continuum lumping kinetics of complex reactive systems

Chemical Engineering Science 76 (2012) 154–164

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Chemical Engineering Science

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Continuum lumping kinetics of complex reactive systems

Mustafa Adam a, Vincenzo Calemma b, Francesca Galimberti b, Chiara Gambaro b,Johan Heiszwolf c, Raffaella Ocone a,n

a Chemical Engineering, Heriot-Watt University, Edinburgh EH144AS, UKb ENI, Centro Ricerche di San Donato Milanese, Via F. Maritano 26, 20097 San Donato Milanese (MI), Italyc Catalyst Technology Group, Albermarle Catalysts Company BV, 1022 AB Amsterdam, The Netherlands

a r t i c l e i n f o

Article history:

Received 16 December 2011

Received in revised form

5 March 2012

Accepted 25 March 2012Available online 5 April 2012

Keywords:

Continuum lumping

Hydrocracking

n-paraffins

Modelling

Middle distillates

Species-types distribution

09/$ - see front matter & 2012 Elsevier Ltd. A

x.doi.org/10.1016/j.ces.2012.03.037

esponding author. Tel.: þ44 131 4513777; fa

ail address: [email protected] (R. Ocone).

a b s t r a c t

Continuum lumping is a methodology widely used to describe the kinetic or thermodynamic behaviour

of complex reactive mixtures of various components all undergoing similar types of reactions (e.g.,

cracking, pyrolysis, oligomerisation). The methodology is particularly convenient when the number of

species involved is very large and the species can be characterised by properties that can be measured

in a continuous fashion, as for instance in a chromatogram. By applying the lumping methodology,

a large reactive system can be reduced to a simpler and more tractable one. In the present paper,

the continuum lumping methodology is applied to the hydrocracking of Fischer–Tropsch waxes

(n-paraffins), extending a previous model by Laxminarasimhan et al. (1996). Specifically, the role of

the type-distribution function is investigated by employing two expressions of such function and by

studying how it affects the model predictions. The effect of the operating conditions (namely

temperature, pressure and hydrogen to feed ratio) on the hydrocracking process is also investigated.

The model parameters are estimated by using experimental data obtained from a bench scale trickle

bed reactor for specified ranges of operating conditions.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

The hydrocracking process is largely employed in the petro-chemical industry for production of high quality middle distillates(Handwerk, 1984); the process improves the octane number inthe gasoline fraction and raises the ratio of iso-butane to n-butanein the butane fraction. Hydrocracking involves complex chemistryand a large variety of reactions, such as isomerisation,hydrogenation–dehydrogenation, C–C bond scission, hydrogentransfer, ring saturation, and dealkylation (Laxminarasimhanet al., 1996). The feedstock for the hydrocracking process isformed by heavy components, among others by aromatic cycleoils and coker distillates (Govindhakannan, 2003). Informationabout the effect that the operating conditions have on theselectivity and conversion and information on the type of feed-stock are essential to understand and operate the hydrocrakingprocess (Elizalde et al., 2009; Elizalde et al., 2010). This informa-tion may be obtained from experiments or through modelling.Modelling is inexpensive and faster for preliminary processdesign in contrast to experimental information which is costlyand time consuming. On the other hand, modelling the

ll rights reserved.

x: þ44 131 4513129.

hydrocracking of paraffins and heavy oils is not an easy task,since many data for the feed and the products are needed.

Models are useful tools to predict the yields of variousproducts and can be successfully employed for optimisation,new design, analysis, and control of the reactive system. Reliablemodels, describing the complex hydrocracking kinetics, have arole in the refinery industry to predict the yields of desired andundesired products at different operating conditions for a givenfeedstock; to control and design the process; to select theappropriate catalyst. A model for hydrocracking kinetics couldavoid running expensive experiments in a laboratory or pilotplant. Ideally, the model, to be used as a predictive tool, shouldinclude all the reactions that each single component in the feedundergoes; however, the large number of reactive species andreactions poses a serious problem for modelling and simula-tion. Indeed, if one wants to take into consideration thecomplex reactive network in its entirety, the problem soonbecomes untreatable. The need for devising simpler, effectiveand reliable models is evident and it has been identified as realindustrial need.

Different methods have been used to describe the hydrocrack-ing and they have been recently reviewed (Ocone, 2009; Okinoand Mavrovouniotis, 1998). The three main modelling strategiesapplied to hydrocracking are: the single-event kinetics method,the discrete lumping method, and the continuum lumping

