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Computers and Chemical Engineering 36 (2012) 68– 78

Contents lists available at ScienceDirect

Computers and Chemical Engineering

j ourna l ho me pag e: w ww.elsev ier .com/ locate /compchemeng

more efficient simulator of particle size distribution in slurry phaseolyolefin systems

ohn T. McCoy, Randhir Rawatlal ∗

epartment of Chemical Engineering, University of Cape Town, Corner Ring Road and South Lane, Upper Campus, Cape Town, Western Cape 7701, South Africa

r t i c l e i n f o

rticle history:eceived 29 October 2010eceived in revised form 10 May 2011ccepted 4 June 2011vailable online 12 June 2011

a b s t r a c t

The Segregation Approach is used as an alternative to Population Balance Modelling (PBM) to predictparticle size distributions in olefin polymerisation reactors. The Segregation Approach affords simpler for-mulation, more explicit integration of mixing characteristics, particle growth and initial size distribution,and easier extension to unsteady-state conditions and complex kinetics.

The evolution of a non-uniformly sized feed of catalyst particles in a polymer reactor system is simu-

eywords:opulation Balance Modellingegregation Approacharticle size distributionlefin polymerization

lated. While the two approaches yield identical predictions of the particle size distribution, in terms ofcomputation time the Segregation model (0.09–1.7 s) far outperforms the PBM (1.8–6.2 s). This is becausethe sets of differential equations required by the PBM are avoided by the Segregation Approach.

Based on this result, the Segregation model is described in the context of developing a model whichcan be used for the real-time model-based control and grade transition trajectory optimisation of olefin

.

odel-based control polymerisation processes

. Introduction

Polyolefins such as high density polyethylene (HDPE) are com-only produced on an industrial scale in continuous flow slurry

hase reactors using heterogeneous Ziegler–Natta catalysts. Theechanical properties of the polymer are highly dependent on a

ariety of factors, including reactor operating conditions like tem-erature and pressure, reactant concentrations and ratios, catalystctivity and reactor mixing characteristics.

To understand the interplay among these competing phenom-na, it is convenient to consider three scales of perspective: micro,eso and macro (McKenna & Soares, 2001; Ray, 1986, 1988). This

aper will focus on the mesoscale.The microscale (more correctly nanoscale, but the term has

ecome accepted in the literature) is concerned with the kinet-cs of the catalysed polymerisation reaction, including catalystite activation, chain propagation and termination and copoly-er incorporation. Some authors have proposed the existence ofultiple active site types, each of which produce polymer with dif-

erent properties at different rates, to explain observed phenomenauch as catalyst activity profiles and otherwise unexpected poly-er property distributions (Chen & Fan, 2006; Kissin, Nowlin, &

ink, 1993; Kissin, Liu, Pollick, Brungard, & Chang, 2008; Soares

Hamielec, 1995a). Other workers have suggested that catalystctivity is more closely related to titanium site oxidation states,

∗ Corresponding author. Tel.: +27 21 650 4431; fax: +27 21 650 5501.E-mail address: randhir.rawatlal@uct.ac.za (R. Rawatlal).

098-1354/$ – see front matter © 2011 Elsevier Ltd. All rights reserved.oi:10.1016/j.compchemeng.2011.06.003

© 2011 Elsevier Ltd. All rights reserved.

with only one (or at most two) oxidation states being activefor polymerisation (Bhaduri, Mukhopadhyay, & Kulkarni, 2006;Fregonese, Mortara, & Bresadola, 2001; Han-Adebekun & Ray, 1997;Han-Adebekun, Hamba, & Ray, 1997; Soga, Chen, & Ohnishi, 1982),rather than multiple sites polymerising at different rates. Polymerproperty distributions can be explained in terms of differing ter-mination, rather than propagation, rates (Floyd, Heiskanen, Taylor,Mann, & Ray, 1987; Rawatlal, 2004; Touloupides et al., 2010).

On the mesoscale, particle property dynamics can be modelledfrom the physical-kinetic expressions available; properties of inter-est include particle size and mass and heat transfer rates.

On the macroscale, reactor hydrodynamics and vapour liquidequilibria must be quantified. There are published attempts at mod-elling the transfer of monomer from gas to liquid phase, and theeffects of reactor mixing type, residence time and agitation rate(Bhagwat, Bhagwat, & Sharma, 1994; Floyd, Hutchinson, & Ray,1986; Soares & Hamielec, 1995b; Zacca, Debling, & Ray, 1996, 1997).

As mentioned, the focus of this paper is on mesoscale effects,and especially on particle size. During heterogeneous polymeri-sation, catalyst particles become encased in a growing layer ofpolymer, affecting particle morphology and the transfer of heat andmass (Floyd, Choi, Taylor, & Ray, 1986; Kittilsen & McKenna, 2001;Kittilsen, McKenna, Svendsen, Jakobsen, & Fredriksen, 2001).

