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    Chemical Engineering Science 58 (2003) 33373351

    www.elsevier.com/locate/ces

    Implementation of the quadrature method of moments in CFDcodes for aggregationbreakage problems

    Daniele L. Marchisio, R. Dennis Vigil, Rodney O. Fox

    Department of Chemical Engineering, Iowa State University, 2114 Sweeney Hall, Ames, IA 50011-2230, USA

    Received 10 January 2003; received in revised form 7 April 2003; accepted 11 April 2003

    Abstract

    In this work the quadrature method of moments (QMOM) is implemented in a commercial computational uid dynamics (CFD)code (FLUENT) for modeling simultaneous aggregation and breakage. Turbulent and Brownian aggregation kernels are considered in

    combination with dierent breakage kernels (power law and exponential) and various daughter distribution functions (symmetric, erosion,

    uniform). CFD predictions are compared with experimental data taken from other work in the literature and conclusions about CPU time

    required for the simulations and the advantages of this approach are drawn.

    ? 2003 Elsevier Ltd. All rights reserved.

    Keywords: Population balance; Computational uid dynamics; Quadrature method of moments; Aggregation; Breakage; Particulate systems

    1. Introduction

    Aggregation and breakage play an important role in anumber of important chemical processes such as precipita-

    tion, crystallization, separation processes, and reaction in

    multiphase systems. Modeling and simulation of these pro-

    cesses is complicated due to the diculties inherent in de-

    scribing the evolution of a distribution of particle sizes and

    because of the incomplete understanding of the mechanisms

    by which aggregation and breakage occur, including the role

    of hydrodynamics. This latter problem is often neglected,

    despite considerable evidence that aggregation is strongly

    inuenced by mixing. For example, Brown and Glatz

    (1987) investigated the eect of the operating conditions on

    particle size established during breakage of protein particlesprepared under isoelectric precipitation in an agitated ves-

    sel. They found that the aggregation rate increases with both

    particle concentration and shear rate. In another study, the

    inuence of the type of ow on the aggregation rate was in-

    vestigated using a TaylorCouette reactor, a pipe-ow reac-

    tor, and a at-bottomed tank reactor (Krutzer, van Diemen,

    & Stein, 1995). These experiments showed that at equal

    Corresponding author. Tel.: 515-294-3186; fax: 515-294-2689.E-mail address: [email protected] (D. L. Marchisio).

    energy dissipation rates, the aggregation rate is higher for

    isotropic turbulent ows than for non-isotropic ows.

    Raphael and Rohani (1996) investigated the eect of ag-gregation on the particle size distribution (PSD) during sun-

    ower protein precipitation and found that the maximum size

    of the aggregates is determined by the hydrodynamics of the

    reactor and the mean residence time of the particles in the re-

    actor. Serra, Colomer, and Casamitjana (1997) investigated

    aggregation and breakup of particles in a TaylorCouette

    reactor. In their experimental work they analyzed the eect

    of particle concentration, shear rate, and particle initial di-

    ameter. Their results showed the existence of three regions

    determined by particle concentration and type of ow es-

    tablished (laminar or turbulent). Moreover, they found that

    the nal aggregate diameter in the turbulent regime is inde-pendent of monomer size and is instead controlled by the

    Kolmogorov micro-scale, whereas in laminar ow the nal

    mean particle size decreases as the diameter of the primary

    particles is increased.

    One means for characterizing the morphology of clus-

    ters of particles formed by aggregationbreakage processes

    is the fractal dimension, which provides an indication of

    the compactness of aggregates. The relationship between

    fractal dimension and physical properties has been studied

    from both the experimental (Serra & Casamitjana, 1998a),

    and theoretical viewpoints (Filippov, Zurita, & Rosner,

    2000; Jlang & Logan, 1991). Hansen and co-workers used

    0009-2509/03/$ - see front matter? 2003 Elsevier Ltd. All rights reserved.

    doi:10.1016/S0009-2509(03)00211-2

    mailto:[email protected]:[email protected]
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    3338 D. L. Marchisio et al./ Chemical Engineering Science 58 (2003) 3337 3351

    a TaylorCouette reactor to study orthokinetic aggregation

    for monodisperse and bidisperse colloidal systems (Hansen,

    Malmsten, Bergenstahl, & Bergstrom, 1999). Particle ag-

    gregation was investigated by direct observation using a

    CCD camera, and the observed aggregates were character-

    ized by a high fractal dimension, suggesting that clusters

    are rearranged and densied by the shear.Aggregation and breakage are often the last steps of a

    complex sequence of phenomena, such as nucleation, fast

    reactions, combustion, and molecular growth (Marchisio,

    Barresi, & Garbero, 2002; Rosner & Pyykonen, 2002;

    Baldyga & Orciuch, 2001). Such systems inevitably lead

    to non-negligible spatial heterogeneities, and therefore a

    method of modeling these processes that accounts for hy-

    drodynamics is crucial for predicting reactor performance.

    One approach to account for non-ideal mixing is to use

    CFD methods. In such an approach the reactor is represented

    by a computational grid and the continuity and Navier

    Stokes equations are solved over the computational domain.

    When dealing with turbulent ows, the set of equations is

    unclosed and turbulence models are used to solve the clo-

    sure problem. In addition to these equations, the population

    balance equation for the solid phase has to be solved.

    The population balance is a continuity statement writ-

    ten in terms of the PSD and has consistently received

    attention since Smoluchowski (1917) introduced the math-

    ematical formalism nearly a century ago. A compre-

    hensive overview of the mathematical issues involved,

    the numerical methods available, and possible develop-

    ments for the future have been given by Ramkrishna

    (1985, 2000) and Ramkrishna and Mahoney (2002).

    A general form of the mean-eld population balance for aspatially extended system can be written as follows (repeated

    indices implies summation):

    @n(; x; t)

    @t+ ui @n(; x; t)

    @xi

    @@xi

    (t + )

    @n(; x; t)

    @xi

    = @@j

    [n(; t)j] + h(; t); (1)

    where (1; : : : ; n) is the property vector that speciesthe state of the particle, n(; x; t) is the number density func-

    tion, ui is the Reynolds-average velocity in the ith direc-tion, xi is the spatial coordinate in the ith direction, is the

    molecular diusivity and t is the turbulent diusivity. For

    turbulent ows t and thus it is commonly assumedthat t + t. The ux in -space is denoted by

    j djdt

    ; j 1; : : : ; N ; (2)

    and h(; t) represents the net rate of introduction of new

    particles into the system (Hulburt & Katz, 1964).

    The main problem in solving the above equation is the

    presence of the extra variables i, which dene the particle

    size, shape, etc. In most CFD codes it is possible to intro-

    duce user-dened scalars by using user-dened subroutines,

    but these scalars must only be functions of time and space.

    Hence, in order to reduce the dimension of the problem,

    several methods have been developed.The discretized population balance (DPB) approach is

    based on the discretization of the internal coordinates (i.e.,

    the components of the property vector). A detailed compar-

    ison of the performance of the most popular DPB methods

    has been carried out by Vanni (2000a). The principle ad-

    vantage of the DPB method is that the PSD is calculated di-

    rectly. However, in order to maintain reasonable accuracy,

    a large number of scalars (i.e., particle classes) are required.

    As a consequence, the DPB approach is computationally in-

    tractable for spatially heterogeneous systems and therefore

    not suitable for CFD applications.