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M. Adam et al. / Chemical Engineering Science 76 (2012) 154–164 155

method. Lumping is one of the methodologies used to reduce alarge reactive system to much simpler and tractable one. Systemreduction can involve both the reactive mechanisms and thedimension (i.e., number of components) of the system. In redu-cing a system to one of lower dimensionality, one may followrules based on experience, trial and error or on mathematicalconstraints; all of these concepts have been followed. Brazill et al.(1997) used a discrete lumping model to study the reactionsoccurring during the hydrocracking of heavy oil and residue andlumped the product and feed compounds into cuts characterisedby the distribution of the boiling temperature. The compoundscan undergo parallel or series reactions. A discrete three lumpmodel with three reactions for the catalytic cracking of oil gas wasproposed by Weekman and Nace (1970). This model wasextended to ten lumps with 17 reactions by Jacob et al. (1976).Common to all these modelling efforts are underlying mathema-tical rules guaranteeing the conditions under which the reductionis carried out; also, the representative pseudo-components (i.e.,the components mimicking each lump) must still be able todescribe the behaviour of the original mixture (Ancheyta et al.,2005). The choice of the representative lumps depends on thefeedstock and the main advantage of the lumping technique isthat only a small amount of data is required aimed essentially toestimate the model parameters (Elizalde et al., 2010).

In contrast to the discrete lumping, the continuum lumpingassumes that the properties of each individual component (e.g.,reactivity, concentration, volatility) are described through suita-ble component indexes, such as the boiling point or the molecularweight: the methodology is particularly useful when the numberof components is large (usually greater than twenty). In con-tinuum lumping, it is convenient to introduce an index, x, and toidentify the ‘‘species’’ AðxÞdx as the sub-mixture whose index (e.g.,retention time, boiling point, etc.) lies in the range ðx,xþdxÞ. Inother words, we substitute some sort of ‘‘lumped’’ reactants to thereal mixture and their collective behaviour is determined. It isimportant to notice that the label is a continuous variable, andtherefore one is not constrained to take values that are integers.De Donder (1931) was the first to introduce the theory ofcontinuum lumping; Astarita and Ocone (1988) presented aprocedure for lumping nonlinear kinetics using the continuumapproach.

The hydrocracking reaction of n-paraffins over bifunctionalcatalysts has been studied intensively by Martens et al.(1986);Schulz and Weitkamp (1972). They investigated the pro-duct distribution and the isomer composition. Later, Froment(1987) developed the ‘‘lumped kinetic models’’ for the hydro-cracking studying the reaction of different pure n-paraffins. Inthese kinds of models, the reacting products were divided intomain classes, or lumps, which correspond to n-paraffins, iso-paraffins (sometimes split into mono and multi branched) andcracked products. Laxminarasimhan et al. (1996) applied thecontinuum lumping theory to the hydrocraking of vacuum gasoil. The model is based on the feedstock true-boiling point (TBP)data; the authors introduced a yield distribution function tomimic the reactive system. The approach has been applied withsuccess to a number of case studies (e.g., Lababidi et al. (2011)).

The main objective of this work is to investigate the robustnessof the continuum lumping to describe the product composition ofhydrocracking process of paraffins (Fischer–Tropsch waxes) in acatalytic reactor. The low temperature Fischer–Tropsch (FT)process leads to the formation of a range of n-paraffins (490%)with small percentages of alcohols and olefins. The FT productsare characterised by a wide distribution of molecular weightswhich can be described through the Anderson–Schulz–Florymodel (Dry, 2002). A large fraction of FT products is characterisedby a boiling point higher than 370 1C (waxes) while the middle

distillates (MD) (150–370 1C) cut, although characterised by veryhigh cetane numbers, shows very poor cold flow properties (i.e.,relatively high melting point) that hamper their use as transpor-tation fuel. The hydrocracking process leads to an increase of MDyield and to the formation of iso-paraffins. The increase of iso-paraffins results in a strong improvement of cold flow propertieswhile the decrease of cetane number associated with isomerisa-tion is rather limited (Calemma et al., 2005; Collins et al., 2006).

2. Continuum lumping model

2.1. Model formulation

Labelling the reactants represents the starting point forperforming the continuum lumping methodology. The label (orindex) can be any particular characteristic which unequivocallyidentifies the reactants. For examples, one can choose the kineticconstant, the boiling point, the mass, etc, provided that a uniquerelationship between the components and the chosen label exists.The label, say x, is taken over the interval ½0,1Þ and, if the variableof interest is the concentration, the initial concentration of thespecies within the interval ðx,xþdxÞ is given by:

cðx,0Þdx¼ c0hðxÞdx ð1Þ

where c0 is the total initial concentration and hðxÞ is a distributionfunction which must be normalised so that the mass conservationis assured:Z 1

0cðxÞhðxÞdx¼ 1 ð2Þ

If such a correspondence between components and label canbe established, and a distribution function is introduced, then thelumped (global) variables, at each time, can be obtained by therelevant integration over the label. As an example, let us considerthe chosen label being the kinetic rate constant k, then the global(lumped) concentration of the mixture, C(t), can be defined and itcan be calculated as follows:

C tð Þ ¼

Z 10

cðk,tÞDðkÞdk ð3Þ

where c(k,t) is the concentration of the species with reactivity k atthe considered time t and D(k) is the reactant-type distributionfunction. The reactant-type distribution function depends on thefeed and describes the mixture with respect to the particular labelchosen. In the case at hand, having chosen the reactivity as thelabel (expressed in terms of kinetic constant, k), D(k) representsthe distribution of reactivity of the various components. Conse-quently, D(k)dk represents the reactant-type distribution of com-ponents with rate constant between k and kþdk.