Figs. 1 and 2 show scanning electron microscope (SEM) imagesof an ethylene/1-butene copolymer produced on a silica-based

Ziegler–Natta catalyst, at different residence times in a lab-scalebatch reactor. Fig. 1 shows polymer particles after 15 min of poly-merisation, having achieved an average size of ∼100 �m. Theincrease in residence time to 60 min (shown in Fig. 2) therefore

J.T. McCoy, R. Rawatlal / Computers and Ch

Fc

yea

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Ptb

(intt

Fc

ig. 1. SEM image of silica-based Ziegler–Natta-catalysed ethylene/1-buteneopolymer after 15 min of polymerisation.

ields an expected increase in particle mean size to ∼200 �m. Thisxperimentally-observed phenomenon must be accounted for inny simulation or analysis of polymerisation reactors.

Mean particle size and the particle size distribution (PSD) canffect not only the rate of polymerisation but also product han-ling in post-reactor steps such as transport, storage and blendingChoi, Zhao, & Tang, 1994; Zacca & Debling, 2001). It is thereforelear that the size distribution of the polymer particles produced inolymerisation reactors must be quantified.

The most commonly used mathematical tool for investigatingSDs (and distributions in general) is the population balance equa-ion. The general, unsteady state, multiple-property populationalance equation (PBE) is given by Eq. (1) (Hulbert & Katz, 1964).

ı�

ıt+

∑i

ı

ı˛i

(ı˛i

ıt�

)= B − D (1)

In Eq. (1), � is the time-varying number-density of a populationof particles or fluid elements, for example) which is distributed

n the properties ˛i, each of which can also vary with time. Theumber density is defined such that �(t, ˛1, ˛2, . . .)�˛1�˛2 . . . ishe number fraction of particles or fluid elements having proper-ies in the ranges [˛1, ˛1+ �˛1], [˛2, ˛2 + �˛2], . . . at time t. Size

ig. 2. SEM image of silica-based Ziegler–Natta-catalysed ethylene/1-buteneopolymer after 60 min of polymerisation.

emical Engineering 36 (2012) 68– 78 69

is a typical property of interest; others could be age, temperatureor catalytic activity. If size were the property of interest, then thenumber density function, �, would represent the number-basedparticle size distribution (PSD). The functions B and D representthe distribution of the rates of birth and death of the members ofthe population. The birth and death functions are generally relatedto the flow of material in and out of a reactor, but can also referto processes such as nucleation (birth) or fragmentation (death). Inorder to find the distribution of n properties within a population,one must solve an (n+1)-dimensional Partial Differential Equation(PDE): one dimension for each property, plus one for time.

Application of Population Balance Modelling (PBM) in predict-ing steady-state PSD in polymerisation processes is well-developedin fluidised-bed gas-phase reactors (Choi et al., 1994; Hatzantonis& Kiparissides, 1998; Hatzantonis, Goulas, & Kiparissides, 1998;Khang & Lee, 1996) and emulsion polymerisation reactors (Vale& McKenna, 2005). Some more general models, based on popu-lation balance methods, have been developed to investigate theinfluence of residence time on steady state reactor performance(Soares & Hamielec, 1995b; Zacca & Debling, 2001; Zacca et al.,1996, 1997). These models consisted of n-dimensional PDEs, sincethe time dimension was not investigated in these steady state sys-tems.

There is growing interest in developing dynamic reactor mod-els for the eventual goal of model-based control (Alexopoulos& Kiparissides, 2005; Alexopoulos, Roussos, & Kiparissides,2004; Harshe, Utikar, & Ranade, 2004; Meimaroglou, Roussos, &Kiparissides, 2006; Neto, Freitas, Nele, & Pinto, 2005; Roussos,Alexopoulos, & Kiparissides, 2005). The major focus of much ofthis work is on finding the most efficient numerical method for thesolution of the PDEs that naturally result when developing PBMs(Attarakih, Drumm, & Bart, 2009; Chakraborty & Kumar, 2007;Dorao & Jakobsen, 2006; Kumar, Peglow, Warnecke, & Heinrich,2008; McCoy, 2002; McCoy & Madras, 2003; Pinto, Immanuel, &Doyle, 2008; Qamar, Warnecke, & Elsner, 2009). As such, it is oftenreported that even though the population balance approach canvery accurately predict PSD, computational expense is still veryhigh (Ahmadzadeh, Arastoopour, Teymour, & Strumendo, 2008;Alexopoulos, Roussos, & Kiparissides, 2009). This makes these mod-els unsuitable for use in real-time process control.

In engineering applications, it can be sufficient to determineonly the mean (and possibly the variance) of a property, rather thanthe entire distribution. For example, in the polymer field, it is notalways necessary to know the chain length distribution: a poly-mer can be characterised by the mean chain length (Mn) and chainlength variance (indicated by the Polydispersity Index) (Rawatlal,2004).

To find the mean of the i-th property, ˛i, the full distribution, �,must be determined from the solution of the PDEs that result fromthe associated PBM. Only once this distribution is known, can themean can be found as the first moment of the distribution (Eq. (2)).

¯ i =∫

˛i�(˛)d (2)

The Segregation Approach has been proposed as a compu-tationally efficient alternative to the PBM (Rawatlal, 2009). TheSegregation Approach is a method of predicting the mean (andother moments) of a distribution without having to determine thewhole distribution first. Again considering the i-th property, ˛i, ofa distribution, the current value of the property can in general be

expressed in terms of the initial value of the property, ˛i, and age,�. For example, particle size can be found in terms of initial particlesize and particle age, related through expressions for the dynamicsof particle growth.