    An alternative to PBE approaches is to implement a

    stochastic analog via a Monte Carlo algorithm (Smith &

    Matsoukas, 1998; Lee & Matsoukas, 2000; Rosner & Yu,

    2001). These methods have the advantage of satisfying

    mass conservation as well as correctly accounting for uc-

    tuations that arise as the system mass accumulates in a small

    number of large aggregates. However, incorporation of

    these methods into CFD codes is also not computationally

    tractable because of the large number of scalars required.

    In contrast to the DPB or stochastic approaches, the mo-

    ment method (MM) is suitable for use with CFD codes be-

    cause the internal coordinates are integrated out such that

    solution only requires a small number of scalars (i.e., 4

    6 moments of the PSD) at each grid point. Of course thevast reduction in the number of scalars, which makes im-

    plementation in CFD codes feasible, comes at the cost of a

    less-detailed description of the PSD. However, provided that

    the PSD function is suciently simple (e.g., monomodal or

    bimodal), a low-order moment description may be sucient.

    The method was rst proposed many years ago by Hulburt

    and Katz (1964), but it has not found wide applicability due

    to the diculty of expressing the transport equations for the

    moments of the PSD in terms of the moments themselves.

    More recently, several approaches for contending with this

    closure problem have been developed, and a discussion

    of these can be found in Diemer and Olson (2002).When the population balance is written in terms of one

    internal coordinate (e.g., particle length or particle volume)

    the closure problem has been successfully solved with the

    use of a quadrature approximation (McGraw, 1997), where

    weights and abscissas of the quadrature approximation can

    be found by using the product-dierence (PD) algorithm de-

    scribed by Gordon (1968). The so-called quadrature method

    of moments (QMOM) has been validated for several prob-

    lems (e.g., molecular growth, aggregation, breakage) and

    by using dierent internal coordinates (Marchisio, Pikturna,

    Fox, Vigil, & Barresi, 2003a; Barret & Webb, 1998). More-

    over, the method has been extended to the study of aerosol

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    D. L. Marchisio et al./ Chemical Engineering Science 58 (2003) 3337 3351 3339

    dynamics by using population balances with two internal

    coordinates: particle volume and surface area (Wright, Mc-

    Graw, & Rosner, 2002).

    The main purpose of this work is to implement the

    QMOM approach to solving the population balance equa-

    tion in a commercial CFD code and to verify the feasibility

    of these calculations for practical applications. The spe-cic problem considered is aggregation and breakage in

    turbulent TaylorCouette ow.

    The CFD-based predictions for the particle size distribu-

    tions are generated by using several dierent combinations

    of conventional expressions for the aggregation and breakup

    kernels and are compared with experimental data from Serra

    et al. (1997) and Serra and Casamitjana (1998a, b). The

    paper is organized as follows: governing equations are dis-

    cussed, aggregation and breakage models are presented, and

    the results of the case study considered, particularly with re-

    spect to the number of scalars involved in the calculations,

    the quality of the predictions, and the CPU time in compar-

    ison with alternative approaches.

    2. Governing equations

    In this section the QMOM is presented and explained.

    Particular attention will be devoted to its implementation in

    FLUENT.

    2.1. Quadrature method of moments

    In this work, we consider a number density function de-ned in terms of particle length (1 L). The resultingpopulation balance is (Randolph & Larson, 1988)

    @n(L; x; t)

    @t+ ui @n(L; x; t)

    @xi @

    @xi

    t

    @n(L; x; t)

    @xi

    = @@L

    [G(L)n(L; x; t)] + B(L; x; t) D(L; x; t); (3)

    where G(L) is the growth rate, and B(L; x; t) and D(L; x; t)

    are, respectively, the birth and death rates due to aggrega-

    tion and breakage. The moments of the PSD are dened as

    follows:

    mk(x; t) =

    0

    n(L; x; t)Lk dL; (4)

    and thus the transport equation for the kth moment is

    @mk(x; t)

    @t+ ui @mk(x; t)

    @xi @

    @xi

    t

    @mk(x; t)

    @xi

    =(0)kJ(x; t) +

    0

    kLk1G(L)n(L; x; t) dL

    +Bak(x; t)

    Dak(x; t) + B

    bk(x; t)

    Dbk(x; t); (5)

    where J(x; t) is the nucleation rate and where

    Bak(x; t) =1

    2

    +0

    n(; x; t)

    +0

    (u; )

    (u3 + 3)k=3n(u; x; t) du d; (6)

    Dak(x; t) =

    +

    0

    Lkn(L; x; t)

    +

    0

    (L; )

    n(; x; t) d dL; (7)

    Bbk(x; t) =

    +0

    Lk+

    0

    a()b(L|)n(; x; t) d dL; (8)

    Dbk(x; t) =

    +0

    Lka(L)n(L; x; t) dL (9)

    are the moments of the birth and death rates, respectively,

    due to aggregation and breakage. The aggregation kernel

    (L; ) represents the rate coecient for aggregation of two

    particles with lengths L and whereas the breakage kernel

    a(L) denes the rate coecient for breakage of a particle of

    length L. The fragment distribution function for the break-

    age of a particle of size L is given by b(L|). A detailedderivation of these equations can be found elsewhere (see

    Marchisio, Vigil, & Fox, 2003b); discussion of the choice

    of aggregation and breakage kernels can be found in

    the following sections.

    The QMOM is based on the following quadrature approx-

    imation

    mk =+

    0n(L)Lk dL

    Ndi=1

    wiLki ; (10)

    where weights (wi) and abscissas (Li) are determined

    through the product-dierence (PD) algorithm from the

    low-order moments (Gordon, 1968); a detailed explanation

    of the method can be found in Appendix A. By using the

    PD algorithm, a quadrature approximation with Nd weights

    and Nd abscissas can be constructed using the rst 2Ndmoments of the PSD. For example, if Nd = 3 only the rst

    six moments (m0; : : : ; m5) are tracked, and the quadrature

    approximation is given by

    mk =

    Nd

    =3i=1

    wiLki = w1L

    k1 + w2L

    k2 + w3L

    k3: (11)

    Knowledge ofwi and Li suces to close the transport equa-

    tions for the moments, and in the specic case of aggrega-

    tion and breakage without nucleation and molecular growth,

    the moment equations become

    @mk(x; t)

    @t+ ui @mk(x; t)

    @xi @

    @xi

    t

    @mk(x; t)

    @xi

    =1

    2

    i

    wi

    j

    wj(L3i + L

    3j )

    k=3ij

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    3340 D. L. Marchisio et al./ Chemical Engineering Science 58 (2003) 3337 3351

    i

    Lki wi

    j

    ijwj

    +

    i

    ai b(k)i wi

    i

    Lki aiwi; (12)

    where ij = (Li ; Lj), ai = a(Li), and

    b(k)i =

    +0

    Lkb(L|Li) dL: (13)

    It has been shown (Marchisio et al., 2003a, b; McGraw,

    1997; Barret & Webb, 1998) that by using this quadrature

    approximation it is possible to track the moments of the PSD

    with very high accuracy.

    2.2. Aggregation kernel

    Aggregation of solids suspended in a uid is a complex

    phenomena that in general depends upon particleparticleand particleuid interactions, particle morphology, as well

    as uid mixing. However, regardless of the specic mech-

    anism by which aggregation occurs, the process can be di-

    vided up into two steps: (1) particleparticle collision and

    (2) sticking or fusion of colliding particles. The aggre-

    gation kernel can then be expressed as the product of the

    sticking coecient and a collision frequency function.