In this study, the feedstock is characterised through the chainlength of its components or, equivalently, their molecular weight;consequently, the various components are unequivocally identi-fied by their carbon number, which is then adopted as themixture label. The molecular weight distribution could substitutethe carbon number distribution which is adopted only for con-venience. The procedure was developed originally byLaxminarasimhan et al. (1996) for the hydrocracking of vacuumgas oil, where they used the boiling point as the continuous label.The normalised chain length of each species is defined as:

y¼L�Ls

Ll�Lsð4Þ

where Ls and Ll represent the shortest and longest chain in thereaction mixture.

M. Adam et al. / Chemical Engineering Science 76 (2012) 154–164156

The concentration of the generic component i can then beexpressed as:

CiðtÞ ¼ Cðy,tÞdy ð5Þ

And, in terms of reactivity, k, as:

Cðy,tÞdy¼ cðk,tÞDðkÞdk ð6Þ

The power law relation is one which has been widely used totransform the k-space into the y-space:

k

kmax¼ y1=a

ð7Þ

where a is a model parameter and kmax represents the rateconstant for the species with the longest chain. The mass balanceequation for the species with reactivity k can be presented by:

dcðk,tÞ

dt¼�kcðk,tÞþ

Z kmax

kPðk,KÞKcðk,tÞDðkÞdk ð8Þ

where P(k,K) represents the yield distribution function anddetermines the amount of formation of the species with reactivityk from the species with reactivity K greater than k. c (k,t) is theconcentration of species with a reactivity k. and D(k) is the speciestype distribution. D(k) was introduced by Chou and Ho (1988) todescribe the transformation from the label space to the kineticspace and consequently it has a well-defined physical meaning.D(k) depends on the type of reactants in the feed and, veryimportantly, it is independent of the feed concentration. D(k)plays an important role in defining the region of validity of thecontinuum approach (Ho and White, 1995). D(k) can be deter-mined by carrying out experiments on a model compound whichmimics the behaviour of the mixture. The model compound isindependent of the concentrations of each species. Conversely,the precise value of the concentration of the reactants, c(k,t), doesnot pose any restriction to the implementation of the continuumapproach, being only the number of components and D(k) thediscriminatory factors for implementing the continuum lumpingmethodology. D(k) determines the nature of the solution of Eq. (3)by inputting information on how the kinetic constants are spacedon the k-space and defining the relative reactivity of the reactivespecies. Consequently, depending on the specific form that D(k)takes, the reactivity of different feeds can be represented.

In this work, D(k) represents the transformation from thespace L (chain length, or carbon number) to the space k. Despitewe define it in correspondence of the discrete points Li, as thenumber of components becomes very large, D(k) can be treated asa continuous function. D(k)dk represents the number of species-type with rate constant between k and kþdk and must satisfy thefollowing relation:

N¼

Z 10

DðkÞdk ð9Þ

where N is the number of species.How D(k) expresses the interdependence between the reactiv-

ity of the various components is clearly seen by considering thefollowing expression:

DðkiÞ ¼Di

Dki¼ðiþ1Þ�i

kiþ1�ki¼

1

Dkið10Þ

From Eq. (10) it follows that, depending on the value of ki, D(k)can be positive, negative or infinite. Dki40 implies that compo-nents with larger carbon number have a larger reactivity; con-versely, if Dkio0 components with larger carbon number have asmaller reactivity. Dki ¼ 0 describes the case where all thecomponents have the same reactivity.

It is useful to express the reactant-type distribution as afunction of the label; the distribution transforms the discretedistribution of hydrocarbons into a continuous distribution,

expressed by the following relation between the generic compo-nent i and y:

D kð Þ ¼di

dydydk¼N

dydk

ð11Þ

Depending on the specific relation chosen to link i(y) and k

(namely the way the kinetics is related to the label), D(k) can beevaluated by using Eq. (10).

Two different situations are analysed in this work, and theycorrespond to different forms chosen for the function D(k): Case Iwill be used to indicate the situation where D(k) is a power-lawrelation, whilst Case II indicates the case where D(k) is expressedthrough a gamma function. (It is worth noticing that Case II is themore general one, being the power-law a special case of thegamma distribution.) Although experimental evidence confirmsthat in hydrocracking the reactivity increases as the carbonnumber increases, therefore Case I (i.e., the power law relation)is the appropriate choice, we report here calculation for Case II aswell. The reason for undertaking comparative calculations using agamma distribution is twofold: firstly, the gamma distribution isvery general and therefore we aim to show the generality of theimplementation of the distribution-kind function; second, byintroducing additional model parameters, we are able to discusstheir influence on the solution. Finally, we will show how thenumerical programme can be modified when a different expres-sion for D(k) is considered.