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If the initial distribution in the i-th property, �0(˛i,0), and thenternal residence time distribution, I(�), of the reaction vessel ofnterest, are also known, then the mean value of the i-th propertyan be found through application of Eq. (3).

¯ i =∫ ∫

˛i(˛i,0, �)�0(˛i,0)I(�)d˛i,0d� (3)

It is clear that Eqs. (2) and (3) are equivalent: they are therst moments of density functions. The difference between Eq. (2)which would be applied to the solution of the PBM) and Eq. (3)the Segregation Approach) is in the steps before the evaluation ofhe integral. In order to evaluate Eq. (2), the function � is required,nd hence the PBM (and the associated multidimensional PDEs)ust be solved. By comparison, evaluation of Eq. (3) is far simpler,

equiring knowledge only of the inlet distribution and residenceime distribution (both of which can be physically measured onhe plant), and the integration of these known functions.

The reduced computational expense associated with the Seg-egation model is its primary benefit over the traditional PBMpproach, a benefit which is significant given that it makes model-ased control feasible.

A second benefit is the ease with which complex or arbi-rary mixing patterns can be incorporated using the Segregationpproach. The complexity of simulating non-ideal mixing with theBM significantly adds to the difficulty of applying the technique.n the Segregation Approach, all the hydrodynamics are includedxplicitly in the form of the residence time distribution (RTD)unction. There are standard methods for dealing with imperfect

ixing using RTD functions, including compartmentalisation orirect measurement by tracer experiments (Fogler, 2005). In addi-ion, this formulation allows for easy incorporation of the recentlyeveloped unsteady state RTD, allowing simulations of unsteadyarticle flows or sudden changes in residence time (Rawatlal &tarzak, 2003). In the PBM, unsteady flows can only be incorpo-ated by introducing real time as an additional dimension in theystem of multidimensional PDEs.

The Segregation Approach has been successfully applied in theeld of polymer reaction engineering, to predict polymer proper-ies (Rawatlal, 2009), and to bioprocess engineering, predicting theerformance of tank bioleach reactors (Kotsiopoulos, Hansford, &awatlal, 2008). These earlier works, however, explored catalystctivity rather than particle size effects, and did not rigorouslyompare results from the two methods in terms of accuracy andimulation run-time.

The aim of the current paper is to show equivalence between theBM and the Segregation Approach in the context of particle size,nd to quantify the extent of the gain in computational efficiencyssociated with the Segregation Approach.

. Model development

The models developed here will predict the particle size distri-ution (PSD) for an olefin polymerisation reactor. The starting point

s the prediction of the rate of propagation of the polymerisationeaction at the active sites, which is considered to be a microscaleroperty. The propagation rate will be incorporated within a modelor a single particle’s size, which is a mesoscale level of detail. These

esoscale particle models will then be used in macroscale modelshich predict the PSD of the reactor.

.1. Polymerisation propagation rate

The polymerisation propagation rate, rp [mol s−1], can bexpressed in terms of kp, the polymerisation rate constantm3 s−1 mol−1], P*(t), the number of moles of polymerising sites at

emical Engineering 36 (2012) 68– 78

time t [mol], and [M], monomer concentration [mol m−3], as givenin Eq. (4).

rp = kpP∗(t)[M] (4)

The number of active sites can be rewritten as a product of CTi,0,the initial concentration of sites in the particle [mol kg−1], the par-ticle’s initial mass, mp,0, voidage, ε, density, �, and the fraction ofsites which are active at a given time, a(t), as shown in Eq. (5).

P∗(t) = CTi,0mp,0

�εa(t) (5)

2.2. Single particle growth model

A single particle’s mass increases as polymer is formed throughthe propagation reaction. Particle mass can also be affected by frag-mentation and agglomeration; these effects will be ignored in thiswork as they are less significant in slurry phase reactors. The massbalance, defining the rate at which particle mass changes in a singlegrowing catalyst/polymer particle, is expressed in Eq. (6).

dmp

dt= kpP∗(t)[M]MMmon (6)

Here, mp is the particle mass [kg], t is particle time (or parti-cle age in the context of a reactor) [s], and MMmon is monomermolar mass [kg mol−1]. Combining Eqs. (5) and (6), the expressionfor particle mass as a function of time can be simplified to Eq. (7).

dmp

dt= Gmp; mp(0) = mp,0 (7)

where mp,0 is the mass of a particle at the inlet to the reactor, beforeany growth has occurred, and G is the growth factor, defined asG = ((kpCTi,0)/(�ε))[M]a(t)MMmon. This growth factor is introducedto simplify the notation.

If it is assumed that the rate constant, kp, concentration of Tisites, CTi,0, monomer concentration, [M], and active sites fraction,a(t), are all constant with time, then the mass of the particle can befound in terms of the growth function, G, and the time of polymeri-sation, t, by solving Eq. (7) to yield Eq. (8).

mp(t) = mp,0(1 + Gt) (8)

It should be noted that the activity of the Ziegler–Natta catalystis generally understood to vary with time, and is correlated withthe oxidation state of the titanium active sites on the particle (Han-Adebekun et al., 1997; Soga et al., 1982). In this article, this effectwill be ignored in order to focus on particle size effects and allow forsimpler comparison of the solution of the two models of interest.

