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Computers and Chemical Engineering 35 (2011) 50–62

Contents lists available at ScienceDirect

Computers and Chemical Engineering

journa l homepage: www.e lsev ier .com/ locate /compchemeng

imension reduction of bivariate population balances using the quadratureethod of moments

olfram Heinekena,c, Dietrich Flockerzia, Andreas Voigtb,∗, Kai Sundmachera,b

Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstr. 1, D-39106 Magdeburg, GermanyOtto von Guericke University, Process Systems Engineering, Universitätsplatz 2, PSF 4120, D-39016 Magdeburg, GermanyFraunhofer Institute for Factory Operation and Automation IFF, PSF 1453, D-39004 Magdeburg, Germany

r t i c l e i n f o

rticle history:eceived 15 May 2009eceived in revised form 21 June 2010ccepted 22 June 2010vailable online 6 July 2010

a b s t r a c t

Crystallization models with direction-dependent growth rates give rise to multi-dimensional populationbalance equations (PBE) that require a high computational cost. We propose a model reduction based onthe quadrature method of moments (QMOM). Using this method a two-dimensional population balanceis reduced to a system of one-dimensional advection equations. Despite the dimension reduction themethod keeps important volume dependent information of the crystal size distribution (CSD). It returns

eywords:rystallizationirection-dependent growthulti-dimensional population balancesuadrature method of moments

the crystal volume distribution as well as other volume dependent moments of the two-dimensional CSD.The method is applied to a model problem with direction-dependent growth of barium sulphate crystals,and shows good performance and convergence in these examples. We also compare it on numericalexamples to another model reduction using a normal distribution ansatz approach. We can show thatour method still gives satisfactory results where the other approach is not suitable.

pwind-MUSCL schemeecond order finite volume scheme

. Introduction

Crystallization processes are often modelled using one-imensional population balances. Models of this kind are able toescribe the distribution of crystals in a reactor with respect tone internal property. However, crystalline particles are in gen-ral of a far more complex shape that may, in addition, changeuring a growth process. Moreover, the speed of crystal growthight be different along the axes of a crystal. In order to sim-

late crystallization processes more accurately, one is interestedn models that are able to cover the complicated structure ofrystal shapes and growth mechanisms. As a step in this direc-ion several authors have included more than one internal crystalroperty and direction-dependent growth speeds, resulting inulti-dimensional population balances. To name a few exam-

les, two-dimensional population balances have been used byuel, Marchal, and Klein (1997) and Puel, Févotte, and Klein2003a, 2003b) for the description of hydroquinone (C6H4(OH)2)

rystallization, and by Ma, Tafti, and Braatz (2002), Gunawan,usman, and Braatz (2004) and Gunawan, Ma, Fujiwara, and Braatz2002) to model the direction-dependent growth of potassiumihydrogen phosphate (KH2PO4) crystals. While these authors

∗ Corresponding author. Tel.: +49 391 6110406; fax: +49 391 6110606.E-mail addresses: voigt@mpi-magdeburg.mpg.de, andreas.voigt@ovgu.de

A. Voigt).

098-1354/$ – see front matter © 2010 Elsevier Ltd. All rights reserved.oi:10.1016/j.compchemeng.2010.06.012

© 2010 Elsevier Ltd. All rights reserved.

track the crystal distribution with respect to two characteristiclength coordinates, (Gerstlauer, Mitrović, Motz, & Gilles, 2001) con-sider a two-dimensional population balance model covering onelength coordinate and the inner lattice strain as a second crystalproperty.

Multi-dimensional population balances are computationallyrather costly. To overcome this problem, several model reduc-tion approaches have been suggested. Some of them are basedon the quadrature method of moments (QMOM), a method thatapproximates the moments of the crystal distribution by finitesums. The method was originally introduced by McGraw (1997)to simplify one-dimensional population balances. Extensions tomulti-dimensional population balances have been proposed byWright, McGraw, and Rosner (2001), Rosner and Pyykonen (2002),Yoon and McGraw (2004), and Marchisio and Fox (2005). However,all these higher-dimensional QMOM approaches are only able toreturn moments of the crystal distribution that are averaged overall internal coordinates. Therefore, using those methods one cannot calculate a crystal volume distribution or other moments thatare functions of a crystal volume coordinate.

On the other hand, Briesen (2006) has introduced a model reduc-tion that keeps information on the volume distribution of crystals.

Briesen approximates the crystal size distribution (CSD) by a nor-mal distribution ansatz. This method is shown to perform well ifthe ansatz is closely met by the CSD of the problem. However, itis questionable if Briesen’s method can produce accurate results ifthe underlying ansatz is not a good approximation of the CSD. This

dx.doi.org/10.1016/j.compchemeng.2010.06.012http://www.sciencedirect.com/science/journal/00981354http://www.elsevier.com/locate/compchemengmailto:voigt@mpi-magdeburg.mpg.demailto:andreas.voigt@ovgu.dedx.doi.org/10.1016/j.compchemeng.2010.06.012

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roblem will arise, e.g., if a multi-modal CSD is considered, i.e. aSD having more than one local maximum.

The model reduction method that we are going to propose isased on QMOM. However, in contrast to the methods mentionedbove, our method computes moments of the CSD that still dependn the crystal volume, one of them being the crystal volume distri-ution. Regarding this aspect, our approach is similar to the methodf Briesen. However, our method will be better suited to deal withrystal size distributions that are not approximately normally dis-ributed, such as multi-modal ones. Such distributions frequentlyccur in crystallization experiments. A number of numerical exam-les will illustrate this issue.

The method presented here is designed to reduce the dimensionf two-dimensional population balances that result from mod-lling crystals growing with different speed in two characteristicirections, e.g. in length and in width. Therefore we will restrict our-elves in this paper to the growth mechanism. Our method could beurther extended to include other processes like crystal nucleation.

The article is structured as follows: In Section 2, the crystal-ization process is modelled using a two-dimensional populationalance. A special coordinate transformation that will allow uso calculate crystal volume distributions is introduced in Section. A number of model problems is presented in Section 4, therst three of them being rather academic examples with a knownxact solution, while the fourth problem is the direction-dependentrowth of barium sulphate crystals. In Section 5, the populationalance equation is transformed into a system of moments of theSD. The normal distribution method of Briesen is described inection 6, while in Section 7 we present our QMOM based dimen-ion reduction method. In Section 8 the numerical performancef Briesen’s normal distribution method and our QMOM basedethod is shown for the given model problems. Numerous plots of

he numerical error depending on the number of quadrature pointsre included.

. The population balance

The growth process of crystals showing direction-dependentrowth can be modelled using a multi-dimensional populationalance. We first consider a simple model where every crystal isssumed to be in the shape of a parallelepiped with length l1 > 0nd equal width and depth l2 > 0 so that the volume of a crystal isiven by

= l1l22. (1)The crystal shall have the growth rate G1(l1, l2) in its length-

irection and the growth rate G2(l1, l2) in the direction of its widthnd depth where G1 and G2 might differ. As indicated, both growthates are allowed to be size-dependent, i.e. they can depend onhe characteristic quantities l1 and l2 of the crystal. The crystal sizeistribution (CSD) is described by a density n(t, l1, l2) defined inuch a way that the number of crystals over [lmin1 , l

max1 ] × [lmin2 , lmax2 ]

t time t ≥ 0 is given by∫ lmax

1

lmin1

∫ lmax2

lmin2

n(t, l1, l2) dl2dl1.