    In certain simplied cases, the aggregation kernel can

    be computed exactly, provided that certain assumptions are

    satised. For example, if long-range particleparticle forces

    are absent, if particles are Euclidean (as opposed to fractal),

    and if the particles are suciently small so that they neitherinuence the uid phase nor are disturbed by uid shear, then

    the collision frequency may be described by the Brownian

    kernel:

    (L; ) =2kBT

    3

    (L + )2

    L; (14)

    where kB is Boltzmanns constant, T is the absolute temper-

    ature, is the uid viscosity, and L and are particle sizes,

    as rst proposed by Smoluchowski (1917).

    When particles are suciently large compared to shear

    gradients, particleparticle collision frequencies are inu-

    enced by uid motion. In the case of turbulent ow, the ra-tio between particle size and the Kolmogorov micro-scale is

    of primary importance. When particles are smaller than the

    Kolmogorov micro-scale the aggregation kernel can be com-

    puted as follows (Adachi, Cohen Stuart, & Fokkink, 1994):

    (L; ) =4

    3

    3

    10

    1=2

    1=2(L + )3; (15)

    where is the turbulent dissipation rate and is the kinematic

    viscosity. In fact, by using Taylors statistical analysis of

    isotropic turbulence and assuming a normal distribution of

    the velocity gradient, it can be shown that the shear rate

    produced by uctuating turbulent velocities is proportional

    to

    =. Saman and Turner (1956) used this result to ex-

    plain the formation of drops in clouds, and Tontrup, Gruy,

    and Cournil (2000) used it for modeling turbulent aggrega-

    tion of titania particles in water.

    For chemically inert systems, aggregation eciency

    is primarily determined by the balance between attrac-tive (e.g. Van der Waals) and repulsive (e.g. electrical

    double layer) forces. This balance is strongly inuenced by

    the ionic strength of the solution. When the ionic strength

    increases, the double layer repulsion is reduced, and at very

    high values it is practically negligible. In these conditions

    the aggregation eciency is usually considered to be equal

    to one. Aggregation eciency can also be related to the

    probability of fragmentation of newly formed aggregates.

    In fact, the fragmentation kernels presented in the next

    section are suitable for breakage of particles that have had

    sucient time to restructure after collision (Gruy, 2001). If

    a new aggregate does not have sucient time for restructur-

    ing and is fragmented into its original two components, it

    can be referred to as null aggregation eciency (Brakalov,

    1987). Another important factor in determining the aggre-

    gation eciency in precipitating particle system is the rate

    of formation of bridges in zones of high supersaturation

    (Baldyga, Jasinska, & Orciuch, 2002; Hounslow, Mumtaz,

    Collier, Barrick, & Bramley, 2001). In the present work,

    we consider only aggregation and breakage of particles in

    chemically inert systems.

    2.3. Breakage

    Particle breakage functions can be factored into two parts.

    The breakage kernel, a(L), is the rate coecient for breakage

    of a particle of size L, and b(|L) denes the probabilitythat a fragment of size is formed from the breakage of an

    L-sized particle. Several expressions for the breakage kernel

    relevant to hydrodynamic breakage mechanisms have been

    derived by dierent authors and they are summarized in

    Table 1.

    Breakage kernel (1) in Table 1 was derived for disruption

    of disperse liquid droplets in turbulent ow under the hy-

    pothesis of local isotropy and assuming that the droplet size

    is within the inertial subrange. A breakage event occurs if the

    kinetic energy transmitted by an eddy to a droplet is greaterthan the droplet surface energy. The breakup frequency is

    derived by considering the fraction of eddies present in the

    system with sucient kinetic energy to cause breakage. This

    can be done by estimating the fraction of drops breaking and

    by estimating the characteristic time required for breakage

    (tb) through the second-order structure function of the uid

    velocity DLL(r). This function represents the covariance of

    the dierence in velocity dierences between two points at

    a given distance r and according to turbulent theory (Frisch,

    1995) it can be written as

    DLL(r) = c2(r)2=3; (16)

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    D. L. Marchisio et al./ Chemical Engineering Science 58 (2003) 3337 3351 3341

    Table 1

    Breakage kernels proposed in literature

    System a(L) Reference

    (1) Liquidliquid a(L) = k1L2=31=3 exp

    k2

    d2=3L5=3

    Coulaloglou and Tavlarides (1977)

    (2) Liquidliquid a(L) = f erfc

    3:5(L

    Ls )5=6

    Narsimhan, Gupta, and Ramkrishna (1979)

    (3) Liquidliquid a(L) = c4

    L2

    1=3 1min

    (1+)2

    11=3exp

    12cf

    d 2=3L5=311=3

    d Luo and Svendsen (1996)

    (4) Solidliquid a(L) = 115

    (

    )1=2 exp

    f

    (=)1=2

    Ayazi Shamlou, Stravrinides, Titchener-Hooker, and Hoare (1994)

    (5) Solidliquid a(L) = c1L Luo and Svendsen (1996)

    Kramer and Clark (1999)Wojcik and Jones (1998)

    where c2 is a numerical constant and is the turbulent dis-

    sipation rate. Assuming that the motion of the daughterdroplets is similar to the relative motion of two lumps of

    uid in a turbulent ow and that the distance r is the droplet

    diameter, we can write thatL

    tb

    2 DLL(L) = c2(L)2=3; (17)

    which can be solved for tb. The nal expression for a(L) is

    a function of the surface tension and the droplet density

    d and is reported in Table 1 [see breakage kernel (1)].

    Breakage kernel (2) in Table 1 was derived by consid-

    ering the frequency that eddies arrive at the surface of a

    disperse phase droplet, whose maximum stable diameter isindicated by Ls. Under the same phenomenological simpli-

    cations of the above mentioned models, and considering the

    dierent contributions of the population of eddies in the in-

    ertial subrange, it is possible to derive an expression for the

    breakage rate without unknown constants [see kernel (3) in

    Table 1], where = =L, is the eddy length-scale, c4 is a

    constant of order unity (c4 =0 :92), =3=2, cf is the increase

    coecient in the surface area [cf = f2=3 + (1 f)2=3 1],

    and f is the volume fraction of one of the two fragments

    (the function is symmetric about f = 0:5).

    As with breakage of liquid droplets, breakage of solid ag-

    gregates depends strongly on the ratio between the particle

    size and the smallest turbulent eddy. When the size of a par-ticle is greater than the size of the smallest eddies, break-

    age is likely to occur by instantaneous normal stresses due

    to pressure uctuations acting on the surface of the parti-

    cle. For particle sizes smaller than the turbulent micro-scale,

    breakage is likely to be caused by shear stresses originat-

    ing from the turbulent dynamic velocity dierences acting

    on the opposite sides of the particle. Under these conditions

    the breakage kernel can be written as a function of the ag-

    gregate strength (f), which can be estimated as follows:

    f =9

    8kcF

    1

    L2o; (18)

    where Lo is the diameter of a primary particle, kc is the co-

    ordination number, and F is the inter-particle force betweentwo primary particles (Ayazi Shamlou et al., 1994). The re-

    sulting breakage kernel is reported in Table 1 [see kernel

    (4)]. The inter-particle force F can be computed as follows:

    F =ALo

    12H2o; (19)

    where A is the Hamaker constant for the liquid-particle sys-

    tem and Ho is the distance between two primary particles.

    The coordination number is based on experimental observa-

    tion and can be calculated as

    kc

    151:2; (20)

    where is the volume fraction of solid within the aggregates.

    This quantity can be determined once the fractal dimension

    df of the aggregates in known

    (L) = C

    L

    Lo

    df3; (21)

    where C = 0:414df 0:211 (Vanni, 2000b).Starting from the same basic considerations, but using a

    simplied approach, Kramer and Clark (1999) developed

    another hydrodynamic breakage model for solid aggregates.