By assuming perfectly spherical particles, particle size and par-ticle mass can be related through Eq. (9).

m = �

6l3�ε (9)

Using the relation between mass and size, and the expressionfor time-dependent mass (Eq. (8)), the particle diameter after poly-merisation time t can be found by Eq. (10).

l(l0, t) = l03√

1 + Gt (10)

It is important to note that l0 is the initial size of a particle, orthe size of a particle as it enters the reactor, and that Eq. (10) canbe used to predict the final size of a particle for which the inlet size

and polymerisation time are known.

The next step is to use this expression for a single particle’s sizeto determine the evolution of the PSD of a population of particleswhich are distributed in initial size and reaction time.

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J.T. McCoy, R. Rawatlal / Computers a

.3. Segregation model

The product PSD of a polymerisation reactor could be repre-ented by the mean particle size, which can be found by using theegregation Approach, as shown in Eq. (11).

=∫ ∫

l(l0, �)fm,0(l0)I(�)dl0d� (11)

Eq. (11) is used to determine the mean particle size, l, in termsf the mass-based initial size distribution, fm,0(l0), RTD function,(�), and the relationship between initial size, age and current size,s developed in Eq. (10). The important factor is this alternativeormulation allows for predicting the mean size without needingo find the internal particle size distribution, fm(l), as opposed tohe requirements when applying the traditional PBM framework.

In the present work the equivalence of the Population Balancend Segregation Models will be rigorously demonstrated. For thiseason alone, the entire mass-based particle size distribution, fm(l),ather than only the mean size, will be determined. Thus a modifiedersion of the Segregation model which will predict this full distri-ution is required. This modified model will represent a worst-casecenario of the performance of the Segregation model since in thisase, far more data is generated than is usually required. However,he modified model also shows the flexibility of the Segregationpproach; it can be extended to predict the same information as theBM, but primarily lends itself to averaged properties at extremelyigh computational efficiencies.

The exit size distribution function, fm(l), and the inlet size dis-ribution function, fm,0(l0), are defined such that fm(l)�l is the massraction of particles in the reactor in the size range [l, l + �l], andm,0(l0)�l0 is the mass fraction of particles in the feed in the ini-ial size range [l0, l0 + �l0]. By multiplying the fraction in that sizeange by the mass feed rate, min, and then dividing by the mass ofach of these particles, (�/6)l30�ε, the number of particles in theeed that are in the initial size range [l0, l0 + �l0] can be found:fm,0(l0)�l0min)/((�/6)l30�ε).

The fraction of particles currently in the reactor in the ageange [�, � + ��] can be found in terms of the internal reactoresidence time distribution I(�), as I(�)��. The mass of parti-les that entered the reactor during this time is given by min��.he number of particles currently in the reactor in the age range�, � + ��] that were in the feed size range [l0, l0 + �l0] is then(fm,0(l0)�l0min��)/((�/6)l30�ε))I(�)��.

The mass of each of these particles can be written in terms ofhe relation between current size and initial size, developed in therevious section (Eq. (10)), as given in Eq. (12).

p = �

6l3�ε = �

6l30(1 + G�)�ε (12)

Multiplying the mass of each particle by the number of sucharticles gives the mass of all particles in the reactor, Ml0,� , which,hile in the feed, were in the feed size range [l0, l0 + �l0] and are

urrently in the age range [�, � + ��]. This is given by Eq. (13).

l0,� = (1 + G�)fm,0(l0)�l0min��I(�)�� (13)

Dividing by the total mass of particles then obviously returnshe mass fraction of particles of interest, ml0,� , as given in Eq. (14).

l0,� = (1 + G�)fm,0(l0)�l0I(�)�� (14)

This function can be interrogated to determine the mass frac-ion of particles of size l or smaller in the reactor; this will be aumulative distribution function. This function, which will be used

o determine the exit PSD, will be referred to as the kernel for the

odel.Fig. 3 shows a typical feed size distribution, residence time dis-

ribution and kernel for this model. The feed size (a) is a normal

emical Engineering 36 (2012) 68– 78 71

distribution with a mean of 50 �m and a variance of 8 �m. Anycurve whose domain runs parallel to the l0 axis through the surfaceof the kernel shown in (c) displays a similar “bell-curve” shape.The residence time distribution (b) is for a CSTR with a mean resi-dence time of 4000s. Any curve whose domain runs parallel to the� axis through the surface of the kernel displays the form of an ini-tial rise and then decay from the maximum. This is caused by theinteraction of the linearly increasing growth term, (1 + G�), and theexponential decay of the RTD function.

It is therefore clear that the kernel to be integrated is of the formthat should be expected from a physical understanding of the pro-cesses that it represents. The ranges of initial sizes and ages thatwill contribute to size l are now sought. In the absence of fragmen-tation, only particles which were initially smaller than size l cancontribute to the mass of particles of interest. Thus only particleswith initial sizes that satisfy the following inequality are of interest:l0 ≤ l.