We assume G(l1, l2) = (G1(l1, l2), G2(l1, l2))T to be a continu-usly differentiable vector field and consider the two-dimensionalrystal growth process modeled by the population balance equa-ion

∂n(t, l1, l2)∂t

+ ∂[G1(l1, l2)n(t, l1, l2)]∂l1

+ ∂[G2(l1, l2)n(t, l1, l2)]∂l2

= 0,

0 < t < tmax, (l1, l2) ∈ ˝, (2)ver a domain ˝ ⊂ R2+. This population balance is a two-imensional advection equation. With the notations ∇n =∂n/∂l1, ∂n/∂l2)

T and ∇ · G = ∂G1/∂l1 + ∂G2/∂l2 we rewrite Eq. (2)

ical Engineering 35 (2011) 50–62 51

in the compact form

∂n

∂t+ G · ∇n + (∇ · G)n = 0. (3)

Remark 2.1. In commonly used crystallization models the growthrate G usually depends also on quantities like the concentrationof solute in the supersaturated liquid, the temperature etc., seee.g. Mersmann (2001). A problem with concentration dependentgrowth will be introduced in Section 4. The model considered hereis restricted to crystal growth and does not cover processes likecrystal nucleation or breakage. We have chosen such a simplemodel to illustrate the dimension reduction method described inthe sequel. Extensions to include crystal nucleation are possible.

For simple growth functions G1,2 the solution of problem (2)is best obtained by the method of characteristics. For more com-plicated growth rates G1,2, e.g. in case the growths functions G1,2depend on n, analytic expressions for the characteristic curves maynot be computable so that the method of characteristics is to becombined with numerical discretization. In such a case one mightprefer to use a finite difference or finite volume scheme for thesolution of the advection Eq. (2). Alternatively, a model reductiontechnique might be applied reducing the computational effort byconverting the population balance equation into a system of equa-tions for certain moments of n. Such moments provide only partialinformation of the solution n, but in many applications this will besufficient.

3. A coordinate transformation

As in Briesen (2006), the coordinates (l1, l2) are transformed tonew coordinates (v, ∗) according to

v = l1l22, ∗ = l2. (4a)The transformed crystal density is defined by

N(t, v, ∗) = n(t, l1, l2)∗2

(4b)

in the variables v and ∗: At time t, the number ofcrystals over [vmin, vmax] × [∗min, ∗max] is now given by∫ vmax

vmin

∫ ∗max∗min N(t, v, ∗) d∗dv.

Throughout this article we will assume the following conditionsto be fulfilled:

Assumption 1. The integrals∫ ∞

0∗iN(t, v, ∗) d∗ exist, are finite and

differentiable for all t ≥ 0, v > 0 and i ∈R. �Assumption 2. There holds lim∗→0∗iN(t, v, ∗) = 0 for all t ≥ 0, v >0 and i ∈R. �

These assumptions are satisfied for all crystal size distributionsconsidered in the model problems we are going to introduce inSection 4. The assumptions are also fulfilled for all processes witha smooth crystal size distribution where the maximum crystal sizestays bounded.

The moments of the transformed crystal size density N aredefined by

Mi(t, v) =∫ ∞

0

∗iN(t, v, ∗) d∗ (5)

for i ∈Z.In particular, the zeroth moment M0(t, v) is the crystal volume

distribution, meaning that the number of crystals having an indi-∫ v2

vidual volume v ∈ [v1, v2] at time t is given by v1 M0(t, v) dv. Thusthe total volume of crystals can be expressed as

Vcr,tot(t) =∫ ∞

0

vM0(t, v) dv. (6)

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2 W. Heineken et al. / Computers and

In the case of some simple growth rates, a model reduction tech-ique leads to a system of partial differential equations for theoments Mi. We will illustrate such reduction techniques with

ome examples in Section 5.

emark 3.1. A model reduction is also possible in terms ofhe original coordinates (l1, l2) leading to the moments M̂i(t, l1) =∞0

li2n(t, l1, l2) dl2. However, the results for the moments M̂i areot so suited for a comparison with experimental data since therystal distributions over the length coordinates li are difficult toeasure in practice. A volume based crystal distribution is much

asier to obtain from experiments.

We define

˜i(v, ∗) = Gi(v/∗2, ∗) = Gi(l1, l2) (7)

or i = 1, 2 and obtain by chain rule the partial derivatives∂n(t, l1, l2)

∂t= ∗2 ∂N(t, v, ∗)

∂t,

∂n(t, l1, l2)∂l1

= ∗4 ∂N(t, v, ∗)∂v

,

∂n(t, l1, l2)∂l2

= 2∗N(t, v, ∗) + 2v∗∂N(t, v, ∗)∂v

+ ∗2 ∂N(t, v, ∗)∂∗ ,

nd

∂Gi(l1, l2)∂t

= ∂G̃i(v, ∗)∂t

,∂Gi(l1, l2)

∂l1= ∗2 ∂G̃i(v, ∗)

∂v,

∂Gi(l1, l2)∂l2

= 2v∗∂G̃i(v, ∗)

∂v+ ∂G̃i(v, ∗)

∂∗ .

Thus the population balance (2) transforms to the equation (seeriesen (2006))

∂N

∂t+

(∗2G̃1 +

2v∗ G̃2

)∂N

∂v+ G̃2

∂N

∂∗

+(

∗2 ∂G̃1∂v

+ 2v∗∂G̃2∂v

+ ∂G̃2∂∗ +

2∗ G̃2

)N = 0. (8)

Introducing the vector

=(

G1G2

)=

(∗2G̃1 + 2vG̃2/∗

G̃2

)(9a)

nd the notation ∇̃ = (∂/∂v, ∂/∂∗)T Eq. (8) can be written in theorm

∂N

∂t+ G · ∇̃N + (∇̃ · G)N = 0. (9b)

In this sense, Eq. (3) is invariant under the coordinate transfor-ation (l1, l2) → (v, ∗).

. Model problems

We will apply the model reduction technique described in thisrticle to the following model problems. The Problems 1–3 areather academic and not related to the crystallization process ofspecific substance. Nevertheless, these problems are useful test

ases for our numerical method with a known exact solution.roblem 4 describes the direction-dependent growth of bariumulphate crystals.

roblem 1 (Unimodal distribution, size-independent growth).or given v0 and vmax satisfying 0 < v0 < vmax, let the domain ˝ beefined by

= {(l , l ) ∈R2 : v < l l2 < vmax}.

