

2

3

∂Uk

∂xk

)

− 2

3ρgkδij

)

+ Sbd

(4.2)3∑

i=1

∂

∂xi(ρg(q + k + cpT )Ui) =

3∑

i=1

∂

∂xi

(

λtot∂T

∂xi− 2

3ρgkδij

)

+ SE (4.3)

3∑

i=1

∂

∂xi

(ρgUik) =3∑

i=1

∂

∂xi

(

(

µg +µturb

σk

)

∂k

∂xi

)

+ Pk − ρgε (4.4)


3∑

i=1

∂

∂xi(ρgUiε) =

3∑

i=1

∂

∂xi

(

(

µg +µturb

σε

)

∂ε

∂xi

)

+ Pε − C2ερgε2

k(4.5)

4.2.2 Radiation model equations

The six-flux De Marco and Lockwood radiation model is used to calculate the radiativeheat exchange. This model prescribes the angular distribution of the radiation, greatlysimplifying the solution of the problem. The model equations (applicable to gray gases)can be written as (De Marco and Lockwood, 1975):

∂

∂x

(

1

ka

∂Cx

∂x

)

= ka(3Cx − Cy − Cz)− kaσT4 (4.6)

∂

∂y

(

1

ka

∂Cy

∂y

)

= ka(−Cx + 3Cy − Cz)− kaσT4 (4.7)

∂

∂z

(

1

ka

∂Cz

∂z

)

= ka(−Cx − Cy + 3Cz)− kaσT4 (4.8)

The local volumetric energy release due to radiation can be written as a function ofthe dependent variables Cx, Cy and Cz and can be substituted in the energy equation(Equation (4.3)) of the flow module as part of the source term SE :

Qrad =4

3ka(Cx + Cy + Cz)− kaσT

4. (4.9)

4.2.3 Combustion model equations

The combustion module consists of the continuity equations for the different reactioncomponents:

3∑

i=1

∂

∂xi(ρgUiyj) =

3∑

i=1

∂

∂xi

(

ρgDt∂yj

∂xi

)

+RjMj. (4.10)

The net production rate RjMj is calculated based on a five-step reaction mechanismfor a common fuel gas (for thermal cracking furnaces) comprising of methane, ethane,propane and hydrogen. Rate coefficients were taken from Westbrook and Dryer (1981).The eddy breakup model of Spalding (1972) accounts for turbulent mixing.

Current research focuses on the application of more detailed (radical mechanism)combustion kinetics and incorporation of NOx formation.

4.2.4 Boundary conditions

All variables except the pressure need to be prescribed at the inlet of the furnace. Atthe furnace outlet, the pressure is imposed. Wall functions are used to bridge the lam-inar boundary layer near reactor and furnace walls, thus replacing the k-e model. The


no-slip condition is imposed at the reactor and furnace walls. With respect to the radia-tion model the reactor and furnace walls are described as gray surfaces (De Marco andLockwood, 1975).

4.3 Numerical methods

The model equations described above are spatially discretised by means of a finite vo-lume method. Using a free-ware grid generation program a 3-dimensional tetrahedralintegration grid is constructed for the computational domain. The choice for an un-structured 3-dimensional tetrahedral grid allows to describe accurately all features ofthe furnace geometry (including for example reactor tubes and part of the convectionsection). For our purposes a typical integration grid contains about 50,000 to 250,000volume cells depending on geometry and symmetry considerations.

For both the flow module and the combustion module the convective terms in theequations (Equations (4.1) to (4.5) and (4.10)) are treated using a first-order flux dif-ference splitting method (Dick, 1988; Dick, 1990; Dick and Steelant, 1997). The basicpolynomial flux difference splitting is done with respect to the primitive variables (r, u,v, w, p, k, e), followed by a transformation to a splitting with respect to the conservativevariables (r, ru, rv, rw, rE, rk, re). Although this upwind technique is not considered tobe the most suitable method for calculating low-speed flows, it is expected to be satis-factory for our purposes for simulating thermal cracking furnaces where the focus is onpredicting the reactor performance. For the viscous terms in these equations a centraldifferencing scheme is used.

The linearization of the source terms in the k- and e-equations is based on Dickand Steelant (1997) to increase the diagonal dominance of the set of equations in theflow module. The treatment of the production term in the energy equation (Equation(4.3)) is described by Detemmerman (1997). This source term containing the reactionenthalpy is distributed over the left- and right-hand side of the equation to increase thediagonal dominance and maintain stability. The remaining source terms (gravitationand volumetric energy release due to radiation) appear in the right hand side of theequation set.

The calculation of the variables in the radiation model (Equations (4.6) to (4.9)) isalso done by means of a central differencing scheme.

For all modules the resulting (linearized) set of equations is solved for each gridpoint by a Gauss elimination method, using a Gauss-Seidel iteration scheme. The lattermeans that at any moment in the calculations the most recent values for the variablesare used, which in general increases the convergence rate.

4.4 Results and discussion

The above equations and calculation procedures were used to calculate the three-di-mensional flow, heat release and temperature fields for a pyrolysis furnace for thermal


cracking. The results are presented in Heynderickx et al. (2001) and Oprins et al. (2001).The first article describes the occurrence of highly asymmetrical flow fields, whereas inthe second article these results are compared to a simulation with asymmetrical fuel gasdistribution over the burners and focuses on the compensating nature of radiative heattransfer.

The presented results were obtained by applying the calculated flow and heat releasefields in the combined furnace-reactor calculations based on the zone method of Hotteland Sarofim. Current research focuses on a direct calculation of the net heat transfer tothe reactor tubes within the CFD software (flow, radiation and combustion modules).When flow calculations are necessary or relevant, this will lead to a faster and possiblymore accurate simulation package for the simulation of thermal cracking furnaces. Theflow calculation software can also be applied for the simulation of furnaces in otherindustrial applications.

4.5 Bibliography

[1] G.J. Heynderickx, G.F. Froment, A Pyrolysis Furnace with Reactor Tubes of EllipticalCross Section., Ind. Eng. Chem. Res., 35, 2183 (1996).

[2] P.M. Plehiers, G.C. Reyniers, G.F. Froment, Simulation of the Run Length of anEthane Cracking Furnace, Ind. Eng. Chem. Res., 29, 636 (1990).

[3] G.C. Reyniers, G.F. Froment, F.-D. Kopinke, G. Zimmerman, Coke Formation in theThermal Cracking of Hydrocarbons, Modeling of Coke Formation in Naphtha Cracking,Ind. Eng. Chem. Res., 33, 2584 (1994).

[4] P. Willems, G.F. Froment, Kinetic Modeling of the Thermal Cracking of Hydrocar-bons, Part 1: Calculation of the Frequency Factors. Ind. Eng. Chem. Res., 27, 1959 (1988a).

[5] P. Willems, G.F. Froment, Kinetic Modeling of the Thermal Cracking of Hydrocar-bons, Part 2: Calculation of Activation Energy. Ind. Eng. Chem. Res., 27, 1966 (1988b).

[6] H.C. Hottel, A.F. Sarofim, Radiative Heat Transfer, McGraw-Hill, New York (1967).

[7] M.V.R. Rao, P.M. Plehiers, G.F. Froment, Simulation of the Run Length of an EthaneCracking Furnace, Ind. Eng. Chem. Sci., 43, 1223 (1988).

[8] P.M. Plehiers, G.F. Froment, Firebox Simulation of Olefin Units, Chem. Eng. Commun.,80, 81 (1989).

[9] K.M. Sundaram, J.V. Albano, Design Pyrolysis Heaters using CFD Models, Hydrocar-bon Process., 7, 79 (1997).

[10] J.J. De Saegher, T. Detemmerman, G.F. Froment, Three-Dimensional Simulation ofHigh-Severity Internally Finned Cracking Coils for Olefins Production, Rev. Inst. Fr. Pet.,51, 245 (1996).


[11] T. Detemmerman, G.F. Froment, Three-Dimensional Coupled Simulation of Fur-naces and Reactor Tubes for the Thermal Cracking of Hydrocarbons, Rev. Inst. Fr. Pet.,53, 181 (1998).

[12] A.G. De Marco, F.C. Lockwood, A New Flux Model for the Calculation of Radiationin Furnaces, Riv. Combust., 29, 184 (1975).

[13] C.K. Westbrook, F.L. Dryer, Simplified Reaction Mechanisms for the Oxidation ofHydrocarbon Fuel in Flames, Combust. Sci. Technol., 27, 31 (1981).

[14] D.B. Spalding, Mixing and Chemical Reaction in Steady Confined TurbulentFlames, Imperial College of Science and Technology, London, p. 649 (1972).

[15] E. Dick, A Flux-Difference Splitting Method for Steady Euler Equations, J. Comp.Phys., 76, 1 (1988).

[16] E. Dick, Multigrid Formulation of Polynomial Flux-Difference Splitting for SteadyEuler Equations, J. Comp. Phys., 91, 1 (1990).

[17] E. Dick, J. Steelant, Coupled Solution of the Steady Compressible Navier-StokesEquations and the k-e Turbulence Equations with a Multigrid Method, App. Num. Math.,23, 49–61 (1997).

[18] T. Detemmerman, Driedimensionale Simulatie van Stroming, Warmteoverdracht en Re-actie in Ovens en Reactoren voor de Thermische Kraking van Koolwaterstoffen, Ph.D. Thesis,Ghent University (1997).

[19] G.J. Heynderickx, A.J.M. Oprins, G.B. Marin, E. Dick, Three-Dimensional Flow Pat-terns in Cracking Furnaces with Long-Flame Burners, AIChE Journal, 47, 2, 388–400(2001).

[20] A.J.M. Oprins, G.J. Heynderickx, G.B. Marin, Three-Dimensional Asymmetric Flowand Temperature Fields in Cracking Furnaces, Ind. Eng. Chem. Res., 40, 5087–5094 (2001).

Chapter 5

Applying the Dynamic Monte Carlo Approach for Modeling Complex

Polymerization Reaction Systems in Isothermal and Non-Isothermal

Environments

T.A.M. Verbrugge1 C.F. Jaap den Doelder1

5.1 Introduction

A general approach for modeling complex polymerization reaction systems at a me-soscale level by means of an off-space dynamic Monte Carlo (DMC) algorithm is pre-sented. The different components associated with such a method are (1) the physi-cal model, which represents the actual relation between the reactants, (2) the (master)equations describing the evolution of the chemical system, and (3) the derived MC al-gorithm to solve the latter. Typical input elements are the initial concentrations of thereactive monomers and their related properties, the involved parameterized reactionmechanisms and the time-temperature profile the system is submitted to.

The presented method differs from other more traditional simulation methods by itsability to generate connectivity information for the different reactants. In particular forpolymerization systems this leads to the advantage of monitoring the molecular weightdistribution (MWD) and the formation of circuits or cycles resulting from intramolecu-lar reactions, by keeping track of the bonds being formed or broken. A combination ofthese assets makes this method very suitable for modeling step growth reactions, likein polyurethane and epoxy polymerization systems.

5.2 Physical Model

Establishing connectivity information is done by means of a graph theoretical approach:full topological information supplied by keeping track of the way the initial monomerend groups are connected to each other, i.e., end group connectivity within the distinctpolymers. The generated information is then extracted by analyzing the connectivity ofthe complete system population after a predefined time step or conversion level.

1Dow Benelux NV, P.O. Box 48, 4530 AA Terneuzen, The Netherlands.

24

T.A.M. Verbrugge and C.F.J. den Doelder 25

5.3 Chemical Reaction Model

The core of this DMC method is formed by a master equation [3,5]. This equation de-scribes how the system under interest evolves over time. Applying a MC method tosolve this equation implies that the master equation is not solved as a differential equa-tion, but is rather investigated by generating an ensemble of realizations of the stochas-tic process and evaluating the quantities of interest as ensemble averages. In the lit-erature, three distinct methods for representing stochastic reaction equations derivedfrom the master equation are found. The first method expresses the reaction rates as afunction of the number of end and/or mid groups [1], the second method calculates thereaction rates by means of the number of monomers [6], while the last method makesuse of the number of monomers and formed polymers [7,8]. The latter method is re-ferred to as the Smoluchowski coagulation equation, because it represents the polymersas clusters merging together or separating during reaction [2]. The most appropriatemethod to use depends on the problem to solve.

5.4 Monte Carlo Algorithm

As has been shown by Gillespie [3] for reactions in homogeneous systems, and is alsoknown for master equations in general, it is possible to take one step to each of thesubsequent moments where the system changes, if the transition probabilities are timeindependent. Later, both Gillespie [3,4] and Jansen [5] showed that this is also possiblewith time dependent transition probabilities.

So, given that the system is in a certain configuration at a certain time, then essen-tially what is needed in order to move the system forward in time are the answers totwo questions: (1) when will the next reaction occur, and (2) what kind of reaction willit be? By the very nature of chemical reactions, the answer to these two questions couldbe described as a stochastic process. Two methods can be distinguished. The VariableStep Size Method considers the reactions to be subdivided among different predefinedreaction types. A time step is then calculated, and the type of reaction to occur is deter-mined. In the First Reaction Method a list of all possible reactions is computed for eachconfiguration that occurs, and for each reaction a time of occurrence is generated. Thelist of all reactions is ordered according to time of occurrence, and the configuration ischanged corresponding to the first reaction in the list. This leads to a new configurationand a new time, and then the whole procedure has to be repeated.

5.5 Application To A Thermoplastic Polyurethane System

A thermoplastic polyurethane (TPU) system is now considered. During an industrialpultrusion process, the solid TPU pellets are melted, impregnated with fibers, andcooled down again. Depolymerization reactions of urethane bonds and side reactions


-OH- + -NCO- ↔ -Urethane--NCO- + -NCO- → -NCN- + CO2

-NCO- + H2O → -NH2- + CO2

-Urethane- → -NH2- + CH2=CH- + CO2

-NCO- + -NH2- → -Urea-

Table 5.1: TPU reaction equations.

Figure 5.1: Mw evolution for different time-temperature profiles

occur mainly at the elevated temperatures applied in the extrusion part. After im-pregnation, temperatures rapidly drop down by exposing the composite to succeed-ing cooling units, favoring polymerization again. Corresponding temperature profilesare presented in Fig. 5.1. To investigate the influence of the extruder residence timeon the final molecular weight build-up, the assumed reaction mechanism as indicatedin Table 5.1 was implemented for the above-developed method. Low order moments(Mn,Mw) were calculated from the distribution curves. From Fig. 5.1, it can be seenthat the residence time will affect the final molecular weight. A shorter residence timefavors a higher molecular weight at the end. This is due to the higher index ratio([-NCO-]/[-OH-]) associated with the shorter residence time. The index ratio is on itsturn determined by the occurrence of side reactions. Snapshots of the weight distribu-tions of the formed polymers at time 25s and 85s are presented in Fig. 5.2.

5.6 Conclusions

Combining the chemical reaction model and the MC algorithm of the DMC methodwith a graph theoretical connectivity approach allows us to create an advanced tool


Figure 5.2: Evolution of weight distribution at time 25s and 85s.

to simulate polymerization/depolymerization reactions. Besides the ability to simu-late the kinetics involved in uni-, bi- and termolecular step polymerization reactions ofpolyfunctional species with possible zero and first shell substitution effects in a time-temperature varying environment, the DMC presented here also offers the possibilityof simultaneously obtaining structural information related to the modeled polymers.The method was illustrated by means of the pultrusion process for TPU.

5.7 Bibliography

[1] C. Dubois, A. Aıt-Kadi and P.A. Tanguy, Chemorheology of polyurethane systems aspredicted from Monte Carlo Simulations of their evolutive molecular weight distribu-tion, Journal of Rheology 42, No. 3, 1998, p. 435–452.

[2] K. Dusek and J. Somvarsky, Chemical clusters in polymer networks, Faraday Discuss.101, 1995, p. 147–158.

[3] D.T. Gillespie, A general method for numerically simulating the stochastic time evo-lution of coupled chemical reactions, Journal of Computational Physics 22, 1976, p. 403–434.

[4] D.T. Gillespie, Monte Carlo simulation of random walks with residence time depen-dent transition probability rates, Journal of Computational Physics 28, 1978, p. 395–407.

[5] A.P.J. Jansen, Monte Carlo Simulations of Chemical Reactions on a Surface with time-dependent reaction-rate constants, Comput. Phys. Comm. 86, 1995, p. 1.

[6] S.E. Rankin, L.J. Kasehagen, A.V. McCormick and C.W. Macosko, Dynamic MonteCarlo Simulation of Gelation with Extensive Cyclization, Macromolecules 33, No. 20,2000, p. 7639–7648.


[7] J. Somvarsky and K. Dusek, Kinetic Monte-Carlo simulation of network formation I.Simulation method, Polymer Bulletin 33, 1994, p. 369–376.

[8] J. Somvarsky and K. Dusek, Kinetic Monte-Carlo simulation of network formationII. Effect of system size, Polymer Bulletin 33, 1994, p. 377–384.

Chapter 6

Chaotic Bursting in an Externally Excited Industrial Reactor

David West1 Guido Smits1 Mark Kotanchek1 Kip Mercure1

6.1 Introduction

High temperature vapor-phase chlorination reactions, such as the reaction of chlorineand methyl chloride, are of considerable industrial importance. Such reactions proceedby a multi-step, free-radical, chain reaction, and are strongly exothermic. Coupling be-tween the fluid flow and the chemical kinetics gives rise to complex dynamic behavior.In this paper we examine the steady state and dynamic behavior of a full scale chlo-rination reactor. The steady-state bifurcation behavior can be explained by a simplemodel which predicts ignition, extinction, and self-excited oscillations near the extinc-tion point. The presence of noise causes chaotic oscillations far from the extinction point.As the noise level is increased beyond some threshold there is a bifurcation giving riseto large amplitude, almost random, bursts. The bursts have a characteristic waveform,and power-law frequency and size distributions. While there is no apparent attract-ing set in the bursting dynamics itself, the frequency and time averaged dynamics hasa low dimensional chaotic attractor. Moreover, short term prediction of the averageddynamics is possible using a low dimensional model derived from experimental timeseries.

6.2 Results and Discussion

Fig. 6.1 shows a schematic diagram of a typical chlorination reactor. Vaporized reactantsare fed into a large open tubular reactor through a small diameter feed pipe at high ve-locity (> 20m/s). The sudden expansion causes flow recirculation in the front part of thereactor. Mixing between the hot products and colder feed stabilizes the reaction withina small zone near the inlet, similar to a flame front. To first approximation, the steady-state and dynamic behavior of a jet-stirred chlorination reactor can be explained byregarding the self-stirred section (see Fig. 6.1) as completely mixed and modeling it as a

1Corporate R&D, The Dow Chemical Company 2301 N. Brazosport Blvd., Freeport, Texas, 77541-3257.

29

D. West, G. Smits, M. Kotanchek and K. Mercure 30

CSTR (West et al., 1999). The reactor typically operates in an autothermally ignited statefar from the extinction point, so the ignited state is usually a stable focus or node. Asfeed flow rate is increased the dynamic response to perturbations becomes increasinglyoscillatory and eventually self-sustained oscillations appear just before the extinctionpoint. These qualitative changes in dynamics are generic features of autothermal CSTRsas described in the classic paper by Uppal et al. (1972). A qualitatively similar transitionfrom stability to extinction is observed in jet stirred chlorination reactors. The appear-ance of sustained oscillations can be used as an early warning sign of incipient reactionextinction, so knowledge of the reactor dynamics is of great practical importance. Reac-tor pressure is easier to measure than other state variables of the system, so we use it todetect the oscillating state. Fig. 6.2 shows the emergence of oscillations prior to extinc-tion in an industrial reactor. The presence of noise or coherent perturbations causes theemergence of chaotic oscillations even relatively far from the extinction point. When thesize of the perturbation exceeds a critical threshold, chaotic bursting emerges, resultingin large amplitude, almost random, pressure oscillations. Fig. 6.3 shows a typical timeseries and a magnified view of one such burst. There is no apparent low dimensionalattractor for the bursting dynamics. However, analysis of the time and frequency aver-ages of the bursting dynamics indicate the averaged dynamics has a low dimensionalchaotic attractor. Fig. 6.4 shows chaotic attractors reconstructed from two different re-actors: one having excitation below threshold (left) and the other with above thresholdexcitation (right). It appears the underlying dynamics are the same for the two systems.In the bursting regime, there is a power law relationship between the size and frequencyof bursts. The burst size distribution gives a straight line on a log-log plot, with negativeslope. There is a continuum of bursts with no preferred amplitude. Such distributionsare reminiscent of noise generated by natural processes in which there are persistentinteractions. In contrast, white noise (generated from an entirely random, uncorrelated,process) has zero slope in such a log-log plot. Processes that generate power-law dis-tributions are believed to arise from excitation of a self-organized system near a criticalpoint. We construct a simple empirical model of the averaged bursting dynamics thatis capable of short term prediction; see Fig. 6.5. The model consists of three ordinarydifferential equations and was obtained by singular value decomposition of time andfrequency averaged experimental time series.

6.3 Conclusions

We examine the chaotic bursting dynamics of an externally excited industrial reactor.Although the bursting dynamics itself appears to be high dimensional with featuressimilar to 1/f noise, the time and frequency averaged dynamics forms a low dimen-sional chaotic attractor. We interpret the bursting dynamics as arising from the excita-tion of a buoyant, turbulent, reaction front leading to almost random, reaction extinctionevents, of various sizes up to global extinction.


Figure 6.1: Schematic diagram of a chlorination reactor.

6.4 Bibliography

[1] Uppal, A.; Ray, W.H.; Poore, A.B. (1974), On the dynamic behavior of continuousstirred tank reactors, Chemical Engineering Science 29, p. 967–985.

[2] West, D.H.; Hebert, L A.; Pividal, K. A. (1999), On-line detection of extinction andinstability in chloromethane chlorination, presented at the National AIChE meeting,Dallas TX, Session No. T2005, Paper No. 46f.


Figure 6.2: Pressure oscillations leading up to extinction.

Figure 6.3: One hour of data showing a series of chaotic bursts (left) and zoom in on onesuch event (right).


Figure 6.4: Experimental phase portraits from two different reactors; left, reactor with-out external excitation; right, reactor with external excitation.

Figure 6.5: Comparison of empirical model (points) with a small section of an experi-mental time series (lines).

Chapter 7

Numerical solutions in moving bed separation simulations

D. Pavone1 S. Louret1

7.1 Abstract

The major part of industrial separations (80%) are distillation processes. However, thereare some cases where distillation is either technically not feasible or uneconomical ascompared to crystallisation membranes or adsorption. Simulated moving beds (SMB)is a counter-current adsorption based process. It is used to achieve separation wheneverdistillation is uneconomical.

SMB processes use competitive adsorption. When the feed (A+B) is injected, thedesired product (A) is adsorbed by the solid phase, undesired product (B) is carriedaway by the liquid phase and withdrawn (raffinate). As the surrounding concentrationin desorbent (C) increases, the desired product (A) is desorbed from the solid phase,replaced by desorbent molecules, and withdrawn (extract). As it is technically difficultto move solids, counter current is simulated by shifting injections and withdrawals.

Because SMB is commonly used to produce fluids at very high purity (99.8%), SMBprocesses require a fine tuning. In addition, surrounding equipment (valves, distribu-tor plates, pump, . . . ) have significant influence on purity and so need to be properlymodelled. There are at least five main operating parameters interacting with each other.Hence the difficulty to find an optimal tuning for a given production rate at a givenpurity.

In a modern approach, a mathematical and a numerical model is developed alongwith the process R&D. The model is facing different purposes from the most obvious,finding the best tuning, to more technical ones such as model based advanced controlor to anticipate possible problems.

Nowadays, our simulator is widely used on industrial units, especially to help thefinal tuning. But, before getting this tool, we faced different numerical difficulties. Thispaper presents the IFP Eluxyl process and its mathematical modelling in a first part.Then, in the second part, the paper describes the way we solved the numerical difficul-ties encountered. Numerical difficulties are threefold :

1Institut Francais du Petrole, Process Engineering Department, BP 3, F-69390 Vernaison, France.