Particle age is slightly more complex, due to the relationshipbetween initial size, current size and age developed in Eq. (10).Rearranging Eq. (10) allows the age range of interest to be definedas � ≤ ((l3/l30) − 1)/G.

By integrating across all relevant initial sizes and ages, the cumu-lative mass fraction, M(l), can be found, as shown in Eq. (15).

M(l) =∫ l0=l

l0=0

∫ �=((l3/l30

)−1)/G

(1 + G�)fm,0(l0)I(�)d�dl0 (15)

Differentiating the cumulative mass distribution with respect tol gives the particle size distribution, fm(l), as shown in Eq. (16).

fm(l) = d

dlM(l) = d

dl

∫ l0=l

l0=0

∫ �=((l3/l30

)−1)/G

(1 + G�)fm,0(l0)I(�)d�dl0

(16)

Successive applications of Leibniz’s Rule result in the simpler Eq.(17).

fm(l) = 3l5

G

∫ l0=l

l0=0

fm,0(l0)I

(� = (l3/l30) − 1

G

)dl0 (17)

As mentioned earlier, the explicit inclusion of the residencetime distribution function, I(�), allows the Segregation Approach tomodel imperfect reactor mixing or unsteady state operation withrelative ease. It is also important to note that this form of the Segre-gation model represents worst-case performance: rather than justpredicting the mean of the distribution, as could be done using Eq.(11), the full distribution has been extracted.

An analogous expression for the Population Balance Model willnow be derived.

2.4. Population Balance Model

The general, unsteady state, multiple-property population bal-ance equation (PBE) is given by Eq. (1). There are two properties ofinterest: size, l, and age, �, at steady state, so the PBE assumes thespecific form shown in Eq. (18).

ı

ıl

(ıl

ıt�(l, �)

)+ ı

ı�

(ı�

ıt�(l, �)

)= B(l, �) − D(l, �) (18)

In the absence of nucleation, aggregation or fragmentation, therate of birth of particles is only due to the flow of particles into thereactor in the feed stream. Similarly, particle death is due only to

the flow of particles out of the reactor in the product stream.

For a perfectly mixed reactor the distribution of the rate of par-ticle birth is simply the distribution in the feed stream, �0, dividedby the mean residence time, �. Similarly, the exit stream is assumed

72 J.T. McCoy, R. Rawatlal / Computers and Chemical Engineering 36 (2012) 68– 78

Fig. 3. Typical inputs to the Segregation model: (a) feed size distribution; (b) residence time distribution; and (c) kernel. See text for a description of the features of thesegraphs.

nd Chemical Engineering 36 (2012) 68– 78 73

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ta

c

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J.T. McCoy, R. Rawatlal / Computers a

o have the same properties as the reactor contents and so the dis-ribution of the rate of particle death is the distribution function, �,ivided by the mean residence time. The birth and death functionsan then be expressed in Eq. (19).

= �0

�= ı(�)fn,0(l0)

�; D = �

�(19)

The feed distribution, �0, can be expressed as the product of theirac-ı function, ı(�), since all entering particles are assumed to bef age zero, and the number-based feed size distribution, fn,0(l0).ote that the feed size distribution is number-based, since this is

he basis for the PBM, in contrast with the Segregation Approach,hich used a mass basis.

The dependence of particle size on age given by Eq. (10) can beifferentiated with respect to age to determine the (ıl)/(ıt) term inq. (18). The (ı�)/(ıt) term is simply unity, as age increases at theame rate as time.

The final PBE for a steady state perfectly mixed reactor withime–constant concentrations and activities can then be expresseds Eq. (20).

l0G

3(1 + G)2/3

ı�

ıl+ ı�

ı�= ı(�)fn,0(l0) − �

�(20)

This relatively simple PDE can be reduced, using the method ofharacteristics, to a set of ordinary differential equations.

d ′(l0, s)ds

= −�′(l0, s)�

for s > 0; �′(l0, s = 0) = fn,0(l0)�

(21)

The applied parameterisation in the variable s gives � = s and = l0

3√1 + Gs. The solution of Eq. (21) is the full distribution of par-icles within the reactor, in terms of initial size and current age.he cumulative number fraction, N(l), is found from the age andnitial size distribution, �′(l0, s), using the same integration limitss for the Segregation model (Eq. (15)). The current size distribu-ion, fn(l), can be found by differentiating the cumulative numberraction, N(l), with respect to l, as shown in Eq. (22).

n(l) = d

dlN(l) = d

dl

∫ l0=l

l0=0

∫ s=((l3/l30

)−1)/G

s=0

�′(l0, s)dsdl0 (22)

Application of the Leibniz’s Rule results in the number-basedSD (Eq. (23)).

n(l) = 3l2

G

∫ l0=l

l0=0

�′ (l0, s = ((l3/l30) − 1)/G)

l30dl0 (23)

A comparison of Eqs. (17) and (23) reveals the differenceetween the two methods: while the PBM requires the solution of awo dimensional PDE, the Segregation model only requires integra-ion over known functions: the inlet conditions and reactor mixingharacteristics.