Case I: If the power law relation is used (Eq. (7)), then thecorresponding expression for D(k) is obtained by substituting thederivative of Eq. (7) into Eq. (11) as follows:

DðkÞ ¼Na

kmaxaka�1

ð12Þ

which clearly shows the role of k on the distribution. N is the totalnumber of species in the mixture. a and kmax are the equationparameters. This function satisfies the following normalisationcriteria for the species type distribution function:

1

N

Z kmax

0DðkÞdk¼ 1 ð13Þ

Case II: An alternative way to express the reaction-typedistribution would be to consider an expression more flexiblethan the power law. A powerful expression, able to accommodatea large number of kinetics is what we call the gamma distribu-tion; we adopt the following distribution form Chou and Ho(Collins et al., 2006; Chou and Ho, 1989):

DðkÞ ¼ qkZe�xk ð14Þ

where q is normalisation constant which determined by:

N¼

Z 10

DðkÞdk¼qGðZþ1Þ

xZþ1ð15Þ

where �1oZo1, x40 and G is the gamma function. N is thenumber of spices and Zand x are the equation parameters.

The yield function (Pðk,KÞ):Following the work by Laxminarasimhan et al. (1996), the

yield distribution function will be described as follows for bothCases I and II:

Pðk,KÞ ¼1

S0

ffiffiffiffiffiffi2pp exp�

ððk=KÞa0�0:5Þ

a1

� �2

�AþB

" #ð16Þ

where

A¼ e��

0:5=a1

�2

ð17Þ

C1 C2 Ci CN

t = t1.....

C1 C2 Ci CN

t = t2

C1 C2 Ci CN

t = t1

C1 C2 Ci CN

t = t0tδ

Fig. 1. Schematic of the model solution. The arrow indicates that the concen-

tration of the component of index higher than i must be employed to calculate the

concentration of the i component.


B¼ d 1�k

K

� �ð18Þ

S0 ¼

Z K

0

1ffiffiffiffiffiffi2pp exp�

fðk=KÞa0�0:5g

a1

� �2

�AþB

" # !DðkÞdk ð19Þ

a0, a1 are model parameters which determine the location of thepeak in the interval kAð0,KÞ; d is a small finite quantity thataccounts for the possibility that P(k,K) could take a small finitevalue when k¼0. The parameters are characteristic of the systemconsidered and usually depend on variables such as the catalysttype, the catalyst activity and the impurities present in the feed. Itis envisaged that such parameters, being constitutive character-istics of the system under scrutiny (i.e., the hydrocracking ofparaffins), will be general for the given system and therefore, oncethe feedstock is changed, if the kinetic is the same, they shouldremain the same (contrary to what happen for the parametersappearing in D(k) which depend on the feedstock only). However,the precise physical meaning of those parameters remains to beinvestigated.

The assumption made in this work is that the reactantsundergo cracking only. Since the label chosen is the carbonnumber (the molecular weight) it results impossible to distin-guish between isomers. Although the model is unable to accountfor isomerisation, it is able to describe the formation of MDderiving from the hydrocracking of longer chain compounds. It isworth noticing that the model presented here, as all models, isable to describe only the features that are modelled and can beused only to investigate the kinetics of hydrocracking. Eachcomponent can be formed by the cracking of all the componentshaving a larger chain length (molecular weight). The distributionfunction, P(k,K) represents the yield distribution function for theformation of the component having reactivity k from the compo-nent of reactivity K(with K4k). The properties of P(k,K) arereadily obtained by considering its physical meaning:

–
The value of P(k,K) has to be zero when k¼K. – Pðk,KÞ ¼ 0 for k4K since dimerisation is not significant in
hydrocracking.
– Pðk,KÞhas to satisfy a material balance, namelyR K
0 P k,Kð ÞD kð Þdk¼ 1
– Pðk,KÞ should always be positive.
2.2. Solution methodology

In the following, the model considers the n-C and iso-C as a singlelump. The balance equation, for each species, in a plug flow reactor, isthen expressed by an integro-differential equation (Eq. (8)), which issolved numerically. Considering the totality of species, a system ofintegro-differential equations must be solved at each time. Theintegration space is the (c,t) plane which is represented schematicallyin Fig. 1. The integration is particularly demanding since the integralin Eq. (8) must be solved ‘‘backwards’’. Consider, as an example, thegeneric component i (of reactivity ki): at a given time t, the integralappearing in Eq., (8) must be solved by considering all the compo-nents with a chain length longer than i, namely it must be solved overthe interval [ci�1,cN] (see arrow in Fig. 1). However, the actual value(at time t) of those components (greater than i) is not known yet. Tosolve this problem, integration needs to be carried out backwards,starting from the component N with the largest chain length. Indeed,the component N is not formed by any other component andtherefore the integral representing the ‘‘production’’ of that compo-nent (see Eq. (8)) becomes null for n¼N¼70. Once the compositioncN of N is found, then the concentration of the component N�1 canbe obtained through Eq. (8). The integration can then proceed

backwards to cN�2 and so forth till the concentration of the genericcomponent i is calculated. This procedure must be repeated at eachtime step. Given the inherent complexity of the backward calculation,an alternative method has been proposed, checked and implemented.The method is based on the assumption that, if the time step isextremely small, then the evaluation of the ‘‘production’’ term, basedon the concentrations at the previous time, does not give significantand appreciable deviation from the ‘‘production’’ calculated throughthe backward method. Consequently, the final numerical programmehas been implemented adopting the ‘‘small time step’’ method.