34

D. Pavone and S. Louret 35

Figure 7.1: Separation using a theoretical True Moving Bed

1. Computation time.

2. Numerical instabilities.

3. Accurate numerical resolution.

1. Computation time was our first concern. At first, simulation time was drasticallyreduced using a double upwind differencing method and a fixed step time solver.Secondly, our model has 88 injection/withdrawal distributors, each of which sig-nificantly impacts on product purity and unit performances. Therefore, we usedmodel reduction techniques to simulate these distributors accurately enough.

2. Our model considers three distinct phases located in different porosities:

• one macro free phase, which is made of the liquid circulating around themolecular sieve

• one micro free phase, made of the liquid inside the molecular sieve, not ad-sorbed as well but with a different composition than the previous one,

• one adsorbed phase in the molecular cages.

Transfer between the macro and the micro free phases is governed by Fick’s law,while the micro free phase and the adsorbed phase are at equilibrium. When closeto equilibrium between the macro and the micro free phase, numerical instabilitiesappear due to fast mass transfers. We suppressed these instabilities using two


different phase transfer models: Fick law or equilibrium. As a result we can covera wide range of transfer rates with acceptable time steps.

3. Accurate numerical resolution of the mathematical model is facing two problems.The first one relates to the accuracy of numerical resolution around the injec-tion point while the second one deals with volumetric balance in the microscopicphases.

For the first item, accurate numerical resolution of the mathematical model aroundinjection/withdrawal points was obtained ‘breaking’ the double-upwind differ-encing scheme by a simple centred scheme. In addition, we precisely located theinjection points dividing the injected flow in two flows injected in two adjacentcells.

For the second item, adsorption kinetics is calculated by a phenomenological lawthat do not guaranty the volumetric balance in the micro free phase. Volumet-ric balance was achieved using an additional flux which, although fairly small,ensures that volumetric fractions sum to 1.

7.2 Eluxyl process presentation


Figure 7.2: Example of Eluxyl SMB simulation


Figure 7.3: Example of Eluxyl SMB simulation presented as for a TMB process

Chapter 8

Contaminant transport with adsorption in unsaturated-saturated

porous media

J. Kacur1

8.1 Abstract

Some new ideas and results in numerical modelling of contaminant transport in un-saturated-saturated porous media will be presented. These results were developed inrecent papers with R. Van Keer and D. Constales (see [1-3, 6-9]).

These contributions include the numerical approximation and convergence analysisof:

• unsaturated-saturated flow in porous media governed by Richard’s equation (see[5, 8-10]);

• transport of contaminant (see [7, 2, 3]);

• adsorption of contaminant by porous media (see [7, 2, 3]);

• calibration of the mathematical models including the determination of geohydro-logical parameters and sorption isotherms (see [1-3]).

The complex model leads to strongly nonlinear and degenerate system of convec-tion-diffusion and adsorption. In the last years a very dynamical development in so-lution of these problems is realized. There are still some open problems in analysisand numerical realization of this system (see [11, 12]). The flow model is governed byRichard’s equation

∂tθ = div(k(h)A(x)∇(h + z)) (8.1)

1Faculty of Mathematics and Physics, Comenius University Bratislava, Mly’nska dolina, 84215Bratislava, Slovakia.

39

J. Kacur 40

where

θ = θ(h) = θr +θs − θr

(1 + (αh)n)m, k(S) = S1/2(1− (1− S1/m)m)2, S =

θ − θr

θ − θr

and k(h) = k(θ(h)) for h < 0 and k(h) = 1 for h ≥ 0. Here θ(h), k(h) are retention andhydraulic permeability curves in the Van Genuchten Ansatz andA(x) = ks = ks(x). Thedata θs, θr, α, m, n, and ks are soil parameters.

Using Kirchhoff’s transformation u := β(h) =∫ h0 k(z)dz, b(u) = θ(β−1(u)), equation

(8.1) is transformed to

∂tb(u) = div(A(x)∇u)− ∂zK(x, b(u)) = 0 (8.2)

where K(x, b(u)) = A(x)k(b(u))e, e being the unit vector in the z-direction.This flow model is coupled with the contaminant transport equation (see [11])

b(u)∂tw + ρ1∂tΨ(w) + ρ2

∫

Λ∂tvdλ+ div(q(u,∇u)w−D(u,∇u)∇w) = G (8.3)

where w is the contaminant concentration and v = v(x, t, λ represents the adsorbedcontaminant per unit volume of porous media (at the point x) and λ ∈ Λ characterizeschemically qualitatively different adsorption sites at the skeleton surface correspondingto this volume.

The kinetics of nonequilibrium adsorption is governed by

∂tv = f(λ, w, v) (8.4)

where f is nonincreasing in v (for fixed λ, u). The Darcy velocity q equals q = ∇u −K(x, b(u)) and the dispersivity matrix D equals

Dij = D0 + αT |q|δij + (αL − αT )qiqj|q| , i, j = 1, 2, 3.

In the numerical realization we consider the special form of (8.3) where

f(λ, w, v)) := d(λ, x)(ϕ(λ, w)− v)

with a rate parameter d and adsorption isotherm ϕ(λ, w). The function Ψ has similarproperties as ϕ and the term ∂tΨ(w) corresponds to the sorption in equilibrium mode,when d→∞.

The mathematical models (8.2)–(8.4) are completed with the corresponding bound-ary and initial conditions.

The numerical approximation is based on a special relaxation method (see [4, 7]) bymeans of which we control the nonlinearity and degeneracy of the parabolic term (b(u),resp. b(u)w + Ψ(w)) and the regularized method of characteristics, by means of whichwe control the convective terms ∂uK(x, u), q(u,∇u) (see [5, 7, 9, 10]).

J. Kacur 41

The solution of inverse problems, when calibrating the mathematical models in(8.2)–(8.4), are based on additional measurements in single injection-extraction wells(see [2]) and dual-well tests in steady state flow regime (see [3]). The soil parameters forvery impermeable materials are determined using the Richard’s flow model in terms ofsaturation and centrifugation, to increase the sensitivity of soil parameters on the wet-ness front evolution (see [1]). Some numerical experiments supporting the theoreticalresults will be presented.

8.2 Bibliography

[1] D. Constales, J. Kacur: Determination of soil parameters via the solution of inverseproblems in infiltration. Computational Geosciences 5 (2001), 25–46.

[2] D. Constales, J. Kacur, R. Van Keer: Parameter identification by a single injection-extraction well. Submitted to Advances in Water Resources Research.

[3] D. Constales, J. Kacur: Parameter identification by means of dual-well tests. Submit-ted to Water Resources Research.

[4] W. Jager, J. Kacur, : Solution of doubly nonlinear and degenerate parabolic prob-lems by relaxation schemes. M2AN Mathematical modelling and numerical analysis 29 n. 5(1995), 605–627.

[5] J. Kacur: Solution of convection-diffusion problems by the method of characteristics.SIAM J. Mumer. Anal. 39 n. 4 (2001), 858–879.

[6] J. Kacur: Solution to strongly nonlinear parabolic problems by a linear approxima-tion scheme. IMA Journal of Numerical Analysis 19 (1999), 119–145.

[7] J. Kacur, R. Van Keer: Solution of contaminant transport with adsorption in porousmedia by the method of characteristics. M2AN Mathematical Modelling and Numericalanalysis.

[8] J. Kacur, R. Van Keer: Solution of degenerate parabolic variational inequalities withconvection. Applied Numerical Analysis.

[9] J. Kacur, R. Van Keer: Numerical approximation of flow and transport system inunsaturated-saturated porous media, SIAM J. Num. Anal.

[10] R. Van Keer, P. Frolkovic, J. Kacur: A numerical method for nonlinear conve ction-diffusion problems in porous media. Numer. Math.

[11] P. Knabner: Finite-Element-Approximation of Solute Transport in Porous Mediawith General Adsorption Processes. In: Flow and Transport in Porous Media, Ed. XiaoShutie, Beijing, World Scientific (1992), 223–292.

[12] P. Knabner, F. Otto: Solute transport in porous media with equilibrium and non-equilibrium multiple-site desorption: Uniqueness of the solution. To appear.

Chapter 9

On a numerical relaxation method for a reaction-diffusion problem

with an instantaneous and irreversible reaction

N. Batens1 R. Van Keer1

9.1 Abstract

In this paper we deal with a numerical method for a free boundary value problem (FBP)in 1D, describing a chemical diffusion problem accompanied by an irreversible and in-stantaneous reaction which gives rise to a moving internal boundary.

Consider a beam shaped volume element in a chemical reactor. The x-axis is takenalong the length axis of the beam. The reactor contains two chemical components A andB that react with each other giving rise to a third chemical component C:

NAA +NBB → NCC, (9.1)

where NA, NB and NC are reaction coefficients.The components enter or leave the volume element through the opposite bound-

aries x = 0 and x = L, the flux being orthogonal to these boundaries and being uni-formly distributed over them. The speed of the reaction is assumed to be so high thatit can be considered to be instantaneous. Further, we also suppose that the reaction isirreversible. This idealisation implies that the reaction takes place at a reaction planethat moves through the liquid, and moreover that reacting components do not coex-ist. Consequently, neither reaction speeds nor reaction terms will enter the governing(simplified) diffusion equations, see also [1,2].

The diffusion process in each of the zones may be considered to take place inone direction only, viz along the length axis of the reactor with constant diffusioncoefficients. The time-varying position of the reaction plane is denoted as x = s(t),which, evidently, is an unknown function of time. The unknown concentration profilesof the chemical components at place x and time t are denoted as CI

A(x, t), CIC(x, t),

CIIC (x, t), CII

B (x, t), the superscript referring to the respective (time-varying) zone, seeFig. 9.1.

1Ghent University, Department of Mathematical Analysis, Galglaan 2, B-9000 Ghent, Belgium.

42

N. Batens and R. Van Keer 43

x

6concentration

AC C B

0NAA+NBB → NCC L

-------

-----

Zone 1 Zone 2

Figure 9.1: Cross-section of the beam shaped volume element, concentration profiles

The governing diffusion equations of this FBP read:

∂tCIA = DA∂xxC

IA, ∂tC

IC = DC∂xxC

IC , 0 < x < s(t), t > 0, (9.2)

∂tCIIC = DC∂xxC

IIC , ∂tC

IIB = DB∂xxC

IIB , s(t) < x < L, t > 0, (9.3)

where the diffusion coefficients, DA,DB andDC are given constants. The accompanyingboundary condition for CI

A is taken to be:

a1DA∂xCIA + a2C

IA = φA(t), x = 0, t > 0. (9.4)

Here φA(t) is a given function of time and a1, a2 are known constants with (a1, a2) 6=(0, 0). Similar BCs are imposed on CI

C at x = 0, and on CIIC and CII

B at x = L.At the moving reaction point, x = s(t), five transmission conditions (TCs) are im-

posed. The first two TCs express the non coexistence of the reacting components at thereaction plane. The third TC guarantees the continuity of the concentration profile of thematerial produced. The last two TCs arise from a mass-balance argument, stating thatthe incoming fluxes of the reacting components are equal and are also equal to the sumof the outgoing left and right flux of the produced component. Explicitly: at x = s(t),for t > 0:

CIA = 0, CII

B = 0, CIC = CII

C , (9.5)

−DA∂xCIA

NA

=DB∂xC

IIB

NB

(9.6)

=DC∂xC

IC −DC∂xC

IIC

NC. (9.7)

The system is completed with initial data, viz the known initial position of the reactionplane at t = 0 and given initial concentration profiles in the respective zones. Basically,the proposed numerical method consists of four steps:

1. a Landau transformation mapping each of the two time-varying intervals on afixed domain, which results in a strongly nonlinear, nonlocal boundary valueproblem (BVP);

2. a central difference method with respect to the space variable, that takes properlyinto account the various transition conditions;

N. Batens and R. Van Keer 44

3. construction from (9.3) of an ODE containing a relaxation parameter ε to representthe movement of the internal boundary, similarly to the TC in a classical Stefanproblem;

4. a time integration of the resulting stiff system of ODEs by suitable computer pack-ages.

The numerical method is evaluated by comparison with an analytical solution for aspecial but non trivial case, and by a mass-balance argument. The presented methodcan be extended to the case of several irreversible and instantaneous reactions.

9.2 Bibliography

[1] G. Froment, K. Bischoff, Chemical reactor analysis and design, Wiley, New York,1990.

[2] R. Garg, S. Nair, A.N. Bhaskarwar, Mass-transfer with instantaneous chemical reac-tion in finite gas-liquid systems, Chemical Engineering Journal 4.2, (2000) 89–98.

Chapter 10

A High-Speed Method for Obtaining Kinetic Data for Exothermic or

Endothermic Catalytic Reactors under non-isothermal Conditions

illustrated for the Ammonia Synthesis

M. Kolkowski1 F. Keil1 C. Liebner2 D. Wolf2 M. Baerns2

10.1 Introduction

The Polythermal Ramping Reactor (PTR), based on the idea of Wojciechowski [1], isa tool to acquire experimental data to determine kinetic parameters of heterogeneouscatalysed reactions. The PTR-method enables the investigation of a heterogeneous catal-ysed reaction in a shorter time than established methods. The investigation can be doneefficiently for a wide range of temperatures, initial reactant concentrations and systempressures. An efficient approach for data evaluation was developed based on experi-mental PTR data.

The method is illustrated for the ammonia synthesis over a commercial iron catalyst(BASF).

10.2 Experimental setup

The experimental setup can be described in the following way: A tube filled with cata-lyst pellets is placed in a closed oven. Both the reactants at the inlet of the tube and thesurrounding of the tube are heated up in the same way. At the outlet of the tube reactorcomposition and temperature are measured.

10.3 Experimental procedure

For a fixed inlet composition a set of, e.g., five different inlet gas streams equivalent tofour different residence times tj = mCat/Vj are chosen. After adjusting the first flow

1Technical University of Hamburg-Harburg, Chemical Reaction Engineering, Eissendorfer Str. 38, D-21071 Hamburg.

2Institute of Applied Chemistry Berlin-Adlershof, Richard-WillstŁtter-Str. 12, D-12489 Berlin.

45

M. Kolkowski, F. Keil, C. Liebner, D. Wolf and M. Baerns 46

Figure 10.1:

rate the temperature of the gas at the inlet of the tube reactor is increased with a certaintemperature program. The resulting composition at the outlet of the tube reactor andits temperature are recorded. The same is done for the other gas streams. It should bestressed that every experiment has to be done with the same temperature program. Theresulting concentration and temperature curves at the outlet of the reactor are analysedwith respect to inlet temperature in two steps to get the reaction rates.

10.4 Obtaining reaction rates

The first step is smoothing the slightly scattering experimental data by an approxima-tion with normalized bicubic basis splines. Outlet concentration and temperature arenow known as a function of the inlet temperature Tinlet and residence time tj . In thesecond step the reaction rate is determined by computing the first derivative of thecomponent balance with respect to the residence time tj.

∑

j

νjirj,Cat =xinlet,inertninlet

Vref

∂

∂τCat,ref

(

xi(Tinlet, τj)

xoutlet,inert(Tinlet, τj)

)∣

∣

∣

∣

∣

Tinlet=const.

.

In terms of b-splines it can be written:

∂

∂τCat,ref

(

xi(Tinlet, τj)

xoutlet,inert(Tinlet, τj)

)

=p∑

h

q∑

j

dhj,iMh,i(Tinlet)∂Nj,i(τCat)

∂τCat

so that∑

j

νjirj,Cat =xinlet,inertninlet

Vref

p∑

h

q∑

j

dhj,iMh,i(Tinlet)∂Nj,i(τCat)

∂τCat.

10.5 Bibliography

[1] Rice, Wojciechowski, Catal. Today, 36, 191–207, 1997


Figure 10.2: Product of smoothed experimental data


Figure 10.3: Derivation of data with respect to residence time


Figure 10.4: Resultant reaction rates

Chapter 11

Mathematical modelling of the unsteady-state oxidation of nickel

gauze catalysts

B. Monnerat1 L. Kiwi-Minsker1 A. Renken1

11.1 Introduction

Nickel catalysts are widely used in many important industrial processes like hydro-genation of oil, hydrogen production through partial oxidation, steam reforming or au-tothermal reforming of hydrocarbons and methanation of syngas. In a previous work[1] we studied an alternative method for hydrogen production by catalytic decomposi-tion of CH4 into hydrogen and carbon over structured nickel catalysts. This process hasbeen performed in periodic mode in which the cycle of CH4 decomposition followed bya catalyst regeneration period under oxidative atmosphere (air). The integral selectivityfor hydrogen was found to depend strongly on the oxidation state of the catalyst.

Over a totally reduced Ni catalyst, only hydrogen and unconverted methane wasobserved, whereas after oxidative regeneration period, CO, CO2 and H2O were formedover the partially oxidised catalyst. The formation of these products can be explainedby the redox reactions of nickel oxide with methane and the decomposition products.

The aim of this work is to obtain more detailed information on the mechanism of Nioxidation reaction by applying the transient response methods with oxygen. Based ontransient experiments, a mathematical model was developed describing the oxidationprocess of the catalyst.

11.2 Experimental

The reactions were carried out in a tubular reactor (ID = 9mm and L = 230mm). Metal-lic nickel in the form of wire gauze was used as a catalyst. The main advantages ofthis kind of catalyst are the regular open structure and the flexibility that enables touse them in structured catalytic beds [1]. In order to increase the specific surface area

1Laboratory of Chemical Reaction Engineering, Swiss Federal Institute of Technology, 1015 Lausanne,Switzerland.

50

B. Monnerat, L. Kiwi-Minsker and A. Renken 51

of the bulk nickel metal, a Raney-type layer was formed on the outer surface of thenickel gauze. The method of preparation was presented elsewhere [1,2]. This catalystcombines the characteristics of metal-gauze structures, the high thermal conductivityof catalytic beds and the high surface area of Raney-metals. The reactor effluents weremonitored continuously by a quadrupole mass spectrometer. Experiments were per-formed with the modified nickel gauze (≈ 210mg), introduced in the middle part of thereactor in a rolled form (length 20 − 30mm) between two inert packings consisting ofquartz beads. Residence time distribution experiments have shown that the behaviourcorresponds to a plug flow reactor. Details of the experimental set-up are described in[1]. Prior to each step of oxygen, the catalyst was reduced during 4 hours in a flow con-taining a molar fraction of hydrogen of 10% at 773K (Q = 75ml(STP)/min, ptot = 1.5bar)in order to remove traces of oxygen. Afterwards the reactor was purged with argon andthe reaction conditions were adjusted.

11.3 Experimental results and mathematical modelling

A typical example of the transient responses of oxygen to a step change from argon tooxygen over initially reduced nickel catalyst is presented in Fig. 11.1 for three differ-ent temperatures. The oxidation reaction proceeds very fast leading to an importanttemperature increase of the catalyst up to 50K as shown in Fig. 11.2. Furthermore, theoxygen response is characterised by an important tailing due to the diffusion of oxygenatoms from the surface into the bulk of the metal.

On the basis of the experimental results obtained at various temperatures, spacetimes and concentrations of oxygen, a mathematical model has been developed. Themass balance for the oxygen in the gas phase in a heterogeneous catalytic reactor withplug flow conditions is given by the equation (11.1):

∂cO2

∂t= −ugg

∂cO2

∂x+ ρcat

(1− ε)

εRO2

, (11.1)

where ug is the gas velocity, ρcat the catalyst bulk density, ε the bed porosity and RO2

the rate of oxygen transformation which is related to the reaction mechanism. The fol-lowing kinetic model is proposed: The first step (11.2) corresponds in a fast dissociativechemisorption of oxygen on the catalytic surface. The second step corresponds in a sub-surface diffusion of oxygen atoms from the surface into the bulk (11.3) that proceedsslower.

O2 + 2Ni → 2O-(Ni) (11.2)

O-(Ni) → (NiO) (11.3)

For step (11.2) a first order for oxygen and a second order towards reduced sites (θv,s) issupposed [3,4,5]:

rox = koxcO2θ2

v,s. (11.4)


Figure 11.1: Comparison between experimental transient responses of oxygen duringthe oxidation of nickel at three different temperatures and the non-isothermal model.Conditions: yO2,0 = 8%, Q = 75ml(TPN)/min and ptot = 1.5bar.


Figure 11.2: Comparison between experimental transient responses of the tempera-ture during the oxidation of nickel at three different initial temperatures of the reac-tor and the prediction of the non-isothermal model. Conditions: yO2,0 = 8%, Q =75ml(TPN)/min and ptot = 1.5bar.


Subsurface oxygen diffusion was suggested in this work to explain the long tailing ofthe oxygen responses observed during the experimental transients. Thus, an additionaldimension has to be introduced: the distance into the nickel layer (z), in which diffusiontakes place. The oxidation of nickel at the surface (z = 1) produces surface oxygenspecies, which diffuses into the subsurface oxygen vacancies, giving rise to a radialprofile of degree of oxidation (qO). Therefore, the degree of oxidation of nickel is afunction of x, z and t [6].

As shown in Fig. 11.2, the nickel oxidation was accompanied by an initial rise intemperature. In order to check whether these temporal temperature variations couldsignificantly affect the transient responses – and thereby the estimated parameter values– the transient responses of oxygen under non-isothermal conditions were simulated byincluding the energy balance for the gas phase.

The system of partial differential equations was solved using the finite difference ap-proximation method, using 10 nodes in the x direction and 40 nodes in the z direction.Numerical integration was carried out using a variable step algorithm (Gear) [7]. Opti-misation of parameters was performed by fitting the calculated molar fraction of O2 andthe reactor temperature to their experimental values, using the Nelder-Mead algorithmand the likelihood function as objective function [7]. The best fitting to experimentaldata was obtained by combining subsurface oxygen diffusion to an exponential activitydistribution under non-isothermal conditions. The comparison between the non-iso-thermal model calculations and the experimental results are presented in Fig. 11.1 and11.2.

11.4 Conclusions

The oxidation of nickel by O2 was investigated by transient responses methods in orderto improve the knowledge on the redox mechanism during the production of hydrogenby decomposition of CH4 in periodic mode. Different models were used to describethe experimental data. For the nickel oxidation, a non-isothermal model combining thesubsurface oxygen diffusion with an exponential activity distribution for the surfacesites provided a good description of the transient nickel oxidation. As a result, theactivation energies and the pre-exponential factors of the rate constants as well as thecharacteristic oxygen diffusion times were determined.

11.5 Bibliography

[1] B. Monnerat, L. Kiwi-Minsker, A. Renken, Hydrogen production by catalytic crack-ing of methane over nickel gauze under periodic operation, Chem. Eng. Sci., 56, (2001),633–639.

[2] M.S. Wainwright, Handbook of Heterogeneous Catalysis, Edition G. Ertl, H. Knot-zinger, J. Weitkamp, Wiley-WCH 1, (1997), 64–72.


[3] J.T. Stuckless, C.E. Wartnaby, Al-Sarraf N., St.J.B. Dixon-Waren, M. Kovar, D.A. King,Oxygen chemisorption and oxide film growth on Ni100, 110 and 111: stickingprobabilities and microcalorimetric adsorption heats, J. Chem. Phys., 106, (1997), 2012–2030.

[4] P.J. Gellings, H.J.M. Bouwmeester, Solid state aspects of oxidation catalysis, Catal.Today, 58, (2000), 1–53.

[5] R.J. Vreeburg, P.K. van Tongeren, O.L.J. Gijzeman, J.W. Geus, A comparison betweenthe reduction kinetics of oxidized Ni(111) and Ni(100) surfaces by hydrogen and deu-terium, Surf. Sci., 272, (1992), 294–298.

[6] H. Randall, R. Doepper, A. Renken, Reduction of nitrogen oxides by carbon monox-ide over an iron oxide catalyst under dynamic conditions, Appl. Catal. B, 17, (1998),357–359.

[7] A. Constantinides, N. Mostoufi, Numerical methods for chemical engineers with MatLabapplications, Prentice Hall International Series in the Physical and Chemical EngineeringSciences, Prentice Hall PTR, New Jersey, (1999).

