Single Event Kinetic Modelling Without Explicit

Single Event Kinetic Modelling without ExplicitGeneration of Large Networks:

Application to Hydrocracking of Long ParaffinsD. Guillaume1*, E. Valéry1§, J.J. Verstraete1, K. Surla1, P. Galtier1 and D. Schweich2

1 IFP Energies nouvelles, Rond-point de l’échangeur de Solaize, BP 3, 69360 Solaize - France2 LGPC, ESCPE-CNRS, 43 bd du 11 novembre 1918, BP 2077, 69616 Villeurbanne - France

§ Current address: Novasep, Pompey – [email protected] - [email protected] - [email protected] - [email protected] - [email protected] - [email protected]

* Corresponding author

Résumé — Modélisation cinétique par événements constitutifs sans génération explicite de grandsréseaux : application à l’hydrocraquage des paraffines longues — Le concept de modélisation parévénements constitutifs permet de développer des modèles cinétiques pour la simulation des procédés deraffinage. Pour des réseaux réactionnels de centaines de milliers d’espèces, comme cela est le cas pour lereformage catalytique, le regroupement rigoureux par nombre d’atomes de carbone et degré deramification a été utilisé efficacement en faisant l’hypothèse de l’équilibre chimique dans chaque groupe.Cette technique de regroupement conduit à un modèle regroupé compact sans perte d’information, maisnécessite tous les détails d’un réseau réactionnel complet, généré de manière explicite.Les techniques classiques de génération de réseaux deviennent inutilisables quand les hydrocarburescontiennent au delà d’environ 20 atomes de carbone à cause de la croissance extrêmement rapide duréseau réactionnel. Ainsi, des techniques implicites de regroupement ont été développées afin de calculerles coefficients de regroupement sans avoir à générer le réseau réactionnel détaillé. Deux approchesalternatives et équivalentes sont présentées, basées soit sur une décomposition en classes structurales, soitune décomposition en chaînes latérales. Ces deux méthodes sont comparées et la méthode dedécomposition en chaînes latérales est appliquée à la modélisation cinétique de l’hydrocraquage desparaffines longues. La méthode de décomposition en chaînes latérales est strictement équivalente à laméthode originelle basée sur la génération explicite du réseau réactionnel détaillé, pour autant que laméthode de contribution de groupe de Benson soit utilisée pour calculer les propriétés thermodynamiquesdans chacune des approches.

Abstract — Single Event Kinetic Modelling without Explicit Generation of Large Networks:Application to Hydrocracking of Long Paraffins — The single event modelling concept allowsdeveloping kinetic models for the simulation of refinery processes. For reaction networks with severalhundreds of thousands of species, as is the case for catalytic reforming, rigorous relumping by carbonatom number and branching degree were efficiently employed by assuming chemical equilibrium in eachlump. This relumping technique yields a compact lumped model without any loss of information, butrequires the full detail of an explicitly generated reaction network.Classic network generation techniques become impractical when the hydrocarbon species contain morethan approximately 20 carbon atoms, because of the extremely rapid growth of reaction network. Hence,

Oil & Gas Science and Technology – Rev. IFP Energies nouvelles, Vol. 66 (2011), No. 3, pp. 399-422Copyright © 2011, IFP Energies nouvellesDOI: 10.2516/ogst/2011118

Chemical Reaction Modelling of Refining Processes Modélisation cinétique des procédés de raffinage

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implicit relumping techniques were developed in order to compute lumping coefficients withoutgenerating the detailed reaction network. Two alternative and equivalent approaches are presented, basedeither on structural classes or on lateral chain decomposition. These two methods are discussed and thelateral chain decomposition method is applied to the kinetic modelling of long chain paraffinhydroisomerization and hydrocracking. The lateral chain decomposition technique is exactly equivalentto the original calculation method based on the explicitly generated detailed reaction network, as long asBenson’s group contribution method is used to calculate the necessary thermodynamic data in bothapproaches.

NOMENCLATURE

Roman

bFoPhysisorption coefficient for lump Fo

bPiPhysisorption coefficient for paraffin Pi

Csat Saturation concentration, i.e. physical adsorptioncapacity of the zeolite

CH+ Concentration of free acid sitesCt Total concentration of acid sitesDENg Langmuir type inhibition term due to physical

adsorption for the zeoliteDENg

acid Langmuir type inhibition term due to chemisorp-tion on the acid sites

Fo Reacting paraffin lumpFp, Fq Produced paraffin lumpsKDHij

Dehydrogenation/hydrogenation equilibriumconstant of paraffin Pi into olefin Oij

Kisom Intrinsic equilibrium constant of olefin isomer-ization (excluding the contributions for rotationsymmetries and chirality)

KPr(m;Oij) Protonation/deprotonation equilibrium constantfor a carbenium ion of type m into olefin Oij

KPr Intrinsic equilibrium constant of olefin protona-tion (excluding the contributions for rotationsymmetries and chirality)

K*g(n,T) Equilibrium coefficient of formation of the

paraffin lump g at temperature TK*

ref,g(n,T) Intrinsic equilibrium coefficient of formation ofreference olefin Oref and of hydrogen from thepure elements at temperature T

kcr Cracking rate coefficientkpcp PCP branching rate coefficientkX Rate coefficient for the elementary step for

reaction X (X = reac, Pr, dep, ...)kX Single event rate coefficient for reaction X

(X = reac, Pr, dep, ...)kreac(m;n) Rate coefficient for reaction reac of a carbenium

ion of type m into a carbenium ion of type nk*

reac(m;u) Re-parameterised composite rate coefficient forreaction reac of a carbenium ion of type m into acarbenium ion of type n

LCisom(m,n)(g,h) Lumping coefficient for isomerization oflump g into lump h by means of a (m,n)type isomerization.

LHSV Liquid Hourly Space Velocitym Type (primary, secondary, tertiary) of the

reactive carbenium ionNisom(m,n) Number of (m,n) type pathways that allow

to transform lump g into lump hn Type (primary, secondary, tertiary) of the

produced carbenium ionnb Number of branchesnc Number of carbon atomsnch Number of chiral carbon atoms among the

three carbon atoms of the 2-electron 3-centerbond

ne Number of single eventsnp Number of carbon atoms in the “main chain”

of a lateral chainnalk,C Number of alkanes in the structural class Cnions,C Number of carbenium ions in the carbenium

ion structural class Cngch,C Minimum number of gauche interactions

for paraffins in structural class Cnp,C Number of primary carbon atoms for paraf-

fins in structural class Cns,C Number of secondary carbon atoms for

paraffins in structural class Cnt,C Number of tertiary carbon atoms for paraf-

fins in structural class Cnq,C Number of quaternary carbon atoms for

paraffins in structural class Cnocc(A-ZA-B) Number of reactions of type (m,u) that

involve activated complex A - ZA - BOij Olefin j resulting from the dehydrogenation

of paraffin Pi

Oref Reference olefinPi Paraffin iPFo

Partial pressure of lump Fo

PH2Partial pressure of hydrogen

PPiPartial pressure of paraffin Pi

Rate of reaction reac, of type (m,u),between lump Fo and lump Fp

r F Freac m u

o p→{ }( , )


D Guillaume et al. / Single Event Kinetic Modelling without Explicit Generation of Large Networks: Application to Hydrocracking of Long Paraffins

401

R Ideal gas constant

R+ik Carbenium ion k resulting from protonation of

olefin Oij

SPLC(nc,nb) Symmetry Property of Lateral Chains with nc

carbon atoms and nb branches

TPLC(nc,nb) Thermodynamic Property of Lateral Chainswith nc carbon atoms and nb branches

T Temperature

W Weight of catalyst inside the reactor

W/F0tot Space time

Y Species (Y = H2, Pi, Oij, R+ik, #ikl, ...)

yi Mole fraction of paraffin Pi in lump Fo

yi Experimental mole fraction of observable i

yi Calculated mole fraction of observable i

Z Group of Benson’s group contribution method(primary carbon atom, secondary carbon atom,tertiary carbon atom, quaternary carbon atom,and minimum number of gauche interactions)

Greek

ΔG0f (Y) Gibbs free enthalpy of formation of species Y

ΔG0f (Y) Intrinsic Gibbs free enthalpy of formation

(rotation symmetries and chirality excluded)

ΔG0f,Z Benson group contribution to the intrinsic

Gibbs free enthalpy of formation (rotationsymmetries and chirality excluded) at tem-perature T for group Z

σY Global symmetry number of species Y

σPi,CGlobal symmetry number for paraffins Pi inthe structural class C

Subscript

ext Refers to the external symmetry

int Refers to the internal symmetry

i Refers to paraffin Pi in lump Fo

ij Refers to the j-th olefin Oij resulting fromdehydrogenation of paraffin Pi

ik Refers to the k-th carbenium ion R+ik resulting

from protonation of olefin Oij

ikl Refers to the l-th activated complex #ikl result-ing from a reaction of carbenium ion R+

ik

r Reactant

reac Type of reaction (PCP branching, methylshift, ...)

# Activated complex

INTRODUCTION

Using the concept of “single events” is an efficient way ofmodelling acid-catalyzed reactions in refining without intro-ducing reductive assumptions on the composition of the feed(Froment, 1999, 2005). The single event methodology hasalready been extensively described in the literature. Thisacid-catalyzed modelling approach has been applied to iso-merization (Guillaume et al., 2003a; Surla et al., 2004,2011), alkylation (Martinis and Froment, 2006), olefinoligomerization (Guillaume, 2006; Shahrouzi et al., 2008),methanol-to-olefins (Park and Froment, 2001a, b, 2004; AlWahabi and Froment, 2004; Froment, 2005), catalyticreforming (Sotelo-Boyás and Froment, 2009; Cochegrue etal., 2011), catalytic cracking (Feng et al., 1993; Dewachtereet al., 1999; Beirnaert et al., 2001; Moustafa and Froment,2003; Froment, 2005; Quintana-Solórzano et al., 2005,2007a, b, 2010) and hydrocracking (Baltanas and Froment,1985; Baltanas et al., 1989; Vynckier and Froment, 1991;Svoboda et al., 1995; Schweitzer et al., 1999; Martens andFroment, 1999; Martens et al., 2000, 2001; Martens andMarin, 2001; Thybaut et al., 2001, 2009; Thybaut and Marin,2003; Laxmi Narasimhan et al., 2003a, b, 2004, 2006, 2007;Chavarría-Hernández et al., 2004, 2008; Kumar andFroment, 2007a, b; Chavarría-Hernández and Ramírez, 2009;Mitsios et al., 2009; Choudhury et al., 2010).