A quadrature algorithm method was used to evaluate theintegral part in the main equation (Eq. (8)) and the differentialequation was solved by using Runge–Kutta method. At t¼t0¼0the component distribution corresponds to the feed distributionand, by using it as the input, the component concentration at t¼t1

is obtained. The experimental feed distribution is used. At t¼t2

the output is obtained by using the results at t1 as the new ‘‘feed’’.Because of the numerical approximation employed, renormalisa-tion is needed at each step to make sure that the percentage of thevarious components rightly furnishes the total mass. The proce-dure is continued until the numerical time corresponds to the realtime that the mixture has spent in the reactor.

An optimisation Toolbox in Matlab was used to determine theminimum of the objective function which depends on the valuesof seven model parameters. The Levenberg–Marquardt algorithm(lsqcurvefit) was applied to get the optimal set of model para-meters. The objective function is expressed as the sum of thesquare difference between the experimental and the computedweight percents:

Min½JðcðtÞÞ� ¼XN

i ¼ 1

½cðtÞexp�cðtÞmodel�2 ð20Þ

where c(t) is the weight percent for the experiment and model.

3. Experimental

The experimental data were used for three different purposes:(i) to tune the model parameters; (ii) to determine the correspon-dence between the numerical time and experimental time; and (iii)to understand the features that could make the model predictive.

The experiments (hydrocracking tests) were carried out in abench scale trickle bed reactor operated in down flow mode asshown in Fig. 2. The reactor was filled with 9 g of powderedcatalyst which crushed previously and sieved to 0.625 mmaverage particle size. The catalyst pellet diameter was reducedin order to approximate plug flow behaviour. Liquid and gasproducts were both analysed by GC. A GC HP-5890 II equippedwith a column injection system, electronic pressure control, andFID (Flame Ionization Detector) detector was used to analysis theliquid products but gaseous fraction of products was analysed bya GC HP-5890 II equipped with a FID detector and automatic

SEPARATION ZONE

ANALYSIS ZONE

REACTION

ZONE

FEEDING ZONE

Hydrocracking of Fischer-Tropsch waxes: lab unit scheme

Sampling

Fig. 2. Schematic representation of the experimental set-up.

0

1

2

3

4

5

0 20 40 60 80

para

ffin

s w

t%

carbon number

Fig. 3. Experimental feedstock composition.

0

1

2

3

4

5

0 0.2 0.4 0.6 0.8 1

k

θ

alpha>1alpha=1alpha<1

Fig. 4. The exponential distribution function.


sampling loop. The effect of operating conditions, temperature,pressure, H2/feed ratio, and weight hourly space velocity (WHSV)were investigated by second order factorial design, the so calledCentral Composite Design (Froment (1987)).

The feed used throughout the tests was a Fischer–Tropsch waxmixture whose composition is given in Fig. 3: a mixture a normalparaffins ranging from C5 up to C70. The operating ranges of theconditions were: temperature (324–354 1C); pressure (40–60 bar);H2/feed ratio (0.06–0.15 kg/kg), WHSV (1–3 h�1). The catalyst was atypical bifunctional system made up of Platinum (0.6%) loaded withamorphous silica–alumina.

The vapour–liquid equilibrium data are not used in the model.The model, by making use of the carbon number as the lumpinglabel, does not take into account the different phases; theequilibrium conditions are not included.

4. Results and discussion

The change of k with y, for a41, ao1 and a¼1, is illustratedin Fig. 4; the graph is obtained using Eq. (4). The treatment

followed in this work assumes that a is positive as in most of theliterature. Physically, this translates into considering that thereactivity of the compounds increases with increasing the chainlength.

Figs. 5 and 6 show the behaviour of D(k) versus the chainlength (Li) for various values of a by using Case I and Case II,respectively. The comparison between Case I and Case II showsthat Case I can only accommodate monotonically increasingvalues of Dki and physically this implies that the reactivityincreases as the carbon number increases. Case II gives theflexibility to accommodate the probability that a given compo-nent has higher reactivity of other components regardless of thelabel (carbon number).

The model contains a number of parameters which have beentuned through an optimisation procedure. A sensitivity analysisstudy has been undertaken to determine how the various para-meters change with the operating conditions. The variation of theparameters with the variables is obtained and such relations areinserted into the numerical programme to calculate the para-meters for a value of the operating variables different from theone attempted in the experimental runs. In the following theinfluence of operating conditions is reviewed briefly.