1 2 0 1 2We consider the population balance (2) with size-independent

rowth, i.e. G1(l1, l2) ≡ G1, G2(l1, l2) ≡ G2 with positive constants1, G2. Initial and boundary conditions are set according to the

unction.

ical Engineering 35 (2011) 50–62

n(t, l1, l2) = e−ı((l1−G1t)2+(l2−G2t)2) (10)

with ı > 0 being the solution. �

Problem 2 (Bimodal distribution, size-independent growth).Let ˝, G1 and G2 be defined as in Problem 1. Initial and boundaryconditions are set according to

n(t, l1, l2) = e−ı((l1−l1,0−G1t)2+(l2−G2t)2) + e−ı((l1−G1t)2+(l2−l2,0−G2t)2)

(11)

with parameters ı, l1,0, l2,0 > 0 being the solution. �

Problem 3. Let ˝, G1 and G2 be defined as in Problem 1. The initialcondition n(0, l1, l2) = n0(l1, l2) is set according to

a(v) = exp(−˛(v − v1)2), �(v) = 3√

v, �(v) = ˇ 3√v,

N0(v, ∗) =

⎧⎨⎩

a(v)√2��(v)

exp

(− (∗ − �(v))

2

2�(v)2

), v > 0, ∗ > 0,

0, otherwise.n0(l1, l2) = l22N0(l1l22, l2)

where ˛ > 0, ˇ > 0 and v1 > 0 are given parameters. Boundaryconditions are set according to the desired solution n(t, l1, l2) =n0(l1 − G1t, l2 − G2t).

We note that Problem 3 has been designed such that initially thecrystal size distribution n is a normal distribution with respect to ∗along the curves v = const. The same kind of problem is consideredin Briesen (2006). �

Problem 4 (Direction-dependent growth of barium sulphatecrystals). We consider the growth of barium sulphate crystalsunder batch conditions in a supersaturated solution. To simplify theproblem, additional processes like crystal nucleation are excluded.The following assumptions are made. �

• Every crystal has the shape of a prism with hexagonal base.Crystals of this particular form have been observed in precipi-tation experiments, see Voigt and Sundmacher (2007), Niemannand Sundmacher (2007) or Voigt, Heineken, Flockerzi, andSundmacher (2008). Fig. 1 (left) shows crystals of hexagonalshape that we have observed in precipitation experiments. Thedimensions of the hexagonal base of the prism are indicated inFig. 1 (right). The thickness of the prism is denoted by L2.

• All hexagonal bases of the crystals are self-similar, i.e. the rela-tions L3 = ˇL1 and L4 = �L1 hold for all crystals with givenconstants ˇ > 0 and � > 0. We set ˛ = ˇ + � .

• The thickness L2 of the crystals is assumed to be always less thana given value Lmax > 0. All crystals have a volume greater than aminimum volume v0 > 0.

• The growth of L1 is given by the growth rate dL1/dt = Ĝ1(c) :=G(c), and the growth of L2 by the growth rate dL2/dt = Ĝ2(c, L2) :=G(c)(1 − L2/Lmax) where c is the concentration of barium sulphatein solution and G is defined, according to Bałdyga, Podgórska, andPohorecki (1995), as

G(c) = kD(

c − csat + c∗ −√

c2∗ + 2c∗(c − csat))

, c ≥ csat

with kD = 4 × 10−8 (m/s)(m3/mol), csat = 1.05 × 10−2 mol/m3,

and c∗ = 0.345 mol/m3. The CSD n̂(t, L1, L2) is a number densityin L1 and L2 and satisfies the population balance

∂n̂

∂t+ ∂Ĝ1n̂

∂L1+ ∂Ĝ2n̂

∂L2= 0. (12)

W. Heineken et al. / Computers and Chemical Engineering 35 (2011) 50–62 53

in a p

•

dL1d

V

whacd

c

˝

wi

n

w√

d≤ v∗

and L

iw

Fig. 1. Left: Hexagonal barium sulphate crystals observed

An initial CSD n̂(0, L1, L2) is given.The concentration c decreases since barium sulphate from thesolution is needed to grow the crystals:

c(t) = c(0) − �molVreac

(Vcr,tot(t) − Vcr,tot(0))

= c(0) − �molVreac

∫ ∞v0

∫ ∞0

Vcr(L1, L2)(n̂(t, L1, L2) − n̂(0, L1, L2))

Here, Vreac is the reactor volume, �mol = 1.928 × 104 mol/m3 isthe molar density of crystalline barium sulphate, and Vcr(L1, L2) =˛L21L2/2 is the volume of a crystal. An initial concentration c(0) >csat is given. c(t) > csat shall hold for all t ≥ 0.

As an example we set Lmax = 3 × 10−6 m, c(0) = 2 mol/m3,reac = 10 m3, ˇ = 2, � = 1.5, which gives ˛ = 3.5.

We introduce the notations

l1 = L2, l2 =√

˛

2L1, G1 =

(1 − l1

Lmax

)G(c), G2 =

√˛

2G(c),

n(t, l1, l2) =√

2˛

n̂(t, L1, L2),

ith the coordinates l1 for the length and l2 for the geometry of theexagonal base so that G1 and G2 are of the form G1 = G1(v, ∗, c)nd G2 = G2(c) respectively. Then Eq. (2) holds and the volume of arystal satisfies Vcr(L1, L2) = l1l22 (cf. (1)). Moreover, n is a numberensity in the variables l1 and l2 and Eq. (13) becomes

(t) = c(0) − �molVreac

∫ ∞v0

v(M0(t, v) − M0(0, v)) dv. (14)

We consider the problem on the domain

= {(L1, L2) ∈R2 : v0 < Vcr(L1, L2) < vmax}ith v0 = 10−20 m3 and vmax = 2 × 10−16 m3. The initial condition

s given by

ˆ(0, L1, L2) = d(L1, L2)ae−b((L1−L∗1

)2+(L2−L∗2)2) (15)

ith a = 1025 m−2, b = 2 × 1012 m−2, v∗ = 5 × 10−19 m3, L∗1 =2/˛ 3

√v∗, L∗2 = 3

√v∗, and

(L1, L2) =

⎧⎪⎨⎪

12

cos(

ln Vcr(L1, L2) − ln v∗ln v∗ − ln v0

�)

+ 12

, v0 ≤ Vcr(L1, L2)1, v∗ ≤ Vcr(L1, L2)

⎩ 0, otherwise.

Note that d can be seen as a function of Vcr or even of ln Vcr. Thenitial condition of Problem 4 is a Gaussian distribution multiplied

ith the function d. We have chosen the function d such that the

recipitation experiment. Right: Hexagonal base of crystal.

L2.(13)

and L2 < Lmax,

2 < Lmax

initial condition is zero at the boundary v = v0 and for L2 ≥ Lmax.We impose zero boundary conditions at the inflow boundary thatis given by Vcr(L1, L2) = v0.

Remark 4.1. The growth rate G(c) has been taken from Bałdyga,Podgórska, and Pohorecki (1995) where direction dependentgrowth is not considered. G(c) is derived there as a uniform growthrate for barium sulphate crystals. Since we have observed thinplate-like crystals in the experiment, we introduce the growth ratesĜ1 and Ĝ2 as a simple model for obtaining such flat crystals. Thegrowth rate Ĝ2 is constructed such that it does not allow the lengthL2 to exceed the maximum value Lmax. These growth rates are notbased on empirical measurement; they are just taken to provide asimple test example for our model reduction technique.

It follows from the definitions of d, n(t, l1, l2) and N(t, v, ∗) (cf.(4b)) and from the Eqs. (12) and (15) that n̂(t, L1, L2) vanishes forL2 ≥ Lmax and all t ≥ 0. This is equivalent to

N(t, v, ∗) = 0 for t ≥ 0 and ∗ ≤√

v/Lmax =: ∗(P4)min (v). (16)

5. The system of moments

A PDE system for the moments Mi exists for some simple growthrates, especially for size-independent growth rates as in the Prob-lems 1–3 and for the linear size-dependent growth in Problem 4.