The Segregation model can also account for more complexinetic systems (such as deactivating catalysts) through a simpleodification of the growth function G. To include such proper-

ies in the PBM would require reformulation and the inclusion ofdditional dimensions to the PDE shown in Eq. (20), unnecessar-ly complicating the present efforts to compare the performance ofhe two models. It should be noted that incorporating this influenceould cause the PBM to become an even more inefficient simulatorue to the additional computational expense associated with thextra property dimension (activity). The present approach is there-ore a very conservative investigation into the effectiveness of theegregation Approach over the Population Balance.

It should also be noted that the PBM predicts the number-basedSD, and the Segregation model predicts the mass-based PSD. Aethod for conversion between the number- and mass-based size

istributions is detailed in Appendix A.

Fig. 4. Comparison of output of Population Balance and Segregation Models.

Having developed expressions for the PSD of the product of apolymerisation reactor, the Segregation and Population BalanceModels can be compared through simulation.

3. Results and discussion

3.1. Model equivalence

Before even attempting to compare computational efficiencies,it must first be established that the two methods predict the sameresults. To do so, a continuous stirred tank reactor will be simulated.The reactor has a mean residence time of 4000s; the feed is catalystparticles with a normal distribution of mean size 50 �m and vari-ance 8 �m. The growth function, G, has a value of 3.136 × 10−3 s−1.

The results of this simulation are shown in Fig. 4. The dottedline represents the feed PSD described above. The solid line andpoints represent product stream PSD predictions of the Segrega-tion model and PBM, respectively. Particles leaving the reactor areclearly larger than those entering the reactor, as expected. The dis-tribution has also widened considerably, and displays a slight skew,with a long tail; this is due to the distribution of residence timesused in the simulation.

It is clear that the product PSD predicted by the new Segrega-tion model is identical to that predicted by the more establishedPBM (R2 = 1). It can be shown that this equivalence holds for anycombination of residence time and feed PSD.

It must also be confirmed that the predicted PSDs reflect reality,at least qualitatively. Fig. 5 shows the influence of residence timeon the product PSD, as predicted by the models. It is clear that alonger residence time corresponds to an increase in average particlesize, as expected, and as observed when comparing Figs. 1 and 2. Aquantitative validation of the PSD model is beyond the scope of thiswork; however, it is clear that the models do qualitatively reflectthe physical situation of interest.

3.2. Model efficiency

Having shown equivalence of the results generated by the two

formulations, it must be determined which model yields the moreefficient simulation in terms of an eventual application in the con-trol environment. As such, the time taken to predict the PSD mustbe as low as possible.

74 J.T. McCoy, R. Rawatlal / Computers and Ch

tt

dPtbrro

foPirio

e

ilrTo8c

mra

lPat

vom

Fig. 5. Effect of mean residence time on PSD.

The review of the literature has already highlighted the researchhat has been conducted in order to find the most efficient solutiono the PDEs which result from application of the PBM.

The run-times for the PBM and Segregation models are depen-ent on the resolution required in each of three variables: the feedSD, which specifies the initial conditions, the age distribution, andhe product PSD, which is the output. A balance must be foundetween resolution and efficiency: at low resolution, the modelsun very quickly, but accuracy is lost, while at high resolution accu-acy improves, but the models can take significantly greater periodsf time to converge.

Run time was determined using Matlab’s built-in “tic . . . toc”unction. Average error was quantified by comparing the predictionf each model at each resolution with a reference size distribution,SDref, taken as the benchmark. One reasonable proposal for PSDrefs the average of the predictions of the two models at very highesolution. For a predicted distribution, PSD, at a resolution of anntermediate number of n points, the average error is found in termsf the reference distribution, as shown in Eq. (24).

rror =

√∑ni=1(PSD(i) − PSDref (i))2

n(24)

Accuracy was found to be practically insensitive to resolutionn the feed PSD and resolution in age distribution, and so the reso-utions for the functions f0(l0) and I(�) were set at 250 points. Theesolution in product PSD was varied between 100 and 2000 points.he benchmark, PSDref, was determined at product PSD resolutionf 4000 points, and feed PSD and age distribution resolutions of00 points. The run time and accuracy of each model were thenompared (see Fig. 6).

These results make clear the practical benefits of the Segregationodel over the Population Balance Model. Not only does the Seg-

egation model run an order of magnitude faster, it also has a lowerverage error when the models are run at the same resolution.

The order of magnitude decrease in run time becomes particu-arly significant when considering the ultimate application of theSD model. The Segregation model can give good accuracy even in

control environment, which requires per-second model solutionimes.

More importantly, the PSD is only one of several important stateariable distributions that are needed to fully describe a reactor’speration, which requires all three scales of detail (micro, meso andacro). Within this larger model, particularly in order to predict

emical Engineering 36 (2012) 68– 78

unsteady-state operation, the PSD sub-module may be evaluatedhundreds or thousands of times.

Industrially, polyolefins are commonly produced in multiplereactors in series. Although the results of the comparison of the PBMand Segregation Approach to multiple reactors will not be shownhere, a brief discussion will show that the Segregation Approachremains more efficient.

There are two methods that could be used to apply the PBMto multiple reactors. The first option is to include additional timedimensions in the formulation of the PBM partial differential equa-tions (Eq. (1)). The other option is to use the PBM in series multipletimes, once for each additional reactor, with different values for theparameters relating to hold up. Both of these options will severelyincrease the solution time of the PBM.