0

2

4

6

8

10

12

14

16

18

20

0 10 20 30 40 50 60 70

D(k

i)

Li

alpha>1alpha=1alpha<1

Fig. 6. D(k) as a function of the chain length (Case II).

0

1

2

3

4

5

6

7

0 10 20 30 40 50 60 70

wt%

carbon number

feedexp(T=324C)model(T=324C)exp(T=342C)model(T=342C)exp(T=354C)model(T=354C)

Fig. 7. Case I—Numerical and experimental results at different operating

temperatures.

0

1

2

3

4

5

6

7

0 10 20 30 40 50 60 70

wt%

carbon number


Fig. 8. Case II—Numerical and experimental results s at different operating

temperatures.

0

10

20

30

40

50

60

70

320 330 340 350 360

conv

ersi

on (

wt%

)

temperature (°C)

exp

model

Fig. 9. Effect of the reactor temperature on the conversion (WHSV¼2 h�1,

Pressure¼47.5 bar, and H2/Feed ratio¼0.105 kg/kg).

0

5

10

15

20

25

30

35

0 10 20 30 40 50 60 70

D(k

i)

Li

alpha=1alpha>1alpha<1

Fig. 5. D(k) as a function of the chain length (Case I).


4.1. The effect of temperature

To study the effect of the temperature on the paraffin conver-sion and product distribution, the simulations have been carriedout at three different temperatures (T¼324 1C, 342 1C, and354 1C) whilst the other operating conditions are kept constant(WHSV¼2 h�1, Pressure¼47.5 bar, and H2/Feed ratio¼0.105 kg/kg).The various simulation results are shown in Figs. 7 and 8 for Case Iand Case II, respectively. As the temperature is increased, thegeneral observed trend shows that the concentration of allcomponents with longer chains decreases whilst the concentra-tion of the components with shorter chain length increases. Ahigher temperature results in higher rates of hydrocracking. Theparaffins experimental and modelled conversion is reported inFig. 9. The conversion increases approximately of 26% when thetemperature increases of 22 1C in agreement with evidencesreported in the literature (Weismantel, 1992). The hydrocrackingconversion is defined according to the following equation:

%C22þ conv¼%wtC22þ in�%wtC22þout

%wtC22þ in

� �n100 ð21Þ

The optimised model parameters at each temperature for CaseI and Case II are presented in Table 1. Case I has five independentparameters and two additional parameters are required for CaseII. These parameters were used to predict the yield distributioncurve of the products. Both cases were used to determine theweight percent of species in the product which are compared

with the experimental values. The cumulative weights for thefeed, the experiments and model calculations are reported inFigs. 10 and 11, for Cases I and Case II, respectively.

Table 1The values of the parameters for Case I and Case II at different temperatures.

Model parameters

Temperature (1C) a0 a1 kmax a d Z x

324 8.14 4.68 2.64 0.35 7.05E-07 – –

Case I 342 6.00 3.80 8.08 0.40 9.05E-07 – –

354 5.82 3.60 32.08 0.314 7.05E-07 – –

324 9.66 2.78 2.86 0.50 1.28E-01 65.64 39.80

Case II 342 8.66 3.46 18.28 0.31 1.78E-02 40.64 44.80

354 5.66 2.48 29.28 0.34 8.08E-01 40.64 40.80

0

10

20

30

40

50

60

70

80

90

100

110

0 10 20 30 40 50 60 70

cum

ulat

ive

(wt%

)

carbon number


Fig. 10. Case I—Cumulative weights at different operating temperatures.

0

10

20

30

40

50

60

70

80

90

100

110

0 10 20 30 40 50 60 70

cum

ulat

ive

(wt%

)

carbon number


Fig. 11. Case II—Cumulative weights at different operating temperatures.

0

1

2

3

4

5

6

0 1 2 3 4 5 6

mod

el (

wt%

)

experimental (wt%)

T=324CT=342CT=354C

Fig. 12. Parity plot between model calculations and experimental data for Case I.

0

1

2

3

4

5

6

0 1 2 3 4 5 6

mod

el (

wt%

)

experimental (wt%)

T=325CT=342CT=354C

Fig. 13. Parity plot between model calculations and experimental data for Case II.

-1

-0.5

0

0.5

1

1.5

2

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75

resi

dual

(w

t%)

carbon number

T=324CT=342CT=354C

Fig. 14. Case I—The residual values for the model product.


It can be seen that the experimental data and the results fromthe model are practically indistinguishable for both cases attemperature 324 1C and 342 1C but when the temperature isincreased to 354 1C, results are less good, especially for Case I.