5.1. The system of moments for size-independent growth

Since in the Problems 1–3 the growth rates G1 and G2 do notdepend on l1 and l2, hence G̃1 = G1 and G2 = G̃2 = G2 defined in (7)and (9a) are independent on v and ∗. Due to (9a), G1 depends on vand ∗ too.

Multiplication of (8) with ∗i and integration with respect to ∗from 0 to ∞ leads to

5 Chem

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4 W. Heineken et al. / Computers and

∂

∂t

∫ ∞0

∗iN d∗ +G1∂

∂v

∫ ∞0

∗i+2N d∗ + 2vG2∂

∂v

∫ ∞0

∗i−1N d∗

+G2∫ ∞

0

∗i ∂N∂∗ d∗ + 2G2

∫ ∞0

∗i−1N d∗ = 0.

Integrating the term involving ∂N/∂∗ by parts we get

2

∫ ∞0

∗i ∂N∂∗ d∗ = −iG2

∫ ∞0

∗i−1N d∗.

Note that no boundary terms are present due to the Assumptionsand 2. We finally obtain the system of moments

∂Mi∂t

+ G1∂Mi+2

∂v+ 2vG2

∂Mi−1∂v

+ (2 − i)G2Mi−1 = 0. (17)

The partial differential Eq. (17) hold for all i ∈Z. However, choos-ng i from any finite subset S of Z, the resulting system does not

lose, i.e. it contains always more moments than equations. There-ore, the system (17) with i ∈ S, equipped with initial and boundaryonditions, is not sufficient to determine the moments. In the Sec-ions 6 and 7 we will present two approaches to close this system.

.2. The system of moments for Problem 4

In Problem 4 the growth rate G1 depends linearly on l1. If weultiply the population balance (8) with ∗i and integrate with

espect to ∗ from 0 to ∞, we get the system of moments∂Mi∂t

+√

2˛ vG(c)∂Mi−1

∂v− vG(c)

Lmax2

∂Mi∂v

+ G(c) ∂Mi+2∂v

+(2 − i)√

˛/2 G(c)Mi−1 −G(c)Lmax2

Mi = 0(18)

or i ∈Z. As in the case of size-independent growth in Section 5.1,he system of moments is not closed. In Section 7, an approach topproximate its solution is introduced.

. Closure of the moment equations for size-independentrowth using a normal distribution ansatz

M̃0 = a, M̃1 = a�,M̃3 = a(�3 + 3��2), M̃4 = a(�4 + 6�2�2

∂ a�

∂t+ G1

∂ a(�3 + 3��2)∂v

+ 2vG2∂ a

∂v+ G2a

∂ a(�2 + �2)∂t

+ G1∂ a(�4 + 6�2�2 + 3�4)

∂v+

∂ a(�3 + 3��2)∂t

+ G1∂ a(�5 + 10�3�2 + 15

∂v

In order to close the moment system (17), Briesen (2006) useshe normal distribution ansatz

(t, v, ∗) ≈ NND(t, v, ∗) =a(t, v)

�(t, v)√

2�e

− (∗−�(t,v))22�(t,v)2 . (19)

ical Engineering 35 (2011) 50–62

The function N(t, v, ∗) is extended to negative values of ∗:

NR(t, v, ∗) ={

N(t, v, ∗), ∗ > 0,0, ∗ ≤ 0

We denote by M̃i(t, v) the moments of the ansatz function NND,i.e. M̃i(t, v) :=

∫ ∞−∞ ∗

iNND(t, v, ∗) d∗ for i = 0, 1, 2, . . . . We define k!!for k = −1, 0, 1, . . . in the following way:

k!! =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

k/2∏i=1

2i, k even,

(k−1)/2∏i=0

(2i + 1), k odd,k > 0,

1, k = −1 or k = 0.

The moments M̃i can be expressed analytically in a, � and �as

M̃i(t, v) = a(t, v)i∑

k = 0k even

(ik

)(k − 1)!!�(t, v)i−k�(t, v)k, i = 0, 1, 2, . . .

For i = 0, 1, 2, 3, 4, 5 the moments M̃i are given byM̃2 = a(�2 + �2),

4), M̃5 = a(�5 + 10�3�2 + 15��4).(20)

We consider the Eq. (17) for i = 1, 2, 3 and replace the momentsMi by the moments M̃i of the normal distribution ansatz NND. Using(20) one gets a system of three equations of advection-reactiontype, namely

,

2∂ a�

∂v= 0,

) + 2vG2∂ a(�2 + �2)

∂v− G2a(�2 + �2) = 0

(21)

for the three unknown functions a, � and � of (t, v). With u =(a, �, �)T this system can be written in matrix form as

A(u)∂u∂t

+ B(u, v) ∂u∂v

+ c(u) = 0 (22)

where the entries of A, B and c are easily computed from (21).The advection-reaction system (22) is solved by a second order

splitting scheme, which is described in detail in SupplementaryMaterial, Section A. The scheme consists of a second order upwindscheme for the advection part (cf. Toro (1998)) and a second orderRunge-Kutta method for the source term. Initial and boundaryconditions needed in the numerical scheme are computed fromthe initial and boundary moments M0, M1 and M2: If u(t∗, v∗) =(a(t∗, v∗), �(t∗, v∗), �(t∗, v∗))T is needed at a point (t∗, v∗) of the ini-tial or boundary conditions, it is calculated by

a(t∗, v∗) = M0(t∗, v∗), �(t∗, v∗) = M1(t∗, v∗)/M0(t∗, v∗),�(t∗, v∗) =

√M0(t∗, v∗)M2(t∗, v∗) − M21(t∗, v∗)/M0(t∗, v∗).

(23)

The moments M0(t∗, v∗), i = 0, 1, 2, are derived from knowninitial and boundary values of the crystal size distribution n bynumerical quadrature. The expressions in (23) are only defined ifM0(t∗, v∗) > 0 and M0(t∗, v∗)M2(t∗, v∗) > M21(t∗, v∗) holds. The firstinequality is violated if the function n is zero initially or along a

boundary; it can be avoided by adding a negligible positive per-turbation to n. The second inequality was always fulfilled in thenumerical computations considered in this article. If the solutionu = (a, �, �)T has been computed with the numerical scheme, themoments M̃i can be derived by (20).

Chem

iasidNrtiqi

7m

(GnAdtwitanclmeattopTb

sma

g

ws{

a

∑j ∈ I1

I1

jaj

,

W. Heineken et al. / Computers and

It can be expected that, as long as the function N is well approx-mated by the normal distribution ansatz (19), the moments M̃i arelso good approximations of the moments Mi. In Briesen (2006)uch an example is numerically investigated, and Problem 3 definedn Section 4 is similar to Briesen’s example. However, if the normalistribution ansatz is a rather poor approximation to the function, it turns out that this closure method does not give acceptable

esults. This will be illustrated with the numerical examples in Sec-ion 8. For problems far from normal distribution, as e.g. Problem 2n Section 4, we alternatively propose a moment closure using theuadrature method of moments. This approach will be described

n the next section.