Chapter 12

Diffusion-Reaction Modeling of DNA Hybridization Kinetics on

Biochips

K. Pappaert1 J. Vanderhoeven1 P. Van Hummelen2 G.V. Baron1 G. Desmet1

12.1 Abstract

An analytical expression for the rate of hybridization on biochips is presented. Sincetheir introduction around 1995, DNA chips have gained wide use in analytical chem-istry, with applications in important areas such as gene identification, genetic expressionanalysis, DNA sequencing and clinical diagnostics.

Biochips generally consist of a two-dimensional array of small circular ‘target’-spotsof identical DNA-strands (covalently) linked to the chip surface. The analysis of a mix-ture of unknown probe DNA material is then effectuated by first fluorescently labelingthe probe DNA, and then contacting the chip surface with an aliquot of this mixture(typically 25 to 100ml). When a sufficient amount of the probe DNA strands have hy-bridized to their matching counterpart, confocal laser scan analysis of the fluorescenceintensity of the different target spots indicates the presence or absence of a given genesequence in the sample. Due to the array-format, this information is obtained for thou-sands of sequences in parallel. Despite of their widespread use, the kinetic modellingof the binding events occurring on bio-chips is, however, still in its infancy.

The hybridization process can be described briefly as the following sequence of e-vents: 1) diffusion of a probe DNA strand towards a target strand, 2) collision and for-mation of a nucleation site and 3) either full hybridization (=‘zippering’) or, in case of amismatch, breaking up and diffusing away to make another attempt. Considering thehuge number of possible combinations, it should be noted that each DNA molecule gen-erally passes through a large series of unsuccessful attempts before a successful bindingtakes place.

Thus far, hybridization kinetics are most often interpreted with an (inadequate) em-pirical modification of the kinetic expression for hybridization in free solutions [1,2].

1Department of Chemical Engineering, Vrije Universiteit Brussel, Belgium.2MicroArrayFacility Lab, Flemish Institute for Biotechnology (VIB), Belgium.

56

K. Pappaert, J. Vanderhoeven, P. Van Hummelen, G.V. Baron and G. Desmet 57

The most sophisticated expression proposed thus far [3] is based upon the (inaccu-rate) approximation of a linear concentration gradient and by assuming a constant DNAsource located at a given distance h. The latter approach of course prohibits the repre-sentation of the important depletion effects occurring during the hybridization process.

In the present study, we have tried to establish a complete (i.e., accounting for allgradients) kinetic expression, by solving the complete diffusion-reaction mass balance:

∂Cp

∂t= Dmol

(

∂2Cp

∂x2+∂2Cp

∂y2+∂2Cp

∂z2

)

(12.1)

with

Dmol∂Cp

∂y= kCp, y = 0, x2 + z2 ≤ a2, (12.2)

∂Cp

∂y= 0, y = 0, h, x2 + y2 ≥ a2. (12.3)

Cp = Cp,0, x, z = ±∞. (12.4)

The boundary condition in eq. (12.2) represents the fact that, because of the large num-ber of collisions typically needed before a successful binding occurs, a target spot sur-face (radius=a) has to be treated as a partial reflector, in agreement with the Collins &Kimball approach [4] for diffusion-limited reactions.

With eqs. (12.1)–(12.3), the variation of the concentration H of formed hybrid pairswith time can be calculated from the concentration at the reactive y = 0-plane accordingto:

dH

dt= Dmol

∂C

∂y

∣

∣

∣

∣

∣

y=0

= kC(y = 0). (12.5)

For the simplified 1D-case (only transport in the y-direction), a relatively simple solutioncould be established for eqs. (12.1)–(12.5) using a Finite Fourier Transform technique,yielding:

H(t) = 2hC0

+∞∑

n=1

[sin βn − (Da/βn)(cos βn − 1)]

βn [Da + β2n/Da + 1]

[

1− exp(−β2nDmolt/h

2)]

(12.6)

with

βn tan βn =kh

Dmol= Da. (12.7)

Eq. (12.6) represents the first full exact analytical expression for the rate of DNAhybridization on a DNA micro-array. The dependency on the Damkohler number Dawas expected from the nature of the boundary condition in eq. (12.2). The dependencyon the liquid layer height h has an important technological implication, in that currentDNA hybridization procedures are still always performed without specification for theliquid layer height, leading to a very poor lab-to-lab reproducibility of the results. Es-tablishing an analytical solution for the 3D-problem is cumbersome, but a numericalsolution can easily be obtained.


Apart from solving the general mass balance, we have also attempted to establish amechanistic expression for the reaction rate constant k. Considering that the number ofcollisions per time and unit surface area (N , expressed in moles per second and squaremeter) for a collection of molecules present at a concentration C is given by [5]:

N = A · u · C (12.8)

where u is the product of the mean molecular jumping frequency n and the mean molec-ular displacement length l, and where A is a dimensionless geometric constant which,for the case of pure 1D-collisions, can be calculated to be given by A = 1/

√2p. Noting

that N = k · Cy=0 = D(∂C/∂y)y=0, the desired expression for k can now be directlyobtained from eq. (12.8), yielding:

k =1√2πνλ. (12.9)

Noting from the theory of Brownian motion that the molecular diffusion coefficient inthe 1D-case is given by:

Dmol =νλ2

2, (12.10)

and multiplying the expression in eq. (12.9) with the probability c of a successful colli-sion, we obtain:

k =

√

2

π

Dmol

λc. (12.11)

Inserting eq. (12.11) into eq. (12.6), we now have a complete hybridization rate modelbased on the pure fundamental mass transport and binding probability parameters.

To verify the result of our calculation, we have performed a series of 1-D randomwalk simulations in which imaginary molecules were allowed to make jumps of a dis-tance +l or −l along a line extending between the positions y = 0 and y = h. Thejumping frequency was taken equal to n, and each time the molecule reached the y = 0position it had a probability c to react and a probability 1− c to reflect from the surface.

As can be noted from Fig. 12.1, the result of the analytical solution and the RandomWalk simulations agree very well, thus confirming our procedure for the calculation ofk.

By expanding our random walk technique to the 3-D case, we have also been ableto demonstrate that the formation of so-called ‘doughnut’-hybridization patterns, animportant problem frequently hampering accurate detection, is not due to an artefact inthe target spotting process, but can be explained as the result of a temporary diffusionlimitation.

12.2 Bibliography

[1] U. Christensen, N. Jacobsen, V.K. Rajwanshi, J. Wengel and T. Koch, 2001. BiochemicalJournal 354 (3), 481–484.


i

Figure 12.1: Concentration H of formed hybrid pairs versus time: comparison betweensolution of analytical model and random walk simulations.


[2] S. Vontel, A. Ramakrishnan and A. Sadana, 2000. Biotechnology and Applied Biochem-istry 31 (2), 161–170.

[3] V. Chan, D.J. Graves and S.E. McKenzie, 1995. Biophysical Journal 69 (6), 2243–2255.

[4] F.C. Collins and G.E. Kimball, 1949. Journal of Colloid Science 4, 425–437.

[5] F. Reif, Fundamentals of statistical and thermal physics, Mc-Graw Hill, London, 1984.

Chapter 13

‘State Defining’ Experiment in Chemical Kinetics: Primary

Characterization of Catalyst Activity in a TAP Experiment

Sergiy O. Shekhtman1 Gregory S. Yablonsky1 John T. Gleaves1

13.1 Introduction

Recently, a new approach for characterizing the catalytic activity of technical and modelcatalysts called ‘interrogative kinetics’ (IK) was described by Gleaves and Yablonsky [1].Originally, the IK approach was based on pulse-response TAP experiment. The mainidea of this approach is to combine two types of experiments, called ‘state-defining’ and‘state-altering’ experiments to probe different states of catalyst sample. In a state-defin-ing experiment, the catalyst composition and structure changes insignificantly duringa kinetic test. In a state-altering experiment, the catalyst composition is changed in acontrolled manner.

13.2 ‘State defining’ TAP experiment and kinetics

A typical one-pulse TAP experiment (an experiment with the typically small pulse in-tensity) is the ‘state-defining’ experiment [1]. This experiment kinetically characterizesthe given catalyst state, i.e., the state with given composition and structure. The pre-sented theoretical approach of the state-defining experiment allows:

1. To separate explicitly transport (measuring stick) and reaction in general in TAPdata analysis

2. To identify the primary characteristics measured in TAP experiment (to determinewhich parameters can be extracted from the data, how many parameters are inde-pendent and finally to find these parameters)

3. To relate measured primary characteristics to the detailed mechanisms

1Dpt. of Chemical Engineering, Washington University, Campus Box 1198, One Brookings Drive, St.Louis, MO 63130.

61

S.O. Shekhtman, G.S. Yablonsky and J.T. Gleaves 62

13.3 General model of ‘State defining’ experiment

In general, two main processes take place in a one-pulse TAP experiment. The first oneis a transport governed by Knudsen diffusion. This is a simple process (‘measuringstick’), which is supposed to be known. The second process is, in general, a complexkinetic process that occurs on the particle surface or inside of the particle (in the case ofporous material). The main difference between the second process and the first one isthat the second process doesn’t participate in transport throughout the reactor. In otherwords, once gas molecule gets involved in the second process (adsorbs on the surface,enters the pore, etc.), it no longer moves along the axial coordinate of the reactor.

The model of the most common symmetric three-zone reactor (where the lengths ofthe inert zones are equal) was considered as a basic one. For any gaseous substance(reactant or product), the general three-zone model of a one-pulse TAP experiment canbe presented by the following equations:

εin∂Cg

∂t= Din

∂2Cg

∂x2(13.1)

in the inert zones,

εcat = Dcat∂2Cg

∂x2∓Rg(K,Cg(x, t), θ(x, t)) (13.2)

in the catalyst zone, where Cg is the gaseous concentration; t is time; x is the axialcoordinate; Din and Dcat are diffusivities in inert and catalyst zones, respectively; εin andεcat are bed voidages in inert and catalyst zones, respectively; Rg(K,Cg(x, t), theta(x, t))is an overall reaction rate; K is a set of kinetic parameters corresponding to the catalyststate; θ is the surface concentration.

The right hand side of the equation (13.1) represents the diffusional transport thatis the only process in the inert zones. The right hand side of the equation (13.2) repre-sents both the diffusional transport and the studied kinetic process (reaction or porousdiffusion). The later is described by the rate term . For the given detailed kinetic model,the rate term can be represented as a complex function of concentrations and kineticparameters of this model.

13.4 State-defining kinetics regime

Usually, a complex catalytic reaction includes exchange between gas phase and catalystphase surface as well as processes on the surface. For a state-defining experiment, twokinds of surface substances should be distinguished. Surface substances of the first kind(surface state substances) determine the catalyst state and basically don’t change duringthe experiment. Surface substances of the second kind (surface intermediate) that arisein the course of state-defining experiment are small compared to first kind substances.Consequently, under state-defining kinetic conditions, the following assumptions aboutrates of elementary reactions can be readily made:


(a) all the corresponding rates of elementary reactions do not change as functions ofthe concentrations of the first kind of surface substances;

(b) all the corresponding rates of elementary reactions are linear with respect to smallconcentrations of the second kind of surface substances in the wide class of cat-alytic mechanisms [2].

(c) the rates of interaction between gaseous substances and surface are usually linearwith respect to the corresponding gaseous concentrations [2].

In the TAP reactor, all the diffusional transport occurs only in the interparticle gas phase(not on the surface). Based on this fact and assumptions (a), (b) and (c), the rate term inthe Laplace domain can be significantly simplified as:

R(K,Cg(z, t), θ(z, t)) = r(K, s)Cg(s, x). (13.3)

Equation (13.3) allows to separate explicitly the gaseous concentration factor Cg(s, x),and the reactivity factor, r(K, s). As can be seen from this equation, the overall rate forgas substance has the same x-dependence as the corresponding gaseous concentration.The proportionality coefficient, r(K, s), is the only factor left in the equation that is afunction of the detailed kinetic model, particularly of the parameters of the model. Itcan be termed the ‘reactivity’ of the catalyst state is so far that it relates the rate of thekinetic process to the gaseous concentration at the given catalyst state. Obviously, thedimension of this reactivity is reciprocal second.

The solution obtained in the Laplace domain or Laplace reactivity, r(K, s) is not themeasured quantity itself, but can be directly related to the moments of exit flow. To berelated to the moments, the reactivity was expanded in series with respect to the Laplacevariable, s, as:

r(K, s) = r0(K) + r1(K)s+ r2(K)s2 + r3(K)s3 +O(s4), (13.4)

where the rn(K) are basic kinetic coefficients (kinetic set) that are functions of the cata-lyst state only.

The physical meaning of these coefficients can be understood based upon dimensionconsiderations. The zeroth coefficient, r0(K), has the dimension of reciprocal secondand presents an apparent kinetic constant. The first coefficient, r1(K), is dimensionlessand presents the ‘intermediate capacity’ of the catalyst. The second coefficient, r2(K),has dimension of second and presents an apparent time delay.

The basic kinetic coefficients for each observed gaseous substance, rn(K), particu-larly the first three coefficients, can be calculated analytically from experimental datawith no detailed mechanism assumption. The main properties of the introduced basickinetic coefficients are the following:

1. the coefficients contain the information about kinetics only (no transport proper-ties are present in there);


2. the coefficients are functions of catalyst composition and temperature but not ofconcentration;

3. the coefficients were analytically related to experimentally observed quantitieswithout introducing a detailed kinetic model (the only kinetic assumption beingmade is the linearity of state-defining kinetics)

From this perspective, the basic kinetic coefficients can be viewed as experimentallymeasured primary (apparent) kinetic characteristics, or kinetic set of the studied catalyststate. They can be analyzed independently for any state of the catalyst.

From the other side, based on they properties, the coefficients can be treated in termof detailed kinetic mechanisms. Particularly, they can be calculated as function of rateconstants for each particular detailed mechanism. Then, using experimental values ofbasic kinetic coefficients different detailed mechanisms can be distinguished.

The state-defining approach is illustrated by an analysis of butene and furane oxida-tion over VPO catalyst

13.5 Conclusions

The insignificant perturbation of the catalyst surface by the gas mixture produces thestate-defining kinetic regime that can be readily viewed as the new interesting ‘subdy-namics’ of the complex catalytic reactions. The analysis of state-defining experimentaldata using proposed basic kinetic coefficients approach enables to characterize quantita-tively the catalyst in a variety of its states. For the given mechanism and correspondingkinetic models, the dynamic behavior of the new state-defining regime can be studiedanalytically or in computer experiments as it was previously done for the well-knownpseudo-steady-state regime.

From the general point of view, the state-defining kinetic regime can be realized inany kinetic device that allows to perform a small perturbation of a chemical mixturecomposition.

13.6 Bibliography

[1] J.T. Gleaves, G.S. Yablonskii, Phanawadee, Ph., Y. Schuurman, TAP-2: An Interroga-tive Kinetics Approach, Appl. Catal. A: General, 160 (1997) 55.

[2] G.S. Yablonskii, V.I. Bykov, A.N. Gorban’, V.I. Elokhin, Kinetic Models of CatalyticReactions. Comprehensive Chemical Kinetics. 32, Amsterdam-Oxford-New York-Tokyo,1991, 396pp.

Chapter 14

Simulating Diffusion in Nanoporous Materials

M.-O. Coppens1

14.1 Introduction

Nanoporous materials are extensively used in heterogeneous catalysis, adsorption andto separate mixtures, because of their high internal surface area that can be accessedthrough a network of narrow pores.

A first example consists of zeolites: crystalline microporous aluminosilicates with aregular network of pores, which are typically smaller than one nanometer in diameter.A second important example are amorphous porous materials with a complex irregularnetwork of pores, which may have a variety of shapes and typically have a rough in-ternal surface; the pore size is usually a few nanometers on average, although smallerpores are also possible.

Because the pores are so narrow, transport by diffusion is often the limiting step inheterogeneous catalysis, and may play a significant role in adsorption and separationprocesses. Simulating diffusion in nanoporous materials is therefore of great interest tochemical engineering.

14.2 Methodology

To account for the complexity of the pore space and/or the interactions of the diffus-ing molecules with the internal surface, statistical mechanical simulations can be veryuseful. When the accessibility of molecules to the pore surface is uniform and the porenetwork is well connected, mean-field approximations can be used. When this is notthe case, dynamic Monte-Carlo simulations, in which the trajectories of the moleculesthrough the pores are followed, enable us to investigate the qualitative influence of avariety of physico-chemical parameters as well as geometrical parameters describingthe pore network topology or the surface morphology.

1Department of Chemical Technology, Delft University of Technology, Julianalaan 136, 2628 BL Delft,The Netherlands. E-mail: [email protected]

65

M.-O. Coppens 66

14.3 Diffusion in zeolites

While rather simple to implement, quantitative simulations in macroscopic nanoporousheterogeneous materials (e.g., a zeolite membrane or a catalyst particle a few millime-ters in size) are often impossible to carry out on today’s computers, the reason being thatthe system is very ‘stiff’. In zeolites, for example, the molecules are typically similar insize to the pores, so that their motion constitutes of infrequent hops between adsorptionsites: the molecules linger for a long time around certain locations before crossing theenergy barrier to move to another site.

Especially when there are different types of sites with widely differing adsorptiontimes, the calculation time for simulating this activated process using molecular dy-namics (i.e., the integration of Newton’s equation of motion) is prohibitive. The solu-tion to attack problems like this is to apply a hierarchical simulation approach, com-bining mesoscopic dynamic Monte-Carlo simulations on lattice models with moleculardynamics and transition-state theory on smaller scales. In this way, both self-diffusion(diffusion in the absence of a concentration gradient) and transport diffusion (diffusionunder the influence of an imposed gradient) can be studied [1].

14.4 Diffusion in amorphous mesoporous materials

The second class of nanoporous materials that will be discussed consists of amorphousporous materials, such as common porous SiO2 and Al2O3, with pores a few nanometersin average size. The rough pore surface of these materials often turns out to be notsimply irregular, but fractal [2]. This property allows to make analytical predictions ofthe diffusivity through the pores. Especially Knudsen diffusion is of interest, which isthe diffusion regime where the mean free path of the molecules (distance between twocollisions) is larger than the pore diameter, so that the motion is essentially dominatedby consecutive collisions of the molecules with the surface, and hence becomes sensitiveto the surface morphology.

Dynamic Monte-Carlo simulations can again be performed to study the effect of sur-face roughness on self- and transport diffusion. These simulations reveal that, just likefor zeolites, self- and transport diffusivities differ from each other: self-diffusivity de-pends on roughness, while transport diffusivity is roughness-independent [3]. This hasa significant impact on heterogeneous catalysis, and on the interpretation of experimen-tal measurements of diffusivities using different techniques.

To include the complexity of the pore network topology and/or the chemical hetero-geneity of the pore surface, a hierarchical simulation approach could again be applied.

14.5 Conclusions

In summary, diffusion in nanoporous materials can be studied using a mesoscopicmean-field or, more accurately, a dynamic Monte-Carlo approach, in combination with

M.-O. Coppens 67

microscopic information that is typically provided by short-time molecular dynamicssimulations. This presentation concentrates on the Monte-Carlo approach and discussesthe application of hierarchical simulation techniques to obtain quantitative information.

14.6 Bibliography

[1] F.J. Keil, R. Krishna and M.-O. Coppens, 2000, Modeling of diffusion in zeolites. Rev.Chem. Engng 16, 71–197.

[2] M.-O. Coppens, 1999, The effect of fractal surface roughness on diffusion and reac-tion in porous catalysts – From fundamentals to practical applications. Catalysis Today53, 225–243.

[3] K. Malek and M.-O. Coppens, 2001, Effects of surface roughness on self- and trans-port diffusion in porous media in the Knudsen regime. Phys. Rev. Lett., 87, 125505.

Chapter 15

Determination of adsorption parameters from chromatographic

experiments and their use in catalysis modelling

Joeri F.M. Denayer1 G.V. Baron1

15.1 Abstract

Zeolite catalysis invokes the interplay of several intrinsically complicated phenomena.Factors like diffusion, adsorption, shape selectivity and the activity of the catalytic sitesall contribute to the global, observed reaction patterns and rates.

Although it is generally accepted that adsorption effects are of major importance indetermining reaction orders and selectivities in reactions catalyzed by zeolites, practi-cally no attempts were made to really determine the adsorption properties in catalyticconditions, and to use the adsorption parameters in the modeling of such reactions.

In this work, the knowledge obtained from independent studies of adsorption equi-libria and reaction kinetics were combined in order to gain a better understanding ofshape selectivity and synergy effects in zeolite catalysis.

Several chromatographic techniques in gas and liquid phase were used to determineadsorption properties of hydrocarbon molecules in a broad range of conditions. Ad-sorption equilibrium constants in the Henry domain for pure components are calculatedfrom the first moment of the response to a pulse experiment. Adsorption isothermsor binary adsorption data were obtained from chromatographic experiments in whichthe mobile phase consists of a mixture of an inert gas and an adsorbing component ora mixture of adsorbing components. Theoretical models are needed to extract mean-ingful parameters from such kind of experiments. Besides adsorption constants, masstransfer resistances (axial dispersion, macro- and macropore diffusion) can be obtainedfrom the second moments of the response curves or by fitting of the response curves totheoretical models.

Adsorption properties were determined at low, intermediate and high zeolite load-ing. At low surface coverage, no interactions between the adsorbed molecules occur.

1Department of Chemical Engineering, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Bel-gium.

68

Joeri F.M. Denayer and G.V. Baron 69

It was shown that in such conditions, simple, rational relationships exist between ad-sorption properties (Henry constants, adsorption enthalpies and entropies), adsorbateproperties (molecular weight, size and polarity) and adsorbent properties (pore size,composition and topology). At higher zeolite loading, the competition between dif-ferent molecule types is influenced by adsorbate-adsorbate interactions and shieldingeffects. It was also shown that the slowest step in mass transfer in the studied faujasitesystems is governed by hindered molecular diffusion in the macropores of the pelletsfor alkane molecules.

On faujasites, the adsorption capacity of n-alkanes decreases with increasing carbonnumber and temperature. On ZSM-22, the adsorption capacity of the linear chains ismuch larger than that of the iso-alkanes. Linear alkanes adsorb in the pores of ZSM-22,while iso-alkanes have no access to the pores and adsorb in the pore mouths at low sur-face coverage. At higher partial pressures, n- and iso-alkanes adsorb in a non selectiveway on the relatively large external surface of the ZSM-22 crystals, which contributes toa large extent to the total adsorption capacity. The experimentally determined isothermswere fitted to the Langmuir, the Langmuir-Freundlich, the Langmuir + Interaction andthe bimodal Langmuir adsorption model. For the faujasites, the Langmuir + Interactionmodel gave the best data fitting, while adsorption on ZSM-22 could be well describedwith the bimodal Langmuir model, accounting for adsorption in the pores, pore mouthsand on the external surface.

The adsorption parameters obtained from the adsorption study were used in themodeling of the hydrocracking of C6 − C9 n-alkanes.

As a non-shape-selective system, hydro-isomerisation and -cracking on Pt/H-Y ze-olite catalysts was studied. Kinetic constants for monobranching, multibranching andcracking of alkanes, obtained from fitting of the experimental conversion data, increasemuch less strongly with the carbon number than the adsorption constants. Althoughthe Si/Al ratio of the zeolite has a strong influence on the absolute reactions rates, therelative rates of the different reaction types are not influenced by the composition of thezeolite.

As a shape-selective system, hydro-isomerisation and -cracking on Pt/H-ZSM-22was investigated. A high yield of monobranched alkanes and a strong selectivity for2-methyl branching are typical properties in hydroconversion on Pt/H-ZSM-22. Thehydro-isomerisation and hydrocracking of C6 − C9 n-alkanes on Pt/H-ZSM-22 was an-alyzed and modeled using different adsorption models in order to investigate the in-fluence of the different adsorption modes on the reaction patterns. The competitionfor adsorption in reaction conditions appeared to be much smaller than expected fromthe large differences in adsorption constants for adsorption in the pores and the poremouths at low coverage. n- and iso-alkanes are first adsorbed in a noncompetitive wayon the external surface before they are converted. The skeletal rearrangement and crack-ing reactions occur in the pore mouths at the external surface of ZSM-22, where shapeselective pore mouth catalysis gives rise to the observed reaction products.