Comparisons between experimental data and model calcula-tions are shown in Figs. 12 and 13 for Case I and Case II,respectively. It can be seen that the comparison is very good.The parity plot of the product at different temperatures ispresented in Figs. 14 and 15. The results give a good indicationof which components in the mixtures are described with lessaccuracy. The higher temperature seems to give less good results,especially in correspondence of low and middle length chain

components. On the contrary, the very large chain componentsare described well at high temperatures. Indeed, when thetemperature is high, the reactivity of the longest chain compo-nents should be higher; when the temperature increases, thelonger chain components start cracking faster than what observedthrough the model. At high temperature, the cracking of thecomponents with longer chain is likely to produce more compo-nents with middle length chain (from C15 to C22) than compo-nents with low length chain (from C1 to C15); however theopposite is observed from the model results. This result remainsunexplained at present requiring further investigation.


The optimised sets of parameters corresponding to eachexperimental run are calculated for a limited number of runs.The variation of the parameters with the temperature is showedin Fig. 16(a) for both Case I and Case II. The variation of kmax withthe temperature is well interpolated by a linear relationship(Fig. 16(b)). The activation energy for both Case I and Case II is

-1

-0.5

0

0.5

1

1.5

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75resi

dual

(w

t%)

carbon number

T=324CT=342CT=354C

Fig. 15. Case II—The residual values for the model product.

4

5

6

7

8

9

10

11

320 330 340 350 360

a 0

temperature (°C)

case Icase II

0.2

0.3

0.4

0.5

0.6

320 330

α

temper

ln(kmax) = -30.414*

ln(kmax)

0

0.5

1

1.5

2

2.5

3

3.5

4

1.58 1.6 1.62

ln(k

max

)

1

Fig. 16. (a) Variation of optimised values of the parameters with th

calculated as 150 kJ/mole/K. The parameter d changes very littlewith the operating parameters (temperature, pressure, H2 feedratio and HWSV) for both Case I and Case II.

4.2. The effect of pressure

To study the effect of the pressure on the paraffin conversionand product distribution, the simulations have been carried out atthree different reactor pressures (p¼40 bar, 47.5 bar, and 60 bar)while the other operating parameters are kept constant(WHSV¼2 h�1, T¼342 1C, and H2/Feed ratio¼0.105 kg/kg). Theexperimental and model results are showed in Figs. 17 and 18 forboth Case I and Case II. As the pressure is increased, theconcentration of the hydrogen increases which in turn increasesthe rate of hydrogenation rather than increasing the rate ofcracking. On the other hand, when the pressure increases, theconcentration of the lighter components increases, making suchcomponents susceptible of cracking again. This means thatcomponents having length chain between C13 and C20 are crack-ing again to produce components with lower length chain. Theconversion of C22þ decreases with increasing the pressure due tothe increase of the fugacity of hydrogen that affects negatively thedehydrogenation equilibrium of the feed (Gamba et al., 2009).

2

2.5

3

3.5

4

4.5

5

320 330 340 350 360

a 1

temperature (°C)

case Icase II

340 350 360

ature (°C)

case Icase II

1/T + 51.811

= -29.811*1/T + 51.095

1.64 1.66 1.68

/T(k-1)

case Icase II

e temperature and (b) Variation of kmax with the temperature.

0

1

2

3

4

5

6

0 10 20 30 40 50 60 70

wt%

carbon number

feedexp(40 bar)model(40 bar)exp(47.5 bar)model(47.5 bar)exp(60 bar)model(60 bar)

Fig. 18. Case II—Calculated weight distributions at different operating pressure (bar).

Table 2Model parameters at different values of the pressure.

Model parameters

Pressure (bar) a0 a1 kmax a d g n

40 6.20 3.60 9.08 0.34 9.91E-07 – –

Case I 47.5 6.00 3.80 8.08 0.40 9.05E-07 – –

60 6.40 1.60 7.48 0.324 1.01E-06 – –

40 4.66 4.88 13.26 0.33 1.78E-02 15.64 38.80

Case II 47.5 8.66 3.46 18.26 0.31 1.78E-02 40.64 44.80

60 4.66 4.88 11.46 0.31 1.78E-02 25.64 40.80

0

10

20

30

40

50

60

70

80

90

100

110

0 10 20 30 40 50 60 70

cum

ulat

ive

(wt %

)

carbon number


Fig. 19. Case I—Comparison of predicted and experimental data of cumulative

weight percent.

0

10

20

30

40

50

60

70

80

90

100

110

0 10 20 30 40 50 60 70

cum

ulat

ive

(wt%

)

carbon number


Fig. 20. Case II—Comparison of predicted and experimental data of cumulative

weight percent.

0

1

2

3

4

5

6

0 10 20 30 40 50 60 70

wt%

carbon number

feedexp(0.06 kg/kg)model(0.06 kg/kg)exp(0.105 kg/kg)model(0.105 kg/kg)exp(0.15 kg/kg)model(0.15 kg/kg)

Fig. 21. Case I—Comparison between calculated and experimental data at

different values of H2/feed ratio (kg/kg).

0

1

2

3

4

5

6

0 10 20 30 40 50 60 70

wt%

carbon number


Fig. 17. Case I—Calculated weight distributions at different operating pressures (bar).