Concerning the modelling of the hydrocracking process,good results have been obtained for the modelling of hydroc-racking at lab scale (Baltanas et al., 1989; Vynckier andFroment, 1991; Svoboda et al., 1995; Martens and Froment,1999; Martens and Marin, 2001; Martens et al., 2001;Thybaut and Marin, 2003; Kumar and Froment, 2007a, b;Chavarría-Hernández et al., 2008; Chavarría-Hernández andRamírez, 2009; Thybaut et al., 2009) and at pilot plant scale(Schweitzer et al., 1999; Mitsios et al., 2009). The directapproach of the methodology, as described by Baltanas andFroment (1985) and Vynckier and Froment (1991), becomesunpractical when it is applied to heavy feeds with high car-bon numbers, as the size of the generated network, both interms of species and reactions, increases extremely rapidlywith carbon number, resulting in colossal computing timesand intractable networks. Despite this, Kumar and Froment(2007a, b) were able to apply the direct generation of theentire detailed reaction network up to C40 by using dynamicmemory allocation via linked lists for storing and searchingthe intermediate olefinic and ionic species. While the contin-uously developing computing power and memory availabil-ity will allow to push back the limits of the network sizeattainable by direct generation, this approach will still remaina complex and lengthy task.

In order to circumvent this difficulty, the equationsderived for the single event kinetic modelling method werecarefully inspected. Their reformulation led to two alternativemethods: one based upon the concept of structural classes



(Martens, 2000; Martens and Marin, 2001; Laxmi Narasimhanet al., 2004), the other based upon the concept of lateralchains (Valéry, 2002; Guillaume et al., 2003b; Valéry et al.,2007; Mitsios et al., 2009). Both methods allow to circum-vent the explicit generation of the detailed reaction network.Moreover, they also provide further in the evolution of thereaction network with carbon number.

The method proposed by Martens and Marin (2001),based on structural classes, essentially consists in countingstructures that have the same branching number and globalsymmetry so as to determine the frequency of each structureby carbon number and by degree of branching. In this way,this method does not need any explicit network generation.For a given network, the structural classes have to be enu-merated by the user. Hence, this method seems to need aclassification of the structural classes, which has to be manu-ally updated when the maximum degree of branching of thenetwork is increased.

The method developed by Valéry (2002) is based on alateral chain decomposition approach. The computations areentirely automated. As only input, the highest carbon numberand the highest degree of branching in the network has to bespecified by the user (Valéry et al., 2007; Mitsios et al.,2009).

It should be stressed that both techniques are strictlyequivalent and the lumping coefficients they calculate willhave the same value. Moreover, both methods are strictlyequivalent to the original calculation method based on theexplicitly generated reaction network, as long as Benson’sgroup contribution method (Benson et al., 1969; Benson,1976) is used to obtain the necessary thermodynamic data inall approaches. Of the two alternative methods that do notneed the explicit network generation, the lateral chaindecomposition method seems easier to apply, however. Inthis paper, this lateral chain decomposition approach is usedto calculate these lumping coefficients, which are then subse-quently used to derive a kinetic model for long paraffinhydrocracking. The kinetic parameters were identified basedon n-hexadecane hydrocracking data. Validation is performedby extrapolating the single event kinetic model to highercarbon number paraffins and confronting the model results toexperimental data.

1 A POSTERIORI RELUMPING OF LARGE NETWORKS

The use of computer tools is the route for modelling verylarge reaction networks. As illustrated in the previous papers,the single event modelling methodology is based on threemain tools and concepts:– computer generation of the reaction network (to obtain the

full mechanistic detail);– single event rate coefficients (to reduce the number of

kinetic parameters);

– rigorous regrouping (to limit the number of species with-out losing information).This methodology, as initially developed (Baltanas and

Froment, 1985; Vynckier and Froment, 1991), is not reallyefficient for extremely large networks as those needed todescribe hydrocracking of actual VGO feedstocks. However,alternative rigorous solutions can be developed, as describedin Martens (2000) and Valéry (2002). Both solutions arebased on reformulating the kinetic equations and on examin-ing them from a new perspective. Let us first examine thesolutions from the original equations.

According to Baltanas and Froment (1985), the ratecoefficient of an elementary step can be written as:

where reac stands for the type of reaction (PCP branching /methyl shift / ...), ne is the number of single events, m and ncharacterize the nature of the reactive carbenium ion and ofthe produced carbenium ion. It is assumed that the intrinsicparameter Kreac (m,n) does not depend on the number of car-bon atoms but only on the type of the reaction and on thetype of reactive and produced carbenium ions. The influenceof the geometry of the molecules is limited to the number of

single events, which is equal to , i.e. the ratio of the

symmetry number of the reactive ion and of the activatedcomplex involved in the reaction. The single event conceptallows to drastically reduce the number of rate coefficientsentailed in the kinetic description of the full network.

Vynckier and Froment (1991) introduced a posteriorilumping of species by carbon number and by degree ofbranching in large single event networks. In opposition to apriori lumping technique, which is generally a “blind” lump-ing approach devised to cope with the lack of component-based analytical data, this a posteriori lumping scheme is arigorous relumping approach based on regrouping species atthermodynamic equilibrium inside a single lump. In thissense, it is strictly an exact lumping technique as defined byWei and Kuo (1969), since the composition is known throughthe equilibrium relations. The latter underlying assumptionwas reasonably verified by Schweitzer et al. (1999) andValéry (2002) for the hydrocracking of n-hexadecane.

The main assumptions in the relumping scheme are:

– hydrogenation/dehydrogenation steps are at equilibrium;

– protonation/deprotonation steps are at equilibrium;

– intermediate species (alkenes, carbenium ions) are atpseudo-steady state;

– the adsorption coefficients for physical (Langmuir) adsorp-tion depend only on the number of carbon atoms of themolecules;

– the amount of adsorbed olefins is negligible with respectto the amount of paraffins.

ner=

σσ#

k m n n k m nreac e reac( , ) ,= ⋅ ( )�



403

The second assumption (protonation/deprotonation atequilibrium) also implies that the hydride shift reactionsbetween carbenium ions are also at equilibrium. It is furtherassumed that gas-liquid equilibrium prevails. The latterhypothesis enables expressing the rate laws using the partialpressures instead of the concentrations of the dissolvedspecies (Martens and Marin, 2001).

Let Fo, Fp and Fq be lumps, i.e. groups of paraffins. Ineach lump, all hydrocarbons have the same number of carbonatoms and they are in chemical equilibrium. Let “reac” be anelementary step, such as a PCP or PCB isomerization step(Fo → Fp), or a β-scission cracking step (Fo → Fp + Fq). Thelumped rate expression for the disappearance of lump Fo isgiven by (Vynckier and Froment, 1991):

(1)

where PH2is the partial pressure of hydrogen,

PF0is the partial pressure of lump Fo,

bF0is the physical adsorption coefficient oflump Fo,

Csat is the physical adsorption capacity of thezeolite (assumed to be lump independent),

Ct is the total concentration of acid sites,DENg is the inhibition term due to physical adsorp-

tion for the zeolite,DENg

acid is the inhibition term due to chemisorptionon the acid sites, i.e. protonation of olefins,

yi is the mole fraction of paraffin Pi in lump Fo,m, u are the type (secondary or tertiary) of the

reactant and product carbenium ions,kreac(m;n) is the rate coefficient for reaction reac of a

carbenium ion of type m into a carbeniumion of type n,

KDHijis the dehydrogenation/hydrogenation equi-librium constant of paraffin Pi into olefinOij,

KPr(m;Oij) is the protonation/deprotonation equilib-rium constant for a carbenium ion of typem into olefin Oij.

Index “i” refers to paraffin Pi in Fo, “ij” to the j-th olefinOij resulting from dehydrogenation of paraffin Pi, “ik” to thek-th carbenium ion R+

ik resulting from protonation of olefinOij, and “ikl” to the l-th activated complex #ikl resulting froma reaction of carbenium ion R+

ik (Fig. 1). Paraffin “i”, olefin“ij”, carbenium ion “ik”, and activated complex “ikl” havethe same number of carbon atoms within lump Fo.

r

K m O K

reac m uF F F

ij DH

o p q

ij

( , )

Pr ;

→ +( )⎧⎨⎩

⎫⎬⎭

=

( ) ⋅ ⋅⋅ ⋅⎡⎣

⎤⎦

∈

k m u yreac i

iklreac m uF

( ; )( , )

� oo p q

o o

F F

t sat F FC C b P

P

→ +( )⎧⎨⎩

⎫⎬⎭

∑ ⋅

⋅ ⋅ ⋅�

HH2⋅ +DEN DENg

acidg

For the three rate coefficients, the following relationsapply:

(2)

where kX is the rate coefficient for the elementary stepfor reaction X (X = reac, Pr, dep, ...),

kX is the single event rate coefficient forreaction X,

σY is the global symmetry number of species Y(Y = Oij, R

+ik, #ikl, ...),

Kisom is the intrinsic equilibrium constant of olefinisomerization (excluding the contributionsfor rotation symmetries and chirality),

KPr is the intrinsic equilibrium constant of olefinprotonation (excluding the contributions forrotation symmetries and chirality),

ΔG0f (Y) is the Gibbs free enthalpy of formation of

species Y,ΔG0

f (Y) is the intrinsic Gibbs free enthalpy of for-mation (rotation symmetries and chiralityexcluded).