Chapter 16

The multi-scale mathematical model of the oscillatory chemical

reaction proceeding over a porous catalyst

E.S. Kurkina1 E.D. Kuretova1

16.1 Abstract

The present work is aimed at the modeling of a heterogeneous chemical reaction occur-ring over supported catalysts. Such catalysts consist of small crystallites of a catalyti-cally active component embedded in porous support pellets. Such complicated systemsare rather difficult for model description, and usually for real simulations some approx-imations are used, for example, kinetic or diffusion ones, as well as models of idealmixing reactor, etc. For this purpose, various criteria are calculated, which allow esti-mating which factors are essential for the dynamical behavior of the considered systemand which ones could be neglected. However, one can reliably apply this approach forstationary reactions only, in which case the relationships between the rates of all pro-cesses are well known. However, a lot of reactions proceed in the oscillatory mode. Inthese cases such estimates are rather difficult or, sometimes, impossible. Therefore, itseemed necessary to work out a quite complete mathematical model, taking into ac-count all basic physical and chemical factors.

Presented is the distributed mathematical model of the reaction proceeding in theoscillatory mode over the micro-clusters of metal catalyst embedded in the porousmedium. The detailed description is presented in [1]. The model takes into accountthe passing of the reactant flow through the granular catalyst layer, diffusion in poresof support pellets, reaction on the surface of embedded metal clusters, the heat effectof the reaction, and heat and mass transfer across the free volume of the catalyst layer.The model is formulated according to the hierarchical principle and comprises threebasic levels of description corresponding to different spatial scales: (1) the level of anindividual metal cluster; (2) the level of an individual catalyst pellet; and (3) the levelof the whole layer of the catalyst. Each level can be detailed if necessary, taking intoconsideration additional factors and incorporating these into the complete model.

1CM&C Department, Lomonosov Moscow State University, 119899, Moscow.

70

E.S. Kurkina and E.D. Kuretova 71

Figure 16.1:

The suggested model constitutes a non-linear PDE system complemented by initialand boundary conditions. To perform the simulations, we suggested the method ofspatial approximation of the differential operators, which enabled us to reduce it to anODE system of high dimensionality. For this purpose the spatial grid was constructedwith a certain method, which permits to take into account the processes of heat andmass transfer both within the 3D free volume of the catalyst layer, and within eachcatalyst pellet.

The model was investigated on the basis of CO oxidation reaction over a Pd-zeolitecatalyst [2-4]. Earlier several mathematical models of this reaction were developed in[5,6]. However, each of these models considers the process of the reaction in an approx-imation accounting for the influence of only a few factors and neglecting the other ones.The suggested model has a more general character and corresponds to the physical con-ditions of the experiments.

The simulations performed allowed obtaining various types of reaction rate oscil-lations; in particular, chaotic and mixed-mode regimes similar in shape to those ob-served in the experiments [4] were found in a wide range of external parameters. Fig.16.2 demonstrates several examples of the oscillations obtained. Results of these stud-ies were partly presented in [1]. Besides, within the framework of the comprehensivemodel the limits of applicability of several commonly used approximations have beenestablished.

One of the issues we have investigated was the effect of the diffusion rate along the‘vertical’ direction within the free volume of the catalyst disk. Besides, the system oftwo catalyst disks located one under another was considered (see Fig. 16.3). Each diskis very thin (about 10−3cm) and consists of several layers of catalyst pellets.


Figure 16.2: Calculated time series at (a) flow rate F = 2.75cm3/s, external CO pres-sure PCO,0 = 3.05Torr, palladium mass fraction g = 0.35%; (b) F = 2.5cm3/s, PCO,0 =0.311Torr, g = 0.5%; (c) F = 2.5cm3/s, PCO,0 = 1.95Torr, g = 0.4%.


Figure 16.3:

Depending on the size of the gap between the disks, different types of their cooper-ative behavior were obtained. If the gap is rather small, each disk influences the otherone. When the distance increases, the mutual influence of the disks to each other be-comes weaker. In this case, the observed oscillations on the second disk are practicallyidentical to those on the first one, but they are shifted in time with respect to each other.

16.2 Acknowledgment

This work is supported by Russian Fund of Basic Researches (Grant no. 00-01-00587).

16.3 Bibliography

[1] E.S. Kurkina, E.D. Tolstunova, Appl. Surf. Sci., 2001, 182/1-2, p. 77–90.

[2] N.I. Jaeger, K. Moller, P.J. Plath, J. Chem. Soc. Faraday Trans. Pt II, 1986, 82, p. 3315–3330.

[3] M.M. Slin’ko, N.I. Jaeger, P. Svensson, J. Catal., 1989, 118, p. 349–359.

[4] M. Liauw, P.J. Plath, N.I. Jaeger, J. Chem. Phys., 1996, 104 (16), p. 6375–6386.

[5] E.S. Kurkina, N.V. Peskov, M.M. Slin’ko, Physica D, 1998 118, p. 103–122.

[6] M.M. Slin’ko, E.S. Kurkina, M.A. Liauw, N.I. Jaeger, J. Chem. Phys., 1999 111 (17), p.8105–8114.

Chapter 17

Dynamic Monte Carlo simulation of diffusion-reaction processes in

rough fractal pores

K. Malek1 M.-O. Coppens1

17.1 Abstract

The study of transport phenomena in porous media is a long-standing research subjectwith many industrial and technological applications. The improvement of modellingtechniques for the description of diffusion-reaction in heterogeneous catalysis providesa real challenge, mostly due to limitations of the classical representations (Froment andBischoff, 1990; Sahimi et al., 1990). In recent years considerable effort has been dedicatedto the investigation of diffusion and reaction in complex geometries. The role played bythe pore shape and the local surface morphology on the overall diffusivity and reactivityof the catalyst remains to be understood. It is known that the shape and irregularity ofthe pore walls could affect diffusivity. This effect is more significant in the Knudsen dif-fusion regime, involving molecular motions between points on the wall surface, whichcould be influenced by surface roughness (Coppens, 1999). Using dynamic Monte Carlosimulations and analytical calculations, we investigated the effect of pore surface rough-ness on Knudsen diffusion with and without reaction on the pore surface. Simulationsin the absence of reaction reveal conceptual differences between the effect of roughnesson diffusion in the presence of a concentration gradient (transport diffusion) and gradi-entless (self- or tracer) diffusion (Malek and Coppens, 2001). Self-diffusion is the morefundamental diffusion mechanism that is of direct relevance to, e.g., catalysis. Self- andtransport diffusivities are presented as a function of roughness in Fig. 17.1 for a three-dimensional model fractal pore (part of this pore has been shown in the insert). Theirregularity or roughness factor of a pore at any level of the fractal generator, ξ, is char-acterised by the ratio of the pore boundary length (2D) or surface area (3D) to the poreboundary length or surface area of an unperturbed smooth pore with the same cross-section. There clearly is a large effect of roughness on self-diffusion: the self-diffusivityis a direct function of the individual molecular trajectories, the total trajectory length

1DelftChemTech, Delft University of Technology Julianalaan 136, 2628 BL Delft, The Netherlands.

74

K. Malek and M.-O. Coppens 75

Figure 17.1: Self-diffusivity (circle) and transport diffusivity (square) as a function of theroughness ξ. DK0 is the Knudsen diffusivity in a smooth pore. Insert shows a segmentof a 3D pore with a random Koch surface.


or residence time, and therefore decreases significantly as the roughness factor, ξ, in-creases. On the other hand, the transport diffusivity does not vary with ξ, i.e., it doesnot depend on roughness. We recently conducted simulations of Knudsen diffusion andfirst order reaction in rough fractal pores. Using classical reaction engineering (Fromentand Bischoff, 1990), the conversion in the pore can be estimated by:

x =2DΦ

Lu sinh Φ(cosh(Φ)− 1) =

2D

Lu(1

η− Φ

sinh Φ), (17.1)

where D is the diffusion coefficient, 2L is the pore length, η is the pore effectivenessfactor and u is the average molecular velocity. In this formula, the Thiele modulus isgiven by:

Φ = 2L

√

√

√

√

CΩ√ΩaN2

(δ′N2)Dads−2

ηs

Dk. (17.2)

In this equation, CΩ is the proportionality constant in the Mandelbrot area-perimeterrelationship, Ω is the area of a pore cross-section, aN2

is the nitrogen BET-area, δ′N2is

the reduced effective diameter of nitrogen molecules, ηs is the surface effectiveness fac-tor, Dads is the fractal (adsorption) dimension of the surface, and k is the reaction rateconstant (Coppens and Froment, 1995). For Monte-Carlo simulations, it is useful towrite the reaction rate constant as a function of the reaction rate probability, p, when amolecule hits the surface:

k =

(

p

1− p

)

Nr, (17.3)

where Nr is the number of reacting molecules per unit time. Fig. 17.2 shows simulatedand calculated conversion results for the 3D pore shown earlier. For this pore, Cω =13/5, Ω = 100nm2, δ′N2

= 4 × 10−11, Dads = 2.33, and ηs = 1. When we substitute Din eq. (17.1) by the self-diffusivity, the analytically calculated conversion perfectly fitsthe simulation results. This is not true for the transport diffusivity: there is a poor fit,except for a smooth pore where the self- and transport diffusivities are approximatelythe same. The residence time distribution does not have any effect on the overall flux,because the molecular directions are distributed in the same way, following a cosinedistribution with respect to any real or virtual plane within the pore volume, whateverthe pore shape. As a result, whether a molecule remains for a long or a short time ina pore does not have any effect on the outlet through which it leaves the pore. Thelatter implies that the transmission probability and therefore the transport diffusivityremain constant. This result is in good agreement with our analytical calculations andresolves seemingly contradicting results in the literature. Like in zeolites the distinctionbetween self- and transport diffusion has important consequences for the interpretationof experimental diffusion measurements, as well as for the study of diffusion-limitedreactions in nanoporous catalysts.


Figure 17.2: Comparison of the conversions calculated by MC (points) to analyticalresults (lines) for a first order reaction in a smooth pore (circle) and in rough pores[ξ = 1.44 (diamond), ξ = 2.09 (triangle)]. The full line corresponds to analytical calcula-tions where the self-diffusivity is used (excellent fit), while the dotted line correspondsto analytical calculations where the transport diffusivity is used (bad fit except for thesmooth pore).


17.2 Bibliography

[1] G.F. Froment and K.B. Bischoff, Chemical Reactor Analysis and Design, 1999, 2nd ed.,Wiley, New York.

[2] M. Sahimi, G.R. Gavalas and T.T. Tsotsis, Chem. Engng Sci. 45, 1990, p. 1443–1502.

[3] M.-O. Coppens, Catalysis Today 53, 1999, p. 225–243.

[4] K. Malek and M.-O. Coppens, Phys. Rev. Lett. 87, 2001, 125505.

[5] M.-O. Coppens and G.F. Froment, Chem. Engng Sci. 50, 1995, p. 1027–1039.

Chapter 18

Fingering of chemical fronts in porous media

A. De Wit1

18.1 Abstract

Chemical fronts between fresh reactants and products may become hydrodynamicallyunstable because of density differences between the two phases. In the gravity field, thefront may in particular lead to density fingering if the denser solution is placed on topof the lighter one. In this context, we have provided a quantitative theoretical descrip-tion of recent experimental studies of density fingering of miscible reaction-diffusionfronts of the iodate-arsenious acid reaction [1]. The experimental system consists in atwo-dimensional Hele-Shaw cell (two thin glass plates separated by a thin gap width)filled with the fresh reactants. The reaction is triggered at the bottom of the cell. Theresulting chemical front moves upwards invading the fresh reactants, and leaving theproducts behind it. The density of the reactant solution is higher than that of the prod-uct solution. Hence, the upwards travelling front is buoyantly unstable and developsdensity fingers in the course of time. The system is modelled using Darcy’s law describ-ing the hydrodynamic flow in porous media or in thin Hele-Shaw cells coupled to areaction-diffusion-advection evolution equation for the concentration of the solute (io-dide) ruling the density of the solution. Using these macroscopic evolution equations,we have analyzed the stability of the system as a function of the various parameters suchas the gap of the cell, the density difference between the two solutions or the chemicalconcentrations. We have obtained good quantitative agreement with dispersion curvesmeasured experimentally for small gap widths. We have next analysed the non linearregime in order to understand the long-time nonlinear evolution of the density fingers.There we find using a numerical integration of the full nonlinear evolution equationsthat the asymptotic regime of the system is one single finger spanning the entire widthof the system. This final finger travels at a higher speed than the initial planar chemi-cal front. We have characterized this dynamics as a function of the various parametersof the system. We have in parallel developed an analog study of the density fingering

1Center for Nonlinear Phenomena and Complex Systems, CP 231, Universite Libre de Bruxelles, 1050Brussels, Belgium.

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A. De Wit 80

instability of another chemical system: the chlorite-tetrathionate system. This redox re-action also produces planar chemical waves that may become unstable due to buoyancyeffects in the gravity field. By studying this system, we have underlined to what extentthe stability properties of reaction-diffusion fronts with regards to hydrodynamical fin-gering instabilities depend on the chemical kinetics [2].

18.2 Bibliography

[1] A. De Wit, Fingering of Chemical fronts in Porous Media, Phys. Rev. Letters, 87,054502 (2001).

[2] J. Yang, A. D’Onofrio, S. Kalliadasis and A. De Wit, Rayleigh-Taylor Instability ofreaction-diffusion chlorite-tetrathionate front, submitted to J. Chem. Phys.

Chapter 19

Modeling the NO Reduction by Hydrogen on a Pt Field Emitter Tip

Y. De Decker1 F. Baras1 G. Nicolis1 N. Kruse2

19.1 Abstract

Within the past decades, the investigation of solid-gas heterogeneous catalytic reactionsat the nanoscale has revealed a rich variety of complex and nonlinear phenomena [1].The most familiar description of such processes rests on the mean-field (MF) approxi-mation for the reaction kinetics. It leads to a first understanding of the origins of theobserved behaviors. The validity of this description is however limited: the spatialconstraints induced by the support may lead to the emergence and amplification ofinhomogeneous fluctuations. In this respect, kinetic Monte Carlo (MC) simulations ap-plied to surface reactions become particularly relevant, because they incorporate thestochasticity of reactive events and the restricted geometry of the surface in an intrinsicmanner. These two methods are used here to investigate a complex surface reaction.The reaction considered here is the reduction of NO by H2 as observed on Platinum byField Ion Microscopy (FIM) [2]. With this technique, reactions can be followed in real-time on metal tips whose geometry and size approach those of the clusters that can befound in real-world catalysts. Specific patterns associated with the nonlinear characterof the considered surface reaction are seen to appear on the tip (Fig. 19.1). The observedcatalytic cycle is associated with a burst-like increase of the reaction rate and emptyingof the surface. The process repeats itself in a non-periodic way.

In an attempt to understand the rapid catalytic ignition seen in Fig. 19.1, a two-species autocatalytic process has been proposed to be in operation [3]. We here presenta minimal (one-species) model for this reaction that still retains the main characteristicsof the observed phenomenon such as the explosive behavior. The surface reaction isexpected to proceed according to the following mechanism:

NOg + S ↔ NOads (19.1)

1Center for Nonlinear Phenomena and Complex Systems, Universite Libre de Bruxelles, CampusPlaine, C.P. 231, B-1050 Brussels, Belgium.

2Chemical Physics of Materials, Universite Libre de Bruxelles, Campus Plaine, C.P. 243, B-1050 Brus-sels, Belgium.

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Y. De Decker, F. Baras, G. Nicolis and N. Kruse 82

Figure 19.1: ‘Surface explosion’ during NO reduction by H2 on a Pt tip (PNO = 2.5 ×10−3Pa; PH2

= 4× 10−3Pa, F = 8.7V boxnm−1).

NOads + 3S → 4S. (19.2)

where S denotes an empty site on the surface. The first equation corresponds to adsorp-tion and desorption of the reactant and the second step represents reaction. The NOads

decomposition is the limiting step for the reaction at the surface level. The release offour free sites induces the strong autocatalytic character of the process.

19.2 Mean-Field Results

The Field Ion Microscope is considered as a flow reactor with a constant pressure of ni-trogen monoxide. If we suppose that the maximum coverage of the surface correspondsto a monolayer, the rate equation associated with this scheme reads (in the limit of anideal system)

∂x

∂t= PNO(1− x)− x(1− x)3 − kx +D∇2x, (19.3)

where x is the NOads coverage, and with PNO, k and D being respectively the (rescaled)NOg partial pressure, desorption constant and diffusion coefficient. Within an appropri-ate range of the parameters, this system admits a bistability between a highly-coveredinactive state and a less-covered reactive state. A temporal explosive behavior is ob-served in the vicinity of one of the turning point of the corresponding hysteresis, thatretains the main features of the experiments (Fig. 19.2). If diffusion is considered, wavescan be observed that correspond to the displacement of the less stable state by the otherone. Because of the small size of the facets where the experimental phenomenon isobserved, the effects of fluctuations on the dynamics should also be considered.


Figure 19.2: Explosive transition between the two states, where a) is the NOads coverageand b) is (5×) the reaction rate. x0 = 0.8, PNO = 0.0175, k = 0.001.


19.3 Monte-Carlo Simulations

When performing simulations of this system on a (square) lattice, numerous deviationsfrom the MF behavior are observed:

• The steady-state coverage and the stability domains can be strongly modified.

• The ignition time for the explosions are statistically distributed, leading to a tran-sient bimodality in the coverage (Fig. 19.3).

• For small systems, reversible transitions are observed between the two stablestates forming the bistability.

• The propagation of waves between the two stable states is altered by a reaction-induced propagation term.

• The spatial development of the surface explosions occurs via a spontaneous nu-cleation and growth of the reactive state within the inactive state.

These insufficiencies of the MF predictions can be traced back to an interplay betweenthe stochastic aspect of the reaction, the restricted geometry of the support and the hard-core repulsion between adsorbed particles. The analysis reported here is restricted toan isolated facet. The complex behavior of the experimental system could be also berelated to the many interacting facets forming the tip. Each facet has different size andreactivity. A further development should thus be to model the surface as a patchwork ofcommunicating facets, coupled to each other via diffusion. In such an extended modeli-sation, one should be able to understand the origin of the sustained character of surfaceexplosions.

19.4 Bibliography

[1] R. Imbihl and G. Ertl, Chem. Rev. 95, p. 697 (1995).

[2] C. Voss, N. Kruse, Appl. Surf. Sci. 87/88, p. 127 (1994).

[3] Y. De Decker, F. Baras, N. Kruse and G. Nicolis, submitted to J. Chem. Phys.


Figure 19.3: Probability distribution in the vicinity of one of the turning points after a)1, b) 1000 and c) 20000 time steps. Obtained from 104 realizations on a 10 × 10 squarelattice, with x0 = 0.8, PNO = 0.0175, k = D = 0.001.

Chapter 20

Entropic structure and singular perturbation analysis of reactive

gaseous mixtures in the limit of partial equilibrium reduced chemistry

M. Massot1

20.1 Introduction

Numerical simulations of multicomponent reactive flows raise several difficulties cre-ated by the large number of unknowns and the wide range of temporal scales due tolarge and complex detailed chemical kinetic mechanisms. One way to address the prob-lem of the stiffness and to reduce the range of time scales to be solved, is to use reducedmechanism for the complex chemistry. It is valid when the quickest time scales are dueto very fast reactions and when, most of the time, the solution is on the so-called ‘slowmanifold’. There has been a tremendous effort rt in creating efficient and predictivenumerical methods in order to define and solve reduced system of equations for com-bustion and air pollution modeling applications (among other studies [5], [6], [7] andthe large bibliography in [8]). Most of the time, the question of the the compatibilityof the reduction step on the ‘slow’ manifold with the entropy production due to chem-ical reaction is not investigated, except in [8] for an isothermal homogeneous reactor,where a nonlinear and nonconstant projection onto the partial equilibrium manifoldis proposed, but it does not preserve the original entropic structure. Finally, a singularperturbation analysis is performed in [1] in a mathematical framework without thermo-chemical assumptions and thus without entropic structure for the original dynamicalsystem.

20.2 Entropic Structure of the PDE’s system of equations

In this contribution, we investigate the system of equations modeling multicomponentreactive flows with detailed transport and complex chemistry in the limit of partial equi-librium. The reduced system is obtained using a projection step compatible with the

1MAPLY–UMR 5585, Laboratoire de Mathematiques Appliquees de Lyon, Universite Claude Bernard,Lyon 1 69622 Villeurbanne Cedex, France.

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M. Massot 87

chemical entropy production. The reduced multicomponent transport and convectionfluxes are shown to be compatible with the mathematical entropy, thus providing asymmetric form as well as normal forms for the reduced system [3,4]. This yields acharacterization of the mathematical structure of the system on the partial equilibriummanifold; however the singular perturbation analysis is not conducted.

20.3 Singular Perturbation Analysis for a homogeneous reactor

We then investigate the singular perturbation analysis on the simpler problem with-out hydrodynamics nor diffusion. We consider a homogeneous reactor containing agaseous mixture at constant density and internal energy, where the temperature canencounter strong variations. We consider an arbitrarily complex network of reversiblereactions, the equilibrium constant of which are compatible with thermodynamics, thusproviding an entropic structure [3]. We extend [3], established for an isothermal reac-tor, and prove the existence of a global in time solution and of an asymptotically stableequilibrium state [2]. We then assume that a subset of the reaction network is consti-tuted of fast reactions. We define a linear and constant projection compatible with theentropy production due to chemical reactions in order to obtain the reduced system onthe partial equilibrium manifold. The well-posedness and the entropic structure of thereduced system is proved and the distance between the full system and the reduced oneis evaluated through a singular perturbation analysis, globally in time [2].

20.4 Conclusion

We show that, starting from a comprehensive model describing multicomponent reac-tive flows, with an entropic structure naturally inherited from the kinetic theory, it ispossible to define a reduced chemistry model in the limit of partial equilibrium, com-patible with the entropy production. Partial equilibrium constraints are shown to belinear in the entropic variable so that the reduction of the system onto the partial equi-librium manifold can be done using a constant orthogonal projection. Finally the singu-lar perturbation analysis is conducted in the homogeneous case globally in time. Thiswork actually provides basic tools in order to understand and study the effects of smallscales in numerical methods such as operator splitting techniques for the resolution ofconvection-reaction-diffusion systems.

20.5 Acknowledgement

The present research was done thanks to the support of Universite Claude Bernard,Lyon 1, through a BQR grant (M. Massot) and to the support of a CNRS Young Investi-gator Award (M. Massot, V. Volpert).

M. Massot 88

20.6 Bibliography

[1] P. Duchene, P. Rouchon, Kinetic scheme reduction via geometric singular perturba-tion techniques, Chemical Engineering Science, 51, 20 (1996) p. 4661–4672.

[2] M. Massot, Singular perturbation analysis for the reduction of complex chemistryin gaseous mixtures using the entropic structure, Preprint No 332 of the LaboratoryMAPLY2 (2001).

[3] V. Giovangigli, M. Massot, Asymptotic Stability of Equilibrium States for Multicom-ponent Reactive Flows, Math. Mod. Meth. Appl. Sci., 8 (1998) p. 251–297.

[4] V. Giovangigli, M. Massot, Entropic Structure of Multicomponent Reactive Flowswith Partial Equilibrium Reduced Chemistry, Preprint No 326 of the LaboratoryMAPLY3 (2001).

[5] S.H. Lam and D.A. Goussis, The csp method for simplifying kinetics, Int. J. of Chem.Kin., 26 (1994).

[6] U. Maas, S.B. Pope, Simplifying chemical kinetics : intrinsic low dimensional mani-fold in composition space, Combustion and Flame, 88 (1992) p. 239–264.

[7] M.D. Smooke (Ed.), Reduced Kinetic Mechanisms and Asymptotic Approximationsfor Methane-Air Flames, Lecture Notes in Physics, Springer Verlag, 384 (1991).