The Segregation Approach will require the use of a joint RTDfunction in order to simulate multiple reactors. This will impacton the calculations performed to set up the kernel for integra-tion, but will not have an impact on the integration steps. There isthus a minimal effect on the computational cost of the SegregationApproach.

4. Case study: varying model parameters

Having demonstrated the equivalence of the Population Balanceand Segregation Models, model sensitivity will now be investi-gated more rigorously. In order to optimise the code, the algorithmswhich offer the biggest delays must be identified. This is done byseparating the calculation steps for each model into three cate-gories:

• Numerical solution of the ordinary differential equations of Eq.(21) in the PBM (the Segregation Approach avoids the need forODE solution),

• Interpolation and integration steps for the solution of Eq. (17) and(23), and

• Sundry calculation steps, such as vector initialization and distri-bution basis conversion.

Performing ODE solution and interpolation and integrationsteps requires significantly more computing power than the sundrysteps; the former are measured in seconds, but the latter in mil-liseconds. For this reason, the investigation is limited to the impactof resolution on ODE solution and interpolation and integrationprocesses. The results of this investigation are shown in Figs. 7–9.

Fig. 7 shows the effect of product PSD resolution on the solutiontimes for the calculation steps. The clearest result from this plot isthat ODE solution is independent of product PSD resolution. Thisis to be expected, as the solution of the ODEs in the PBM (Eq. (21))involves only the feed distribution and age variables.

The second result from Fig. 7 is that the integration and interpo-lation steps for the solution of Eq. (23) (the PBM) are more sensitiveto product PSD resolution than the solution of Eq. (17) (the Segre-gation model). This is because the interpolation for the PBM is overa two-dimensional matrix (in initial size and age), but the Segre-gation model requires interpolation in only one dimension (initialsize).

Fig. 8 shows how feed PSD resolution affects solution times. Itis clear that the ODE solution is very sensitive to feed resolution.This is because the number of discrete ODEs in Eq. (21) is directlydetermined by the resolution in feed PSD.

The influence of feed resolution on interpolation and integra-

tion is slightly less than the impact of product resolution, as can beseen by comparing Figs. 7 and 8; however, PBM interpolation is stillmore sensitive than Segregation interpolation, as it is still in twodimensions.

J.T. McCoy, R. Rawatlal / Computers and Chemical Engineering 36 (2012) 68– 78 75

ror fo

ibM

tisloh

gains a large computational advantage over the traditional PBM.

Fig. 6. Comparison of run time and average er

The deviations from linearity of the results for integration andnterpolation in Fig. 8 (and Fig. 9) are completely reproducible, andelieved to be caused by interventions by switching functions inatlab’s built-in interpolation routines.Fig. 9 reveals the impact of resolution in age, which determines

he resolution in the ODEs in Eq. (21) for the PBM, and the resolutionn age distribution in Eq. (17) for the Segregation model. The ODEolution time is, as expected, sensitive to resolution in age. Interpo-ation in the PBM is still sensitive to age resolution, since this is one

f the dimensions in the matrix over which interpolation occurs;owever, interpolation in the Segregation model is independent

Fig. 7. Effect of product PSD resolution on model solution.

r Segregation and Population Balance Models.

of age resolution, since the interpolation is only in the initial sizedimension.

What is clear from all three figures is that the time required forthe solution of the ODEs in the PBM is always a significant contrib-utor to the total solution time for the model. Even if the differencesbetween the integration and interpolation steps for the two mod-els could be eliminated, the PBM would still be significantly lessefficient. By avoiding the solution of ODEs, the Segregation model

It should also be noted that as more complex systems are simu-lated, the benefits of the Segregation Approach over the PopulationBalance Model increase. Two typical additions to these models

Fig. 8. Effect of feed PSD resolution on model solution.

76 J.T. McCoy, R. Rawatlal / Computers and Ch

wta

ags(a

onr

gctteic

sc

5

tpgdP

atmstaa

i

∫ ∞ = ∫ ∞ (28)

Fig. 9. Effect of age distribution resolution on model solution.

ould be to simulate a catalyst which displays activity decay ando simulate particles which undergo fragmentation and agglomer-tion.

Terms relating to particle breakup and coalescence and catalystctivity profiles can be included by extending the single particlerowth model. This would lead to a modified version of the expres-ion describing particle size as a function of initial size and ageEq. (10)). Modification of the particle growth function to includectivity decay is developed in Appendix A.

This extended particle size function would be evaluated onlynce for the Segregation model (in order to generate the kernel forumerical integration), having a minor effect on the computationalequirements for this approach.

The impact on the computational cost of the PBM will be muchreater, however. At best, the formulation of the PBM will nothange significantly; computational cost will still increase, becausehis more complex function must be evaluated at each step ofhe numerical solution of the differential equations. At worst, thextension of the particle size function will lead to non-linear effectsn the PBM equations, making their solution significantly moreomputationally demanding.

Since this work attempts to present a conservative compari-on between the two approaches, such complexities have not beenonsidered.

. Conclusion

The Segregation Approach has been extended to the predic-ion of the particle size distribution of the population of growingolymer particles in an olefin polymerisation reactor. The Segre-ation model avoids the need to solve multidimensional partialifferential equations, the major drawback to the currently-usedopulation Balance Modelling approach.