The two inhibition terms combine the contributions of theadsorption of the various species:

(3)

where PFqis the partial pressure of lump Fq,

PPiis the partial pressure of paraffin Pi,

bPiis the physical adsorption constant of paraffin Pi.

The mole fraction of paraffin Pi in lump Fo is given by:

(4)

ye

e

i

G P

RT

G P

RT

f i

f i

=

−⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

−⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

Δ

Δ

0

0

( )

( )

PP Fi o∈

∑

DEN b P

DEN C b K

gF F

q

acidg

sat P

ion

DH

q q

i ij

= +

= ⋅ ⋅

∑

∑∀

1

PP O

K O O

i ij

isom ij refOr

⇔ +( ) ⋅

⇔( ) ⋅

H

2

�σ

eef

ik

i

Rref PK m O P

σ +

⋅ ( ) ⋅�Pr ,

k m u n k m u kreac e reacR

reacik

ikl

( , ) ( , )#

= ⋅ = ⋅+� �

σ

σ(( , )

( , ) ( , )

(

Pr Pr

Pr

m u

K m O K m O

K

ijO

Rij

ij

ik

= ⋅+

σ

σ�

� mm O K m O K O Oij ref isom ij ref, ) ( , )Pr= ⋅ ⇔( ) =� �

�

� �

K m O eref

G O G Of ref f ij

Pr

( ) ( )

( , ) ⋅−

−Δ Δ0 0

RRT

G O

RTe

f ij

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

−⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

Δ � 0 ( )

== ⋅−

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

σO

G O

RT

ij

f ij

e

Δ 0 ( )



Finally, the equilibrium constant for dehydrogenation ofparaffin Pi into olefin Oij can be calculated from the Gibbsfree energy of formation of the various species:

(5)

In the original method, the sums in equation (1) are calcu-lated by browsing through all the available reactions obtainedthrough explicit generation of the detailed reaction networkand adding up the appropriate terms. Hence, Vynckier andFroment (1991) proposed to reorganize the rate equationabove in order to separate the rate coefficients from the fac-tors that only depend on the reaction network and the choiceof the lumps in order to arrive at a coarser a posteriorirelumped reaction network. For the case of gas-phase iso-merization of lump g into lump h, they finally arrived at thefollowing rate expression:

K P O eDH i ij

G O G G

ij

f ij f f

⇔ +( ) =−

+ −

H2

HΔ Δ Δ0 02

0( ) ( ) (PP

RTi )⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

(6)

As earlier work (Svoboda et al., 1995; Martens andFroment, 1999; Martens and Marin, 2001) showed that thetotal concentration of carbenium ions could be neglectedcompared to the total concentration of acid sites, the forwardreaction rate for the gas-phase reaction is therefore reducedto:

(7)r g h

k g h C b P

P b Pisom

isomL

sat g g

Fq

( ; ),

=( ) ⋅ ⋅ ⋅

⋅ +H21 FF

qq∑

⎛

⎝⎜⎜

⎞

⎠⎟⎟

r g hk g h C b P

P b P

isomisomL

sat g g

Fq

( ; ),

=( ) ⋅ ⋅ ⋅

⋅ +H21 FF

q

c F sat F F

qq q q q

K C b P∑ ∑⎛

⎝

⎜⎜

⎞

⎠

⎟⎟+ ,

+

+H

+

+

+

+

+

+

+

+

H

+

H

+

H

+

H

HH

H

H

HH

HH

HH

Paraffin P(index i)

Olefin O(indices i,j)

Carbenium ions R+

(indices i,k)Activated complexes #

(indices i,k,l)

Figure 1

Definitions of the symbols and indices of paraffins, olefins and activated complexes illustrated for 2,3 dimethyl heptane structures.



405

In this equation, the rate coefficient for the forward reactioncan be decomposed into (Vynckier and Froment, 1991):

(8)

where LCisom(m,n) (g,h) is the lumping coefficient forisomerization of lump g into lump h by means of an (m,n)type isomerization.

In the original method (Vynckier and Froment, 1991), thelumping coefficients are calculated as a sum over all isomer-ization reactions of type (m,n) that transform a carbenium ionof type m obtained from a hydrocarbon of lump g into acarbenium ion of type n that corresponds to a hydrocarbon oflump h: See Equation (9).

As explained elsewhere (Vynckier and Froment, 1991;Martens and Marin, 2001; Cochegrue et al., 2011), the calcu-lation of the lumping coefficient only requires the knowledgeof the detailed reaction network of elementary steps and ofthe choice of the lumps, but does not require the knowledgeof the single event rate coefficients, as these have beenexplicitly isolated in the expressions above. Hence, the lump-ing coefficients need to be calculated only once. The lumpingcoefficients do, however, still depend on temperature.Indeed, yi represents the equilibrium mole fraction of ahydrocarbon in a given lump, which evidently depends ontemperature. The same holds for the olefin isomerization andthe dehydrogenation equilibrium constants. The resultinglumping coefficients exhibit a van’t Hoff type temperaturedependency, which is mainly due to the hydrogenation/dehy-drogenation equilibrium constant, since both the values forthe olefin isomerization equilibrium constant and the equilib-rium mole fractions vary only slightly with temperature. Inconclusion, the advantage of isolating the rate coefficients isthat, for each temperature, the lumping coefficients need tobe calculated only once for a given reaction network.

When dealing with heavy molecules, however, the numberof species and the number of reactions grow very rapidlywith carbon number, leading to immense reaction networksand resulting in excessive computation times and colossalmemory requirements. Once the entire reaction network ofelementary steps is finally generated, inspection of this vastnetwork allows calculating the lumping coefficients thateffectively summarize this profusion of detail into an a poste-riori relumped reaction network.

k g h LC g h K s OisomL

isom s s ref, , ,( , ) Pr( ) = ( ) ⋅ ( ) ⋅� �kk s s C

LC g h K s O

isom t

isom s t ref

( , )

, ,( , ) Pr

⋅ +

( ) ⋅ (� )) ⋅ ⋅ +

( ) ⋅

�

�

k s t C

LC g h K t O

isom t

isom t s r

( , )

, ,( , ) Pr eef isom t

isom t t

k t s C

LC g h K t

( ) ⋅ ⋅ +

( ) ⋅

�

�

( , )

,( , ) Pr ,, ( , )O k t t Cref isom t( ) ⋅ ⋅�

Besides the original method (Vynckier and Froment, 1991),which proceeds through inspection of the detailed reactionnetwork, two alternative ways of computing these lumpingcoefficients without explicitly generating the entire reactionnetwork have been developed in the literature and are pre-sented below.

2 STRUCTURAL CLASS METHOD

To avoid the explicit generation of the complete reactionnetwork of elementary steps, Martens and Marin (2001)proposed a method based on structural classes. The main ideais based on the fact that hydrocarbon molecules can bedescribed by combining a limited number of structures.Hence, in this approach, these structures that have the sameintrinsic Gibbs energy of formation and symmetry numberare listed: such a structure is a class. The sums in Equation(9) therefore reduce to sums over the classes weighted by thenumber of reactions that involve each species of a class. Thestructural class method is an enumeration technique thatcounts the number of species in each class. It thereforerequires listing the classes and the members of the class.However, this is much faster than the exhaustive generationof all the species.

Following the structural class approach (Martens andMarin, 2001), the lumping coefficient defined by Vynckierand Froment (1991) is reorganized as follows:

(10)

The factor Nisom(m,n) represents the number of (m,n) typepathways that allow to transform lump g into lump h. It istherefore a weighted sum of the single event numbers of each(m,n) reaction linking both lumps, corrected for the globalsymmetry number of hydrogen:

(11)

The factor K*ref,g(n,T) is related to the intrinsic equilibrium

coefficient of formation of the reference olefin and of hydro-gen from the pure elements:

(12)

The factor K*g(n,T) corresponds to the equilibrium coefficient

of formation of the paraffinic lump and is calculated as the

K n T eref g

G O G

RT

f ref f

,* ,( ) =

−( )+ ( )Δ Δ� �0 0

2H

Nn

isom m ne ikqr

Rr hq hk gi g ik

( , ),=⋅+

∈∈∈∈

∑∑∑∑ σ σH2

LC g hN K n T

Kisom m n

isom m n ref g

g( , )

( , ) ,*

*,

,( ) =

⋅ ( )nn T,( )

LC g h n K Oisom m n e ikqrO

Risom

ij

ik

( , ) ,,( ) = ⋅ ⋅+

σ

σ�

iij ref DH i ij i

r hq hk gi

O K P O yij

⇔( ) ⋅ ⇔ +( ) ⋅∈∈∈

∑∑∑ H2∈∈

∑g

(9)



sum of the equilibrium coefficients of formation of allparaffins Pi of lump g:

(13)

2.1 Computation of K*g(n,T )

In order to calculate K*g(n,T), one needs to compute the Gibbs

enthalpy of formation of each paraffin in lump g. As no reli-able data are available in the literature for high molecularweight species, the Gibbs enthalpy of formation is obtainedby means of the Benson group contribution method (Bensonet al., 1969; Benson, 1976). In this group contributionmethod, the values of the contributions to the enthalpy offormation, the entropy of formation and the specific heatcapacity of paraffins only depend on the type (i.e., primary,secondary, tertiary or quaternary) of each carbon atom in themolecule, on the global symmetry number and on the num-ber of gauche interactions between the carbon-carbon bondsof the molecule. Molecules with the same number of occur-rences of each property in Benson’s group contributionmethod will therefore have the same Gibbs enthalpy of for-mation. More precisely, paraffins can be sorted by theirdecomposition into Benson groups, their global symmetrynumber and their number of gauche interactions. Each com-bination of these properties is called a structural class. As aconsequence, all paraffins within a lump can easily be subdi-vided into several structural classes with exactly the samenumber of occurrences of each Benson property. For paraf-fins, each structural class is therefore characterized by thenumbers of primary np,C, secondary ns,C, tertiary nt,C, andquaternary nq,C carbon atoms, the minimum number ofgauche interactions ngch,C, the number of alkanes nalk,C, andthe global symmetry number σPi,C

for paraffins in the struc-tural class C. The factor K*

g(n,T) can hence be written as: SeeEquation (14).

with ΔG0f,Z is the Benson group contribution to the intrinsic

Gibbs free enthalpy of formation (rotation sym-metries and chirality excluded) at temperature Tfor group Z (Z = primary carbon atom, secondarycarbon atom, tertiary carbon atom, quaternarycarbon atom, and minimum number of gaucheinteractions).