[8] B. Sportisse, Contribution a la modelisation des ecoulements reactifs : reduction desmodeles de cinetique chimique et simulation de la pollution atmospherique, Ph.D. The-sis, Ecole Polytechnique (1999).

2http://maply.univlyon1.fr/publis/publiv/2001/publis.html3http://maply.univ-lyon1.fr/publis/publiv/2001/publis.html

Chapter 21

The origin of oscillations in the imitation model of CO + O2 reaction

on a palladium surface

E.S. Kurkina1 N.L. Semendyaeva1

21.1 Abstract

The present study is aimed to investigate the origin and the detailed driving mechanismof oscillations in imitation model of CO + O2 occurring on a palladium catalyst surface.The reaction of CO oxidation is one of the widely investigated processes of heteroge-neous catalysis displaying oscillatory kinetics. Experimental researches of this reactionincluding single crystal experiments and experiments with polycrystalline catalysts re-vealed different types of oscillations: regular, quasi-periodic and chaotic ones. It wasshown that the mechanism of kinetic oscillations in this reaction might be connectedwith the ability of Pd to be oxidized and reduced during the reaction. This mechanismwas first proposed in [1] for the description of kinetic oscillations observed over Pdwire. In compliance with the proposed mechanism, the mathematical model of the COoxidation reaction was constructed in [2] (the ‘STM-model’). Last years another math-ematical models taking into account the ability of oxygen to penetrate to a subsurfacelayer of palladium were suggested [3,4]. Some imitation models of this reaction dis-playing the oscillatory behavior were also developed [5]. However the detail analysisof the driving mechanism for simulated oscillations was not studied.

In the present work two models are considered to investigate the oscillations in theCO + O2 reaction: the imitation model using dynamic Monte Carlo algorithms and thedeterministic model of the ideal adsorbed layer constituted the system of three ordi-nary differential equations (ODE). Both models are based on the famous kinetic scheme[1]. The description of the kinetics of CO oxidation reaction in the stochastic modelis carried out in the framework of multi-component lattice gas model and the theoryof absolute reaction rates. A palladium surface is approximated by a two-dimensionallattice of equivalent sites. In spite of expectation the proposed mathematical models

1Department of Computational Mathematics and Cybernetics; Moscow State University, Moscow,119899, Russia.

89

E.S. Kurkina and N.L. Semendyaeva 90

demonstrate the qualitatively different behavior of the CO + O2 reaction. While thedeterministic model has one or several steady states and displays a hysteresis, the imi-tation model demonstrates oscillations in the wide range of CO pressure at the same setof parameters.

The oscillations of two types were founded and studied. In the first case there isone stable stationary state in the deterministic model which is characterized by a highconcentration of subsurface oxygen and low reaction rate. The special arrangement ofthe nullclines of the reaction kinetics in the phase space supplies the excitable dynamicsof the model. It is well known that the excitable media described by reaction-diffusionequations exhibit nonlinear traveling waves. In this case the stochastic model with finitemigration rate of adsorbed particles demonstrates concentration waves spontaneouslyarising and propagating on a catalyst surface. The spatio-temporal fluctuations in theadsorbed layer lead to oscillations. Our investigations detected that the form and theproperties of oscillations in the imitation model (Fig. 21.1) correspond to the pulse solu-tions in the reaction-diffusion model. The influence of the lattice size and the migrationrate has been studied.

In the second case there are three steady states in the system of ODEs. One stablestationary state corresponds to high CO concentration closed to 1, another stable statecorresponds to low coverage of CO molecules and high concentration of subsurfaceoxygen. The third state is unstable one of saddle type. The investigations of the phasespace of the model reveal the existence of trajectories in the nearest neighborhood ofthe second stable state, which depart the system from this state and approach it to thefirst state. Also there are trajectories not far from the first fixed point which approachthe system to the second fixed point. The oscillations are not observed in this model.However, the imitation model demonstrates the oscillation behavior under the sameconditions in the case of the absence the migration of adsorbed particles over a surface.The oscillations look like the spontaneous transitions from one stationary state to an-other one. These states correspond to the fixed points of deterministic model (Fig. 21.1).The most part of the time the system spends in the first state with high coverage of COmolecules. The superthreshold perturbation of this state causes the propagation of the‘oxygen wave’. As a result the catalyst surface is covered mainly by oxygen and thesystem approaches the second state. The small fluctuations switch the system back tothe first state. We have investigated the influence of CO pressure on a period of oscilla-tions and determined the region of such oscillations. The effect of the lattice size on theoscillatory kinetics has been studied.


This work was supported by the Russian Fund of Fundamental Researches (Grant no.00-01-00587).

E.S. Kurkina and N.L. Semendyaeva 91

Figure 21.1:

21.3 Bibliography

[1] B.C. Sales, J.E. Turner, M.B. Maple, Surf. Sci., 1982, 114, p. 381–394.

[2] J.E. Turner, B.C. Sales, M.B. Maple, Surf. Sci., 1981 109, p. 591–601.

[3] N. Hartmann, K. Krischer, R. Imbihl, J. Chem. Phys., 1994 101, No. 8, p. 6717–6727.

[4] E.S. Kurkina, E.D. Tolstunova, Appl. Surf. Sci., 182/1-2 (2001) p. 77–90.

[5] A.L. Vishnevskii, E.I. Latkin, V.I. Elokhin, Surf. Rev. Lett., 1995 2, No. 4, p. 459–469.

Chapter 22

Reaction rate is an eigenvalue: polynomial elimination in chemical

kinetics

Mark Lazman1

22.1 Introduction

Non-linearity presents one of the biggest challenges in analytical and numerical studiesof complex chemical processes.

The typical questions are: what is the number of solutions? what is an efficientmethod to find the solution of the mathematical model related to the assumed reactionmechanism? This paper answers these questions using powerful methods of algebraicgeometry. We show that the steady state kinetic problem can be treated as an eigenvalueproblem. This approach results in an efficient computational method for the analysis ofthe corresponding kinetic model.

We apply our methods to the Quasi Steady State Approximation (QSSA) algebraicsystem.

QSSA Classic chemical kinetics assumes the Mass Action Law (MAL) for the rate wof every reaction stage

w = k∏

cαi

i (22.1)

where ci is the concentration of the i-th reagent, αi is its stoichiometric coefficient, kis the reaction constant. Although it is well known that most of the systems behavenon-ideally, MAL kinetic models are widely applied. Practical methods of numericalsolution of non-ideal problems often use the MAL as approximation.

The material balance of QSSA intermediates can be presented as

w(z) = NW , L(z) = 0. (22.2)

where z is the vector of intermediate concentrations, w(z) is the vector of rates of el-ementary reactions (i.e., difference of rates in the forward and reverse directions cal-culated by MAL), L(z) is a vector of B linear balances of intermediate concentrations,

1AEA Technology Engineering Software, Hyprotech Ltd, 707–8th Avenue SW Suite 800, Calgary, Al-berta, Canada T2P 3V3; e-mail: [email protected]

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M. Lazman 93

the matrix N is composed of P = S − (J − B) vectors of stoichiometric numbers νps ,

(s = 1, . . . , S; p = 1, . . . , P ), S is the number of reactions, J is the number of intermedi-ates. Each v column vector is called a reaction path; vector W is composed of P reactionpath rates. Linearly independent reaction paths form the stoichiometric basis. System(22.2) is the null-space representation of the standard QSSA equations.

22.2 Kinetic polynomial

The MAL steady state (or quasi-steady-state) model is a system of multivariable poly-nomial equations. The solution of polynomial systems is the subject of elimination the-ory, developed by Bezout, Sylvester, Cayley, Macaulay, Kronecker, Hurwitz. The recentrenaissance of almost forgotten elimination theory is concerned with the Newton Poly-hedra approach initiated by works of Arnold’s seminar and developed further in [1].

Polynomial systems allow variable elimination. Our previous studies [2] were con-cerned with the understanding of the resultant of system (22.2) in the reaction path rateW . The resultant (kinetic polynomial) is a polynomial in terms of W . The vanishing ofresultant is necessary and sufficient condition of system (22.2)’s solvability. The resul-tant in terms of W allows QSSA representation in terms of the experimentally measur-able variable W . This equation is symmetric in terms of the reaction parameters and ithas a thermodynamic interpretation. We have proved that the resultant’s constant termalways contains the multiplier (cyclic characteristic)

C =S∏

s=1

f νs

s −S∏

s=1

rνs

s (22.3)

where fs, rs are the reaction weights (i.e., reaction rates at unit intermediate concentra-tions) of the s-th reaction in the forward and reverse directions. The equation C = 0 isequivalent to the thermodynamic equilibrium condition for net reaction. The stoichio-metric numbers entering formula (22.3) are mutually prime (i.e., the resultant’s constantterm corresponds to the net reaction equation obtained with minimal integral stoichio-metric numbers). The further development of theory resulted in explicit formulas forall resultant coefficients. This approach allowed a computer algebra implementation[3]. Variable elimination proved to be an effective tool in applications ranging from theinverse kinetic problem to bifurcation analysis [1,3].

22.3 Multidimensional resultant formulation

The Bezout theorem gives the simplest estimate on the number of all (complex) isolatedzeroes of system (22.2). Define the reaction order d as the maximum of the orders offorward and reverse reactions. Let us assign the index µ to the elementary reactionwith non-zero stoichiometric coefficient that has smallest reaction order (i.e., the minor

M. Lazman 94

reaction). Let

Lµ =S∏

s6=µ

ds.

For a single-path mechanism we have:

Proposition 1 If P = 1 number of system (22.2) complex zeroes in is lesser or equal Lµ.

In the general case we have

Proposition 2 If P > 1 number of system (22.2) complex zeroes is lesser or equal Lµ, wheresubscript m is defined as

µ = arg maxp

mins

(ds : νps 6= 0).

The estimate in Proposition 2 depends on a particular stoichiometric basis. There existsa basis that produces the sharpest estimate.

The explicit resultant approach (for instance, kinetic polynomial) reduces the solu-tion of system (22.2) to the root finding of a single polynomial in one unknown. Otherunknowns can be found either by using resultant properties or by iteration of the elim-ination procedure. Groebner bases can be applied too. Implementation of all theseapproaches requires computer algebra. Technically, it results in a slow, unstable, andunreliable procedure. Heterogeneous software requirements do not help effective im-plementation too.

However, no explicit resultant expression is required to solve the system (22.2) nu-merically. A multidimensional resultant matrix can be built instead. This matrix formu-lation allows the solution of system (22.2) by pure linear algebra. The solution of system(22.2) is reduced to an eigenproblem. In particular, the reaction rate can be found as so-lution of an order Lµ eigenproblem. Stable floating point methods of numerical linearalgebra can be applied here instead of computer algebra.

Our matrix formulation follows the classical Macaulay method. It allows the locationof all roots counted by Bezout theorem.

The following example illustrates the matrix and its application to the solution ofsystem (22.2).

22.4 Example

One of the possible Macaulay matrices for a two stage impact mechanism system

22.4f1z21 − r1z

22 = W, f2z2 − r2z1 = 2W, z1 + z2 = 1, (22.4)

is

M(W ) =

0 r2 −f2 2f1 0 −2r1−1 1 1 0 0 00 −1 0 1 1 00 0 −1 0 1 1

2W r2 −f2 0 0 00 2W 0 r2 −f2 0

.

M. Lazman 95

The condition detM(W ) = 0 is necessary for W to be a zero of system (22.4). We canrewrite it as

(A+WB)u = 0, (22.5)

where A and B are (6 × 6) matrices corresponding to the representation of M(W ) asa sum of a matrix that depends on W and a matrix without W terms; the vector u isnonzero (6× 1) column vector.

Condition (22.5) is a generalized eigenproblem in the reaction rate W . It can bereduced to the regular eigenproblem by matrix operations. The reaction rate W canbe found as a solution of an order 2 eigenproblem derived from matrix M(W ), whicheigenvalue is λ = −1/(2W ). The (2× 2) matrix is

1

2(r1r22 − f1f 2

2 )

(

(r2 + f2)2 + 4r1r2 + 2f2(f1 + f2) 2(f1 − r1)(r2 + f2)

(r2 + f2)(f2 + 2r1) 2f2(f1 − r1)

)

.

22.5 Cayley trick, circuits and cyclic characteristic

Although the classic Macaulay method allows the effective solution of system (22.2), themost promising approach is concerned with the sparse formulation [1]. Our first resultin this direction is the new proof of the kinetic polynomial’s thermodynamic property(22.2) based on A-discriminant theory [1]. The Cayley trick reduces the problem to theanalysis of the discriminant of the specific polynomial. The monomials of this polyno-mial form circuit. This circuit explains the appearance of the cyclic characteristic (22.3)in the constant term of the kinetic polynomial.

22.6 Conclusions

The steady state kinetic problem can be reduced to an eigenvalue problem. This ap-proach allows an efficient computational method for the analysis of the correspondingkinetic model. New estimates on the number of zeroes of QSSA equations are presented.A geometric interpretation of the QSSA resultant’s structure is found.

22.7 Bibliography

[1] I.M. Gelfand, M.M. Kapranov, A.V. Zelevinsky. Discriminants, Resultants and Multi-dimensional Determinants, Birkhauser, Boston, 1994.

[2] M.Z. Lazman, G.S. Yablonskii. Kinetic polynomial: a new concept of chemical kinet-ics, Patterns and Dynamics in Reactive Media, Springer-Verlag 1991, p.117

[3] V.I. Bykov, A.M. Kytmanov, M.Z. Lazman, Elimination methods in polynomial computeralgebra, Kluwer Academic Publishers, 1998.

Chapter 23

The principle of critical simplification in chemical kinetics: a case

study

G.S. Yablonski1 I. Mareels2 M. Lazman3

23.1 Abstract

We identify a simple methodology by which kinetic mechanisms and reaction rate pa-rameters may be readily identified from an experimentally obtained bifurcation dia-gram (reaction rate vs. reaction parameter). The method exploits the presence of a dom-inant competitive reaction mechanism as well as critical phenomena.

23.2 Introduction

Understanding critical phenomena in chemical kinetics, such as the multiplicity ofsteady states, oscillation, chaotic behavior, is still an attractive area for the developmentof physicochemical principles.

In fact the existence of critical phenomena, in particular the multiplicity of steadystates in chemical systems is well known, with contributions since the 1930s. The re-sults obtained in the this field by Frank-Kamenetskii, Zel’dovich and Semenov are sum-marized in the seminal monograph by Frank-Kamenetskii [1]. For example, it is wellknown that in a non-isothermal continuously stirred reactor (CSTR) the S-shaped de-pendence of the rate of heat generation on the temperature and the linear dependenceof the heat removal on temperature, leads naturally to several distinct operating points.More surprising was the observation that such critical effects could actually be foundin isothermal reaction systems. An outstanding role was played here by the discoveryof self-oscillatory reaction mechanisms of Belousov-Zhabotinski. This reaction served

1Department of Chemical Engineering, Washington University in St Louis, D. Campus Box 1198, OneBrookings Drive, St. Louis, MO 63130-4899, USA.

2Department of Electrical and Electronic Engineering, University of Melbourne, Melbourne, VIC 3010,Australia.

3AET, Engineering Software, Hyprotech, Calgary, Canada.

96

G.S. Yablonsky, I. Mareels and M. Lazman 97

as the starting point for the development by Prigogine of the ‘new thermodynamics’.Prigogine and his co-workers [2] proposed and analyzed the so called ‘Brusselator’and ‘Oregonator’ models which made it possible to account for the appearance of self-oscillations of the reaction rate in isothermal regime. The history of the theoretical andexperimental results in this area are presented in great detail in the book [3], see also [4]for a more recent overview. The following questions play a key role

• Under what circumstances, expressed as in domains of parameters, can multiplesteady state solutions co-exist. What is the multiplicity of solutions? Where doesthis multiplicity change?

• How do the characteristics of a steady state depend on the parameters?

• How can system properties and parameters be readily related to simple observa-tions from system behavior (experiments)?

The goal of this paper is to formulate one possible approach, based on the so calledprinciple of critical simplification. It is our opinion that this principle will prove tobe useful in the analysis of complex dynamical behavior of chemical systems, as forexample represented by mass-action-law models. This paper is a further developmentof ideas first proposed by Lazman and Yablonski in the early 80’s. The idea of criticalsimplification was first enunciated in 1996 by Yablonski and Lazman [5], using a modelof carbon monoxide oxidation over platinum. From a mathematical point of view theprinciple is a powerful combination of singular perturbation (reduces complexity) andbifurcation analyses (identifies critical phenomena).

23.3 The principle of critical simplification

Under some mild assumptions, which will be discussed shortly, the principle of criticalsimplification can be formulated as:

At the point of critical change in kinetic behavior, ignition or extinction, the rate of thechemical transformation is uniquely determined by the kinetic parameters of a particular set ofslow reactions.

The boundary of the critical domain is determined by, and determines, the differentslow reaction kinetic parameters.

The assumptions under which the principle holds are

1. There is at least one non-trivial (nonlinear) interaction between two intermediates.This reaction is considered fast as compared to the other reactions, otherwise thereis no simplification.

2. If the remaining reaction mechanisms (after elimination of the fast reaction fromthe kinetics) are linear or more general monotonic then there is a unique equilib-rium and no critical phenomena appear.

Some remarks are in order


Remark 1.

The first assumption indicates an important simplification in the kinetics. The fast in-teraction mechanism does not determine the steady state behavior, but only affects thetransients. The last assumption is obvious in that critical phenomena are only possiblein a non-linear context. Thus besides the fast nonlinear interaction one other non-trivialnonlinear reaction mechanism must be present.

Remark 2.

The term critical simplification is coined to indicate that at criticality, the reaction mech-anism is essentially determined by a simple mechanism in the overall chemical kinetics.Moreover, given an observation of the critical behavior, the kinetic parameters associ-ated with the mechanism can be uniquely determined from this observation.

23.4 Mass action model with two reactants and one product

In this section we discuss the simplest possible case, a mass action model with tworeactants and one product. We assume that there is a fast competitive reaction betweenthe reactants together with one more nonlinear reaction mechanism. This validates theuse of the principle of critical simplification.

Let the concentration of the reactants be represented by x, y ∈ (0, 1) and the producthas concentration 1− x− y. The chemical kinetics are represented by

d

dt

(

xy

)

=

(

f(x, y)g(x, y)

)

=

(

(1− x− y)p1 − a1xp2 + a2y

p3 − dxpyq

a3(1− x− y)p4 + a4xp5 − a5y

p6 − dxpyq

)

. (23.1)

The coefficients ai, i = 1, . . . , 5 are positive. All exponents are positive integers, pi ∈1, 2, i = 1, . . . , 6. It is assumed that d >> 1 is much larger than any of the othercoefficients. The parameter d represents the reaction rate of the competitive term.

Notice that the term (1− x − y)p1 does not involve a coefficient. This can always beachieved via an appropriate re-parameterization of time. In effect the rate at which theproduct with concentration (1 − x − y) is formed determines the clock with which wetime the reaction.

It is easily verified that under the stated conditions, the dynamics (23.1) leaves thephysically relevant part of the state space x > 0, y > 0 and (1 − x − y) > 0 positivelyinvariant.

The equation (23.1) represents a chemical reaction mechanism involving two reac-tants with concentrations x, y, with fast interaction dynamics (the competition termbeing the dxpyq term). The outcome of the reactions is a product with concentration(1−x−y). Both x and y are consumed in the formation of this product, at different ratesa1 > 0 and a5 > 0, the product decomposes to replenish the reactants as do the reactantsthemselves a2 > 0, a4 > 0 and a3 > 0.


Because the fast interaction term will dominate the reaction, for as long as the prod-uct xy is nonzero, it can expected that the dynamics can be decomposed in a fast conver-gence to the surface xy = 0, followed by a slow (as compared to the competition term)settling into an equilibrium. The relevant slow dynamics in the chemical reaction areconfined to the surface xy = 0, 0 < x < 1, 0 < y < 1. In this situation, the slow dynamicsare one-dimensional, and hence asymptotically only equilibria are feasible. Moreover,either x or y will be zero at the equilibrium, indicating that either one of the reactants isdepleted.

In order to study the slow dynamics, those dynamics not affected by the fast compe-tition, consider the coordinate transformation

w = xy, z = x− y.

On the physical domain x, y, (1− x− y) ∈ (0, 1) this is a one-to-one mapping.The dynamics of the z variable become independent of d and will describe the slow

dynamics. The latter are given by:

d

dtz = f(x, y)−g(x, y) = (1−x−y)p−1−a1x

p2 +a2yp3−a3(1−x0y)p4−a4x

p5 +a5yp6, (23.2)

where we should appropriately replace x, y by w, z.The presence of the interaction term indicates that w disappears fast, and hence the

(relevant) slow dynamics are either described by

d

dtx = f(x, 0)− g(x, 0) = (1− x)p1 − a1x

p2 − a− 3(1− x)p4 − a4xp5 , (23.3)

(y = 0, and z = x ∈ (0, 1)) or

− d

dty = f(0, y)− g(0, y) = (1− y)p1 + a2y

p3 − a3(1− y)p4 + a5yp6, (23.4)

(x = 0, and −z = y ∈ (0, 1)) depending on which component x or y disappears from thereaction mechanism.

Remark 3.

Notice that the equation (23.3) does not leave the set x ∈ (0, 1) invariant, nor does theequation (23.4) leave the set y ∈ (0, 1) invariant. The invariance is with respect to thevariable z, z = x − y ∈ (−1, 1), not with respect to its local representation, x or y. Ifthe dynamics in (23.3), which are valid for z = x, indicate that x, or better z wouldbecome negative, one has to interpret this as x goes to zero, and z becomes z = −y. Itis easily verified that z = x − y ∈ (−1; 1) is an invariant set, as (dz/dt)z=x=1 < 0 and(dz/dt)z=−y=−1 > 0.

The slow reaction mechanism described in either equation (23.3) or equation (23.4)can now be further analyzed using relatively simple bifurcation ideas to delineate the


Figure 23.1: Bifurcation parameter plane p1 = 2.

multiplicity of equilibria (the only feasible asymptotic behavior in this situation). A typ-ical bifurcation parameter plane is given in Fig. 23.1. Fig. 23.2 represents a bifurcationdiagram we can use for experiment design or experiment interpretation. The Figurescorrespond to the situation with a single nonlinear term p1 = 3 (all other pj = 1 forj 6= 1)

The relevance of the bifurcation analysis after the singular perturbation is that we areable to infer from the diagrams as Fig. 23.1 and Fig. 23.2, not only qualitatively, but evenquantitatively, what happens in the original, much more complex reaction mechanisms.This is the real power of the idea of critical simplification. Moreover the bifurcationdiagrams provide us with a powerful tool to exploit experimental information to eithercharacterize the reaction mechanism, or reaction rate parameters.

23.5 Generalization

The ideas can be adapted to discuss reaction mechanisms involving several fast reac-tions as well as slow reactions. The remaining slow reaction mechanisms, of which


Figure 23.2: Bifurcation diagram a3 versus rate.


several different versions may exists, depending on which fast reaction mechanismshave been eliminated, can all be subject to a bifurcation analysis to reveal the criticalpoints. How this works, and how the analysis of the simplified reaction mechanismsreveals behavior of the complex dynamics, will discussed in the full paper.


The first two authors wish to acknowledge the support from the National Universityof Singapore, where both authors spent a sabbatical, respectively at the Department ofChemical and Environmental Engineering and The Department of Electrical and Com-puter Engineering. Without NUS’s support this collaboration would not have existed.

23.7 Bibliography

[1] D.A. Frank-Kamenetskii, Diffusion and heat transfer in chemical kinetics, 4th edition,Nauka, Moskva, 1987, (Plenum Press, New York, 1969)

[2] G. Nicolis, I. Prigogine, Self-organisation in non-equilibrium systems, Wiley-Interscience, New York, 1977

[3] G.S. Yablonski, V.I. Bykov, A.N. Gorban, V.I. Elokhin, Kinetic models of catalytic reac-tions, Comprehensive Chemical Kinetics, Vol 32, Elsevier, Amsterdam, New York, 1991.