. Closure of the moment equations using the quadratureethod of moments (QMOM)

The quadrature method of moments was introduced by McGraw1997). In this method the ODE system of moments is closed usingaussian quadrature. An algorithm for the inversion of momentseeds to be carried out in every time step of the numerical solver.modification of this method that avoids the moment inversion

uring the computation was given by Motz (2003). Here the sys-em of moments is replaced by an ODE system for abscissae andeights of the Gaussian quadrature rule. The idea of directly track-

ng abscissae and weights instead of the moments is also the basis ofhe direct quadrature method of moments (DQMOM) of Marchisiond Fox (2005). The quadrature method of moments was origi-ally given for one-dimensional population balances, describingrystals that are only characterized by a single length coordinate. There exist a number of higher-dimensional extensions of this

ethod, the first one given by Wright et al. (2001). Other suchxtensions were presented by Rosner and Pyykonen (2002), Yoonnd McGraw (2004), and Marchisio and Fox (2005). However, allhese higher-dimensional QMOM approaches deal with momentshat are averaged over all crystal size coordinates and only dependn time. In contrast to these approaches, our method is able to com-ute moments of the CSD that still depend on the crystal volume.herefore, it is still possible to evaluate a crystal volume distri-ution using our method. Regarding this aspect, our approach is

imilar to the method of Briesen described in Section 6. Yet, in ourethod we directly compute abscissae and weights, following the

pproach of Motz mentioned above.The systems of moments (17) and (18) can be written in the

eneral form

∂Mi∂t

+∑j ∈ I1

aj∂Mi+j

∂v+

∑j ∈ I2

(bji + dj)Mi+j = 0, (24)

nQP∑k=1

i∗̃i−1k w̃k

⎡⎣∂∗̃k

∂t+

⎛⎝

+nQP∑k=1

∗̃ik

⎡⎣∂w̃k

∂t+

⎛⎝∑

j ∈

here I1, I2 ⊂ Z are certain index sets. In particular, in the case ofize-independent growth with moment system (17) we have I1 =−1, 2} and I2 = {−1} with

−1 = 2vG2, a2 = G1, b−1 = −G2, d−1 = 2G2

ical Engineering 35 (2011) 50–62 55

while for Problem 4 with moment system (18) we have I1 ={−1, 0, 2} and I2 = {−1, 0} with

a−1 =√

2˛vG(c), a0 =−vG(c)Lmax

, a2 = G(c)

and

b−1 = −√

˛

2G(c), b0 = 0, d−1 =

√2˛G(c), d0 =

−G(c)Lmax

.

The quadrature method of moments is based on Gaussianquadrature rules. For a given number of quadrature points nQP ∈Nthe abscissae ∗k(t, v) and weights wk(t, v) of a Gaussian quadraturerule satisfy

Mi(t, v) =nQP∑k=1

∗k(t, v)iwk(t, v), i = 0, . . . , 2nQP − 1, (25)

where the Mi(t, v) are the moments defined in (5). This is a nonlinearsystem for the unknown functions ∗k and wk. From the theory ofGaussian quadrature one has a unique solution ∗k, wk of this systemif Mi(t, v) > 0 holds for all i = 0, . . . , 2nQP − 1. This solution satisfies∗k(t, v) > ∗min(v) (26)

for t ≥ 0, v > 0, k = 1, . . . , nQP in the case of Problem 1–4, where∗min(v) is defined according to

∗min(v) :={

0, for Problems 1–3,∗(P4)min (v), for Problem 4.

(27)

with ∗(P4)min (v) from (16)). The system (25) can be solved using thePD algorithm of Gordon (1968), see also McGraw (1997). Thisalgorithm is included in Supplementary Material, Section B. TheGaussian quadrature rule approximates an integral of the form∫ ∞

0f (∗)N(t, v, ∗) d∗ by the finite sum

∑nQPk=1f (∗k(t, v))wk(t, v). In

particular, the moments Mi(t, v) for arbitrary i ∈Z are approxi-mated by

∑nQPk=1∗k(t, v)

iwk(t, v). By (25), the moments Mi(t, v) fori = 0, 1, . . . , 2nQP − 1 are evaluated exactly by the quadrature rule.For all other moments the quadrature is in general not exact.

In the quadrature method of moments, the moments Mk insystem (24) are formally replaced by the corresponding Gaussianquadrature expressions. In addition, system (24) is only consideredfor i = 0, 1, . . . , 2nQP − 1. The resulting system is

aj ∗̃jk

⎞⎠ ∂∗̃k

∂v+

∑j ∈ I2

bj ∗̃j+1k

⎤⎦

∗̃j−1k

⎞⎠ w̃k ∂∗̃k∂v +

⎛⎝∑

j ∈ I1

aj ∗̃jk

⎞⎠ ∂w̃k

∂v+

⎛⎝∑

j ∈ I2

dj ∗̃jk

⎞⎠ w̃k

⎤⎦ = 0

(28)

for i = 0, 1, . . . , 2nQP − 1. Here we denote the abscissae by ∗̃k andthe weights by w̃k in order to distinguish them from the abscissaeand weights defined in (25) since, in general, they are not equal.The reason is that the moments Mk in (24) are replaced by quadra-ture expressions that are only approximations. We introduce the2nQP × nQP matrices P = (Pik), Q = (Qik) and the nQP-dimensionalrow vectors r = (r1, . . . , rnQP ), s = (s1, . . . , snQP ), whose elementsare given by

Pik = i∗̃i−1k w̃k, Qik = ∗̃ik, rk =∂∗̃k∂t

+

⎛⎝∑

j ∈ I1

aj ∗̃jk

⎞⎠ ∂∗̃k

∂v+

∑j ∈ I2

bj ∗̃j+1k

⎛ ⎞ ⎛ ⎞ ⎛ ⎞

sk =

∂w̃k∂t

+⎝∑j ∈ I1

jaj ∗̃j−1k

⎠ w̃k ∂∗̃k∂v +⎝∑j ∈ I1

aj ∗̃jk⎠ ∂w̃k

∂v+⎝∑

j ∈ I2

dj ∗̃jk⎠ w̃k,

for i = 0, . . . , 2nQP − 1, k = 1, . . . , nQP.

5 Chem

(aet

frf

w̃k∑j ∈

a

c jk,

w

c

�

t

∗̃

fv

M

sM1oib

cttwttb

RS∗

6 W. Heineken et al. / Computers and

Then the system (28) can be written in matrix form asP | Q)(r | s)T = 0. Thus, the matrix (P | Q) is regular provided the ∗̃ks well as the w̃k are pairwise different. So the system (28) is almostverywhere equivalent to the system (r | s) = 0 which we are goingo solve now.

