entropic lattice boltzmann method for crystallization processes
TRANSCRIPT
ARTICLE IN PRESS
Chemical Engineering Science 65 (2010) 3928–3936
Contents lists available at ScienceDirect
Chemical Engineering Science
0009-25
doi:10.1
� Corr
E-m
journal homepage: www.elsevier.com/locate/ces
Entropic lattice Boltzmann method for crystallization processes
Aniruddha Majumder a, Vinay Kariwala a,�, Santosh Ansumali a,b, Arvind Rajendran a
a Division of Chemical and Biomolecular Engineering, Nanyang Technological University, Singapore 637459, Singaporeb Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore 560 064, India
a r t i c l e i n f o
Article history:
Received 13 May 2009
Received in revised form
9 February 2010
Accepted 19 March 2010Available online 24 March 2010
Keywords:
Crystallization
Dynamic simulation
High resolution method
Lattice Boltzmann method
Particulate process
Population balance
09/$ - see front matter & 2010 Elsevier Ltd. A
016/j.ces.2010.03.030
esponding author. Tel.: +65 6316 8746; fax:
ail address: [email protected] (V. Kariwala).
a b s t r a c t
A lattice Boltzmann method (LBM) is introduced for accurate simulation of crystallization processes
modelled using one-dimensional population balance equations (PBEs) with growth and nucleation
phenomena. LBM for PBEs with size independent growth is developed by identifying their similarity
with the advection equation. To obtain an efficient method for PBEs with size dependent growth, a
coordinate transformation scheme is introduced, which can handle processes with size independent
and size dependent growth rates in the same framework. The performance of the proposed scheme is
verified using benchmark examples drawn from literature, which shows that LBM provides at least the
same level of accuracy, while requiring lower computation time than the well-established high
resolution finite volume method.
& 2010 Elsevier Ltd. All rights reserved.
1. Introduction
Population balance equations (PBEs) are widely used to modelcrystallization processes (Randolph and Larson, 1988; Ramkrishna,2000). These equations are hyperbolic partial differential equations(PDEs), which account for various ways in which particles of aspecific state can either form or disappear from the system. For mostpractical problems, PBEs do not have analytical solutions and thusneed to be solved numerically. The simplest methods available forsolving PBEs are various method of moments like standard methodof moments (MOM) (Hulburt and Katz, 1964), quadrature method ofmoments (QMOM) (McGraw, 1997) and direct quadrature methodof moments (DQMOM) (Marchisio and Fox, 2005). These methodsare computationally efficient, but only provide information aboutthe evolution of the moments, from which the distribution needs tobe reconstructed. A weakness of these methods is that the numericalerrors in a fitted distribution can be large if the assumed distributiondoes not accurately parameterize the actual distribution. On theother hand, various discretized methods are available, which solvefor the crystal size distribution (CSD) directly. Such methods includefinite difference method (Kumar and Ramkrishna, 1996; Immanueland Doyle, 2003; Alopaeus et al., 2007; Kumar et al., 2008), finiteelement method (Mahoney and Ramkrishna, 2002) and highresolution finite volume method (Ma et al., 2002; Gunawan et al.,2004, 2008; Qamar et al., 2008). Some other methods which
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solve for CSD are method of characteristics (Kumar and Ramkrishna,1997; Qamar et al., 2008) and Monte Carlo method (Ramkrishna,2000). The large computation time required by many of thesemethods hinder their application for online model based controlof crystallization processes which requires fast repetitive solutionof PBEs.
In this paper, we introduce LBM as an efficient tool forsimulation of crystallization processes modelled using PBEs. Inthis method, fictitious particles resembling groups of moleculesare considered in a lattice with finite set of velocities (Succi, 2001;Wolf-Gladrow, 2000; Karlin et al., 2006). These particles collide atthe lattice nodes and propagate in such a way that themacroscopic behavior of the system is recovered in the long-wavelength and long-time limit. In the past few years, LBM hasbeen applied to solve various problems of practical interest; seee.g., Chen and Doolen (1998), Mantle et al. (2001), Chen et al.(2003), Freund et al. (2003), Theodoropoulos et al. (2004), andSullivan et al. (2005). In spite of the ability of LBM to provide fast,accurate and easily implementable numerical scheme, however,to the best of our knowledge, it has not been used to solve PBEs.