[4] G.S. Yablonski, V.I. Elokhin, Kinetic models of heterogeneous catalysis, Perspective ofCatalysis, ed. Zamaraev, Thomas Blackwell Science, 1992.

[5] G.S. Yablonskii, M.Z. Lazman, New correlations to analyse isothermal critical phe-nomena in heterogeneous catalytic reactions, ‘critical simplification’ and ‘hysteresisthermodynamics’, iReact. Kinet. Catal. Lett. 56 no. 1, p. 145–150, 1996.

Chapter 24

Stochastic model of reaction rate oscillations during CO oxidation

over zeolite supported Pd catalysts

N.V. Peskov1 M.M. Slin’ko2 N.I. Jaeger3

24.1 Introduction

One of the most intensely studied oscillatory systems is the CO oxidation on Pd,whichhas been studied under UHV conditions over single crystal surfaces [1] and at atmo-spheric pressure over zeolite supported Pd catalysts [2]. The microporous structure ofa zeolite host allows the preparation of various metal dispersions with narrow particlesize distributions within the matrix and recently the dependence of the dynamic behav-ior of the catalytic CO oxidation upon the size of Pd particles has been reported [2].

The goal of the paper is to develop a model key elements of which will be (i) thedescription of the dynamic behavior of CO oxidation over a single nm-sized particle,(ii) the consideration of the experimentally observed dependence of the reaction rateupon the particle size and (iii) the inclusion of intrinsic fluctuations, since due to thesmall number of surface atoms and reactant molecules an effect upon the reaction ratecan be expected.

24.2 Particle size effect

The effect of the size of the palladium crystallites on the activity and the dynamic be-havior of the catalysts has been studied under shallow bed conditions in a CSTR. Theactivity and the dynamic behavior of the system have been analyzed under similar ex-perimental conditions for pre-oxidized catalysts with the same Pd loading, equal to

1Department of Computational Mathematics & Cybernetics, Moscow State University, 119899Moscow, Russia.

2Institute of Chemical Physics, Russian Academy of Science, Kosygina Str. 4, Moscow 117334, Russia.3Institut fur Angewandte und Physikalische Chemie, FB 2, UniversitŁt Bremen, PF 330440, 28334

Bremen, Germany.

103

N.V. Peskov, M.M. Slin’ko and N.I. Jaeger 104

0.05%, the same surface area of 15.1cm2 but different size of the Pd particles. For cat-alyst A the size of Pd particles is equal to 10nm, while for catalyst B the size of Pdparticles is equal to 4nm. It was demonstrated that:

• In the case of catalyst B with the smaller Pd particles the activity is higher, the re-gion of oscillations more extended and the amplitudes larger compared to catalystA loaded with 10nm Pd particles.

• The size of the Pd particles has a significant influence upon the long transientperiods during which a stationary oscillatory state is attained. The reaction rateslowly increases during the slow reduction of the catalyst and the transient periodis shorter in the case of 4nm particles as compared to 10nm particles.

• While for a small region of CO inlet concentration (0.3 − 0.32%) regular periodicoscillations could be established in the case of catalyst A, this was not possible forcatalyst B containing 4nm particles for any region of CO concentrations.

24.3 Point model for the reaction on a catalyst particle

The point model was developed on the basis of the model proposed by Sales, Turnerand Maple (STM) [3], which was modified to consider the formation not only of thesubsurface oxygen, but also processes of oxidation and reduction involving oxygen inthe bulk of the Pd particle. The variable w, denoting the concentration of the bulk oxidewas introduced basing on the following assumptions:

1. The surface reaction between chemisorbed CO and O can occur only over non-oxidised Pd. This is considered by the factor (1−w)2 in the rate of CO+O reaction.

2. The process of oxidation involves the entire bulk of a particle and the increase ofthe concentration of the bulk oxygen is proportional to the oxygen pressure PO2

.

3. The rate of the reduction process is supposed to be proportional to the CO partialpressure and to the factor s, which denotes the ratio of the number of surfaceatoms to the total number of atoms in the particle.

The variation of surface coverages with CO (x), with O (y), the concentration of thesubsurface oxygen O* (z) and the fraction of the bulk oxide (w) can be described by thesystem of equations (24.1):

x = PCOk1(1− x− y)− k−1x− k3(1− w)2xy − k5xz

y = PO2k2(1− x− y)2(1− w − z)2 − k3(1− w)2xy − k4y(1− w − z)

z = k4y(1− w − z)− k5xz

w = PO2k6(1− w)− k−6PCOsw. (24.1)

Assuming that the Pd particles have the shape of an octahedron form, the factor s wascalculated for particles of 4nm, s4 = 0.451 and for particles of 10nm, s10 = 0.221.


24.4 Mesoscopic stochastic model

The number of CO and O adsorption centres on the particle surface varies fromNs = 400to 10000 with the increase of the particle size from 4nm to 20nm. During the oscillatoryregime the number of adsorbed atoms of each species NCO and NO, and the number ofsubsurface oxygen atoms NO*, can vary in the range between 0 and Ns. At any momentof time these numbers can be regarded as a set of random values with unknown proba-bility distribution P (NCO, NO, NO*; t). In this case the equations (24.1) may be consideredas the equations for averaged values NCO, NO, and NO* divided by Ns. However, theseequations can be accurate only in the thermodynamic limit when Ns → +∞. At finiteand not so large values ofNs statistical fluctuations of random numbers of species atomscan be significant and equations (24.1) may become incorrect. In order to take into ac-count a finite number of species atoms participating in the reaction and their statisticalfluctuations as well as their mutual correlations a Markovian model is proposed.

The Markovian model simulates the process for the steady state value of the con-centration of the bulk oxide. Therefore it has three discrete variables NCO, NO, andNO* that can take nonnegative integer values in the range [0, Ns], and NCO + NO ≈ Ns,NO* ≈ (1 − w)Ns. The elementary steps in the reaction scheme correspond to the ac-ceptable transitions in the Markovian model. The CO oxidation on the catalyst particleis simulated as a random sequence of acceptable transitions under standard conditions,which imply that multiple transitions are forbidden, any transition can occur at an arbi-trary moment of time and is instantaneous. Each transition has its own probability thatdepends only upon the current state of the model.

We define the transition probability per unit time, or transition rate n and then theprobability of transition during the time interval ∆t will be equal to n∆t. For compati-bility with the model (24.1) the transition rates are defined as follows:

v1(N) = k1PCO(Ns −NCO −NO)

v−1(N = k−1NCO

v2(N) = k2PO2

(

1− w − NO*

Ns

)2 1

Ns

(Ns −NCO −NO)(Ns −NCO − (NO + 1))

v3(N) = k3(1− w)2NCONO

Ns,

v4(N) = k4

(

1− w − NO*

Ns

)

NO

v5(N) = k5NCONO*

Ns,

where N is the triple NCO, NO, NO*, which defines the current state of the model.Under the conditions described the time evolution of the state probability distribu-

tion P(N; t) will be ruled by the master equation

dP(N; t)

dt= −

5∑

i=−1

vi(N)P(N; t) +5∑

i=−1

vi(N(i))P(N(i); t), (24.2)


where N(i) denotes the state, from which the state N can be reached via one i-th transi-

tion. Note that with the standard averaging technique the evolution equations for theaverage values of the model variables can be derived from equation (24.2). For example,the equations for x(t) = 〈NCO/Ns〉 have the form

x = k1PCO(1− x− y)− k−1x− k3(1− w)2xy − k5xz

− 1

N2s

[k3(1− w)2〈(NCO − 〈NCO〉)(NO − 〈NO〉)〉 − k5〈(NCO − 〈NCO〉)(NO* − 〈NO*〉)〉

where the angular brackets 〈a〉 denote the mathematical expectation of a random valuea. Thus one can see that the equations for the average values of the Markovian modelconsist of all the terms of the deterministic equations (24.1) and additional correlationterms. In the limit Ns →∞ the correlation terms are assumed to vanish, but at finite Ns

they can play an essential role in the dynamics of the system.The trajectories of the Markovian process governed by Eq. (24.2) are generated with

the help of one of the Monte Carlo algorithms with continuous time.

24.5 Conclusions

The deterministic model (24.1) is able to simulate important experimental trends,namely the dependence of the catalytic activity and the waveform of the oscillationsupon the particle size and the pretreatment of the catalyst as well as the larger regionof oscillations for the catalyst B with smaller Pd particles. The higher activity of thesmaller particles can be explained by the attainment of a more reduced state of the Pdin smaller particles in the course of the reaction.

The results of simulations with a stochastic model demonstrate a clear differencein the oscillatory behaviour of the reaction rate for particles of various sizes. For thesame parameters and experimental conditions the region of oscillations for 4nm sizeparticles is larger than for 20nm particles due to the much larger effect of internal fluc-tuations upon in the case of the small particles. The comparison of oscillations for 4nmand 10nm particles demonstrates that 4nm sized particles produce more complex andirregular oscillations in a larger region of CO partial pressures compared to 10nm par-ticles in accordance with the experimental data. The cause of the drastic increase of theoscillatory region for the catalyst with the smallest particles is connected to the pres-ence of noise-induced oscillations, i.e., oscillations which can be produced only by thestochastic model and that are absent in the deterministic limit.

24.6 Bibliography

[1] M.R. Basset and R. Imbihl, J. Chem. Phys., 1990, 93, 811.

[2] M.M. Slin’ko, A.A. Ukharskii, N.V. Peskov and N.I. Jaeger, Cat. Today, 2001, in press.

[3] B.C. Sales, J.E. Turner and M.B. Maple, Surf. Sci., 1982, 114, 381.

Chapter 25

Analytic and Numeric Methods in Microkinetic Modeling

De Chen1 Rune Lødeng2 Erlend Bjørgum1 Kjersti Omdahl1 Anders Holmen1

25.1 Abstract

Microkinetic analysis is an examination of catalytic reactions in terms of elementarychemical reactions that occur on the catalytic surface and their relation with each otherand with the surface during a catalytic cycle [1]. Microkinetics has, for the most part, fo-cused on analysis for understanding the reaction mechanism. The approach, however,also holds the promise of being used as an aid in the synthesis of new materials. Themicrokinetic approach has been applied in many catalytic processes, and it has recentlybeen reviewed by Stoltze [2], and Broadbelt and Snurr [3]. It has been shown that mi-crokinetic modeling based on knowledge about elementary steps and their energetics,is a very powerful tool for a detailed understanding of catalytic processes.

The starting point for microkinetic modeling is a detailed reaction mechanism, basedon studies of surface science, theoretical calculations and others. Thus, while a conven-tional kinetic model is formulated as the rate for an apparent gas-phase reaction, the sur-face species are explicitly included in a microkinetic model. However, for a proposed re-action mechanism, stoichiometric and thermodynamic consistency must be fulfilled. Itis quite normal that many elementary reaction steps are involved in the reactions, evenfor a very simple gas-phase reaction, and consistency tests are therefore not straight-forward. Hence, it would be highly desirable if the formulation and investigation of amicrokinetic model for a proposed reaction mechanism could be done automatically.

The reaction mechanism is represented by the adsorption steps of gaseous compo-nents, surface reaction steps and desorption steps of intermediates. The microkineticmodel is a mathematical model for rates of consumption or formation of gaseous com-ponent and intermediates. If the reaction mechanism consists of G gaseous componentswith E elements, I intermediates and site vacancies, the microkinetic model shouldconsist of G− E reactor design equations (ordinary differential equations for plug flow

1Department of Chemical Engineering, Norwegian University of Science and Technology (NTNU),N-7491 Trondheim, Norway.

2SINTEF Applied Chemistry, N-7465 Trondheim, Norway.

107

De Chen, R. Lødeng, E. Bjørgum, K. Omdahl and A. Holmen 108

reactor, (ODE)), E elemental mass balance equations, I − 1 steady state equations andone site conservation equation. Consequently, there are G−E ODE equations and E+Ialgebraic equations in the microkinetic model. The rates of adsorption, surface reac-tion and desorption have usually rather different orders of magnitude. In addition, theresidence time of intermediates on the surface has often very different time scale. Thistypically results in difficult solvable stiff differential algebraic equations (DAE). A so-phisticated mathematical method is therefore needed in order to obtain fast solutions ofthe microkinetic model.

The present work deals with mathematics in the formulation and investigation ofa proposed reaction mechanism, including stoichiometric consistency test, calculatingrates with thermodynamic consistency, as well as the mathematical method in solvingstiff DAEs. The microkinetic modeling of steam reforming is selected as a case studyin the present work. Steam reforming on Ni catalysts plays a key role in synthesis gasand hydrogen production from natural gas. Aparicio [4] has developed a microkineticmodel for the steam reforming over a Ni(25wt%)/MgO − MgAl2O4 catalyst. Recently,we developed an extended microkinetic model with energetics of elementary reactionsteps based on the Bond Order Conservation-Mose Potential (BOC-MP) theory [5] forsteam methane reforming including carbon formation and deactivation [6,7].

The reaction mechanism can be represented by the matrix equation

AX = 0 or

a1,1 a1,2 · · · a1,G a1,G+1 a1,G+2 · · · a1,n−1 a1,n

a2,1 a2,2 · · · a2,G a1,G+1 a1,G+2 · · · a1,n−1 a1,n...

......

......

. . ....

...am,1 am,2 · · · am,G am,G+1 am,G+2 · · · am,n−1 am,n

X1

X2...Xn

= 0,

(25.1)whereX is the matrix of species including reactants and products in gas phase, interme-diates on the surface and site vacancy, A is the stoichiometric matrix, m is the numberof the elementary reaction steps and n is the total number of species (n = G+ I + 1).

As all the reaction steps have been written as adsorptions, surface reactions or des-orption and as the reaction mechanism must be cyclic with respect to adsorbates, theanswer on whether the mechanism is stoichiometrically consistent depends then onwhether the following equation has a solution:

A−1s β = 0 or

a1,G+1 a1,G+2 · · · a1,n−1

a2,G+1 a2,G+2 · · · a2,n−1...

.... . .

...am,G+1 am,G+2 · · · am,n−1

β1

β2...βm

= 0 (25.2)

where βr is the stoichiometric number of reaction r. If the rank number of the matrixA−1

s equals m, the equation has a unique solution.The rate equation for the elementary reaction step j is described by mass action ki-

netics, where the preexponential factor can be estimated by transition state theory, andthe activation energy can be obtained from reported experimental values or from the-oretic calculations. In the microkinetic model for steam reforming reported previously,


the activation energies were estimated by a BOC-MP theory. However, the numericalvalues of the kinetic parameters are constrained by two thermodynamic relations [1].

∑

i

σi(Ei,for)−∑

i

σi(Ei,rev) = ∆H0net, (25.3)

where the σi are the stoichiometric numbers of the elementary steps in the reaction path,and

∏

i

(

Ai,rev

Ai,for

)σi

= exp

(

∆G0net −∆H0

net

RT

)

(25.4)

where ∆H0net and ∆G0

net are the standard heat and free Gibbs energy of the net reaction,which can be represented by the matrix equation

BX = 0. (25.5)

Normally, the kinetic values estimated theoretically are not good enough to perfectlymeet the thermodynamic consistency. At least Ai and Ei of one of the steps should beadjusted based on equations (25.4) and (25.5) to fit the consistency. However, for a com-plex reaction mechanism it is not straightforward to find the stoichiometric numbers(σi) of the independent elementary steps in the reaction path from reactants to prod-ucts. The following equation can be used to solve these stoichiometric numbers:

A−1s = Bi (25.6)

whereBi is the stoichiometric matrix for a given net reaction in which the stoichiometricnumbers are set to be zero for the adsorbates and site vacancy. The Equation (25.6) beinga set of linear algebraic equations can be solved by Gauss Elimination method.

The reaction rate is calculated as turnover frequency. The rate for a reaction stepincluding forward and reverse reaction rR is:

rR,j =n∑

i=1

aijAi exp(−E/(RT ))G∏

j=1

paij

j

G+I+1∏

j=G+1

θaij

j (25.7)

The net reaction rates for all the species are:

R = A−1rR =

[

RG

RI

]

, (25.8)

whereRG andRI are matrix of reaction rate for terminal gaseous reactants and products,and intermediates, respectively.

As mentioned above, the surface coverages and gaseous composition can be calcu-lated by simultaneously solving steady state equations (algebraic, Eq. 25.9) and reactordesign equations (ODE, Eq. 25.10).

dθ

dt= RI = 0 (25.9)


dx

d(W/F0)= ρARG, (25.10)

where θ and x are the matrix of surface coverage and conversion, ρA is the density ofactive sites, W is the amount of catalyst, and F0 is the flow rate of the reactant. It isoften faster to solve a stiff system of ODEs and thus it can be useful to convert a systemof DAEs to ODEs. A concept of ‘flowing surface species approximation’ (8) is usedin the present work to overcome the difficulties in solving stiff DAEs. The Eq. 25.9 istransferred to:

dθ

d(W/F0)= ρARI . (25.11)

By this way the stiff DAEs are transferred to stiff ODEs. Eqs. (25.9) and 25.11 can theneasily be solved by the solver ode15s in MATLAB. Simulation using a series CSTR modelcombined with Eq. (25.9 was also performed, and the simulation results were used asdata base for an evaluation of the ‘flowing surface species approximation’. A CSTRnumber of 100 was used in the present work. The results showed that both approachesgave almost identical gaseous composition and surface coverage, except for CPU time.The CPU time for the ‘flowing intermediates approximation’ approach is only 3 seconds,while it is about 2–5 hours for the series CSTR model.

It can be concluded that the conversion of a set of DAEs to ODEs using ‘flowing sur-face species approximation’ results in considerable time savings without loss of accu-racy. The stoichiometric and thermodynamic consistency can easily be made by simplematrix calculation, which can also be implemented in MATLAB.

25.2 Bibliography

[1] J.A. Dumesic, D.F. Rudd, L.M. Aparicio, J.E. Rekoske and A.A. Trevino, The microki-netics of heterogeneous catalysis, ACS, Washington, 1993.

[2] P. Stoltze, Prog. Surf. Sci., 65 (2000) 65.

[3] L.J. Broadbelt and R.Q. Snurr, Appl. Catal., 200 (2000) 23.

[4] L.M. Aparicio, J. Catal., 165 (1997) 262.

[5] E. Shustorovich, Adv. Catal., 37 (1990) 101.

[6] D. Chen, R. Lødeng, A. Anundskas, O. Olsvik and A. Holmen, Chem. Eng. Sci., 56(2001) 1376.

[7] D. Chen, R. Lødeng, K. Omdahl, A. Anundskas, O. Olsvik and A. Holmen, Stud.Surf. Sci. Catal., 139 (2001) 93

[8] P.V. Joshi, A. Kumar Tahmid, T.I. Mizan and M.T. Klein, Energy & Fuels, 13 (1999)1135

Chapter 26

Asymptotic Analysis and Flow through a Network of Channels

W. Jager1

26.1 Abstract

This lecture gives a survey on recent results obtained by methods of asymptotic analysisin deriving model equations for flow through media with special microscopic structuresfor instance through filters or through systems of thin channels, in channels with roughor porous boundaries. Multi-scale techniques are applied to derive from microscopicstructures macroscopic equations approximating the flow. The parameters of the ef-fective equations can be computed from the microscopic cell problems. Two situationsof deriving model equations will be discussed: homogenisation and reduction of di-mension. The analysis of multi-scale convergence and of layers arising at interfaces,boundaries or bifurcations is playing an important role. The derivation of effectivetransmission or boundary conditions is studied. Problems arising in flow and trans-port through systems of thin channels with solid or flexible walls are discussed. Thelecture is mainly based on results obtained in joint work with A. Mikelic and researchin the Applied Analysis Group in Heidelberg.

1Universitat Heidelberg, Interdisziplinares Zentrum fur Wissenschaftliches Rechnen, Im Neuen-heimer Feld 294, D-69120 Heidelberg.

111

Chapter 27

Nonlinear Parameter Estimation from Single and Multiresponse Data.

Discrimination and Criticism of Mechanistic Models Optimal

Experimental Design for Model Discrimination and Parameter

Estimation. Theory and Demonstrations.

M. Caracotsios1 K. Vanden Bussche1

27.1 Abstract

Process investigations often generate sets of mechanistic models as well as experimentaldata for testing the adequacy of the proposed candidates. Scientific advancement via amechanistic model is particularly useful in the development of new processes. Thechances of meaningful extrapolations with mechanistic models are far greater than theempirical model approach. The process of building and selecting a mechanistic modelinvolves the following general stages (a) identification (b) numerical solution of theproposed model combined with parameter estimation and (c) diagnostically testing theadequacy of the fit and model candidate.

In our paper we discuss methods for modeling experimental data by emphasizingparameter estimation strategies and demonstrating the concepts with available soft-ware. Our experiences in multiresponse modeling are reviewed for several chemicalengineering problems that encompass a wide range of lumped and distributed param-eter systems. We are also going to review state-of the art numerical algorithms for therobust integration of such systems

Our paper then continues with a rigorous treatment of criteria used to identify apreferred model and test its adequacy. Our analysis will focus on the following ques-tions: Which model from a set of candidate models is the most probable according to thedata. Second, do any of the models represent the data adequately. Finally how can aninvestigator design new experiments for parameter estimation and for discriminationamong mechanistic models. Our review will consist of rigorous theoretical concepts

1UOP LLC, Engineering Science Skill Center, 25 East Algonquin Road, Des Plaines, Illinois 60017,USA.

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M. Caracotsios and K. Vanden Bussche 113

based on Bayes’ theorem and live demonstrations using the software package AthenaVisual Workbench.

Chapter 28

Kinetic Data Provision and Evaluation in High-Throughput Testing

for Catalytic Materials in the Oxidative Dehydrogenation of Propane

N. Dropka1 D. Wolf1 M. Baerns1

28.1 Summary

High-throughput testing of materials for their catalytic performance has become an es-tablished technique which was recently extended to provide also the kinetic data for thematerials. Kinetics is an alternate and more comprehensive tool of characterising cat-alytic materials for their performance than testing at one reference condition. Moreover,based on kinetic data, optimum operating conditions for the various catalytic materialsand the occurrence of transport limitations can be predicted. Thus, kinetics is a pre-requisite for accessing optimal reactor design. For the oxidative dehydrogenation ofpropane, a kinetic model comprising 8 reaction steps was considered. This kinetic reac-tion scheme describes the change of all reaction components although there is an over-estimation of propene and CO concentrations at high residence times and temperatures.As a result of the kinetic characterisation and optimisation of experimental conditionsby simulation, isothermal reactor operation at low temperature would be the most ap-propriate. Best catalysts have high performance at low temperatures (630 − 760K) andlow propane concentration (< 5kPa).

28.2 Introduction

Since the 1980s, the catalytic dehydrogenation of propane is a topic of commercial aswell as scientific interest, as the better understanding of these reactions would allowexploitation of cheap and abundant resources of propane for selective production ofpropene. An alternative process to endothermic steam cracking is oxidative dehydro-genation of propane (ODP), being exothermal and not limited by equilibrium. Theconcept of ODP based on redox-type catalysts which are already active below 500Chas been presented elsewhere (see, e.g., [1]). Over these catalysts, propene is formed

1Institute of Applied Chemistry Berlin-Adlershof, Richard-WillstŁtter-Str. 12, D-12489 Berlin.

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N. Dropka, D. Wolf and M. Baerns 115

via a heterogeneous Mars-van-Krevelen-Mechanisms. The search for catalysts of highperformance, i.e., high propene selectivity and propane conversion is still a challenge.High-throughput experimentation is presently applied to achieve this goal. In applyingparallel testing, catalysts are usually exposed to one or at best to a few standardisedconditions for identifying the ‘best’ catalyst composition. The best conditions for trulybest catalyst composition may not have been found. In the present paper a method isdescribed to overcome this hurdle. Kinetics are determined for various catalytic ma-terials and the parameters, i.e., partial pressures of the reactants and temperature ofreaction are optimised for a given type of reactor model based on kinetics. Hereby, thebest catalysts can be identified.