K n T ee

g

G P

RT

i g

G Pf i

f i

*

( )

,( ) = =−

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

∈

−

∑Δ Δ0 0� (( )

∈

∑RT

Pi g iσ

Once all the classes corresponding to one lump areknown, K*

g(n,T) can be computed as a function of the num-ber of carbon atoms n. By means of example, the structuralclasses for single branched alkanes with a methyl or anethyl group are illustrated in Table 1 (Martens and Marin,2001). In total, there are 6 structural classes per carbonnumber in which these single branched paraffins can beclassified. All di- and tribranched alkanes with methyl andethyl groups can also be sorted in a limited number ofclasses (Martens, 2000).

This method can easily be extended to cycloalkanes(Martens, 2000) by introducing the additional Bensongroups for ring structures and providing for the additionalstructural classes. The structural class method was thenapplied to alkanes with up to three branches (methyl and/orethyl), to monocyclic alkyl cyclopentanes with up to foursubstituents and only one substituent longer than methyl,and to non-pericondensed alkyl cyclohexanes with up tofour six-membered rings and up to four substituents withonly one substituent longer than methyl (Martens, 2000;Martens and Marin, 2001).

TABLE 1

Structural classes for monobranched alkanes (moPn) carrying one methylor one ethyl group

Alkane Class C np,C ns,C nt,C nq,C ngch,C nalk,C

3 n-4 1 0 27 1 1

3 n-4 1 0 27 2 n even: 1

n odd: 0

3 n-4 1 0 27/2 2 n even: (n-6)/2

n odd: (n-5)/2

3 n-4 1 0 27 3 1

3 n-4 1 0 27 3 n even: 0

n odd: 1

3 n-4 1 0 27/2 3 n even: (n-8)/2

n odd: (n-9)/2

K n T

ne eg

alk C

P C

n G

RT

n

i

p C f p s C

* ,

,

,, , ,

( ) = ⋅− −

σ

Δ Δ� 0 �� G

RT

n G

RT

n G

RTf s t C f t q C f q

e e e, , , , ,

0 0 0

⋅ ⋅ ⋅− −

Δ Δ−−

∑n G

RT

C

n gch C f gchclasses , ,Δ � 0

(14)

Cx

Cx Cx

Cx Cy

Cx

Cx Cx

Cx Cy

σ P Ci ,



407

TABLE 2

Structural classes of carbenium ions contributing to NPCP(s,t)(diPn;moPn)

Carbenium ion class C ne,ikqr nions,C

243 18 1 3/81

81 12 n-6 6(n-6)/81

81/2 6 1 6/81

81 6 1 3/81

81/2 6 n-9 6(n-9)/81

n ne ikqr ions C

Rik

, ,⋅

⋅+σ σH2

Cx

+

CxCy

+

Cx+

Cx+

CxCy

+

σRik

+

2.2 Computation of Nisom(m,n)

In order to calculate Nisom(m,n) for a given reaction, oneneeds to compute the global symmetry number of all car-benium ions that are able to react according to this path-way, together with its corresponding number of singleevents for this reaction, i.e. the ratio of the global symme-try number of the reacting carbenium ion to the globalsymmetry number of the corresponding transition state.According to the same procedure, Martens and Marin(2001) used the concept of structural classes for the varioustypes of carbenium ions. For example, considering the sec-ondary to tertiary PCP isomerizations between dibranchedparaffins and monobranched paraffins, they found 5 struc-tural classes of secondary dibranched carbenium ions thatare able to react according to a PCP step with formation ofa tertiary monobranched carbenium ion. Table 2 lists these5 classes of carbenium ions, together with their globalsymmetry number σR+

ik, their number of single events ne,ikqr

for the secondary to tertiary PCP isomerization and thenumber of carbenium ions nions,C in each structural class C.Hence, the factor Nisom(m,n) for secondary - tertiary PCP iso-merization from dibranched paraffins to monobranchedparaffins can be computed by added up the various contri-butions of each structural class:

(15)N diP moP

n nPCP s t n n( , ) ;( ) =

−=

−( ) +12 78

81

12 7 6

81 for 10n ≥

This expression is valid only for n ≥ 10, as not all structuralclasses contribute to this reaction for lower carbon numbers.

Again, this approach can easily be extended to isomerizationof the other paraffin lumps and to the various isomerizationreactions of cycloalkanes (Martens, 2000) by introducingthe corresponding structural classes of carbenium ions.Moreover, the approach has also been extended to crackingreactions, as illustrated by Martens and Marin (2001) for thecracking of the lump of branched monocycloalkanes into alump of lighter monocycloalkanes and a lump of dibranchedalkanes.

As mentioned above, the method of structural classesavoids the excessive calculation times required for theexplicit generation of the entire reaction network of elemen-tary reaction steps by directly calculating the lumping coeffi-cient of the various types of reactions. For small degrees ofbranching in the molecules, this approach is extremely inter-esting because it allows obtaining an analytical formulationof the lumping coefficient without the need to enumerate toomany different structural classes.

3 LATERAL CHAIN METHOD

To avoid the explicit generation of the complete reaction net-work of elementary steps, Valéry (2002) proposed a methodbased on lateral chain decomposition. The main idea is basedon the fact that hydrocarbon molecules and their intermedi-ates can be decomposed into lateral chains and active centres.Focusing on the intermediate activated complex, Valéry(2002) used an alternate method to predict the properties ofthese complexes and especially the sum of reciprocal of thesymmetry numbers. This recursive method was also appliedfor the evaluation of the sum of energy terms. In the lateralchain decomposition method, the calculation of the lumpingcoefficients is based on an efficient computer algorithm thatonly needs to know the maximum number of carbon atomsand the maximum degree of branching of the species in thereaction network.

Rearranging Equation (1) in a similar way as above yields:

(16)

with the lumping coefficient LCreac(m,u) (Fo → Fp (+ Fq))defined as in Equation (9).

In the lateral chain decomposition method (Valéry, 2000,2002; Valéry et al., 2007; Mitsios et al., 2009), the lumping

r LC F Freac m uF F F

reac m u o p

o p q

( , ) ( , )→ +( )

⎧⎨⎩

⎫⎬⎭

= → ++( )( ) ⋅

( ) ⋅

F

K m O k m u

q

ref reac � �Pr , ( , ) ⋅⋅

⋅ ⋅ ⋅

⋅ +

C C b P

P DEN DEN

t sat F F

gacidg

g o

H2



coefficient defined by Vynckier and Froment (1991) is reor-ganized as follows:

(17)

In this expression, three factors can be distinguished. Thefirst factor is related to the equilibrium coefficient of forma-tion of the reference olefin and of hydrogen from the pureelements. This factor only depends on the number of carbonatoms of the reacting lump. The second factor in Equation(17) represents the inverse of the equilibrium constant of for-mation of the paraffinic lump and is calculated as the sum ofthe equilibrium constants of formation of all paraffins Pi oflump Fo. The third factor in Equation (17) represents thenumber of (m,n) type pathways that allow to transform lumpFo according to this lumped reaction.

With respect to the structural class approach, the firstfactor corresponds to (σH2

)–1·K*ref,g(n,T), the second factor

corresponds to the reciprocal of K*g(n,T), while the third

factor corresponds to the reciprocal of σH2·Nisom(m,n). This

clearly illustrates that both the structural class approach andthe lateral chain decomposition method, although developedindependently, regroup in a very similar manner the variousfactors included in the lumped coefficients of Equation (9)that were originally defined by Vynckier and Froment(1991). The main difference between both methods thereforeresides in the way of calculating the various factors.

LC F F F ereac m u o p q

G O Gf ref f

( , )

( ) (

→ +( )( ) =−

+Δ Δ� 0 0 HH

2

0

1

)

( )

RT

G P

RT

P F

e

f i

i o

⋅−

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

∈

Δ

∑∑∑⋅

∈→ +( )

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

1σ#( , ) ikl

o p qikl

reac m u

F F F

⎛⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

For the lumping coefficients calculated by the lateral chaindecomposition method (Eq. 17), the first two factors areenergy terms that are obtained from Benson’s group contri-bution method (Benson et al., 1969; Benson, 1976) in orderto maintain the necessary consistency for high molecularweight species. The last factor to be calculated for the lump-ing coefficient is the sum over the symmetry numbers of allactivated complexes occurring in these transformations. It isimportant to note that, in this lateral chain decompositionapproach, this last factor is independent of the produced ionsR+

qr, but only depends on the reacting ion R+ik and on the vari-

ous activated complexes that result from it. This means thatcalculating a lumping coefficient requires only knowledge ofthe ions involved, and thus of the resulting activated com-plexes, instead of a list of the elementary steps. However, thesum extends over all the steps, and some may involve thesame activated complex #ikl. To calculate the last factor, wewill therefore first take a closer look at the representation ofthe activated complexes, before describing the recursive for-mulae for the computation of the factors in the lumping coef-ficients. For the case of comparison with the structural classmethod, it is assumed in what follows that the branches in thelateral chains are only methyl or ethyl substituents.

3.1 Description of an Activated Complex

An activated complex # can be decomposed into three parts:– an “activated zone”, ZA, that carries the positive charge

and is characteristic of the elementary step considered;– a first lateral chain, A;– a second lateral chain, B.