    They classied failure modes by dening manifestation, in-

    duction and location of the failure, and stated that aggregatesbreak when the maximum eigenvalue of the stress tensor is

    greater than the aggregate strength (f). Using the assump-

    tion that the number of aggregate bonds with a strength at or

    below the failure strength f is not a linear function of the

    strain rate and that size-dependency should also be consid-

    ered in a breakup model, the resulting form of the breakage

    kernel is the one given as expression 5 in Table 1.

    By invoking semi-theoretical considerations and tting to

    experimental data, several values of the exponent have

    been proposed = 0; 1=3; 2=3; 1 (Luo & Svendsen, 1996).

    Pandya and Spielman (1982) found that =1, whereas Peng

    and Williams (1994), by tting the model to experimental

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    3342 D. L. Marchisio et al./ Chemical Engineering Science 58 (2003) 3337 3351

    Table 2

    Fragment distribution functions: Lo is the size of the primary particles

    Mechanism b(L|) b(k)i

    Symmetric

    fragmentation

    2 if L = 21=3

    0 otherwise2(3k)=3Lki

    Erosion

    1 if L = Lo

    1 if L = (3 L3o)1=3

    0 otherwise

    Lko + (L3i L

    3o)

    k=3

    Uniform

    6L2

    3if 0 L

    0 otherwiseLki

    6k+3

    data, found that can assume values ranging between 1 and

    3 and that = 1=2.

    Serra and Casamitjana (1998b) t experimental data andfound that the relationship between the breakage kernel and

    the turbulent dissipation rate depends on the solid frac-

    tion. When the solid fraction is low = 0:9, whereas for

    higher values of the solid fraction is much higher. This

    can be explained by the fact that particleparticle collisions

    are more eective if the solid fraction is higher. Their results

    conrm ndings by Spicer and Pratsinis (1996), who used

    data from Oles (1991) and found = 0:8 for low values of

    solid concentration.

    In their investigation of nucleation, growth, agglomera-

    tion and disruption kinetics for calcium oxalate, Zauner and

    Jones (2000) found a linear dependency between the break-age rate and the turbulent dissipation rate (i.e., = 1) and

    the breakage kernel was size-independent (=0). In another

    work (Wojcik & Jones, 1998) using a size-dependent break-

    age kernel, a linear relationship between the kernel and the

    turbulent dissipation rate ( = 1) was also found.

    The form of the fragment distribution function, b(L|),depends upon many factors including the particle properties,

    such as strength and morphology, as well as the breakup

    mechanism. Because the fragment distribution function is

    likely to be highly system-specic, we consider a variety of

    possibilities suggested by experiments on disruption kinetics

    of animal cell aggregates undergoing hydrodynamic break-

    age (Moreira, Cruz, Santana, Aunins, & Carrondo, 1995).

    Some of these distribution functions include erosion of pri-

    mary particles, formation of two equal fragments, formation

    of two fragments with xed mass ratio (e.g., one to four),

    and uniform distribution of fragments. The distribution func-

    tions corresponding to these cases are reported in Table 2.

    3. Computational details and case study

    The experimental data used in this study are taken from

    Serra et al. (1997) and Serra and Casamitjana (1998a, b).

    The reactor used in their work is a TaylorCouette device,

    which consists of a uid conned to the annular region be-

    tween two concentric cylinders (outer cylinder stationary,

    inner cylinder rotating). The wetted diameters of the inner

    and outer cylinders are d1 = 193 mm and d2 = 160 mm, re-

    spectively, thereby giving an annular gap D = (d1d2)=2 =16:5 mm. The reactor length is H=360 mm, resulting in anaspect ratio A = H=D 22. Latex particles were used inthe experiments, and in order to avoid sedimentation eects,

    the density of the solution was set at the same value as that

    of the latex particles (s = 1:055 g=cm3) by adding an inor-

    ganic salt to the solution (KCl). Samples were continuously

    taken from the bottom of the reactor and passed through the

    optical cell of a particle analyzer, and then pumped back

    to the top of the reactor. The ux in the tubes was laminar

    so that disruption of aggregates formed in the reactor could

    be minimized. The residence time of the aggregates in the

    tubes was also kept short in comparison with the time spent

    in the reactor (about 300 times shorter).

    Although aggregationbreakage experiments were car-

    ried out with inner cylinder rotational speeds in the range

    40200 rpm, we compare simulation results only with data

    in the turbulent ow regime (N=80200 rpm). An overview

    of the various hydrodynamic regimes in the reactor can be

    found in the review by Kataoka (1986), whereas validation

    of ow eld predictions in turbulent regimes has been car-

    ried out in our previous work (Marchisio, 2002).

    In the absence of an applied axial ow, the hydrodynam-

    ics in a TaylorCouette reactor depend upon the azimuthal

    Reynolds number

    Re =r1!1D

    ; (22)

    where r1 = d1=2 is the radius of the inner cylinder, !1 is

    the angular velocity of the inner cylinder, D is the annular

    gap, and is the kinematic viscosity of the uid. When the

    rotational speed of the inner cylinder is very small, circular

    Couette ow is established. If the rotational speed of the in-

    ner cylinder is increased past a critical value, this circular

    ow becomes unstable and laminar toroidal vortices (Tay-

    lor vortices) are formed. The critical azimuthal Reynolds

    number for Taylor instability Rec, also depends upon the

    specic reactor geometry and can be calculated by using

    well known correlations (Kataoka, 1986). Other instabilitiesoccur at higher values of Re, and the azimuthal Reynolds

    number ratio Re=Rec, can be used to determine which hy-

    drodynamic resides in the reactor.

    In this work, the ow eld in the reactor was modeled

    by using the commercial CFD software FLUENT 6.0. Since

    the solid particles are smaller than 2030 m, and the solid

    concentration is smaller than 0.1%, the inuence of the solid

    phase on the ow eld can be neglected and single-phase

    turbulence models can be used. Thus, the Reynolds Stress

    Model with standard wall function was used to model the

    ow eld. In our previous work (Marchisio, 2002) this com-

    bination of turbulence model and near-wall treatment was

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    D. L. Marchisio et al./ Chemical Engineering Science 58 (2003) 3337 3351 3343

    Fig. 1. Sketch of the TaylorCouette reactor and of the computational

    domain used in the simulation (grey section).

    found to give the best agreement with experimental data.

    Simulations were carried out in two dimensions under the

    hypothesis of axial-symmetry. Moreover, since the ow was

    found to be symmetrical about the center of the reactor, only

    one half of this section has been modeled.

    A sketch of the reactor and of the computational do-

    main used in the simulations is shown in Fig. 1. Several

    grids were tested in order to verify that the solution was

    grid-independent, and it was found that a grid with 18 nodes

    in the radial direction and 181 in the axial direction for a to-tal of 3060 computational cells was sucient. In our previ-

    ous work (Marchisio, 2002) a comparison between 2D and

    3D simulations was also carried out. It was found that with-

    out axial ow and in a range of operating conditions similar

    to this work no appreciable dierence was detected between

    2D and 3D predictions.

    The QMOM was implemented with three nodes, and as a

    consequence the rst six moments were tracked. This choice

    is justied by previous ndings (Marchisio et al., 2003b)

    where Nd = 3 was found to be the best conguration in

    terms of accuracy and computational costs. The transport

    equations for the rst six moments were implemented by

    using compiled user-dened functions, and details can be

    found in Fluent Inc. (2002).