∂∗̃k∂t

+

⎛⎝∑

j ∈ I1

aj ∗̃jk

⎞⎠ ∂∗̃k

∂v+

∑j ∈ I2

bj ∗̃j+1k

= 0, (29a)

∂w̃k∂t

+

⎛⎝∑

j ∈ I1

jaj ∗̃j−1k

⎞⎠ w̃k ∂∗̃k∂v +

⎛⎝∑

j ∈ I1

aj ∗̃jk

⎞⎠ ∂w̃k

∂v

+

⎛⎝∑

j ∈ I2

dj ∗̃jk

⎞⎠ w̃k = 0 (29b)

or k = 1, . . . , nQP. It is a system of 2nQP equations of advection-eaction type in the 2nQP unknown functions ∗̃k(t, v) and w̃k(t, v)or k = 1, . . . , nQP. If we set

u = (∗̃1, . . . , ∗̃nQP , w̃1, . . . , w̃nQP )T ,

B =(

B1 0B2 B1

), B1 = diag

k=1,...,nQP

⎛⎝∑

j ∈ I1

aj ∗̃jk

⎞⎠ , B2 = diag

k=1,...,nQP

⎛⎝

nd

= (g1, . . . , gnQP , h1, . . . , hnQP )T , gk =∑j ∈ I2

bj ∗̃j+1k

, hk = w̃k∑j ∈ I2

dj ∗̃

e can write system (29a) in matrix form as

∂u∂t

+ B(u, v) ∂u∂v

+ c(u) = 0. (29c)

The PDE system (29a) is equipped with initial and boundaryonditions. With

:= ({0} × [v0, vmax]) ∪ ([0, ∞) × {v0})he initial and boundary conditions are given by

k(t∗, v∗) = ∗k(t∗, v∗), w̃k(t∗, v∗) = wk(t∗, v∗) (30)or (t∗, v∗) ∈ � and k = 1, . . . , nQP where the initial and boundaryalues ∗k(t∗, v∗) and wk(t∗, v∗) are defined by

i(t∗, v∗) =nQP∑k=1

∗k(t∗, v∗)iwk(t∗, v∗), i = 0, . . . , 2nQP − 1,

ee (25). They can be calculated from the moments0(t∗, v∗), . . . , M2nQP−1(t∗, v∗) using the PD algorithm (Gordon,

968; McGraw, 1997) – see Section B in Supplementary Materialf this article. The moments Mi(t∗, v∗) are derived from the knownnitial and boundary values of the crystal size distribution n(t, l1, l2)y numerical quadrature.

If we assume that for the Problems 1–4, the initial and boundaryonditions for ∗̃k are given by smooth functions and that the solu-ion ∗̃k(t, v) of (29c) is smooth for (t, v) ∈ [0, tmax] × [v0, vmax] =: ˝∗,hen advection is directed always towards increasing v. This resultill be proved in Section D in Supplementary Material. It follows

hat the boundary conditions (30) are set correctly since at v = v0here is an inflow boundary and at v = v there is an outflow

maxoundary.

emark 7.1. In some of the numerical applications studied inection 8 the PD algorithm returns a zero or negative abscissak(t∗, v∗) ≤ 0 due to numerical error. Then the advection matrix

ical Engineering 35 (2011) 50–62

I1

jaj ∗̃j−1k

⎞⎠ ,

B is not defined since it contains negative powers of ∗k. We havereported the occurrence of such problems in Section 8.1.

Remark 7.2. The system (29c) is solved numerically using thesame second order splitting scheme as for (22), see Section A inSupplementary Material of this article. The solution of system (29a)is used to approximate the moments Mi(t, v) defined in (5) by theexpression

M̃i(t, v) =nQP∑k=1

∗̃k(t, v)iw̃k(t, v), i ∈Z. (31)

Remark 7.3. If Problem 4 is treated, the coefficients aj , bj and dj in(29a) depend on the concentration c and the Gi in (42) do dependon c too. Since the exact concentration is not known, we work withthe estimate

c̃(t) = c(0) − �molVreac

∫ vmaxv0

v(M̃0(t, v) − M0(0, v)) dv (32)

which we get by replacing M0(t, v) with M̃0(t, v) in (14). The integralin (32) is evaluated by quadrature.

In Problem 4, the CSD and therefore all moments Mi are zeroat the inflow boundary. With zero moments it is not possible toevaluate the boundary condition of system (29a). We overcomethis problem by adding a negligibly small positive perturbation tothe CSD at the inflow boundary:

n̂(t, L1, L2) = 10−8ae−b(L1−2L2)2,

where the constants a and b are the same as in (15).

8. Numerical investigations

In this section we present numerical results using the modelproblems introduced in Section 4 and the methods described inSections 6 and 7. For Problem 4, an exact solution is not known.Therefore we compare the QMOM results with a reference solutionthat has been obtained numerically from a discretization of thetwo-dimensional population balance (12) with a high resolutionfinite volume scheme.

We introduce the following short notation for the methods used:

• ND is the moment closure with normal distribution ansatz asdescribed in Section 6.

• For i = 2, 3, . . . , the notation Qi stands for the QMOM closuredescribed in Section 7 with nQP = i.

• FV2D stands for the finite volume method to solve the two-dimensional population balance (12) modelling Problem 4.

In Table 1 we have listed the problems and the correspond-ing parameters that have been studied. The parameters that arenot listed in the table have been introduced with the problems inSection 4. For Problem 1–3, the numerical scheme operates on an

equidistant grid with mesh size v. For Problem 4, the numeri-cal scheme uses a non-equidistant grid where v0 is the smallestmesh size. Details are given in Section A of Supplementary Materialof this article. The method parameter nQP is introduced in Section7.

W. Heineken et al. / Computers and Chemical Engineering 35 (2011) 50–62 57

Table 1Parameters used in the simulation of Problems 1–4.

Problem problem parameters method parameters

1 G1 = G2 = 1; tmax = 1; v0 = 0.1; vmax = 1; ı = 2, 5 v = 0.005, 0.007, 0.01, 0.014, 0.02; nQP = 2, . . . , 82 G1 = G2 = 1; tmax = 1; v0 = 0.1; vmax = 1; ı = 2, 5; l∗1 = l∗2 = 1 v = 0.005, 0.007, 0.01, 0.014, 0.02; nQP = 2, . . . , 83 G1 = G2 = 1; tmax = 0.4; v0 = 0.1; vmax = 1; ˛ = 50; ˇ = 0.1; v∗ = 0.1 v = 0.005, 0.007, 0.01, 0.014, 0.02; nQP = 2, . . . , 84 tmax = 400 s v0 = 2.5 × 10−22 m3; nQP = 2, . . . , 8

Fig. 2. Problem 1, ı = 2: error err(M̃0) (left) and err(M̃1) (right) over mesh size v for the moment closure methods ND and Q2, Q3, Q7. The results for methods Q4–Q6 areomitted to enhance visibility, they are very close to Q7.

F v for to

8

c

•

••

w

stsoq

way:

err(M̃i) =∫ tmax

0

∫ vmaxv0

|M̃i(t, v) − Mi(t, v)| dvdt∫ tmax0

∫ vmaxv0

|Mi(t, v)| dvdt, i = 0, 1. (33)

Table 2Numerical difficulties with Problems 1–4 for all v (except for the case explicitelystated).

Problem ı ND Q6 Q7 Q8

1 2 PDbreak1 5 absc.≤ 0 PDbreak2 2 ADVbreak PDbreak

ig. 3. Problem 1, ı = 5: error err(M̃0) (left) and err(M̃1) (right) over mesh size mitted to enhance visibility, they are very close to Q6.

.1. Numerical problems

The numerical methods described in this article were not appli-able if one of the following problems occurred:

In the advection scheme, i.e. in Algorithm 1, Step 1 given inSupplementary Material, the matrix C̄i+1/2 used therein has non-real eigenvalues, or not a full-rank eigenspace, see Remark A.1in Supplementary Material. In this case, the numerical scheme isnot applicable.The PD algorithm breaks down due to division by zero.The PD algorithm produces a non-positive abscissa ∗i ≤ 0. SeeRemark 7.1.