In Figure 2, this is illustrated for β-scission and PCPisomerization reactions. As both lateral chains lie outside the

Activated complex for β-scission steps Activated complex for PCP steps

Lateralchains

BBA A AZAZ

+

H

+

Figure 2

Illustration of the two types of activated complexes.



409

activated zone that is characteristic of the elementary step,the actual lateral chains do not depend on the reaction type,nor do they influence the reaction.

3.2 Recursive Algorithm for Calculating the OverallNumber of (m,n) Type Pathways

The overall number of (m,n) type pathways that allow totransform lump Fo according to the lumped reaction is givenby the sum of reciprocal of the global symmetry numbers ofthe activated complexes σ#, as shown is Equation (17). Asdescribed above, in the lateral chain decomposition method,the activated complex is represented by an activated zone andtwo lateral chains. The global symmetry number σ# of theactivated complex # is therefore given by:

σ# = σA· σZA· σB· σext(A - ZA - B) (18)

where σZA is the partial symmetry number for theactivated zone,

σA, σB are the partial symmetry numbers for thelateral chains,

σext(A - ZA - B) is the external symmetry number of theactivated complex.

Hence, the sum of reciprocal of the global symmetry num-bers of the activated complexes σ# now becomes:

(19)

where{R} is the set of the activated complexes, A-ZA-B,involved in reactions of type (m,u) goingfrom lump Fo to lump Fp (and Fq in the caseof a β-scission),

1

σ #( , )

( )

iklikl

react m u

F F Fo p q∈

→ +

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

∑ =

nocc A ZA B

A ZA B ext A ZA B

( )

(

− −

− −⋅ ⋅ ⋅σ σ σ σ ))A ZA B R− − ∈ { }∑

nocc(A-ZA-B) is the number of reactions of type (m,u) thatinvolve activated complex A - ZA - B.

For β-scission, a given activated complex is only involvedin a single cracking step because the complex is an “interme-diate hybrid” between the reacting carbenium ion, and theresulting ion and olefin (Fig. 3). As a consequence nocc = 1for any β-scission step. For PCP-isomerizations, the sameactivated complex is linked to four different carbenium sions,as shown in Figure 4. Given the assumptions used in the aposteriori relumping scheme, the carbenium ions (I) and (II)belong to the same lump, as do the two carbenium ions (III)and (IV). In total, there are six possible pathways linkingthese four carbenium ions. However, the isomerizationbetween ions (I) and (II) does not appear in the relumped net-work, as both ions pertain to the same lump. The same holdsfor the isomerization between ions (III) and (IV). Hence, 4different PCP isomerizations with a change of degree ofbranching between these different lumps and involving thesame activated complex still remain, and nocc = 4. However,in case A = B and L1 = L2, these four different possibilitiesdegenerate into a single isomerization step (nocc = 1). If inaddition L3 = L4, this isomerization step involves an activatedcomplex that has an external symmetry of 2.

For a given relumped reaction, several pathways may befollowed to proceed from the reacting lump to the productlump(s). For each path, the activated complex needs to beidentified in order to be able to define the set of the activatedcomplexes {R} involved in the reaction. Figure 5 illustratesthe set of the 30 activated complexes A - ZA - B involved inthe (s,s) β-scission of a tri-branched hexadecane into a mono-branched secondary hexyl carbenium ion and a mono-branched decene. From this Figure, it can be observed thatfor this reaction, two different activated zones ZA are presentin the set, one activated zone with 4 carbon atoms containinga methyl substituent and an activated zone with 5 carbonatoms containing an ethyl substituent. For a given activatedzone, the various structural isomers of the two lateral chainsappear.

L4

C+

L5

BA

L1

L3

L2

+A

C+

B

L1

L2 L3

L4 L5

+

A

B

L1

L2 L3

L4 L5

Figure 3

Reaction pathway and activated complex for a β-scission step of a carbenium ion into a smaller carbenium ion and an olefin. L1 to L5 aresubstituents that are part of the activated zone, while A and B are lateral chains, which do not participate in the cracking reaction.



(III)

(IV)

(I)

(II)

A B

L1 L2

H

+

L3 L4

A

C+

B

L1 L2

L3 L4

H

A

C+

B

L1 L2

L3 L4

H

A

C+ B

L1

L2

L3 L4

H

A C+

B

L1

L2

L3 L4

H

8,1

...3,1

...

8,1

...

4,1

... 8,1

...

4,1

...

Figure 4

Reaction pathway and activated complex for a PCP-isomerization step of a carbenium ion. For a given activated complex, several PCP-isomerization steps are possible unless the external symmetry of the reacting ion causes the steps to be identical. L1 to L4 are substituents thatare part of the activated zone, while A and B are lateral chains, which do not participate in the PCP rearrangement.

Figure 5

Set of the 30 activated complexes involved in the (s,s) β-scission of a tri-branched hexadecane into a monobranched secondary hexylcarbenium ion and a monobranched decene



411

In order to evaluate Equation (19), it is therefore furtherdecomposed into:– a sum over the various activated zones ZA,– a sub-sum over lateral chains A that have a given number

of carbon atoms, nc,A, and a given number of lateralbranches, nb,A.In a given lump, choosing an activated zone with nc,ZA car-

bon atoms and with nb,ZA branches determines the number ofcarbon atoms and the number of lateral branches that are stillavailable for the lateral chains A and B. A set of lateral chainsA with a given carbon number nc,A and branching degree nb,A

will be termed “equivalent lateral chains”. Once the class ofequivalent lateral chains A and the activated zone areselected, the class of equivalent lateral chains B (i.e., nc,B andnb,B) is fixed because the total number of carbon atoms andthe total number of branches are determined by the lump.Activated complexes are said to be “homologous activatedcomplexes” when they have the same activated zone andwhen the lateral chains A (or B, equivalently) belong to thesame equivalence class.

Figure 6 illustrates the concepts of equivalent lateralchains and of homologous activated complexes. As shown inFigure 5, there are 2 structures for the activated zoneinvolved in the (s,s) β-scission of a tri-branched C16 paraffininto a monobranched secondary hexyl carbenium ion and amonobranched decene, one activated zone with 4 carbonatoms containing a methyl branch and one with 5 carbonatoms containing an ethyl branch. For the activated zone with4 carbon atoms, there are 2 possible monobranched lateralchains with 4 carbon atoms and 10 possible monobranchedlateral chains with 8 carbon atoms (Fig. 6). There are there-fore 2 × 10 = 20 homologous activated complexes with anactivated zone of 4 carbon atoms and containing a methylbranch. For the activated zone with 5 carbon atoms contain-ing an ethyl branch, there is a single monobranched lateralchain with 3 carbon atoms and there are again 10 possiblemonobranched lateral chains with 8 carbon atoms, resultingin 10 homologous activated complexes. Hence, the total setof the activated complexes involved in the (s,s) β-scission ofa tri-branched hexadecane into a monobranched secondaryhexyl carbenium ion and a monobranched decene amounts to30, as was already shown in Figure 5.

In what follows, we will exemplify the lateral chaindecomposition approach for β-scission reactions. For detailsconcerning PCP-isomerization steps, the reader is referred toValéry (2002). For β-scission reactions, the following simpli-fications apply:– there is only one β-scission reaction for a given activated

complex, i.e. nocc = 1;– as the activated zone contains a 2-electron 3-center bond

without flexibility for free rotation around this bond, there isno external symmetry due to its rigid structure, i.e. σext = 1;

– the activated complex of a β-scission reaction has thesame number of carbon atoms and the same degree ofbranching as the reacting paraffin;

– the equivalence class of lateral chains are defined by thelumps of the product ion and olefin.After introducing these simplifications and decomposing

into sub-sums, Equation (19) can be rewritten as follows:

(20)

In Equation (20), the various homologous activatedcomplexes are the individual contributions to each term ofthe sum over ZA. The sum over the lateral chains and the sumover the complementary lateral chains in Equation (20) canbe calculated by a recursive method. Let np be the number ofcarbon atoms of the “main chain” of a lateral chain. The mainchain is easily identified, as it is the longest linear chain ofthe lateral chain. The lateral chain is therefore composed of aterminal CH3 group and successive elementary patterns cho-sen among -CH2-, -CHMe-, -CHEt-, -CMeMe-, -CMeEt-,and -CEtEt-. Each of these patterns contributes to the lengthof the main chain. Figure 7 shows the six possibilities forincreasing the length of a lateral chain starting from CH3-. IfσL is the symmetry number of the lateral chain with npcarbons along the main chain, then the symmetry numberafter inserting an additional pattern is σL times the symmetrynumber of that pattern. Let US(np,nc,nb) be defined by:

(21)

The recursion formula based on “pattern insertion” (Fig. 7)then becomes: see Equation (22).

This formula is valid for np > 3. For low values of nb or nc,the second or third argument of the function US may becomenegative; in this case the contribution to US is set to 0.Ignoring chirality, the symmetry number (1, 3 or 9) of theinserted pattern is defined by the structure of the allowed lat-eral branches (-H, -CH3, -C2H5 in this example, as explainedabove). Chirality is possible as soon as the chain on the leftof the inserted pattern contains more than 3 carbon atoms(i.e., different from -H, -CH3 and -C2H5): the central carbonis chiral when its two remaining substituents are different. Inthis case, the global symmetry number is divided by 2.Further details and formulae for np ≤ 3 are given in Valéry

U n n nS p c b

A

A

Length of main chain n

Numbep

, ,( ) =

∈

=

1

σrr of carbon atoms n

Number of branches nc

b

=

=

⎧

⎨⎪

⎩⎪

⎫

⎬⎬⎪

⎭⎪

∑

1 1 1

σ σ σ σ σA ZA BA ZA B R ZAZA AAn carbonc A

⋅ ⋅= ⋅

− − ∈ ∈

∑ ∑, ss

n branches

sum over lateral ch

b A,{ }∑

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

aains

BBn n nc B c tot c

� ��

⋅

∈ = −

1

σ, , ,ZZA c A

b B b tot b ZA b A

n carbonsn n n n branch

−= − −

,

, , , , ees

sum over complementary lateral

{ }∑

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

cchains� ��



U n n n U n n nS p c b S p c b( , , ) ( , , )= − − ×1 1 1 -CH - inserti2 oon

-CHMe- inse

H2σC

S p c bU n n n

=( )+ − − − ×

1

1 2 12

3( , , ) rrtion

-CHE

HMeσC

S p c bU n n n

=( )

+ − − − ×

3 2

1 3 12

3

/

( , , ) tt- insertion HEtσC

S p c bU n n n

=( )

+ − − − ×

3 2

1 3 2

/

( , , )11

99

1 4

-CMeMe- insertion MeMeσC

S p cU n n

=( )

+ − −( , ,, ) /

(

n

U

b C

S

− × =( )

+

22

99 2-CEtMe- insertion EtMeσ

nn n np c b C− − − × =1 5 21

9, , ) -CEtEt- insertion EtEtσ 99( )

(22)

+

+

Lateral chain BLateral chain A

Activated zone ZA

Equivalence class Equivalence class

nc,B = 8, nb,B = 1nc,A = 4, nb,A = 1nc,ZA = 4nb,ZA = 1

Figure 6

Example of homologous activated complexes and equivalent lateral chains.