28.3 Methodology

For evaluation of kinetic data, obtained in a multi-channel reactor module used for cat-alytic screening, an isothermally operated one-dimensional fixed bed reactor model wasapplied. Assuming steady-state conditions for the reaction and the validity of the per-fect gas law, the differential material balance for the gas-phase species (C3H8, O2, C3H6,C2H4, CO, CO2, H2, and H2O) in the control volume of the reactor is:

dpi

d(mcat/V 0)= RT

∑

i

vijrj. (28.1)

This system of differential equations was integrated numerically. Initial estimates ofkinetic constants and energies of activation were obtained by a ‘genetic’ algorithm [2]and further optimised by a Nelder-Mead algorithm. The applied model consisting ofpseudo-elementary reaction steps is shown in Table 28.1. No rate-determining reactionstep was assumed. For the steady-state of the reactor, the mass balance of the surfacespecies yields a system of nonlinear equations since the differential term for surfacespecies must be zero. The surface coverage θi (for [O], [OH], []) results from the numer-ical solution of this set of nonlinear equations and the balance of normalised surfacecoverage qθii of the oxide surface. Concentration of the adsorbed species CH2,ads wascalculated based on the quasi-steady-state assumption (Bodenstein’s principle). Thegenetic algorithm was also used as a tool for finding optimal reaction conditions. Trans-port limitations were checked by determination of the Weisz and Mears criterions. Ifthe Weisz-criterion ¡ 0.1 and the Mears criterion ¡ 0.15, there is no transport limitation.

28.4 Results and discussion

The reaction scheme (see Table 28.1), consists of the primary heterogeneous formationof propene (step 1) and its consecutive degradation (steps 4 to 6) due to the stepwiseinteraction with lattice oxygen.The primary combustion of propane on adsorbed oxygenspecies is described by step 7. Reaction orders correspond to stoichiometric coefficientsexcept in reaction 7, where the exponent for pO2

was the result of a fitting procedure.


Reaction no. Reaction rateC3H8 + 2[O] C3H6 + 2[OH] r1 = k1 pC3H8 qO22[OH] [ ] + [O] + H2O (2) r2 = k2 qOH22[ ] + O2 2[O] (3) r3 = k3 pO2 qvac2C3H6 + 2[O] C2H4 + CO + 2[ ] + H2O (4) r4 = k4 pC3H6 qO2C2H4 + 2[O] CH2,ads + CO + 2[ ] + H2O (5) r5 = k5 pC2H4 qO2CH2,ads + 2[O] CO + 2[ ] + H2O (6) r6 = k6 pCH2,s qO2C3H8 + 5O2,ads 3CO2 + 4H2O (7) r7 = k7 pC3H8 pO20.2O2 + * O2,ads (8) equilibrated

Table 28.1: Reaction scheme for model 1. Surface species are denoted by brackets [. . .].

In principle, the kinetic reaction scheme and parameters shown for three selected cat-alysts (V0,28Mg0,59Ga0,1, V0,42Mg0,37Mo0,05Ga0,09 and V0,3Mg0,42Mo0,13Fe0,15)describe the change of all reaction components although there is an overestimation ofthe propene and CO concentration at high degree of conversion (Fig. 28.1). Kinetic pa-rameters (preexponential factors and energy of activation) for three selected catalystsare shown in Table 28.2, which also shows that no transport limitations occurred .

The kinetic model can be used to derive optimal reaction conditions for each catalystby solving the following task:

YC3H8= f(T, pC3H8

, pO2, τ) → MAX.

This search for optimal conditions was done within the following boundaries: T =623− 793K ; pC3H8

= 1− 90kPa; pO2= 1− 90kPa and τ = 1− 1000kgs−1m3.

From the numerical solution of the above equation, the optimal temperature, com-position and residence time for maximum propene yield was derived. Results are sum-marised in Table 28.3 and compared with ranking and yields of propene measured atstandard conditions (T = 773K, pC3H8

= 40kPa, pO2= 20kPa and X(O2) > 99%).

28.5 Conclusions

The 8-step kinetic reaction scheme model approximates the changes of all reaction com-ponents adequately.

Under optimal conditions much higher yields can be obtained compared to standardconditions, and the ranking of the catalysts concerning yield is changed. The differencesare surprisingly high. Results on experimental verification will be reported during theWorkshop.

Best catalysts have highest performance at low temperature (630K−760K) and lowerpropene concentration (< 5kPa) than corresponding to standard conditions.


k01 9.56E+00 7.17E+00 4.69E+00k02 6.34E+02 4.22E+03 6.00E+04k03 2.54E+00 3.65E+01 4.30E+01k04 7.40E+00 9.22E+00 6.28E+01k05 3.22E+03 2.55E+01 1.46E+04k06 9.38E-04 1.93E-03 5.47E-05k07 3.03E-02 8.63E-02 6.36E-02EA1 9.97E+04 8.40E+04 5.87E+04EA2 7.34E+04 8.05E+04 9.17E+04EA3 7.68E+04 9.29E+04 1.01E+05EA4 8.47E+04 7.27E+04 5.94E+04EA5 1.10E+05 6.69E+04 8.07E+04EA6 9.36E+04 7.20E+04 5.92E+04EA7 9.33E+04 9.79E+04 9.61E+04Weisz Modulus 3.4E-08 1.1E-05 7.4E-06Mears Criterion 1.2E-06 4.1E-04 2.7E-06

Table 28.2: Kinetic parameters (energy of activation and preexponential factors (EA inJ/mol; koi in m3kg−1s−1Pab ). The Weisz Modulus and Mears Criterion data are basedon the rate of oxygen conversion.

Rank Propene Yield in % T in K τ in skg/m3 check dim pC3H8in kPa pO2

in kPaopt. std. opt. std. opt. opt. opt. opt.1 1* 41.2 10.0 658 9.9E2 2.2 97.72 3** 34.1 8.8 630 1.0E3 2.2 97.73 2*** 10.3 9.6 740 1.1E3 1.7 1.2

Table 28.3: Propene yields for simulated optimal and standard experimental condi-tions; composition of the 3 catalysts carried out: ∗ = V0,28 Mg0,59Ga 0,1; ∗∗ =V0,42Mg0,37Mo0,05Ga0,09; ∗ ∗ ∗ = V0,3Mg0,42Mo0,13Fe0,15.


Figure 28.1: Comparison between experimental and calculated selectivity to propeneand conversion of propane (catalysts ranks 1, 2 and 3 std).


28.6 Bibliography

[1] M. Baerns, O.V. Buevskaya, 2000. Erdol-Erdgas-Kohle Heft 1 116, 25.

[2] NAG Group Ltd., NAG Fortran Workstation Library, 1986.

Chapter 29

Approximate Model for Diffusion and Reaction in a Porous Pellet and

an Effectiveness Factor

Miroslaw Szukiewicz1 Roman Petrus1

29.1 Introduction

The widely reported way of simplifying of heterogeneous models of catalytic processesis to use an effectiveness factor concept. This concept reduces the equation set of theheterogeneous model and makes the solution much easier to obtain. Unfortunately, formost cases it is a difficult and/or lengthy task, to find the proper value of the mentionedfactor. Another way to simplify the mathematical model is to develop an approximatemodel (that is a proper ODE, to substitute a former PDE in the mass balance of a pellet).In this approach, investigation of transient diffusion and reaction processes is possible.

In the present work the approximate model of single reaction in a catalyst pellet wasused for calculation of an effectiveness factor value. Obtained values were comparedwith those obtained by numerical solution of the exact model and by using approximatemethods reported in literature. It turned out that in most investigated cases the approxi-mate model is more accurate for calculating effectiveness factor values than the reportedearlier in literature methods and furthermore it does not fail in multiple steady-state re-gion.

29.2 Exact model

Transient diffusion and reaction in a porous spherical particle can be described by

∂c

∂τ=

(

∂2c

∂x2+

2

x

∂c

∂x

)

− φ2RA(c) (29.1)

IC: τ = 0 : c(x, 0) = cin (29.2)

1Rzeszow University of Technology, Department of Chemical Engineering and Process Control, al.Powstancow Warszawy 6, 35-959 Rzeszow, Poland.

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M. Szukiewicz and R. Petrus 121

BC: τ > 0 : c(0, τ) = extremum (29.3)

c(1, τ) = cs (29.4)

29.3 Approximate model

∂cav

∂τ= 3ψ(cs − cav)− φ2RA(cav) (29.5)

IC: τ = 0 : cav = cin (29.6)

RA,c > 0 : ψ =(√

φ2RA,c cosh√

φ2RA,c − sinh√

φ2RA,c)φ2RA,c

(φ2RA,c sinh√

φ2RA,c − 3(√

φ2RA,c cosh√

φ2RA,c − sinh√

φ2RA,c)(29.7)

RA,c = 0 : ψ = 5 (29.8)

RA,c < 0 : ψ =(√

−φ2RA,c cos√

−φ2RA,c − sin√

−φ2RA,c)φ2RA,c

(φ2RA,c sin√

−φ2RA,c − 3(√

−φ2RA,c cos√

−φ2RA,c − sin√

−φ2RA,c)(29.9)

where

RA,c =∂RA

∂c

∣

∣

∣

∣

∣

c=cs

.

This model was derived in [1].

29.4 Results

It is easy to show that the effectiveness factor can be calculated can be expressed as

η = RA(cav,ss),

where cav,ss is the steady-state value of an average concentration, which can be calcu-lated from the approximate model.

Selected results are presented in Fig. 29.1 and 29.2. In the first Figure the effective-ness factor values calculating by various methods are compared. It is easy to observethat the approximate model gives the most accurate results in the most important inpractice range of small and intermediate Thiele modulus values. In the range of highThiele modulus values better results are given only by the method described in [2], othermethods (described in [3] and [4]) are worse. Within the range of multiple steady-states(where methods described in [2], [3] and [4] fail) the approximate model makes possiblecalculating of effectiveness factor value with good accuracy, as is shown in Fig. 29.2.


Figure 29.1: Effectiveness factor values calculated by various methods.

Figure 29.2: Effectiveness factor values in multiple steady-states region.


29.5 Conclusions

The LDF approximation of diffusion with chemical reaction processes (e.g., heteroge-neous catalysis) can be recommended for various nonlinear kinetic rates and for a broadrange of parameter values for calculating of the effectiveness factor value. Good accu-racy has been observed for any kinetic equation for small and intermediate Thiele mod-ulus values. For larger F accuracy in many cases is satisfactory. It should be pointedout that the method does not fail in the multiple steady-states region.

29.6 Bibliography

[1] M. Szukiewicz, An Approximate Model for Diffusion and Reaction in a Porous Pel-let, Chemical Engineering Science, accepted for publication

[2] H.W. Haynes Jr., An explicit approximation for the effectiveness factor in porousheterogeneous catalysts, Chemical Engineering Science 41, No 2, p. 412–415, 1986

[3] S. Wedel, D. Luss, A rational approximation of the effectiveness factor, Chem. Eng.Commun. 7, p. 245–259, 1980

[4] M.J.G. de la Rosa, V.T. Garcia, O.J.A. Tapia, Evaluation of isothermal effectivenessfactor for nonlinear kinetics using an approximate method, Ind. Eng. Chem. Res. 37, p.3780–3781, 1998

Chapter 30

The principle of the maximum of the entropy production rate for

stationary nonequilibrium processes and self-organizing systems

M.A. Ivanov1

30.1 Abstract

We propose a new principle to describe the wide class of the stationary nonequilib-rium irreversible processes and self-organizing systems. This principle consists in thefollowing: the open nonequilibrium systems, which have different and alternative sta-tionary nonequilibrium states, at specified value of the externals thermodynamic forcesis self-organizing in such a way that there is choosing the state, which corresponds tothe maximum of the entropy production rate. At that it is succeeded to consider and todescribe a large amount of processes as in the physical and chemical kinetics, as in theother scope of nature.

In the thermodynamic theory of the nonequilibrium processes it is a well knownthe principle of the minimum entropy production rate, introduced by I. Prigogine [1-3]. This principle consists in that at the transition from nonstationary nonequilibriumprocesses to stationary ones, the entropy production rate is slowing down, and thisnonequilibrium process, which answered to minimum of the entropy production rateare stable (the Prigogine theorem). But such principle tells nothing about a system be-havior in the case, when at given value of external thermodynamic forces virtual differ-ent stationary nonequilibrium processes taking place.

The principle, that proposed by us, consist in, that in the presence of different sta-tionary nonequilibrium processes it must be realized that one, which correspond to themaximum of the entropy production rate. In that way, there is elaborating a peculiarprinciple of the minimax: at first, there is occur transitions from nonstationary nonequi-librium processes to stationary, and such transitions correspond to lowering of the en-tropy production rate; then transitions are taking place (obviously, by the fluctuatingway) from one stationary nonequilibrium process to another, so, at the end, that one

1Institute of Metal Physics, Nat. Acad. Sci. of Ukraine, Department of the Theory of Nonideal Crystals,Vernadskogo Prosp. 36, Kiev, 03142 Ukraine. E-mail: [email protected]

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M.A. Ivanov 125

is established, which corresponded to the maximum of the indicated rate. Each transi-tion from one stationary process to another is, in a certain way, the breaking of systemsstability, and all process of such transitions exactly constitutes self-organization of thedissipative structures. The thermodynamic approach, that is proposed here, is equi-table when there are sufficient large quantity of the possible instabilities in the system,for example separated in the space. The choice of maximum entropy production ratecorresponds to such way, for which the number of possible systems stations grow withmaximum rate, i.e., this principal enough good conforms to general approach in thestatistical physics. We should note also, that it is a question of the entropy produc-tion rate of the system as a hole, including the environment. Consequently, if there isdedicated subsystem with prescribed value of temperature and pressure, the proposedprinciple replies to maximum of the Gibbs thermodynamic potential degradation rateof this subsystem.

Wide class of the dissipative nonequilibrium physical and chemical structures andprocesses is described by using the proposed principle. Here we will concern, first ofall, on the principle, that is determine the alternative choice of the new growing phaseon the surface of the initial solid phase (or between two initial condensed phases). Thequestion about, how exactly the phase is growing in this case, is one of the most dis-cussion problems in physical and chemical kinetics. In fact in this case the alternativekinetics takes place, because the growth of one phase is hindering and, at the end, for-bidding the growth of other possible phases. Within the scope of the proposed approachis consisting in, that there is growing that phase, for which the rate of degradation offree energy of the whole system is the largest. At that it is necessary to take into accountas the rate of change of this energy in the bulks of each of the initial states, as the rate ofthe ‘chemical’ reaction on the interfaces between the new and the initial phases. At theresult the new phase on the initial stage may grow not stable phase, which correspond toequilibrium phase diagram, but metastable one. And only on the latest stages will growthe stable phase, nevertheless. Very important role in physics and chemistry plays theprocesses of crystallization and kinetics of different first-order phase transitions. Exactlythese processes are the most simple and obvious cases of the self-organization. It wasfound, particularly, possible to describe the process of rising of the macroscopic spaceperiodic structure with the crystallization of the eutectic composition liquid. There wasfound parameters of this structure. It was shown, that at the relatively small concentra-tion of one of the component and strict constancy of the overcooling on the front of thecrystallization, there should appear a triangle ‘lattice’, ‘cells’ of which, consists of thedifferent phase elements, that is looking like the Vigners lattice. The size of the elemen-tary cell is decreasing with the growth of overcooling (or velocity of the crystallization).If the temperature of supercooling is fluctuating on the front of the crystallization, thenit is founding possible the peculiar ‘melting’ of this lattice.

There was clarified the reason of the appearance defects in the structure of crystalsand it was found dependence of the defects concentration from the velocity of the crys-tals growth. Within the scope of proposed approach it was enabled to explain also somepeculiarities of the martensitic transition in crystals and predict some new dependen-

M.A. Ivanov 126

cies, which should take place in such transitions. There was shown, that the origin ofthe space structure in the Benar effect, particularly right hexagonal cell, also bond withthe proposed principle.

We could also suppose, that proposed principle may be useful not only for the de-scription of the dissipative structure, that are considered in physics and chemistry ofthe nonequilibrium processes, but also in many others natural sciences and maybe insocial sciences.

30.2 Bibliography

[1] I. Prigogine, Non-Equilibrium Statistical Mechanics, Wiley-Interscience, New York,1962.

[2] S.R. de Groot, P. Masur, Non-equilibrium Thermodynamics, North Holland, Amster-dam, 1962.

[3] P. Glansdorff, I. Prigogine, Thermodynamics Theory of Structure, Stability and Fluctua-tions, Wiley, New York, 1971.

Chapter 31

Calculation of the molecular mass distribution in free-radical

polymerization with numerical inverse Laplace of the probability

generating function.

T. De Roo1 J.M. De Keyser1 G.J. Heynderickx1 G.B. Marin1

31.1 Abstract

The free-radical polymerization of vinylchloride is mainly carried out in batch suspen-sion polymerization reactors. During the polymerization the reactor contains a waterphase, a monomer and a polymer phase and a gas phase. The water phase serves as aheat management medium. The vinylchloride monomer is dispersed in the water phaseby stirring and by adding suspension stabilisers. The polymer (PVC) is insoluble in themonomer and it forms a separate phase in the monomer phase. Polymerization takesplace in the monomer and in the polymer phase. The important reactions are initiation,propagation, chain transfer to the monomer and termination. Because of the differentreaction conditions in the two phases, a mass balance for each radical and polymermolecule of a certain chain length has to be solved in the monomer and in the polymerphase. As the maximum chain length is very high, the resulting set of differential equa-tions is high dimensional and stiff. In general, this set of equations cannot be solved an-alytically while the direct numerical integration imposes considerable difficulties. Theprobability generating function (pgf) method is used to reduce the large amount of massbalances. This method is the discrete equivalent of the Laplace transformation method.The concentrations of the radical and polymer molecules are transformed into probabil-ity generating functions. The equations in the transformed domain are easily integratedover the reaction time since the number of equations is seriously reduced. By numericalinversion of the resulting probability generating functions the corresponding concentra-tions of radical and polymer molecules for the desired chain lengths are obtained. Forthe inversion, the method of de Hoog for the numerical inversion of Laplace transformsis used (Asteasuain et al., 2002). It gives accurate results over the whole range of poly-mer chain lengths, is easy to use and is available from standard programming libraries

1Laboratorium voor Petrochemische Techniek, Krijgslaan 281 (S5), B9000 Ghent, Belgium.

127

T. De Roo, J.M. De Keyser, G.J. Heynderickx and G.B. Marin 128

Figure 31.1: Calculated molecular mass distribution for the suspension polymerizationof vinylchloride at low conversion. MM = molecular mass of the polymer molecules[g/mol]. Wr = mass fraction of polymer molecules of chain length r [-].

(IMSL, 1999). Because the method of de Hoog uses complex instead of real values of thetransformed variable for the numerical inversion, twice as many pgf balances have tobe solved. The use of the pgf method introduces no limitations on the complexity of thereaction scheme.

The result of a first calculation of the molecular mass distribution is shown in Fig.31.1. A total of 50 log-plot equidistant values of the distribution are shown. The amountof inversion points used by the de Hoog method is 300. The calculation of the molec-ular mass distribution is based on the reaction kinetics and the polymerization modelof the process. The resulting molecular mass distribution is therefore a reflection of thecomplete polymerization process. Therefore, the characteristics of the molecular massdistribution are expected to give information on the importance of the different reac-tions in the polymerization reaction scheme.

T. De Roo, J.M. De Keyser, G.J. Heynderickx and G.B. Marin 129

31.2 Bibliography

[1] M. Asteasuain, A. Brandolin, C. Sarmoria, Recovery of molecular weight distribu-tions from transformed domains, Part II, Application of numerical inversion methods,Polymer 2002, 43 p. 2529-2541.

[2] IMSL Fortran 90 MP Library, Version 4.01, 1999.

Chapter 32

Method of Invariant Manifold in Chemical Kinetics

A.N. Gorban1

32.1 Abstract

The goal of nonequilibrium statistical physics is the understanding of how a systemwith many degrees of freedom acquires a description with a few degrees of freedom.This should lead to reliable methods of extracting the macroscopic description from adetailed microscopic description. Meanwhile, this general problem is still far from thefinal solution, it is reasonable to study simplified models, where, on the one hand, adetailed description is accessible to numerics, on the other hand, analytical methodsdesigned to the solution of problems in real systems can be tested.

Models of complex reactions in thermodynamically isolated systems often demon-strate evolution towards low dimensional manifolds in the phase space. Typical indi-vidual trajectories tend to manifolds of lower dimension, and further proceed to theequilibrium essentially along these manifolds. Thus, such systems demonstrate a di-mensional reduction, and therefore establish a more macroscopic description after sometime since the beginning of the relaxation.

For this class of models, we suggest a direct method to construct such manifolds,and thereby to reduce the effective dimension of the problem. The approach realizesthe invariance principle of the reduced description, it is based on iterations rather thanon a small parameter expansion, it leads to tractable linear problems, and is consistentwith thermodynamic requirements. Earlier the method has been applied to a set ofspecific problems of classical kinetic theory based on the Boltzmann equation.

The goal of the present study is twofold. The first goal is to verify the method of in-variant manifold on a representative class of nonlinear dissipative systems. The secondgoal concerns a more practical issue, namely, the problem of an effective description ofcomplex reactions. The approach is tested with a model of catalytic reaction. There aretwo intuitive ideas behind our approach, and we shall now discuss them in-formally.

Objects to be considered below are manifolds (hypersurfaces) in the phase space ofthe reaction kinetic systems (the phase space is usually a convex polyhedron in a finite-

1Institute of Computational Modeling RAS, Krasnoyarsk, Russia.

130

A.N. Gorban 131

dimensional real space). The ideal picture of the reduced description we have in mindis as follows: A typical phase trajectory c(t), where t is the time, and c is an element ofthe phase space, consists of two pronounced segments. The first segment connects thebeginning of the trajectory c(0) with a certain point c(t1) on the stable invariant manifoldV . The second segment belongs to V , and connects the point c(t1) with the equilibrium.Thus, the manifolds appearing in our ideal picture are patterns formed by the segmentsof individual trajectories.

There are two important features behind this ideal picture. The first feature is theinvariance of the manifold : Once the individual trajectory has V reached , it does notleave V anymore. The second feature is the projecting: The phase points outside V willbe projected onto V . Furthermore, the dissipativity of the system provides an additionalinformation about this ideal picture: Regardless of what happens on the manifold , thefunction G was decreasing along each individual trajectory before it reached V .

This ideal picture of the decomposition of motions is certainly too exaggerated, butit is a useful guide to extract the reduced description. The main advantage is that it iscompletely geometrical, allows for a direct and fairly simple formalization, and makesit possible to apply rapidly convergent iteration methods of solution.

Chapter 33

Macroscopic Kinetics of Diffusion-Induced Noise

Maria K. Koleva1 L.A. Petrov1

33.1 Introduction

The subject of the present work is the macroscopic evolution of open spatially homo-geneous catalytic systems at steady external constraints far from unstable macroscopicstates. Recently one of us proposed [1] a new driving mechanism of fluctuations, calleddiffusion-induced noise, that gives rise to macroscopic effect at any value of the controlparameters and even at asymptotically stable states. On macroscopic scale the involv-ing of these fluctuations is expressed by a system of stochastic differential equations ofspecial type. The major property of these equations is that the amplitude of the stochas-tic terms is bounded. The physical arguments about the boundedness are presented inthe next section. The advantage of the boundedness is that it ensures an arbitrarily long-term stable evolution of any natural system since the later permanently stays within thethresholds of stability of the system. It turns out [2] that asymptotically (as the time ap-proaches infinity) the solutions of the kinetic equations comprises two parts: one com-ing from a system of ODE (ordinary differential equations), called ‘deterministic’ andthe other one that comes from the stochastic terms. It has been established analyticallythat the stochastic part of the solution has certain universal properties, namely:

1. the power spectrum is a continuous band that uniformly fits the shape 1/fα(f) ,where α(f) → 1 as f → 1/T (T is the length of a time series) and α(f) monotoni-cally increases to the value p (p > 2) as the frequency approaches infinity

2. the phase trajectories are confined in a non-homogeneous strange attractor withnon-integer correlation dimension;

3. the Kolmogorov entropy is finite.