We report in Table 2 the model problems and parameter valueshere we have observed such difficulties.

Result. If the ND method was used for Problem 2, the advection

cheme was in some cases not applicable. If QMOM was applied,he PD algorithm broke down and negative abscissae occurred inome cases for nQP ≥ 6, and always for nQP = 8. In general it can bebserved that algorithmic problems increase with the number ofuadrature points.

he moment closure methods ND and Q2, Q6. The results for methods Q3–Q5 are

8.2. Comparison of closure methods for size-independent growth

In this section we compare the accuracy of the methods ND andQMOM using as examples the Problems 1–3 given in Section 4.The problem and method parameters are chosen as in Table 1. Theaccuracy of the numerical solution is measured by computing arelative error of the moments Mi that is defined in the following

2 5 ADVbreak for v = 0.02 PDbreak3 – absc.≤ 0 PDbreak PDbreak4 – – absc.≤ 0 PDbreak PDbreak

Legend: ADVbreak – advection scheme breakdown; PDbreak – PD algorithm break-down, absc.≤ 0–abscissa ≤ 0.

58 W. Heineken et al. / Computers and Chemical Engineering 35 (2011) 50–62

Fig. 4. Problem 2, ı = 2: error err(M̃0) (left) and err(M̃1) (right) over mesh size v for the moment closure methods Q2–Q7.

Fig. 5. Problem 2, ı = 5: error err(M̃0) (left) and err(M̃1) (right) over mesh size v for the moment closure methods ND and Q2–Q7.

F moment closure methods ND and Q2, Q5. The results for methods Q3, Q4 are omitted toe

trt

Trtnfa(rvmw

ig. 6. Problem 3: error err(M̃0) (left) and err(M̃1) (right) over mesh size v for thenhance visibility, they are very close to Q5.

In Figs. 2–6 we plot the error err(M̃0) and err(M̃1) over v. Notehat no result could be displayed if one of the numerical errorseported in Table 2 occurred. Fig. 7 shows for Problem 2 (with ı = 5)he exact and numerically obtained moment M0 at time t = 0.3.

Results.Problem 1: The results for Problem 1 are shown in Figs. 2 and 3.

he initial condition of Problem 1 is a normal distribution withespect to the coordinates l1 and l2. However, it is not a normal dis-ribution with respect to ∗ along the lines v = const. Therefore, theormal distribution ansatz of Briesen’s method ND is not exactly

ulfilled, even not at time t = 0. The method ND shows accept-ble results for ı = 5 (see Fig. 3), but it is not convergent for ı = 2

see Fig. 2). However, also for ı = 5 the methods Q3–Q6 give betteresults than ND (Fig. 3). The method Q2 has a rather irregular con-ergence behaviour. For some mesh sizes v, Q2 approximates theoment M0 unexpectedly well, i.e. better than the QMOM methodsith higher nQP (Fig. 2, left). This seems to be an error cancellation

Fig. 7. Problem 2, ı = 5: moment M0 (exact) and numerical approximations M̃0(obtained with Q2, Q7, ND) at time t = 0.3.

W. Heineken et al. / Computers and Chemical Engineering 35 (2011) 50–62 59

; the

emvmeQt

TdttcNm0dmaaP

ttaas

.

Fz

Fig. 8. Problem 4: crystal size distribution n̂ in m−2. eft: initial condition

ffect. The QMOM methods for nQP ≥ 3 converge smoothly as theesh size v is decreased, but the theoretical second order con-

ergence of the upwind scheme is not achieved. For ı = 2, Q7 is theost accurate method (Fig. 2) and should be used if one is inter-

sted in highly accurate results. In the case of ı = 5, the methods3 – Q6 have nearly the same accuracy (Fig. 3). Therefore it seems

o be sufficient to use Q3 in this case.Problem 2: The results for Problem 1 are shown in Figs. 4 and 5.

he solution of this problem has a double peak, and the normalistribution ansatz of the ND method is a poor approximation inhis case. As expected, the ND method is not well suited to solvehis problem. For ı = 2, the ND method leads to a system whichan not be solved by the upwind scheme. Therefore, results for theD method are not shown in Fig. 4. For ı = 5, the system of the NDethod can be used with the advection scheme except for v =

.02, but the accuracy is very poor, see Fig. 5. The method Q2 isivergent as v gets small (see Figs. 4 and 5, left), but the QMOMethods with higher nQP converge. The order of convergence inv increases with increasing nQP, but second order convergence is

gain not reached. As expected, Q7 is the most accurate method. Asn example, we show the moments M0 and M̃0 at time t = 0.3 forroblem 2 with ı = 5 in Fig. 7.

Problem 3: See Fig. 6 for results with Problem 3. The solution ofhis problem is initially a normal distribution with respect to ∗ along

he lines v = const. This problem is well suited for the ND methodnd for the QMOM methods with nQP ≤ 5. For higher nQP somelgorithmic problems occurred, see Section 8.1. ND and Q2–Q5 areecond order convergent with respect to the mesh size v. For fixedv, ND and Q2–Q5 have nearly the same accuracy.

ig. 9. Problem 4. Moment M0 computed with FV2D and approximation M̃0 obtained wioom into left figure, with Q3, Q4 included.

bold curves mark Vcr = v0 and Vcr = v∗ . Right: solution at time t = 400 s.

8.3. Direction-dependent growth of barium sulphate crystals

The quadrature method of moments has been applied to Prob-lem 4 that models the direction-dependent growth of bariumsulphate crystals. For this problem an exact solution is not known.Therefore we compare the results of our QMOM based closuremethod with a reference solution of the CSD n̂ that has been cal-culated numerically using a discretization of the two-dimensionalpopulation balance (12) with a high resolution finite volumescheme. This scheme is described in Supplementary Material, Sec-tion. The scheme has been taken from Gunawan et al. (2002). Fromthe reference solution n̂ we compute the moments Mi. In the QMOMsimulations we have used the parameters given in Table 1.

Remark 8.1. In the QMOM computations of Problem 4 the lengthsl1 and l2 as well as the crystal size distribution n need to bescaled in order to prevent the occurrence of extremely high or lownumeric values which would severely damage accuracy. With con-stants l̄ and n̄ we introduce the scaled variables l̃1 = l1/l̄, l̃2 = l2/l̄,ñ(t, l̃1, l̃2) = n(t, l1, l2)/n̄, which should be of moderate order ofmagnitude. Thus Eq. (2) transforms to

∂ñ(t, l̃1, l̃2)∂t

+1l̄

∂G1(l̄ l̃1, l̄ l̃2) ñ(t, l̃1, l̃2)

∂l̃1+1

l̄

∂G2(l̄ l̃1, l̄ l̃2) ñ(t, l̃1, l̃2)

∂l̃2= 0

which is then solved with the quadrature method of moments asdescribed in Section 7. Of course, domain, initial condition, bound-ary condition, moments, etc. have also to be scaled in the obviousway. In the example of Problem 4 we have used l̄ = 10−6 m andn̄ = 1025 m−2 for scaling.

th Q2, Q5. Results for Q3, Q4 are omitted to enhance visibility. Right figure shows

60 W. Heineken et al. / Computers and Chemical Engineering 35 (2011) 50–62

F ned wz

aIocFpcswcr

i

eaei

∂∗̄k

Fp

ig. 10. Problem 4. Moment M1 computed with FV2D and approximation M̃1 obtaioom into left figure, with Q3, Q4 included.