413

et al. (2007). We finally define the Symmetry Property ofLateral Chains SPLC(nc,nb) as:

(23)

This is exactly the sum within square brackets in Equation(20). The starting value of index np in this sum is obtained

SPLC n n

U n n n

c b

S p c b

AA nc carbonsb bra

,

, ,

( ) =

( ) =∈

1

σnnchings

n n n

n n

p c b

p c

{ }= −

=

∑∑2

when all the branches are ethyl groups (np = nc – 2nb), themaximum value when the lateral chain is linear (np = nc).

For β-scission reactions, Equation (20) can therefore bewritten as:

(24)

In this expression, the symmetry number of the activatedzone still remains to be determined. It is derived from simple

1

1

σ σ σ

σ

A ZA BA ZA B R

ZA

c ASPLC n

⋅ ⋅=

⋅

− − ∈

∑

, ,, ,, , ,n SPLC n nb A

ZA

c B b B( ) ⋅ ( )∑

CH3

R2

R1

R4

R3

Rn

Rn-1

CH2 #

CH3

R2

R1

R4

R3

Rn

Rn-1

CH #

Me

CH3

R2

R1

R4

R3

Rn

Rn-1

CH #

Et

CH3

R2

R1

R4

R3

Rn

Rn-1

C #

Me

Me

CH3

R2

R1

R4

R3

Rn

Rn-1

C #

Et

Me

CH3

R2

R1

R4

R3

Rn

Rn-1

C #

Et

Et

CH3 #

R2

R1

R4

R3

Rn

Rn-1

- - - - - -

Chain L σL

- - - - - -

- - - - - -

- - - - - -

- - - - - -

- - - - - -

- - - - - -

σ = σL*1

σ = σL*3/2

σ = σL*3/2

σ = σL*9

σ = σL*9/2

σ = σL*9

Figure 7

Concept of the recursive “pattern insertion” method for calculating the symmetry number of a lateral chain.



properties of the activated complex of Figure 3. Let nLi= 0

when Li is a hydrogen substituent and nLi= 1 otherwise.

The internal symmetry number for each Li group is thenσintLi

= 3nLi. Chirality is born by three carbon atoms involvedin the 2-electron 3-center bond. These carbon atoms are chi-ral when their substituents are different. Note that when thelateral chains A and B are more than 2 carbon atoms long, thechirality of the activated zone does no longer depend on thesechains as in this example L1 to L5 are at most ethyl branches.Finally, the symmetry number of the activated zone is there-fore given by:

(25)

where nch is the number of chiral carbon atoms among thethree carbon atoms of the 2-electron 3-center bond. InsertingEquation (25) into Equation (24) therefore allows calculatingthe last factor of the expression for the lumping coefficientdefined by Equation (17) by means of an elegant method thatdoes not require the exhaustive generation of the reactionnetwork.

3.3 Recursive Algorithm for Calculatingthe Equilibrium Constant of Formationof the Paraffinic Lump

As mentioned above, the second factor in the expression forthe lumping coefficients (Eq. 17) represents the reciprocal ofthe equilibrium constant of formation of the paraffinic lump.This equilibrium constant calculated as the sum of the equi-librium constants of formation of all paraffins Pi of lump Fo:

(26)

where ΔG0f (Pi) is the intrinsic Gibbs energy of formation of

paraffin Pi, whose value can be calculated using Benson’sgroup contribution method. Here also, a recursive algorithmcan be used. For the sake of simplicity, the gauche interac-tions and the external symmetry of the paraffins will beignored in our example. The method does, however, accountfor these symmetrical molecules, and the reader is referred toValéry (2002) for further details.

The method is once again based on the concept of lateralchains as defined above: a given paraffin can be decomposedinto two zones, as has been done with activated complexes,but this time without an activated zone. In this approach, aparaffin with np carbon atoms along the main chain is there-fore the concatenation of a lateral chain A with np,A carbonatoms and a lateral chain B with np,B (= np – np,A) carbonatoms along their respective main chains. The symmetrynumber of the concatenated chains is the product of the

e

eG P

RT

P F

G P

RTf i

i

f i

−( )⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

∈

−( )⎛

⎝

⎜

∑ =

ΔΔ

0

0

0�

⎜⎜

⎞

⎠

⎟⎟

∈

∑σ PP F ii 0

σ ZA

n n n n n

n

L L L L L

ch=

+ + + +3

2

1 2 3 4 5

symmetry numbers of the both chains (ignoring externalsymmetry), while the intrinsic Gibbs energy is the sum of theindividual Gibbs energies (ignoring gauche interactions). Theexponential of the normalized intrinsic Gibbs energy is thenthe product of the exponentials of the two lateral chains.Hence, Equation (26) is rewritten as:

(27)

The factor 1/2 results from the fact that chains A and B canbe interchanged, i.e. a paraffin can be considered either as A-Bor as B-A. The length np,A of chain A is arbitrary and remainsto be chosen. For reasons of chirality (Valéry et al., 2007),we set np,A = 3. The properties of chain B can then be calcu-lated using the “pattern insertion” method discussed in theprevious section. Let:

(28)

We now define the Thermodynamic Property of LateralChains TPLC(nc,nb) as:

(29)

Again, the minimum value of index np in this last sum isobtained when all the branches are ethyl groups (np = nc –2nb), while the maximum value is given when the lateralchain is linear (np = nc).

When choosing np,A equal to 3, Equation (27) becomes:see Equation (30).

Finally, using the “pattern insertion” concept and based inFigure 7 and Equation (22), UT can be obtained recursivelyas: see Equation (31).

Further details and the initial UT(3,nc,nb) are given inValéry et al. (2007).

TPLC n ne

c b

G A

RT

AA n carbon

f

c

,

*

( ) =

−( )⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

∈

Δ

σaatomsn lateral branches

T p

b

U n n

{ }∑ =

, cc b

n n n

n n

np c b

p c

,( )= −

=

∑2

U n n ne

T p c b

G A

RT

AA

n leng

f

p

, ,( ) =

−( )⎛

⎝⎜⎜

⎞

⎠⎟⎟

∈

Δ � 0

σtth of main chainn carbon atomsn lateral branche

c

b ss

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

∑

e

G P

RT

PP FP A B

f i

ii

i

−( )⎛

⎝⎜⎜

⎞

⎠⎟⎟

∈= −

∑ = ⋅

Δ � 0

0

1

2σ

e e

G A

RT

A

G B

RTf f−( )⎛

⎝⎜⎜

⎞

⎠⎟⎟ −

( )⎛

⎝⎜⎜

⋅

Δ Δ� �0 0

σ

⎞⎞

⎠⎟⎟

− ∈

∑

⎛

⎝

⎜⎜

σ Bequivalent B chainssuch that A B F0

⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟A chains with n

carbons along themai

P A: ,

nn chain

∑



415

As a conclusion, Equations (22-25) and (29-31) allow tocalculate the lumping coefficients much faster than the origi-nal method, which needs to generate the complete reactionnetwork and to determine the lumping coefficients byinspecting the complete list of single events in a generatedreaction network. The lateral chain decomposition method isa quite flexible method, as it can easily be extended toinclude different type of lateral branches, such as n-propyl,i-propyl, n-butyl, i-butyl, t-butyl, etc. To this aim, the appro-priate insertion patterns simply need to be added and therecursion formulae of Equations (22) and (31) need to bemodified correspondingly, together with the initial values.The pattern insertion method also allows calculating otherproperties of lateral chains recursively. For examples of this,the reader is referred to Valéry et al. (2007).

Compared to the structural class methodology, the lateralchain decomposition approach is slightly more complex forlow branching numbers but it is much more efficient. Indeed,the use of recursion formulae avoids accounting for excep-tions. This allows calculating the various series that consti-tute the lumping coefficients from the lowest carbon numberto the highest carbon number, while special treatment isneeded for low carbon numbers with the structural classapproach. Moreover, the lateral chain decomposition methodallows for a much easier extension to high branching num-bers. As mentioned above, the results of both methods arestrictly equivalent, since they use the same hypotheses.

4 MODEL VS EXPERIMENTATION

4.1 Experimentation

Experiments were performed in an isothermal tubular fixedbed reactor in upflow mode at a total pressure of 150 bar andtemperature of 375°C. The reactor was loaded with 80 g ofan industrial NiMo on a Y-zeolite based catalyst. The catalystwas sulfided with a hydrogenated LCO to which 2 wt% ofDiMethylDiSulfide (DMDS) and 0.1 wt% of aniline wasadded. The pretreating feed was injected at a LHSV of 1 h-1,with a volumetric H2/HC ratio of 1 000 Nl/l. The temperaturewas progressively increased to 350°C. Sulfiding was contin-ued for at least 12 h and until the H2S composition in theeffluent remained constant. Before carrying out the kineticexperiments, a line-out period of 150 h at 375°C with thesulfiding feed was applied in order to reach a stable catalystperformance.