    The quadrature approximation can be determined by

    nding the roots of the polynomial of order Nd belonging

    to the sequence of orthogonal polynomials with respect

    to the measure n(L) (see for details Dette & Studden,

    1997). Since root nding of a polynomial is a notori-ously ill-conditioned problem, weights and abscissas of the

    quadrature approximation can be calculated by using the

    PD algorithm. Following this procedure a tridiagonal matrix

    of rank Nd is calculated from the rst 2Nd moments, and

    the eigenvalues and the square of the rst component of

    the eigenvectors of this matrix are respectively the abscis-

    sas (Li) and weights (wi) of the quadrature approximation.

    The eigen-value/eigen-vector problem was implemented in

    FLUENT via user-dened-subroutine by using the C sub-

    routine TQLI (Press, Teukolsky, Vetterling, & Flannery,

    1992).

    The computational procedure was as follows. First, the

    ow eld was solved until a steady-state solution was

    reached. The convergence criteria required that all normal-

    ized residuals be smaller than 106. Next, the moments were

    determined (m0; : : : ; m5). Their initial values were dened to

    be uniform throughout the computational domain and then

    a time-dependent simulation was initiated with a xed

    time step of 10 s. The convergence criteria for all scalars

    required that the normalized residuals be smaller than 106.

    Several cases were investigated using dierent combina-

    tions of breakage kernels and fragment distribution func-

    tions. In all the cases investigated, aggregation was modeled

    using an aggregation kernel consisting of the summation of

    the Brownian and hydrodynamic kernels reported in Eqs.(14) and (15). As already mentioned, experiments were con-

    ducted in saline solutions in order to match the uid density

    with that of the latex particles. Since the double-layer re-

    pulsion is very sensitive to the value of ionic strength [and

    at high salt concentration is practically negligible (Serra

    et al., 1997)], the aggregation eciency can be consid-

    ered to be unity. Knowledge of the solid volume fraction

    v =2:5105 and of the diameter of the latex primary par-ticles Lo =2106 m is sucient for determining the initialconditions of the six moments (m0 =3:1251012 m3; m1 =6:25 106 m2; m2 = 12:5 m1; m3 = 2:5 105; m4 =5:0 10

    11

    m; m5 = 1:0 1016

    m

    2

    ). A summary of theaggregation and breakage functions used in each of the dif-

    ferent cases is reported in Table 3.

    The exponents of the power-law breakage kernel [see

    cases (1; 2; 3) in Table 3 or kernel (5) in Table 1] were found

    through a dimensional analysis, forcing a linear dependence

    for the particle size (i.e., = 1). The resulting exponent for

    the turbulent dissipation rate ( = 3=4) is in good agree-

    ment with the value found by tting with experimental data

    ( = 0:51). Moreover, direct experimental evidence seems

    to conrm the linear dependence with respect to the particle

    size ( = 1). As concerns the exponential breakage kernel

    we rst need to identify some quantities such as the size of

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    3344 D. L. Marchisio et al./ Chemical Engineering Science 58 (2003) 3337 3351

    Table 3

    Cases tested for the comparison with experimental data

    Case a(L) b(L|)

    (1) c13=45=4L Symmetric fragmentation

    (2) c13=45=4L Erosion

    (3) c13=45=4L Uniform

    (4) cA1=2 exp

    cB

    (L)2:2

    1=2

    Symmetric fragmentation

    (5) cA1=2 exp

    cB

    (L)2:2

    1=2

    Erosion

    (6) cA1=2 exp

    cB

    (L)2:2

    1=2

    Uniform

    For all cases investigated, Lo = 2 106 m and the solid volumefraction v = 2:5 105.

    the primary particles (Lo = 2 106 m), the fractal dimen-sion of the particles df = 2:6 (Serra & Casamitjana, 1998a;

    Sonntag & Russel, 1986), the value of the Hamaker constant

    (A = 1021 J, for details see Fitch, 1997), and the interparti-

    cle distance that can be approximated asHo =24109 m.Hence, the breakage constants [compare kernel (4) in Table

    1 and cases (4; 5; 6) is Table 3] are given by cA = 1=

    12 258:2 and cB = (f

    1=2)=(2:2) 17:9. However, since Aand Ho are properties of the system that are dicult to esti-

    mate accurately, the constant cB will be expressed in terms

    of the parameter group A=H2o as follows

    cB = 2:24 105 A

    H2o : (23)

    As already mentioned, although the numerical value of the

    group A=H2o can be calculated by using values of A and Hofound in the literature (Ayazi Shamlou et al., 1994; Fitch,

    1997), the eect of changes in A=H2o on the nal mean aggre-

    gate size will be investigated. It should be highlighted here

    that this approach presents an inconsistency due to the fact

    that the QMOM in the form presented in this work, assumes

    non-fractal aggregates. However, since these aggregates are

    compact (i.e., df is very close to 3), the application of the

    QMOM (assuming df =3) and the calculation of the break-

    age kernel (assuming df = 2:6) seems acceptable. Indeed,the introduction of the fractal dimension as an independent

    parameter in the QMOM formulation represents one of the

    next steps of our work, but it requires the formulation of the

    population balance using at least two internal coordinates.

    In Fig. 2 breakage frequency rates as a function of particle

    size for the exponential and power-law kernels are shown

    for two rotational speeds, N=125 and 165 rpm (Re=Rec =96

    and 127), which correspond to volume-averaged turbulent

    dissipation rates = 0:035 and 0:070 m2 s3, respectively.

    It is evident that the exponential kernel exhibits a much

    stronger dependency on particle size. In fact, it scales with

    the sixth power for low and with the third power for high .

    L/Lo

    0 10 20 30 40

    N = 165 rpm

    N = 125 rpm

    L/Lo

    a(L/L

    o)

    0 10 20 30 4010

    -3

    10-2

    10-1

    100

    101

    10

    a(

    L/L

    o)

    10-5

    10-3

    10-1

    101

    Fig. 2. Comparison between breakage kernels for = 0:035 m2 s3

    (N = 125 rpm; Re=Rec = 96) and for = 0:070 m2 s3 (N = 165 rpm;

    Re=Rec = 127); open squares: exponential kernel with c1 = 6:0 104,

    open circles: power-law breakage kernel with df = 2:6.

    axialcoordinate,m

    0

    0.02

    0.04

    0.06

    0.08

    0.1

    0.12

    0.14

    0.16

    0.18

    radial coordinate, m

    Fig. 3. Velocity vectors in a meridian section of the TaylorCouette

    reactor at several rotational speeds of the inner cylinder. From left toright: 75, 125, 165 and 211 rpm (Re=Rec = 58, 96, 127, 162).

    4. Results and discussion

    In Fig. 3, the mean velocity vectors in a meridian section

    of the TaylorCouette reactor are shown for 75, 125, 165,

    and 211 rpm. For the four rotational speeds reported above

    we nd that Re=Rec falls in the range between 50 and 160,

    which corresponds to the Turbulent Vortex Flow regime

    (see Marchisio, 2002).