In Fig. 8 we present the initial CSD, see (15), and the CSDt time t = tmax = 400 s obtained with the finite volume scheme.n Figs. 9 and 10 we compare the moments Mi(tmax, v), i = 0, 1btained from the reference solution with the moments M̃i(tmax, v)alculated from the QMOM based moment closure method. Inig. 11 (left), the concentration c from the reference solution is com-ared with the concentrations c̃ calculated with the QMOM basedlosure method. The mesh size L = 5 × 10−9 m has been choseno small that the graphs for FV2D would not visibly change if Lere decreased any further. The relative error of the moments M̃i

an only be estimated by comparison with the moments Mi of theeference solution. We define the estimated relative error

(M̃i, t) =∫ vmax

v0|M̃i(t, v) − Mi(t, v)| dv∫ vmaxv0

Mi(t, v) dv.

Fig. 11 (right) shows the estimated relative error (M̃i, tmax) for= 0, 1, 2, 3 and nQP = 2, 3, 4, 5.

Result. It is visible in the figures that Q2 is less accurate,specially at the peak of the solution, while Q3–Q5 are in quite

good agreement with the solution obtained from FV2D. Asxpected, the estimated error can be decreased by increas-ng the number of quadrature points nQP from 2 to 5. For all

∂∗̄k +

⎛⎝∑aj ∗̄jk

⎞⎠

∂tj ∈ I1

∂v

∂w̄k∂t

+

⎛⎝∑

j ∈ I1

jaj ∗̄j−1k

⎞⎠

ig. 11. Problem 4. Left: concentration c computed with FV2D and approximation c̃ obtalot. Right: estimated relative error (M̃i, 400 s) for i = 0, 1, 2, 3 and Q2–Q5.

ith Q2, Q5. Results for Q3, Q4 are omitted to enhance visibility. Right figure shows

QMOM methods tested, the concentration c̃ is in a very goodagreement with the concentration c obtained with FV2D. Increas-ing the number of quadrature points nQP beyond five was notpossible due to numerical breakdowns mentioned in Section 8.1.To summarize, the QMOM approximation with nQP ≥ 3 seemsto be well suited to compute the crystal growth modelled withProblem 4.

8.4. Other transformations of the CSD

The transformation (4b) of the CSD was defined such that N isagain a number density. However, one can also use other transfor-mations of the CSD for the derivation of a QMOM based dimensionreduction method. We have observed that the accuracy of ourmethod is in some cases severely affected by the particular choiceof this transformation. In this section, we will study this behaviouron a simple example. We generalize the transformation (4b) toN̄(t, v, ∗) = ∗s−2n(t, l1, l2), where s ∈Z is a parameter. Note that s = 0gives the original transformation. We now construct a dimensionreduction method along the lines of Section 7 which is based on N̄instead of N. The moments of N̄ are M̄i(t, v) =

∫ ∞0

∗iN̄(t, v, ∗) d∗ =Mi+s(t, v), and we replace the QMOM ansatz (25) by M̄i(t, v) =∑nQP

k=1∗̄k(t, v)iw̄(t, v). The system (29a) changes to

+∑

bj ∗̄j+1k

= 0,

j ∈ I2

w̄k∂∗̄k∂v

+

⎛⎝∑

j ∈ I1

aj ∗̄jk

⎞⎠ ∂w̄k

∂v+

⎛⎝∑

j ∈ I2

(bjs + dj)∗̄jk

⎞⎠ w̃k = 0,

(34)

ined with Q2–Q5. The graphs are so close that they cannot be distinguished in the

W. Heineken et al. / Computers and Chemical Engineering 35 (2011) 50–62 61

F t). In tv

wIM

cTft

toat(

Mithlbcfepnettret

9

mtbawcspbwin

of two-dimensional population balances based on the quadrature method of

ig. 12. Error err(M̃0) depending on s for Problem 1, ı = 5 (left) and Problem 3 (righery close to Q4.

here the coefficients aj , bj , and dj are defined as in Section 7.nitial and boundary conditions are calculated from the moments¯ 0, . . . , M̄2nQP−1. From the numerical solution of this system we

onstruct the moments M̃i according to M̃i =∑nQP

k=1∗̄k(t, v)i−sw̄(t, v).

he numerical error is defined as in (33). For Problem 1, ı = 5 andor Problem 3, Fig. 12 shows how the numerical error depends onhe parameter s ∈Z.

Result. The investigated accuracy of M̃0 depends strongly onhe parameter s. There is little accuracy if s is either very larger very small. The “optimum” value of s, which gives the highestccuracy, depends on the problem and on the number of quadra-ure points nQP. Since the initial and boundary conditions of system34) are derived from the moments M̄0, . . . , M̄2nQP−1, the moment˜ 0 is initially and at the inflow boundary exact if s ∈Z is in thenterval [1 − 2nQP, 0]. It is remarkable that for Problem 1, ı = 5,he optimal s is outside this interval for Q3–Q6. This means thatere a choice of s that introduces initial and boundary error of M̃0

eads to better results for M̃0 than a choice of s without initial andoundary error of M̃0. This unexpected result is due to an error can-ellation effect. For Problem 3, the optimal parameter was s = −2or all methods. In this case M̃0 has indeed no initial and boundaryrror. The optimum value of s depends strongly on the particularroblem and the numerical methods used. Based only on the fewumerical examples we have studied, we are not able to give gen-ral rules regarding the optimal choice of s. Of course, many otherransformations of the CSD would also be possible. The intention ofhis section is only to indicate by a simple example that the accu-acy might be improved by a transformation of the CSD. A detailedrror analysis would be required in order to decide which kind ofransformation could be a suited to reduce the numerical error.

. Conclusion

We have proposed a QMOM based dimension reductionethod for the numerical simulation of direction-dependent crys-

al growth. The method reduces a two-dimensional populationalance to a small system of advection equations. The method isble to retain the volume-dependence of the crystal distribution. Itas shown to perform well for a number of model problems. Good

onvergence was observed also for problems that are not well-uited for the normal distribution method of Briesen, especially for

roblems with bimodal crystal size distribution. The method coulde extended to cover additional processes like crystal nucleation. Itould also be a valuable extension to consider the flow of crystals

n a reactor that is not ideally mixed. For this purpose, the Eq. (29c)eed to be coupled with a CFD code.

he right figure, the results for Q3 and Q5 are omitted to enhance visibility, they are

Appendix A. Supplementary data

Supplementary data associated with this article can be found, inthe online version, at doi:10.1016/j.compchemeng.2010.06.012.

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Dimension reduction of bivariate population balances using the quadrature method of momentsIntroductionThe population balanceA coordinate transformationModel problemsThe system of momentsThe system of moments for size-independent growthThe system of moments for Problem 4

Closure of the moment equations for size-independent growth using a normal distribution ansatzClosure of the moment equations using the quadrature method of moments (QMOM)Numerical investigationsNumerical problemsComparison of closure methods for size-independent growthDirection-dependent growth of barium sulphate crystalsOther transformations of the CSD

ConclusionSupplementary dataSupplementary data