As primary feedstock, pure n-hexadecane (C16H34) waschosen (purity > 99%) to which DMDS and n-butylaminewere added to maintain the catalyst in a sulfided state andunder industrial conditions with respect to the presence ofH2S and NH3. As n-hexadecane freezes at 18°C, all equip-ment and lines in pilot plant were maintained above 25°C.Kinetic data were collected at various LHSV ranging from0.15 h-1 to 1 h-1. The partial pressure of H2S and of NH3amounted approximately to 1 bar and 0.1 bar respectively.The reactor effluent was separated in a flash drum at ambient

eU n n

G P

RT

P

T c B b B

f i

i

−( )⎛

⎝⎜⎜

⎞

⎠⎟⎟

= ⋅ ( ) ⋅

Δ *

, ,, ,σ1

23 TTPLC n n n nc c B b b B

n

n Min n

c B

c B c

− −( )=

= −(

, ,

,

,,

,

3

1 9))

=

= ( )

∈

∑∑∑n

n Min n

P F b B

b B b

i ,

, ,

0

4

0

U n n n U n n n eT p c b T p c b

G f

( , , ) ( , , )= − − × ×−

(

1 1 1

0Δ � CH2 ))

−(

+ − − − × ×

RT

T p c b

G

U n n n ef

-CH -2

CH

( , , )1 2 12

3

03Δ � ))+ ( )

+ − − − × ×

Δ �G

RT

T p c b

f

U n n n

0

1 3 12

3

CH

-CHMe-

( , , ) eeG G G

RTf f f−( )+ ( )+ ( )

+

Δ Δ Δ� � �03

0 0CH CH CH2

-CHEt-

UU n n n eT p c b

G Gf f

( , , )− − − × ×−

( )+

1 3 21

9

2 03

0Δ Δ� �CH C(( )

−+ − − − × ×

RT

T p c b

G

U n n n ef

-CMeMe-

( , , )1 4 22

9

2 0Δ � CCH CH C2

-CEtMe-3

0 0

1

( )+ ( )+ ( )

+ −

Δ Δ� �G G

RT

T p

f f

U n( ,, , )n n ec b

G Gf f

− − × ×−

( )+ ( )+

5 21

9

2 203

0Δ Δ Δ� �CH CH2��G

RTf0 C

-CEtEt-( )

(30)

(31)



temperature. The gas fraction was analysed on-line, while theliquid effluent was sampled and analysed on a GC equippedwith a PONA column. For components with 8 carbon atomsor less, the GC method allows to obtain the full moleculardetail. For components with more than 8 carbon atoms, theavailable analytic detail allows to distinguish components bycarbon numbers and degree of branching, which correspondsto the lumps of the above a posteriori relumped models.

To validate the single event kinetic model developed forn-hexadecane, additional experiments were carried out with aFischer-Tropsch wax, a mixture of heavy paraffins with 20 to33 carbon atoms (Tab. 3). This Fischer-Tropsch wax wasalso separated by distillation into 4 smaller boiling pointranges. Again, DMDS and aniline were added to all thesefeeds to maintain the catalyst in a sulfided state and underindustrial conditions with respect to the presence of H2S andNH3. During these experiments, all equipment and lines inpilot plant were maintained above 60°C. The experimentswere carried out under the same conditions and kinetic datawere collected at various LHSV ranging from 0.15 h-1 to 1 h-1.

TABLE 3

Composition of the C19-C33 Fischer-Tropsch wax (in mol%)

Carbon number n-Paraffins Iso-paraffins Total

19 0.15 0 0.15

20 0.86 0 0.86

21 2.32 0 2.32

22 4.70 0.03 4.73

23 8.10 0.16 8.26

24 11.79 0.52 12.31

25 12.96 0.96 13.92

26 12.68 1.58 14.26

27 10.39 2.02 12.41

28 8.12 2.01 10.13

29 6.46 1.86 8.32

30 4.11 1.55 5.66

31 2.56 1.31 3.87

32 1.33 0.60 1.93

33 0.60 0.27 0.87

Total 87.13 12.87 100

Finally, experiments with squalane (2, 6, 10, 15, 19, 23-hexamethyltetracosane; C30H62; 99% purity; HaltermannChemicals) were also carried out in order to look at the effectof the degree of branching on the model predictions. Asbefore, DMDS and a nitrogen-containing compound (eithern-butyl amine or aniline) were added to the feed in order tomaintain the catalyst in a sulfided state and under industrial

conditions with respect to the presence of H2S and NH3.Again, the experiments were carried out under the same con-ditions and kinetic data were collected at various LHSVranging from 0.5 h-1 to 1.53 h-1 (Valéry et al., 2007).

4.2 Parameter Identification

The single event kinetic model was incorporated in a plugflow reactor model. As gas and liquid were supposed to be atequilibrium inside the reactor, the compositions of the twophases were calculated by means of a Grayson-Streed flashmodule in each of the differential elements of the reactor.This then leads to a system of Ordinary DifferentialEquations (ODE):

(32)

These equations were reorganized and integrated accord-ing to space time, W/F0

tot, where W is the weight of catalystinside the reactor and F0

tot the total molar flow at the inlet ofthe reactor. The resulting differential equations of the reactorwere solved with LSODE (Hindmarsch, 1980, 1983).

Parameters were identified by minimizing the sum of the

squared errors initially with a simulated

annealing algorithm (Goffe et al., 1994) to screen theresponse surface, while a Levenberg-Marquardt algorithm(Levenberg, 1944; Marquardt, 1963) was used to obtain thefinal parameter estimates. The parameters were estimatedbased on 160 observations of the various responses obtainedduring the experiments with n-hexadecane. As the modelappeared to be insensitive to the values of the protonation/deprotonation equilibrium constants for secondary and tertiarycarbenium ions, these equilibrium constants and reaction ratecoefficients were re-parameterized into composite reactionrate coefficients as follows:

(33)

During the identification of the 6 remaining parameters, italso appeared that the composite parameters k*

pcp(s;t) andk*

pcp(t;t) were highly correlated. To solve this, it was arbitrarychosen not to estimate k*

pcp(t;t) and its value was put to zero.This led to the final parameter shown in Table 4. The sec-ondary-tertiary cracking rate coefficient k*

cr(s;t) was found tobe extremely low. Moreover, according to the statisticalanalysis, this parameter was found to be statistically non-sig-nificant. This is most certainly due to coupling effects or toindiscernability between (s,t) and (t,s) cracking on the lumps.Hence, the composite rate coefficient k*

cr(s;t) was thereforeput to zero.

k m u K m O k m ureac ref reac*

Pr( ; ) ( ; ) ( ; )= ⋅� �

y yi i

i

nobs

−( )=

∑ ˆ ,2

1

dF

dWRi

i=



417

TABLE 4

Final parameter values for the hydrocracking of n-hexadecane

Parameters Best value Standard deviation t-Value

k*pcp(s,s) 3.08 × 102 0.03 × 102 99.3

k*pcp(s,t) 6.07 × 103 0.22 × 103 27.6

k*cr(s,s) 1.89 × 103 0.17 × 103 11.1

k*cr(s,t) 2.73 × 10-3 8.70 × 103 0

k*cr(t,s) 2.28 × 104 0.98 × 104 2.3

k*cr(t,t) 9.07 × 105 1.30 × 105 7.0

With the final parameters, the model behaves quitesatisfactorily. The overall conversion of n-hexadecane is wellpredicted, as is the prediction of total amount of isomerizedproducts and of cracked products (Fig. 8). Within theisomerized products of the n-hexadecane feed, the evolutionof the main simulated lumps (normal, mono- and multi-branched hexadecanes) is relatively close to the experimentaltrend (Fig. 9). With respect to cracking, the relumped singleevent model retains enough detail to predict a detailed distri-bution of the cracked products. As shown in Figure 10 for thelinear paraffins, both the orders of magnitude and the trendsof the product distribution of the cracked products are quitewell predicted, especially when taking into account that themodel only contains 6 parameters.

To validate the model, the values of the parameters obtainedfrom the experiments on n-hexadecane were used to simulatethe experiments obtained on the Fischer-Tropsch wax and onits sub-fractions obtained by distillation. As these feeds arecomposed of paraffins with 20 to 33 carbon atoms, this com-parison allows to directly evaluate the extrapolation withrespect to the number of carbon atoms. Figure 11 comparesthe predicted yields for the linear C20+ paraffins (nC20+), for

the isomerized C20+ paraffins (iC20+) and for the crackedproducts to the experimental data obtained during cracking ofFischer-Tropsch wax. It can be noticed that the decrease ofthe nC20+ paraffins is well predicted, indicating that the valueof the k*

pcp(s;t) rate coefficient is well estimated and allows tocorrectly extrapolate to higher carbon numbers. However, theyield of isomerized C20+ paraffins is somewhat overestimatedat low space times, while the yield of cracked products withless than 20 carbon atoms is slightly underestimated. Thisindicates that the rate of cracking is underestimated to someextent when the degree of branching is relatively low. Athigh space times, the predicted yield of cracked products ishigher than its experimental yield, indicating that the reactionrate is too high when the degree of branching is high. For thesub-fractions obtained by distillation of the Fischer-Tropschwax, the comparison model-experiments of the yields ofnC20+, iC20+ and cracked products is shown as a parity plot in

Space time (gcat . h/mol)

0.1

02500 500 750 1000 1250 1500 1750

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Fra

ctio

nal m

olar

yie

ld (

mol

/mol

)

ConversionIsomerizationCracking

Figure 8

Conversion, isomerization and cracking of n-hexadecane as afunction of space time (points: experimental, lines:simulated).

0.10.20.30.40.50.60.7

0.80.91.0

02500 500 750 1000 1250 1500 1750

Mol

e fr

actio

n (m

ol/m

ol)

n-C16

mono-C16

multi-C16

Space time (gcat . h/mol)

Figure 9

Composition of the C16 fraction as a function of space time(points: experimental, lines: simulated).