The Segregation model and PBM were applied to the same situ-tion of a perfectly mixed reactor with a size-distributed feed andime–constant reactor conditions and catalyst properties. The two

odels were compared through simulation, and it was demon-trated that the Segregation model predicts the same results ashe PBM. It was also shown, through comparison of run time and

ccuracy, that the Segregation model solves more quickly and moreccurately at the same distribution resolutions.

Through further analysis of the solution steps in each model,t was shown that the PBM is less efficient than the Segregation

emical Engineering 36 (2012) 68– 78

model not only because of the required solution of a set of ODEs(avoided by the Segregation model), but also because interpolationsteps are performed over two-dimensional matrices in the PBM,and only one-dimensional matrices for the Segregation model.

The Segregation Approach has been demonstrated to be consid-erably more computationally efficient than the Population BalanceApproach due to differences in formulation. It has also been men-tioned that the Segregation Approach is significantly more flexibleand easy to implement, again due to differences in formulation. Thisapplies particularly to the inclusion of complex reactor mixing pat-terns, unsteady flow conditions and catalyst kinetic models in thePSD framework, and the application of the framework to unsteadystate modes of operation.

It is therefore proposed that this work is a significant steptoward the final overall goals of model-based control and plant-scale optimisation.

Acknowledgements

The authors wish to thank Prof JBP Soares and Dr SYA Shin atthe University of Waterloo, Ontario, Canada, for supplying the SEMimages.

The authors wish to thank Safripol Pty (Ltd) and the University ofCape Town’s Postgraduate Funding Office for their financial support.The financial assistance of the National Research Foundation (NRF)towards this research is hereby acknowledged. Opinions expressedand conclusions arrived at are those of the author and are not nec-essarily to be attributed to the NRF.

Appendix A. Appendix

A.1. Distribution types

Since the PBM predicts the number-based PSD, and the Segrega-tion model predicts the mass-based PSD, a method to convert thebasis of the distribution is required.

The starting point is to define the total mass and number flows inthe stream of interest, Min and Nin, such that Minfm(l)�l is the massof particles in the size range [l, l + �l]; and Ninfn(l)�l is the numberof particles in the size range [l, l + �l]. The mass- and number-based size distribution functions, fm(l) and fn(l), have the followingproperty:∫ ∞

0

fm(l)dl =∫ ∞

0

fn(l)dl = 1 (25)

Using the relation between particle mass and size (Eq. (9)) themass of particles in size range [l, l + �l], using either of the distri-butions, can be expressed with Eq. (26):

Minfm(l)�l = �

6l3�εNinfn(l)�l (26)

The mass of particles up to size l can then be found by integratingthe expression above, to give Eq. (27).∫ l

0

Minfm(l)ıl =∫ l

0

�

6l3�εNinfn(l)dl (27)

The mass fraction of particles up to size l can be found by dividingthe mass of particles up to size l by the total mass of particles of allsizes. This gives Eq. (28).∫ l

0fm(l)dl

∫ l

0l3fn(l)dl

0fm(l)dl

0l3fn(l)dl

Differentiating with respect to l allows the conversion of a cumu-lative distribution into a size distribution function, and gives the

nd Ch

cs

f

m(

f

A

ao

gsb

a

i

l

w

f

Si

Pinrtm

fStm

R

A

A

A

A

A

J.T. McCoy, R. Rawatlal / Computers a

onversion from number-based size distribution to mass-basedize distribution, Eq. (29).

m(l) = l3fn(l)∫ ∞0

l3fn(l)dl(29)

Similar reasoning can be applied to find the conversion fromass-based size distribution to number-based size distribution, Eq.

30).

n(l) = (fm(l))/l3∫ ∞0

((fm(l))/l3)dl(30)

.2. Modified particle growth function

The modification of the particle growth function (Eq. (10)) toccount for catalyst activity profiles is shown below, and the impactf this modification on the efficiency of the models discussed.

If activation and deactivation are assumed to be first order andoverned by rate constants ka and kd respectively, then it can behown that the fraction of active sites will be given by the equationelow:

(t) = ka

ka − kd[exp(−kdt) − exp(−kat)] (31)

Substituting Eq. (31) into Eq. (10) will result in the form shownn Eq. (32).

(l0, t) = l03√

1 + Gf (ka, kd, t) (32)

here

(ka, kd, t)= ka

ka − kd

[1ka

exp(−kat)− 1kd

exp(−kdt) −(

1ka

− 1kd

)](33)

This re-formulation will not affect the solution time for theegregation model, since the kernel is calculated once, before thentegration is performed.

However, the inclusion of activation and deactivation in theBM formulation will complicate the form of the solution with thenclusion of the function f(ka, kd, t). Although the formulation willot change, and the method of characteristics can still be applied,educing the PDEs to a set of ODEs, the evaluation of the integra-ion limits and interpolation steps in the model solution will require

ore computation time.Thus, including deactivation will negatively affect the per-

ormance of the PBM, without affecting the performance of theegregation Approach; this counteracts our attempts to comparehe best-case performance of the PBM to the worst case perfor-

ance of the Segregation Approach.

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