    In all cases, the contour plots show the expected counter-

    rotating vortical structure. Although no experimental

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    D. L. Marchisio et al./ Chemical Engineering Science 58 (2003) 3337 3351 3345

    t, s0 2500 5000

    t, s0 2500 5000

    t, s

    d

    0 2500 50000

    5

    10

    15

    20

    N = 211 rpm

    t, s0 2500 5000

    0

    5

    10

    15

    20

    25

    t, s0 2500 5000

    0

    5

    10

    15

    20

    25

    t, s0 2500 5000

    0

    5

    10

    15

    20

    25

    t, s0 2000 4000 6000 8000

    0

    5

    10

    15

    20

    25

    30

    t, s0 2000 4000 6000 8000

    0

    5

    10

    15

    20

    25

    30

    t, s0 2000 4000 6000 8000

    0

    5

    10

    15

    20

    25

    30

    t, s0 5000 10000 15000

    0

    10

    20

    30

    40

    t, s

    d

    0 5000 10000 150000

    10

    20

    30

    40

    t, s0 5000 10000 15000

    0

    10

    20

    30

    40

    t, s0 5000 10000 15000

    0

    10

    20

    30

    40

    N = 75 rpm

    t, s

    d

    0 2000 4000 6000 80000

    5

    10

    15

    20

    25

    30

    N = 125 rpm

    t, s

    d

    0 2500 50000

    5

    10

    15

    20

    25

    N = 165 rpm

    t, s0 2500 5000

    Fig. 4. Time-evolution of the normalized mean particle size for case 1 (circle: experimental data; dotdashed line: c1 = 0:3 103; dashed line:

    c1 = 0:6 103; continuous line: c1 = 1:0 103).

    velocity data are available for these specic cases, the re-

    sults are consistent with our previous ndings (Marchisio,

    2002). Because the grid used in the CFD simulation only

    represents half of the axial length of the actual reactor,

    the predicted number of vortex pairs in the reactor must

    be multiplied by a factor of two. At the lowest rotational

    speed shown in Fig. 3, the CFD simulation predicts eight

    vortex pairs, whereas at higher rotation rates, seven vortex

    pairs are predicted. It is well-known that the axial wavenumber in the turbulent Taylor vortex regime depends upon

    the azimuthal Reynolds number history, but in general the

    number of vortices decreases to a minimum value as Re

    is increased beyond the critical value for the onset of tur-

    bulent Taylor vortex ow (Lewis & Swinney, 1999). The

    change in the number of vortices in mainly caused by the

    increase in size of the top and bottom vortices, which is

    caused by a well-known end eect, whereas the axial length

    of the central vortices seems to be quite constant. The CFD

    simulations are consistent with this trend.

    Comparison of the aggregationbreakage simulations

    with experimental data is made through the normalized

    mean particle size d, which is the ratio between the mean

    particle size L43 and the size of primary particles Lo. The

    mean size of the aggregate L43 is calculated as the ratio be-

    tween m4 and m3 and corresponds to the volume-averaged

    particle size. Notice that the particle size evolution reported

    in the comparison is the spatial average over the entire

    reactor volume.

    4.1. Power-law breakage kernel

    Case 1. In this case the aggregationbreakage pro-

    cess was modeled by using a power law breakage ker-

    nel and assuming that particles break into two equal

    fragments.

    The comparison between experimental data at four rota-

    tional speeds of the inner cylinder with model predictions

    for case 1 (see Table 3) for dierent values of the con-

    stant c1 is reported in Fig. 4.The model predictions and

    experimental data both show that the steady-state mean

    particle size decreases with increasing rotational speed.

    However, the simulations predict a faster approach tosteady state, especially at the lower rotational speeds in-

    vestigated. The best agreement between simulations and

    experiments is found when c1 = 0:6103. The constant c1is a dimensionless quantity that accounts for physical prop-

    erties of the particles and attractive forces that maintain the

    aggregate.

    Case 2. In this case the aggregationbreakage process was

    modeled by using a power-law breakage kernel and assum-

    ing that particles undergo erosive breakage. The comparison

    with experimental data is shown in Fig. 5. The combina-

    tion of the strong hydrodynamic aggregation kernel with the

    relatively weak erosion fragment distribution function leads

    to simulation predictions of a rapid accumulation of most

    of the system mass into a few very large particles. This is

    not surprising, since erosion of large aggregates results in

    two particles, one of which is a monomer and the other of

    which is still a large aggregate. Changes in the value of c1are unable to reconcile the simulation predictions and ex-

    perimental data, which suggests that erosion cannot be the

    only breakage mechanism at play in the experiments. This

    does not imply that erosion is not playing any role in the

    fragmentation process. It may be possible to explain this

    experimental behaviour in terms of a mixture of breakage

    mechanisms.

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    3346 D. L. Marchisio et al./ Chemical Engineering Science 58 (2003) 3337 3351

    1 10 100 1000 10000t, s

    1

    10

    100

    d

    Fig. 5. Time-evolution of the normalized mean particle size for case

    2 (experimental data: circles N = 75 rpm; squares N = 125 rpm;

    diamonds N = 165 rpm; triangles N = 211 rpm; CFD predictions

    with c1 = 0:3 103: dotdotdashed line N = 75 rpm; dotdashed

    line N = 125 rpm; dasheddashed line N = 165 rpm; continuous lineN = 211 rpm).

    Case 3. In this case the breakage process was modeled

    by using a power-law breakage kernel and assuming that

    aggregates break into fragments with a uniform probability

    distribution of sizes. In this case agreement between experi-

    ment and simulation is quite good for a value of the constant

    c1 in the range between 0:6103 and 1:0103 (see Fig.6). The results are also qualitatively and quantitatively com-

    parable with case 1 (symmetric fragmentation), but in this

    case it seems that the limiting particle size is reached after

    a longer period of time, thereby improving the agreement

    with the experimental data.

    t, s

    d

    0 5000 10000 150000

    10

    20

    30

    40

    50

    N = 75 rpm

    t, s

    d

    0 5000 10000 150000

    10

    20

    30

    40

    50

    t, s

    d

    0 2500 50000

    5

    10

    15

    20

    25

    30

    N = 211 rpm

    t, s

    d

    0 2000 4000 6000 80000

    10

    20

    30

    40

    N = 125 rpm

    t, s

    d

    0 2000 4000 6000 80000

    10

    20

    30

    40

    t, s

    d

    0 2500 50000

    5

    10

    15

    20

    25

    30

    N = 165 rpm

    t, s

    d

    0 2500 50000

    5

    10

    15

    20

    25

    30

    t, s

    d

    0 2500 50000

    5

    10

    15

    20

    25

    30

    t, s

    d

    0 2500 50000

    5

    10

    15

    20

    25

    30

    t, s0 2500 5000

    t, s

    d

    0 2500 50000

    5

    10

    15

    20

    25

    30

    t, s0 2500 5000

    t, s

    d

    0 2000 4000 6000 80000

    10

    20

    30

    40

    t, s0 2000 4000 6000 8000

    t, s

    d

    0 5000 10000 150000

    10

    20

    30

    40

    50

    t, s0 5000 10000 15000

    Fig. 6. Time-evolution of the normalized mean particle size for case 3 (circle: experimental data; dotdashed line: c1 = 0:3 103; dashed line:

    c1 = 0:6 103; continuous line: c1 = 1:0 103).

    4.2. Exponential breakage kernel

    Case 4. This case corresponds to the exponential breakage

    kernel coupled with a fragment distribution function that re-

    quires the formation of two equal-sized fragments. In Fig. 7

    the comparison between experimental data and exponential

    breakage model predictions for the four rotational speedsof the inner cylinder are reported. As was already men-

    tioned, model predictions were calculated using dierent

    values of the group of constants A=H2o , due to the uncertainty

    in determining the Hamaker constant A and the interparticle

    distance Ho. Neither the approach to steady state nor the

    steady-state mean particle size are well predicted by the

    exponential breakage kernel. This could be due to the fact

    that in the beginning of the simulation the aggregates are

    not fractal (i.e., df is practically equal to three) and thus the

    exponential breakage kernel does not give good predictions.

    Case 5. In this case the exponential breakage rate function

    was used with the erosive fragment distribution function.