In other words, increasing the pressure increases the fugacity ofhydrogen that affects the dehydrogenation equilibrium of thefeed and then the total conversion (Kumar and Froment, 2007;Leckel and Liwanga-Ehumbu, 2006).

The model results are in good agreement with the experi-mental data. The model parameters used are reported in Table 2for each experimental run. The effect of the pressure on the rateconstant reduces as the pressure increases. The cumulativecomparison of model results and experimental data, at threedifferent values of pressure for both Case I and Case II is shown inFigs. 19 and 20, respectively. It should be noted that, in contrast

with what happens for the temperature, changes in pressure donot have an appreciable effect on the results, at least for the rangeof values of pressure considered in this study.

Table 3Model parameters at different values of H2/feed ratio.

Model Parameters

H2/Feed ratio(kg/kg)

a0 a1 kmax a d g n

0.06 5.20 1.80 4.284 0.358 1.11E-07 – –

Case I 0.105 6.00 3.80 8.08 0.40 9.05E-07 – –

0.15 6.20 3.60 8.88 0.30 2.01E-07 – –

0.06 4.66 3.88 8.04 0.32 1.78E-02 50.64 38.80

Case II 0.105 8.66 3.46 18.26 0.31 1.78E-02 40.64 44.80

0.15 3.08 4.08 16.64 0.30 1.78E-02 28.64 48.80

0

10

20

30

40

50

0.04 0.08 0.12 0.16

wt %

H2/Feed ratio

C1-C4 (exp) C1-C4(model)C5-C9 (exp) C5-C9(model)C10-C14(exp) C10-C14(model)C15-C22(exp) C15-C22(model)C23+(exp) C23+(model)

Fig. 23. Case II—Comparison between calculated and experimental data for the

5 lumps analysis at different values of H2/Feed ratio.

0

1

2

3

4

5

6

0 10 20 30 40 50 60 70

wt%

carbon number

feedexp(0.06 kg/kg)model(0.06 kg/kg)exp(0.105 kg/kg)model(0.105 kg/kg)exp(0.15 kg/kg)model(0.15 kg/kg)

Fig. 22. Case II—Comparison between calculated and experimental data at

different values of H2/feed ratio (kg/kg).


4.3. The effect of H2/feed ratio

Typical model results, for different values of H2/feed ratio(0.06, 0.105, and 0.15 kg/kg) at T¼342 1C, p¼47.5 bar, andWHSV¼2 h�1, for Case I and Case II, are showed in Figs. 21 and22, respectively. The optimal estimated model parameters wereused to calculate the component productions to compare with theexperimental data. The variations of the parameters, when the H2/Feed ratio is changed, are reported in Table 3. The reactivity of thecomponents with longer chain increases as the hydrogen-to-hydrocarbon ratio increases. Dividing the components into groups(or lumps) based again on the carbon number, namely C1–C4 (fuelgas), C5–C9 (naphtha), C10–C14 (kerosene), C15–C22 (diesel), andC23þ (residue) gives a better understanding of the behaviour ofthe mixture. Fig. 23 shows how the increase of the hydrogen tofeed ratio affects the hydrocraking, when the product is dividedinto five lumps for Case II. Increasing the hydrogen to feed ratio,the weight percent of four lumps (namely, fuel gas, naphtha,kerosene, and diesel) seems to increase, whilst the weight percentof residue decreases.

5. Conclusions

A continuum lumping model, employing two different species-type distribution functions, D(k), was applied to the kinetics ofcatalytic hydrocracking of Fischer–Tropsch waxes (normal paraf-fins from C5 to C70). The kinetics and product distributionparameters for both cases were fine-tuned by using experimentaldata to calculate the weight percent of the products. Two differentexpressions for the reactant-type distribution have been consid-ered, namely a power-law distribution (Case I) and a gamma

distribution (Case II). Although Case II gives a better fitting of theexperimental data, it is believed that Case I is more realistic forthe hydrocracking process and the better results obtained for CaseII are attributed to the larger number of parameters employed.The dependence of the results on the operating variables wasstudied and the trend analysed; in particular, increasing thetemperature and the residence time results in an increase of theerror percentage between the model and experimental data. Oneexplanation could be that the model includes hydrocrackingreaction only.

Nomenclature

A constant, Eq. (11)B constant, Eq. (12)a0, a1 model parametersC(k,t) continuous concentration-reactivity function at time t

D(k) species-type distribution functiongeneric species in mixture

k reaction rate constant, h�1

kmax rate constant of species with longest chainL length chainLl longest chainLs shortest chainN total number of speciesP(k,K) yield distribution functionq model parameter, Case IIt time, t¼ 1

WHSV

T temperature

Greek letters

y normalised length chaina model parameter, Case IZ, x model parameters, Case IIG gamma function

Acknowledgments

The authors would like to EUROKIN for the financial support.AM thanks the Libyan General Peoples Committee Secretary forHigher Education for support towards his studies.


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