2 3 4 5 6 7 8Carbon number (-)

9 10 11 12 13 14

W/F = 1565M

olar

yie

ld (

mol

e %

)

W/F = 213

W/F = 375

W/F = 793

W/F = 1157

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0

Figure 10

Experimental and model predicted distribution of lowercarbon number n-paraffins for n-hexadecane hydrocracking(points: experimental, lines: simulated).



Figure 12. The same conclusions can be drawn: the isomer-ization of the nC20+ paraffins is quite well predicted, the yieldof the iC20+ paraffins is slightly overestimated, while yield ofthe cracked products is somewhat underestimated. The pre-dicted distribution of cracked products is compared to theexperimental product distribution in Figure 13 at a conver-sion of 68 wt% of the nC20+ paraffins of the C25-C31 sub-frac-tion of the Fischer-Tropsch wax. Looking at the feed, anearly perfect agreement is observed for the disappearance ofthe linear paraffins of the C25-C31 feed, while the heaviestisoparaffins (C27-C30) are somewhat overestimated. For thecracked products, the model-predicted distribution profile oflower molecular weight n-paraffins (C3-C24) has the sameprofile as the experimentally observed profile, although theabsolute values are somewhat lower. For lower molecularweight isoparaffins (C4-C24), both the distribution and theabsolute values are very well predicted. In conclusion, a good

overall agreement is obtained when extrapolating the singleevent kinetic model from a low carbon number (C16) tohigher carbon numbers (C20 to C33) without adjusting any ofthe parameters.

To further validate the model, the values of the kineticparameters obtained from the experiments on n-hexadecanewere also used to simulate the experiments with squalane asfeed (Valéry et al., 2007). As squalane is an isoparaffin of 30carbon atoms with 6 methyl side chains, this feed allowsevaluating at the same time an extrapolation with respect tothe number of carbon atoms and with respect to the numberof side chains. While the experimentally observed squalaneconversions varied between 26% and 71% with space time,the model predicted conversions varied between 83% and99% for the same range. The model therefore largely over-predicts the rate of disappearance of squalane. To correctlypredict the conversion, all rate coefficients had to be reducedby a factor of 4.74. With this reduced catalyst activity, themodel correctly predicts both the distribution and the amountof the cracked products as a function of carbon number(Fig. 14). In the experiments described by Valéry et al.(2007), the average degree of branching as a function ofcarbon number was also experimentally determined. Figure15 illustrates that the model is equally able to predict the evo-lution of the degree of branching of the cracked products as afunction of carbon number. As a conclusion, although thecatalyst activity was largely overestimated for the crackingof squalane, the product distribution and the degree ofbranching of the cracked products were well predicted atthe experimental conversion.

Several explanations can be proposed for the high reactivityof the highly branched squalane in the single event kineticmodel. First of all, the single event kinetic model contains an

0.7

0.8

0.6

0.5

0.4

0.3

0.2

0.1

0 0.10 0.2 0.3 0.4 0.5 0.6 0.7 0.8Experimental yield (mol/mol)

Cal

cula

ted

yiel

d (m

ol/m

ol)

Conversion (nC20+)

Isomerization (iC20+)

Cracking (C19-)

Figure 11

Yields of nC20+, iC20+ and cracked products during crackingof the Fischer-Tropsch wax as a function of space time(points: experimental, lines: simulated).

0 500 1000 1500 2000 2500Space time (gcat . h/mol)

Fra

ctio

nal m

olar

yie

ld (

mol

/mol

)

nC20+iC20+C19-

1.0

0.90.8

0.70.60.50.40.3

0.20.1

0

Figure 12

Parity plot for the fractional molar yields of nC20+, iC20+ andcracked products during cracking of the sub-fractions of theFischer-Tropsch wax.

0.12

0.10

0.08

0.06

0.04

0.02

0

Carbon number (-)

Fra

ctio

nal m

olar

yie

ld (

mol

/mol

)

0 5 10 15 20 25 30 35

n-Paraffins

Iso-paraffins

Figure 13

Distribution of the normal and iso-paraffins at a conversionof 68 wt% of the nC20+ paraffins of the C25-C31 sub-fractionof the Fischer-Tropsch wax (points: experimental, lines:simulated).



419

inhibition term that accounts for physical adsorption in thezeolite. In this inhibition term, the physical adsorption coeffi-cient of a lump is considered to be the same for all paraffinswith more 16 carbon atoms. As shown by the validation onthe Fischer-Tropsch wax, this hypothesis seems to be quiteadequate. However, the physical adsorption coefficient isalso considered to be independent of the degree of branching.This hypothesis may therefore be at the basis of the too highreactivity of squalane with respect to linear long-chain paraf-fins. An other explanation of the overprediction of the reac-tivity of squalane may be linked to the chemisorption of thecarbenium ions, as the proposed kinetic model does not con-tain an inhibition term due to chemisorption on acid sites.Indeed, previous literature studies (Svoboda et al., 1995;Martens and Froment, 1999; Martens and Marin, 2001) haveshown that the total concentration of carbenium ions could beneglected compared to the total concentration of acid sites.However, these observations were based on hydrocrackingexperiments on n-C8 to n-C12 paraffins and on mixtures ofn-C8 with 10 mole% of branched C8 paraffins. This may nolonger be the case with very highly branched isoparaffinssuch as squalane. Indeed, squalane has 6 tertiary carbonatoms that may allow for a much higher chemisorption on the

acid phase, hence reducing the apparent catalyst activity. Insuch a case, the chemisorption of highly branched isoparaffinscan no longer be neglected.

Mitsios et al. (2009) also developed a single event kineticmodel for hydroisomerization and hydrocracking, which wasapplied to n-hexadecane, a C20-C30 wax and squalane on anamorphous Pt/SiO2•Al2O3 catalyst. In their single eventkinetic model, they did not account for physisorption, as noexternal or internal limitations were considered to be presentin this macroporous non-zeolitic catalyst. However, theirmodel accounted for the chemisorption of the carbenium ionson acid sites. When extrapolating this model, which does notaccount for physisorption while accounting for chemisorp-tion, from n-hexadecane to the C20-C30 wax and to squalane,the model predicted the correct order of magnitude for theconversion, and for the isomerization and cracking selectivitieson all feeds.

CONCLUSION

In order to apply the single event methodology to acid-catalyzedreactions with high carbon number hydrocarbon fractions,huge calculation times are required for the explicit generationof the complete exhaustive list of molecules and single eventsreactions. Although this approach is straightforward, it requiresgenerating all elementary steps individually and storing thisentire network. In the literature, alternative approaches havebeen developed that directly calculate the lumping coefficientsof a posteriori relumped single event reaction networks. Theabove-described lateral chain decomposition strategy dividesthe various components (molecules, intermediates and activatedcomplexes) into lateral chains and activated zones. Detailedinspection of these various parts allowed deriving a set ofrecursive formulae that calculate the various structural andthermodynamic properties that are needed to determine thelumping coefficients. It should be stressed that the lateralchain decomposition technique is exactly equivalent to theoriginal calculation method based on the explicitly generatedreaction network, as long as Benson’s group contributionmethod is used to obtain the necessary thermodynamic datain both approaches. The advantage of these recursive formu-lae resides in the fact that application of the single eventmodelling methodology to very large reaction networks andvery high degrees of branching is highly simplified, both incomparison to the explicit generation of the entire reactionnetwork and to the structural class approach.

A kinetic model for hydroisomerization and hydrocrackingof n-hexadecane was developed using this lateral chaindecomposition approach. After identification of the 6 remain-ing kinetic parameters, the kinetic model was confronted to theexperimental data and a good agreement between the predictedand measured product distributions was obtained, both withrespect to isomerization and hydrocracking of n-hexadecane.

10

3

5

7

9

Carbon number (-)

Com

posi

tion

(mol

%)

0 5 10 15 20 25 30

Figure 14

Distribution by carbon number of the cracked products at aconversion of 71 wt% of the squalane feed.

6

5

4

3

2

1

0

Ave

rage

num

ber

of b

ranc

hes

(-)

Carbon number (-)0 5 10 15 20 25 30

Figure 15

Evolution of the degree of branching of the cracked productsas a function of carbon number at a conversion of 71 wt% ofthe squalane feed.



To validate the kinetic model, hydrocracking experimentswere carried out on various fractions of a C20-C33 Fischer-Tropsch wax. Without adjusting any of the parameters, thesingle event kinetic model correctly predicts the isomerizationof the linear C20+ paraffins. Although the yield of crackedproducts is slightly underestimated at low conversion at theexpense of the isomerized C20+ paraffins, their product distri-bution as a function of carbon number is very well predicted.In conclusion, a good overall agreement is obtained whenextrapolating the single event kinetic model from a low carbonnumber (C16) to higher carbon numbers (C20 to C33).

Extrapolating the kinetic model with respect to the numberof carbon atoms and with respect to the number of sidechains at the same time was also attempted. Confronting themodel with the squalane experiments (Valéry et al., 2007)showed that the rate of disappearance of squalane was largelyoverpredicted. At the experimental conversion, the selectivitytowards cracked products, the cracked product distributionand their degree of branching were well predicted, however.Several explanations can be proposed for the discrepanciesbetween the model and the experiments, as these seem to indi-cate that the simplifying hypotheses concerning physisorptionand/or chemisorption are no longer valid for highly branchedparaffins.

The lateral chain decomposition method described in thiswork is quite flexible. Indeed, it can easily be extended tomore general lateral chains, as it is straightforward to intro-duce new insertion patterns, to adapt the recursion formulaeaccordingly, and to develop recursion formulae for new lat-eral chain properties.

REFERENCES

Al Wahabi S.M., Froment G.F. (2004) Single Event KineticModeling of the Methanol-to-Olefins Process on SAPO-34, Ind.Eng. Chem. Res. 43, 5098-5111.

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Single Event Kinetic Modelling Without Explicit

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