    Fig. 8 shows that A=H2o = 0:8 104 J=m2 does not givegood agreement with experimental data. In fact, by using

    this value a gelling transition is detected. Gelation occurs

    when mass is no longer being conserved, and in a nite time,

    innite-sized particles are detected. In terms of the moments

    this is signaled by negative values of the zeroth moment.

    By decreasing A=H2o to 0:4 104 J=m2, the gellingtransition is only delayed, and A=H2o has to be decreased to

    0:2104 J=m2 in order not to have gelation within the rst5000 s of the simulation. The existence of a steady-state

    solution for this case is very dicult to be proved, in fact, it

    might be possible that the transition is again only delayed.

    However, even if the solution for A=H2o = 0:2 104 J=m2was the actual steady-state solution, the predicted limiting

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    D. L. Marchisio et al./ Chemical Engineering Science 58 (2003) 3337 3351 3347

    t, s

    d

    0 2500 50000

    5

    10

    15

    t, s

    d

    0 2500 50000

    5

    10

    15

    t, s

    d

    0 2500 50000

    5

    10

    15

    t, s

    d

    0 2000 4000 6000 80000

    5

    10

    15

    20

    t, s

    d

    0 2000 4000 6000 80000

    5

    10

    15

    20

    t, s

    d

    0 2000 4000 6000 80000

    5

    10

    15

    20

    t, s0 5000 10000 15000

    0

    5

    10

    15

    20

    25

    t, s0 5000 10000 15000

    0

    5

    10

    15

    20

    25

    t, s0 5000 10000 15000

    0

    5

    10

    15

    20

    25

    N = 75 rpm

    t, s

    d

    0 2000 4000 6000 80000

    5

    10

    15

    20

    N = 125 rpm

    t, s

    d

    0 2500 50000

    5

    10

    15

    N = 165 rpm

    t, s

    d

    0 2500 50000

    2

    4

    6

    8

    10

    N = 211 rpm

    t, s

    d

    0 2500 50000

    2

    4

    6

    8

    10

    t, s

    d

    0 2500 50000

    2

    4

    6

    8

    10

    t, s

    d

    0 2500 50000

    2

    4

    6

    8

    10

    t, s

    d

    0 5000 10000 150000

    5

    10

    15

    20

    25

    Fig. 7. Time-evolution of the normalized mean particle size for case 4 and for df = 2:6 (circle: experimental data; continuous line: A=H2o = 0:7 10

    4;dashed line A=H2o = 0:8 10

    4; dotdashed line A=H2o = 0:9 104).

    Fig. 8. Time-evolution of the normalized mean particle size for case 5 and

    fordf =2:6 (circle: experimental data; continuous line: A=H2o =0:810

    4;dashed line: A=H2o = 0:4 10

    4; dotdashed line: A=H2o = 0:2 104).

    particle size established is much smaller than what is exper-

    imentally observed.

    Case 6. The last case considered pairs the exponential

    breakage rate with the uniform fragment distribution func-tion. Results in this case are very similar to case 4, but the

    predictions are more sensitive to the value of the parameter

    A=H2o (see Fig. 9).

    4.2.1. Comparison with the homogeneous model

    It is interesting to compare the predictions of the CFD

    model with the predictions of a spatially homogeneous

    model. Fig. 10 shows the CFD predictions of the distribu-

    tion of turbulent dissipation rate over the reactor volume for

    two rotational speeds. Note that there is little spread in the

    distribution, which is one of the most important and attrac-

    tive aspects of turbulent TaylorCouette ow. Nevertheless,

    if simulations are performed neglecting spatial inhomo-

    geneities and using volume-averaged values of turbulent

    properties, signicant dierences arise in the predictions of

    the homogeneous and spatially heterogenous models. For

    example if spatial inhomogeneities are neglected, then the

    mean particle size can be overestimated by 3050%, de-

    pending on the choice of kernels. If the reactor is modeled

    by using a small number of perfectly mixed reactors with

    dierent turbulent properties, according to the hystogramreported in Fig. 10, the agreement may improve without

    resorting to the direct solution of the population balance

    equation in the CFD code. However, if for example the

    solid particles are produced by a fast chemical reaction, or

    if the solid concentration is higher and the ow eld has to

    be described by using a multiphase model, this full-CFD

    approach seems to be necessary (Baldyga et al., 2002;

    Marchisio et al., 2002).

    5. Conclusions

    Simultaneous aggregation and breakage of particles in a

    TaylorCouette reactor was simulated by implementing the

    QMOM in a commercial CFD code (FLUENT). Experi-

    mental data taken from the literature was compared with the

    CFD predictions for a range of operating conditions and for

    several combinations of aggregation and breakage kernels.

    The implementation of the QMOM in FLUENT was

    found to be very convenient. In fact, by tracking only six

    scalars and using the quadrature approximation the mo-

    ments of the PSD were tracked with very small errors. It is

    important to emphasize that achieving the same accuracy

    with a DPB method requires use of at least 2040 classes

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    3348 D. L. Marchisio et al./ Chemical Engineering Science 58 (2003) 3337 3351

    t, s0 5000 10000 15000

    0

    5

    10

    15

    20

    25

    30

    t, s0 5000 10000 15000

    0

    5

    10

    15

    20

    25

    30

    t, s0 5000 10000 15000

    0

    5

    10

    15

    20

    25

    30

    t, s

    d

    0 5000 10000 150000

    5

    10

    15

    20

    25

    30

    N = 75 rpm

    t, s

    d

    0 2000 4000 6000 80000

    5

    10

    15

    20

    N = 125 rpm

    t, s

    d

    0 2000 4000 6000 80000

    5

    10

    15

    20

    t, s

    d

    0 2000 4000 6000 80000

    5

    10

    15

    20

    t, s

    d

    0 2000 4000 6000 80000

    5

    10

    15

    20

    t, s

    d

    0 2500 50000

    5

    10

    15

    N = 165 rpm

    t, s

    d

    0 2500 50000

    5

    10

    15

    N = 165 rpm

    t, s

    d

    0 2500 50000

    5

    10

    15

    N = 165 rpm

    t, s

    d

    0 2500 50000

    5

    10

    15

    N = 165 rpm

    t, s

    d

    0 2500 50000

    2

    4

    6

    8

    10

    N = 211 rpm

    t, s

    d

    0 2500 50000

    2

    4

    6

    8

    10

    N = 211 rpm

    t, s

    d

    0 2500 50000

    2

    4

    6

    8

    10

    N = 211 rpm

    t, s

    d

    0 2500 50000

    2

    4

    6

    8

    10

    N = 211 rpm

    Fig. 9. Time-evolution of the normalized mean particle size for case 6 and for df = 2:6 (circle: experimental data; continuous line: A=H2o = 0:7 10

    4;dashed line: A=H2o = 0:8 10

    4; dotdashed line: A=H2o = 0:9 104).

    turbulent dissipation rate, m2

    s-3

    volume%

    0 0.25 0.5 0.75 10

    20

    40

    60

    80

    N = 165 rpm

    turbulent dissipation rate, m2

    s-3

    volume%

    0 0.1 0.2 0.3 0.4 0.50

    20

    40

    60

    80

    N = 125 rpm

    Fig. 10. Histogram of the distribution of the turbulent dissipation rate

    over the reactor volume at 125 and 165 rpm (Re=Rec = 96 and 127).