The proof of these rather chaotic properties is established by the use of the Wiener-Khinchin theorem on the base of the Lindeberg theorem that states: any bounded irreg-ular sequence has finite mean and finite variance no matter what the distribution of the

1Institute of Catalysis, Bulgarian Academy of Science, 1113 Sofia, Bulgaria.

132

M.K. Koleva and L.A. Petrov 133

stochastic terms is. The properties of the power spectrum provide an easy and powerfulcriterion for discrimination between different types of dynamical regimes. The devia-tions from smoothness of the continuous band serve as a criterion for any change in thekinetic mechanism, surface properties, etc., in the course of time.

33.2 Diffusion-Induced Noise

The diffusion-induced noise is generic for all surface reactions since it appears at theprocess of chemisorption that is typical for any surface reaction. Next we considerthe mechanism of the diffusion-induced noise assuming the ideal adsorption layer ap-proach. The relaxation of an adsorbed (reaction) species at a given active site can beinterrupted by a chemisorbed species that migrates to that active site from the neigh-bourhood. Since no more than one species can be chemisorbed at a single active site,the adsorbed (reaction) species can complete the adsorption (reaction) at another emptyactive site that it reaches by migration. This process, called a diffusion-induced non-perturbative interaction, however, changes non-smoothly the probability for adsorp-tion (reaction). Since a diffusion-induced non-perturbative interaction can happen atany level of the relaxation, the probability for adsorption (reaction) becomes a multi-valued function. A single selection of this multi-valued function, randomly chosenamong all possible, appears at a single adsorption (reaction) event which in turn consti-tutes a ‘stochastic’ behaviour in the course of time. It has been established that at opensystems the probability for undergoing a single diffusion-induced non-perturbative in-teraction is proportional to the concentration of the chemisorbed species. Therefore, farfrom bifurcation points and unstable states, the macroscopic evolution is presented byphenomenological equations in which the probabilities for adsorption and reaction areexpanded in series over the number of the diffusion-induced non-perturbative interac-tions. To the first order of expansion the equations for the reaction mechanism read:

d~n

dt= ~αAdet − ~βRdet + ~αµai(~n)−~bµri(~n), (33.1)

where ~n is the vector of the reaction species; Adet (Rdet) are sums of the probabilities forthe ordinary adsorption (ordinary reaction) and the average values of all the possibleselections at the adsorption (reaction). The subscript i serves to stress that only one se-lection, randomly chosen among all the possible, is realized at a given moment. Thedefinition of µai(~n) and µri(~n) determines a general property that is very important andplays a crucial role in the next considerations. It is that the values of µai(~n) and µri(~n) arebounded in the range [−1, 1] and their average value is zero. The limits of the range areset by the fact that µai(~n) and µri(~n) are probabilities. The possibility for being both pos-itive and negative is set by their definition. The physical foundation of the boundednessof the stochastic terms is traced in the fact that a non-perturbative interaction involvesa finite amount of energy (less than the adsorption energy) and finite amount of matter


(a single chemisorbed species) so that the systems stays permanently microscopicallystable, i.e., no more than one species is chemisorbed at a single active site.

It has been established [2] that the solution of (33.1) comprises two parts: a stochasticone which features are universal and are listed in the Introduction and a ‘deterministic’part that comes from a system of ODE presented through:

d~n

dt= ~αAdet(~n)− ~βRdet(~n). (33.2)

It is obvious that being a system of non-linear ODE eq. (33.2) gives rise to differenttypes of dynamical regimes when the control parameters vary. Therefore, the powerspectrum can serve as a discrimination criterion between different types of stable dy-namical regimes. The asymptotically stable states are characterized by a continuousband of the shape 1/fα(f).The chaotic states in their mathematical sense are not pre-sented since they come from unstable solutions of a system of ODE. It is proven [2]that the simulated dynamical systems has identical mathematical formulation in termsof stochastic differential equations of the type (33.1). The stochasticity arises due tothe inevitable round-off at every point that is enhanced in the time due to the positiveLyapunov coefficient(s). The boundedness is due to the confinement of the phase tra-jectories in a strange attractor. The states of persistent oscillations are characterised bya discrete band superimposed on the continuous one. The multiple steady states areintermittency-like and their power spectrum exhibits temporary deviations from thesmoothness. However, increasing the length of the time series the deviations diminishand eventually disappear as the length of the time series approaches infinity.

33.3 Other impacts on the macroscopic kinetics

The main impact of the diffusion-induced noise on the macroscopic evolution of kineticsystems is through the transformation of the kinetic equations from ODE to stochasticODE (SDE). The changes can be separated into following classes:

• the kinetics becomes non-ideal even in the case of assuming ideal mixture or idealadsorption layer approach. This is substantiated through the modification of theterms of the deterministic part (33.2). They are sums of the terms that come fromordinary relaxation (ideal kinetics) and the average value of all possible selections.

• the presence of the stochastic terms violates the linear dependence among certainsteps and routes of a complex reaction. Therefore, all possible coupled steps androutes that participate a reaction should be presented in the reaction mechanism.

• Temporary changes of the bifurcation diagram of (33.2) are possible due to oc-casional large fluctuations that come from accumulating stochastic terms of thesame sign in the time course. Figuratively, the stochastic terms cause permanentchange in the ratio between different intermediates. Occasional large fluctuations


can cause transitions to a another region of the bifurcation diagram without anychange of the control parameters. These changes result in deviations of smooth-ness of the continuous band in the power spectra.

• The quasi-chemical approach is not relevant because of the impact of the stochasticterms.

33.4 Conclusions

The main influence of the diffusion-induced noise on the macroscopic kinetics is theconversion of the kinetic equations from ODE type to SDE of a special type. The majorproperty of the stochastic terms is their boundedness. This results in new and, to certainextent, unexpected properties of the solutions. The major difference is the presence ofa continuous band of the shape 1/fα(f) in the power spectra. A continuous band ofthat shape has been observed experimentally in the reaction of the catalytic oxidation ofHCOOH over supported Pd catalyst [3] at 80 different values of the control parameters.

33.5 Bibliography

[1] M.K. Koleva, Non-perturbative interactions: a source of a new type noise in opencatalytic systems, Bulg. Chem. Ind. 69 (1998) p. 119–128.

[2] M.K. Koleva and V. Covachev, Common and different features between chaotic dy-namical systems and 1/fα(f) noise, FNL 1 (2001) R131-R149.

[3] M.K. Koleva, A.E. Elyias and L.A. Petrov, Fractal power spectrum at catalytic oxi-dation of HCOOH over supported Pd catalyst in: Metal-Ligand Interactions in Chemistry,Physics and Biology, NATO ASI Series C, vol. 546, Eds. N. Russo and D.R. Salahub,(Kluwer Academic Publishers, Dordrecht, 2000), p. 353–369.

Chapter 34

Multiscale modeling for linking growth, microstructure, and

transport-chemistry properties through inorganic microporous films

D.G. Vlachos1 M.A. Snyder1 R. Lam2 V. Nikolakis1 G. Bonilla1 M. Tsapatsis1 M.A. Kat-soulakis3

34.1 Introduction

This talk focuses on linking growth (synthesis) conditions with microstructure andtransport-reaction properties of microporous materials with the aim of improving ma-terials performance by appropriately controlling synthesis conditions. These are in-herently multiscale problems, with scales ranging from single molecules to macro-scopic scale devices. The specific materials studied are polycrystalline zeolite films of≈ 30microns in thickness and several millimeters in diameter. A close integration ofsimulations with experiments will be presented.

34.2 Multiscale simulations of microporous materials synthesis

The first part of the talk focuses on linking the microstructure with synthesis conditions.Zeolite synthesis can proceed by addition of monomers, small nanoparticles (subcol-loidal particles) of ≈ 2.8nm in diameter, or aggregates of them, all of which have beenseen experimentally. We have used the DLVO theory of colloidal aggregation to modelcrystal growth as a transport-kinetics problem based on the chemical potential formal-ism, with parameters measured experimentally [1]. Atomic force microscopy has beenused to ensure the validity of DLVO theory in describing the mean force of interac-tion. Fig. 34.1 shows comparison of model predictions and experiments (nanometerscale). Our simulations indicate that subcolloidal particles are the major growth units.We have subsequently extended these studies to include surface chemistry in order to

1Department of Chemical Engineering and Center for Catalytic Science and Technology (CCST), Uni-versity of Delaware, Newark, DE 19716-3110.

2Department of Chemical Engineering, University of Massachusetts, Amherst, MA 01003.3Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003.

136

D.G. Vlachos, M.A. Snyder, R. Lam, V. Nikolakis, G. Bonilla, M. Tsapatsis and M.A. Katsoulakis137

Figure 34.1: Comparison of dynamic light scattering data (points) and simulations(lines) at different temperatures ranging from 66C (lower curve) to 90C (upper curve)for MFI zeolite. From [1].

predict particle shape at the micrometer scale. With such validated models, we havedeveloped front tracking and level set techniques to study the evolution of film growth[5]. An example is shown in Fig. 34.2. We have been able to explain for the first time themechanism leading to multiple peaks in x-pole figure analysis and provide guidance onhow to control the preferred orientation of thin zeolite films.

34.3 Mesoscopic simulations for predicting transport-reaction prop-

erties of polycrystalline films

The second part of the talk focuses on predicting transport-reaction properties by in-tegrating quantum and molecular scale information into newly developed mesoscopicmodels. The understanding and ability to predict macroscopic transport characteris-tics for diffusion of interacting molecular species through nanoporous inorganic mem-branes is key to the ultimate development of both traditional applications of this tech-nology, such as separations, and novel ones, such as growth of nanowires withinnanopores [6]. Molecular simulations such as molecular dynamics and Monte Carloalgorithms have emerged in the past few decades as preeminent computational toolsfor science and engineering research. With the advent of enhanced computing capa-


Figure 34.2: Comparison of thin MFI film grown on the computer (left) and cross sectionSEM micrograph (right). From [3].

bilities, these methods can provide unprecedented insights into numerous problemsranging from physicochemical and biological processes, to biomaterials and drug de-sign. Despite their widespread use and the substantial progress in related computa-tional methods, such molecular simulations are limited to short length and time scales,while inorganic films such as zeolite membranes invoke much larger length scales (e.g.,see Fig. and 34.2). A major obstacle in meeting this multiscale modeling challenge is thelack of a rigorous mathematical and computational framework providing a direct linkof atomistic simulations and scales to complex mesoscopic and macroscopic phenom-ena dictated by microscopic intermolecular forces. In this work, a new mathematicalframework is introduced for modeling diffusion of interacting species in nanoporousmaterials over large length scales while retaining molecular scale information typicallycaptured only by molecular simulations. This framework entails the use of newly de-veloped mesoscopic equations derived rigorously from underlying master equations bycoarse-graining non equilibrium statistical mechanics techniques. Comparison of gra-dient Monte Carlo simulations to the solutions of mesoscopic theories shows excellentagreement of the new approach [3,8]. An example is shown in Fig. 34.3. Numericalsolutions of mesoscopic equations are efficiently obtained using finite difference andspectral methods [7,8], and macroscopic transport and thermodynamic laws can be de-veloped [2]. Solution of these equations enables us to perform quantitative comparisonsof molecular flux to laboratory permeation experiments. Modeling of benzene diffusionthrough faujasite membranes will be presented as a comparative example evidencingthe power of this mesoscopic framework as an efficient and accurate tool for multiscalemodeling. Extension to multicomponent species will also be presented. Finally, sincemost nanoporous films are polycrystalline, coarse graining of the mesoscopic equationswill be presented to examine the role of grain boundaries in the overall permeability.For example, we have found an interesting break of symmetry in membrane perfor-mance, i.e., defects affect permeability more strongly when they are located near the


Figure 34.3: Comparison of gradient continuous time Monte Carlo (CTMC) and meso-scopic simulations. From [4].

high-pressure membrane side. Furthermore, we show that due to intermolecular forces,nonlinear behavior is possible in such membranes.

34.4 Bibliography

[1] V. Nikolakis, E. Kokkoli, M. Tirrell, M. Tsapatsis and D.G. Vlachos, Zeolite growth byaddition of subcolloidal particles: Modeling and experimental validation, Chem. Mater.12, p. 845–853 (2000).

[2] M. Katsoulakis and D.G. Vlachos, From microscopic interactions to macroscopiclaws of cluster evolution, Phys. Rev. Letters 84(7), p. 1511–1514 (2000).

[3] G. Bonilla, M. Tsapatsis, D.G. Vlachos, and G. Xomeritakis, Fluorescence confocaloptical microscopy of the grain boundary structure of zeolite MFI membranes made bysecondary (seeded) growth, J. Membrane Sci. 182(1-2), p. 103–109 (2001).

[4] D.G. Vlachos and M. Katsoulakis, Derivation and validation of mesoscopic theoriesfor diffusion of interacting molecules, Phys. Rev. Letters 85(18), p. 3898–3901 (2000).


[5] G. Bonilla, D.G. Vlachos and M. Tsapatsis, Simulations and experiments on thegrowth and microstructure of zeolite MFI films and membranes by secondary growth,Micropor. Mesopor. Mat. 42(2-3), p. 191–203 (2001).

[6] V. Nikolakis, G. Xomeritakis, A. Abibi, M. Dickson, M. Tsapatsis and D.G. Vlachos,Growth of faujasite-type zeolite membrane and its application in the separation of sat-urated/unsaturated hydrocarbon mixtures, J. Membrane Sci. 184(2), p. 209–219 (2001).

[7] D.J. Horntrop, M.A. Katsoulakis and D.G. Vlachos, Spectral methods for mesoscopicmodels of pattern formation, J. Comp. Phys., Accepted (2000).

[8] R. Lam, T. Basak, D. G. Vlachos and M. A. Katsoulakis, Validation of mesoscopic the-ories and their application to computing effective diffusivities, J. Chem. Phys., submitted(2001).

Chapter 35

Group analysis and the inverse problems of chemical kinetics and

thermodynamics

S.Spivak, Bashkir State University, Frunze str. 32, Ufa, 450074, Russia

35.1 Abstract

The inverse problem of chemical kinetics and thermodynamics is analyzed. This prob-lem is defined as a determination of equilibrium or kinetic parameters on the basisof chemical composition measurements under equilibrium, steady-state or non-steady-state conditions. Typically, the accessible information is related to the reactant and prod-uct concentrations, not intermediate ones. As a result, in the general case the describedinverse problem may have several solutions.

The goal of this paper is to answer different questions related to this problem:

• What is a concrete method to solve the problem of the non-uniqueness of the so-lution?

• How to find explicit functional relationships between thermodynamic and kineticparameters and the number of these relationships?

The chosen method of our analysis is the theory of group transformations.The mathematical problem can be written as follows:

kl = 0, kl(t) = kl(t = 0) = const., l = 1, ., P, (35.1)

yj = gj(k, y, u), y(t = 0) = y0, j = 1, ., N, (35.2)

xi = fi(k, y, u), i = 1, ,M, (35.3)

where k = (k1, . . . , kP ) is a P -dimensional model parameters vector (of reaction rateconstants) to be determined; x = (x1, . . . , xM ) is an M -dimensional values vector ofconcentrations, measured during the reaction; y = (y1, . . . , yN) is an N -dimensional val-ues vector of concentrations, describing the reaction development, but nonmeasurable;u = (u1, , uQ) is a Q-dimensional vector of independent input experiment conditions

141

S. Spivak 142

variables; and g(k, y, u) = (g1(k, y, u), , gN(k, y, u)) is an N -dimensional vector functionof right sides written out in accordance, for the example, with the mass-action law.

It is assumed that only a part of the substance concentrations vector (x) is measuredduring the reaction. This is the common situation taking place in complex reactionsmechanisms investigation.

Under the assumption of uniqueness of the inverse problem solution for (35.1-35.3)we have the existence of k∗, k, y∗, y such that for any admissible u we have:

x(k, y(k)) = x(k∗, y∗(k∗)) (35.4)

where y∗ obeys the system (35.5):

y∗ = g(k∗, y∗), y∗(t = 0) = y∗0 (35.5)

or the systemc(k, y) = c(k∗, y∗), q(k, y) = q(k∗, y∗) (35.6)

From this point of view the set of equations (35.1)–(35.3) and (35.4)–(35.5) defines trans-formations of solutions (y, k) into solutions (y∗, k∗) of the form

k∗ = k∗(k) = p(k) (35.7)

y∗ = y∗(k, y) = s(k, y). (35.8)

Thus the transformations (35.7)–(35.8) satisfies all the axioms for a transformationgroup. Indeed, the superposition of the transformations (35.7)–(35.8) plays the roleof the group operation and in the construction of the transformations (35.7)–(35.8) theset (y, k) of the inverse problem solution is transformed into itself; the composition ofthe transformations is associative; the unitary element is identical transformation (i.e.,k∗ = k, y∗ = y, and an inverse element exists (at least locally) in accordance with theinverse function theorem.

The number of admissible transformations of type (35.7)–(35.8) is the number of ele-ments in a correspondent group. This number can be finite or infinite.

Among infinite groups, for our purposes continuous groups will be of great interest,whose elements are settled by a transformation set depending continuously on someparameters so that one can pass continuously from our element over to another onewith an infinitesimal change in the parameters.

The methods of transformation group theory presented in the paper have a suffi-ciently general ability to analyze inverse problem solutions of chemical kinetics. Thesemethods are simple enough to be implemented algorithmically. The simplicity of thealgorithm simplicity permits to try its computer implementation. The existence of ana-lytical calculation systems makes the problem quite tractable.

We have many examples of the analysis of concrete heterogeneous, homogeneous,and enzyme catalysis reactions:

• the cyclooligamerization of butanediene in application of lowvalent of nickel com-plexes;

S. Spivak 143

• the reaction between cyclohexylsulfonyl and cyclohexil radicals;

• CO oxidation on a Pd-containing catalyst;

• anisol synthesis by methanol alkylation of phenol on a zeolite catalyst;

• the cooxidation of arylalkenes and alkyamines;

• the reaction to methane by metagenic symbiotroph;

• the deactivation by nickel catalyst in reactions of benzene hydrogenation;

• the catalyst pyrolysis of n-heptane on alkaline-earth metal chlorides;

• the complex dynamics of the Belousov-Zhabotinskii reaction;

• the Diene Polymerization with Lantanide Systems.

As a rule the number of independent nonlinear functions from kinetic and thermody-namics constants is essentially less than the number of constants. Concrete examples ofanalysis of different types of measurements (stationary, quasistationary, nonstationary)are considered in this work.

Contents

1 V. Balakotaiah and S. Chakraborty: Averaging Theory and Low dimensional Modelsfor Chemical Reactors and Reacting Flows 1

2 V.K. Belnov, N.M. Voskresenskii, S.I. Serdyukov, I.I. Karpov and V.V. Barelko:Mathematical modelling of endothermic reactions in the catalyst unit with ordering ele-ments 8

3 D. Guillaume, K. Surla and P. Galtier: From Single Events Theory to Molecular Ki-netics – Application to industrial process modeling 13

4 A.J.M. Oprins and G.J. Heynderickx: Calculation of Three-Dimensional Flow Fieldsin Cracking Furnaces 18

5 T.A.M. Verbrugge and C.F.J. den Doelder: Applying the Dynamic Monte Carlo Ap-proach for Modeling Complex Polymerization Reaction Systems in Isothermal and Non-Isothermal Environments 24

6 D. West, G. Smits, M. Kotanchek and K. Mercure: Chaotic Bursting in an ExternallyExcited Industrial Reactor 29

7 D. Pavone and S. Louret: Numerical solutions in moving bed separation simulations 34

8 J. Kacur: Contaminant transport with adsorption in unsaturated-saturated porous me-dia 39

9 N. Batens and R. Van Keer: On a numerical relaxation method for a reaction-diffusionproblem with an instantaneous and irreversible reaction 42

10 M. Kolkowski, F. Keil, C. Liebner, D. Wolf and M. Baerns: A High-Speed Methodfor Obtaining Kinetic Data for Exothermic or Endothermic Catalytic Reactors undernon-isothermal Conditions illustrated for the Ammonia Synthesis 45

11 B. Monnerat, L. Kiwi-Minsker and A. Renken: Mathematical modelling of theunsteady-state oxidation of nickel gauze catalysts 50

144

CONTENTS 145

12 K. Pappaert, J. Vanderhoeven, P. Van Hummelen, G.V. Baron and G. Desmet:Diffusion-Reaction Modeling of DNA Hybridization Kinetics on Biochips 56

13 S.O. Shekhtman, G.S. Yablonsky and J.T. Gleaves: ‘State Defining’ Experiment inChemical Kinetics: Primary Characterization of Catalyst Activity in a TAP Experiment 61

14 M.-O. Coppens: Simulating Diffusion in Nanoporous Materials 65

15 Joeri F.M. Denayer and G.V. Baron: Determination of adsorption parameters fromchromatographic experiments and their use in catalysis modelling 68

16 E.S. Kurkina and E.D. Kuretova: The multi-scale mathematical model of the oscillatorychemical reaction proceeding over a porous catalyst 70

17 K. Malek and M.-O. Coppens: Dynamic Monte Carlo simulation of diffusion-reactionprocesses in rough fractal pores 74

18 A. De Wit: Fingering of chemical fronts in porous media 79

19 Y. De Decker, F. Baras, G. Nicolis and N. Kruse: Modeling the NO Reduction byHydrogen on a Pt Field Emitter Tip 81

20 M. Massot: Entropic structure and singular perturbation analysis of reactive gaseousmixtures in the limit of partial equilibrium reduced chemistry 86

21 E.S. Kurkina and N.L. Semendyaeva: The origin of oscillations in the imitation modelof CO + O2 reaction on a palladium surface 89

22 M. Lazman: Reaction rate is an eigenvalue: polynomial elimination in chemical kinetics 92

23 G.S. Yablonsky, I. Mareels and M. Lazman: The principle of critical simplification inchemical kinetics: a case study 96

24 N.V. Peskov, M.M. Slin’ko and N.I. Jaeger: Stochastic model of reaction rate oscilla-tions during CO oxidation over zeolite supported Pd catalysts 103

25 De Chen, R. Lødeng, E. Bjørgum, K. Omdahl and A. Holmen: Analytic and Nu-meric Methods in Microkinetic Modeling 107

26 W. Jager: Asymptotic Analysis and Flow through a Network of Channels 111

27 M. Caracotsios and K. Vanden Bussche: Nonlinear Parameter Estimation from Singleand Multiresponse Data. Discrimination and Criticism of Mechanistic Models OptimalExperimental Design for Model Discrimination and Parameter Estimation. Theory andDemonstrations. 112

CONTENTS 146

28 N. Dropka, D. Wolf and M. Baerns: Kinetic Data Provision and Evaluation inHigh-Throughput Testing for Catalytic Materials in the Oxidative Dehydrogenation ofPropane 114

29 M. Szukiewicz and R. Petrus: Approximate Model for Diffusion and Reaction in aPorous Pellet and an Effectiveness Factor 120

30 M.A. Ivanov: The principle of the maximum of the entropy production rate for station-ary nonequilibrium processes and self-organizing systems 124

31 T. De Roo, J.M. De Keyser, G.J. Heynderickx and G.B. Marin: Calculation ofthe molecular mass distribution in free-radical polymerization with numerical inverseLaplace of the probability generating function. 127

32 A.N. Gorban: Method of Invariant Manifold in Chemical Kinetics 130

33 M.K. Koleva and L.A. Petrov: Macroscopic Kinetics of Diffusion-Induced Noise 132

34 D.G. Vlachos, M.A. Snyder, R. Lam, V. Nikolakis, G. Bonilla, M. Tsapatsisand M.A. Katsoulakis: Multiscale modeling for linking growth, microstructure, andtransport-chemistry properties through inorganic microporous films 136

35 S. Spivak: Group analysis and the inverse problems of chemical kinetics and thermody-namics 141

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MaCKiEŒ2002 - Universiteit Gentdconstal/mackie-workshops/mackie...ii Typeset in LATEX by D. Constales ([email protected]). A special issue of the journal Chemical Engineering Science