    (Marchisio et al., 2003b). For this kind of calculation the

    controlling step is the solution of the convection and turbu-

    lent diusion terms and thus the global CPU time is mainly

    inuenced by the number of scalars to be tracked. Although

    a general rule for nite-volume codes does not exist, it is

    reasonable to expect that the increase in the CPU time with

    the number of scalars is stronger than a linear function. For

    this reason we can infer that an increase in the number of

    scalars from six to 2040 would cause a drastic increase

    in the CPU time, thereby rendering the CFD approach im-

    practical, especially for complex geometries and if coupled

    with multiphase models. A detailed comparison between

    the QMOM and a DPB approach with a multiphase model

    for bubble columns is under investigation, and rst results

    appear to conrm this statement. Moreover, previous results

    showed that by using the QMOM the specic CPU time per

    scalar solved is constant for dierent kernels (Marchisioet al., 2003a). Further conrmation is provided by the fact

    that with all the tested combinations of kernels used in this

    work, the global CPU time was approximately constant (i.e,

    about 34 h of wall time for simulating two hours of real

    experiment). This is not true for most DPB approaches,

    where the discretization of the internal coordinate can

    generate mathematical stiness.

    Although the QMOM like all moment methods, does

    not provide a detailed description of the PSD, recent work

    by Diemer and Olson (2002) shows that knowledge of

    the lower-order moments is sucient to infer the shape of

    the PSD. Moreover, in many applications knowledge of thePSD is not required. It is important to point out that since

    the information needed during the simulation is directly

    obtained from the moments, the PSD reconstruction repre-

    sents a post-processing step at the end of the simulation,

    and thus does not increment the global CPU time.

    Concerning the choice of breakage kernels, the exponen-

    tial kernel is attractive since its formulation does not require

    the invocation of unknown constants. The exponential ker-

    nel can in fact be expressed in terms of physical properties of

    the aggregate and of the suspending solution. The power law

    kernel has no physical basis and proportionality constants

    and exponents can be found only through semi-theoretical

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    D. L. Marchisio et al./ Chemical Engineering Science 58 (2003) 3337 3351 3349

    analysis and comparison with experimental data. However,

    use of the power-law kernel improved the agreement with

    the transient behavior. As already mentioned, this can be

    caused by the fact that for short times the aggregates are

    very compact and small and therefore the fractal description

    underpredicts the breakage rates. Concerning the daughter

    distribution function, it appears that the experimental behav-ior might be explained in terms of a mixture of breakage

    mechanisms, for example erosion for small aggregates and

    symmetric breakage for large ones.

    It is very dicult to assess the suitability of dierent ker-

    nels since the comparison with experiments is made in terms

    of the volume-averaged mean particle size. For this reason,

    the experimental investigation should be carried out using a

    continuous reactor and an in-situ image analysis system to

    gather local information for comparison with CFD predic-

    tions. Such an investigation is under way (Marchisio et al.,

    2002). Moreover the approach has been also applied in a di-

    rect formulation (direct quadrature method of moments) for

    the investigation of multiphase systems (Marchisio & Fox,

    2003). Results from these studies will be reported in greater

    detail in future communications.

    Notation

    a(L) breakage kernel

    A Hamaker constant

    b(L|) daughter distribution functionb

    (k)i kth moment of the daughter distribution function

    for L = LiB(L; x; t) birth term due to aggregation and breakage

    Bak(x; t) kth moment transform of the birth term due to

    aggregation

    Bbk(x; t) kth moment transform of the birth term due to

    breakage

    d dimensionless mean particle size

    d1 diameter of the inner cylinder of the Taylor

    Couette reactor

    d2 diameter of the outer cylinder of the Taylor

    Couette reactor

    df fractal dimension

    D annular gap between the inner and outer cylinder

    of the TaylorCouette reactorD(L; x; t) death term due to aggregation and breakage

    Dak(x; t) kth moment transform of the death term due to

    aggregation

    Dbk(x; t) kth moment transform of the death term due to

    breakage

    F inter-particle force of the fractal aggregate

    G molecular growth rate

    H length of the TaylorCouette reactor

    Ho primary particle distance in the aggregate

    J(x; t) nucleation rate

    kB Boltzmann constant

    kc co-ordination number of the fractal aggregate

    L particle size

    Li abscissa (or node) of the quadrature approxima-

    tion

    Lo primary particle size

    L43 mean particle size

    mk(x; t) kth moment of the PSDn(L; x; t) particle size distribution function

    N rotational speed of the inner cylinder of the

    TaylorCouette reactor

    r1 inner cylinder radius of the TaylorCouette re-

    actor

    Re Reynolds number

    Rec critical Reynolds number for the transition to the

    laminar Taylor Vortex Flow

    t time

    ui Reynolds-averaged velocity in the ith directionwi weight of the quadrature approximation

    x position vector

    xi ith component of the position vector

    Greek letters

    exponent for the kinematic viscosity in the

    power-law breakage kernel

    exponent for the turbulent dissipation rate in the

    power-law breakage kernel

    (L; ) aggregation kernel

    exponent for the particle size in the power-law

    breakage kernel

    molecular diusivity

    A aspect ratio of the TaylorCouette reactort turbulent diusivity

    turbulent dissipation rate

    particle size

    viscosity of the suspending uid

    kinematic viscosity of the suspending uid

    s solid particle density

    f aggregate strength

    (L) volume fraction of solid within the aggregate

    !1 angular velocity of the inner cylinder of the

    TaylorCouette reactor

    Acknowledgements

    This work has been nancially supported the US

    Department of Energy (Project award number DE-FC07-

    01ID14087).

    Appendix A.

    The procedure used to nd weights (wi) and abscissas

    (Li) from the moments is based on the PD algorithm. The

    rst step is the construction of a matrix P with components

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    3350 D. L. Marchisio et al./ Chemical Engineering Science 58 (2003) 3337 3351

    Pi;j starting from the moments. The components in the rst

    column of P are

    Pi; 1 = i1; i 1; : : : ; 2N + 1; (A.1)

    where i1 is the Kronecker delta. The components in the

    second column of P are

    Pi; 2 = (1)i1mi1; i 1; : : : ; 2N + 1: (A.2)

    Since the nal weights can be corrected by multiplying by

    the true m0, the calculations can be done assuming a nor-

    malized distribution (i.e., m0 = 1). Then the remaining com-

    ponents are found from the PD algorithm:

    Pi;j = P1; j1Pi+1;j2 Pi;j2Pi+1; j1;j 3; : : : ; 2N + 1 and i 1; : : : ; 2N + 2 j: (A.3)

    If, for example, N = 2 then P becomes

    P =

    1 1 m1 m2 m21 m3m1 m220 m1 m2 m3 + m2m1 00 m2 m3 0 0

    0 m3 0 0 00 0 0 0 0

    :

    (A.4)

    The coecients of the continued fraction (i) are gener-

    ated by setting the rst element equal to zero (1 = 0), andcomputing the others according to the following recursive

    relationship:

    i =P1; i+1

    P1; iP1;i1; i 2; : : : ; 2N: (A.5)

    A symmetric tridiagonal matrix is obtained from sums and

    products of i:

    ai = 2i + 2i1; i 1; : : : ; 2N 1 (A.6)

    and

    bi = 2i+12i1; i 1; : : : ; 2N 2; (A.7)

    where ai and bi are, respectively, the diagonal and the

    co-diagonal of the Jacobi matrix. Once the tridiagonal ma-

    trix is determined, generation of the weights and abscissas

    is done by nding its eigenvalues and eigenvectors. In fact,

    the eigenvalues are the abscissas and the weights can be

    found as follows:

    wj = m0v2

    j1; (A.8)

    where vj1 is the rst component of the jth eigenvector vj .

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