We consider crystallization processes modelled using one-dimensional (1D) PBEs with growth and nucleation phenomena,where the dimension refers to the number of independent statevariables. For these processes, the PBE is given as
@tnþ@LðGnÞ ¼ B0ðtÞdð0Þ ð1Þ
where n(L,t) is the CSD, G(L,t) represents the growth rate of thecrystals, B0(t) is the rate of nucleation and d is the dirac deltafunction. For a batch crystallization process, mass balance of the
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c
-c 0 c
Fig. 1. A 1D lattice model with three velocities.
A. Majumder et al. / Chemical Engineering Science 65 (2010) 3928–3936 3929
solution is coupled with the PBE to take into account thedepletion of material from solution due to crystal growth andnucleation.
We note that when G is independent of L, the PBE in Eq. (1) isanalogous to the advection equation, which can be handledefficiently by LBM (Karlin et al., 2006; Rasin et al., 2005). With thisknowledge, we demonstrate the potential of LBM by applying it tosimulate 1D crystallization processes with size independentgrowth. We further apply LBM to solve PBEs representing acrystallization process with size dependent growth rate. Althoughsuch processes can be simulated by the standard latticeBoltzmann formulation for the advection equation, the accuracyand efficiency are significantly reduced. To preserve the accuracyand the efficiency of LBM, we introduce a coordinate transforma-tion scheme, which simplifies the problem to a size independentone. This is done by taking advantage of the functional form usedto describe the size dependence of G. Nucleation as a boundarycondition in crystallization is handled in the same manner as openflow boundary condition in a fluid flow problem. A number ofbenchmark problems drawn from literature are used to show theconvergence and accuracy of LBM. In particular, it is found thatLBM is able to provide at least the same level of accuracy as thewell established HR method while the computation time requiredby LBM is smaller.
2. Lattice Boltzmann method
The origin of LBM can be traced back to Frisch et al. (1986),who showed that a simple discrete kinetic model can describe theNavier–Stokes hydrodynamics at appropriate limits. The key ideawas to provide a reduced description of the molecular motion,sufficient to describe the hydrodynamics at desired length scales,by considering pseudo-particle dynamics, where particles areconstrained to move along some fixed discrete directions only.This concept was refined further to obtain a viable hydrodynamicsimulation tool for the Navier–Stokes equations (Higuera et al.,1989; Chen et al., 1992; Qian et al., 1992). The resulting method,known as ‘‘lattice Boltzmann method’’ (LBM), is now a wellestablished numerical scheme for hydrodynamic simulations(Succi, 2001; Wolf-Gladrow, 2000).
In the subsequent discussion, we present an overview of theentropic formulation of LBM for solving the advection equation,which has a similar form as PBEs. As compared to Karlin et al.(2006), who considered advection equation with constantvelocity, we consider that the velocity depends on the positionin the physical space. This extension is necessary to apply LBM toPBEs with size-dependent growth rate.
2.1. Principle
The 1D advection equation is given as
@trþv@xr¼ 0 ð2Þ
where rðx,tÞ, v and x are the density, velocity and coordinate inphysical space, respectively. For the solution of Eq. (2) using LBM,the dynamics of the pseudo-particles should be defined in such away that the original advection equation is obtained in the long-time and long-wavelength limit. To find a set of discretevelocities, which are consistent with the symmetry and theisotropy requirement of macroscopic dynamics, one typicallystarts with a set of velocities in a lattice, based on computationalconsiderations, and checks whether it is possible to construct amodel consistent with long-time (time much larger compared tothe relaxation time) and long-wavelength limit of the kineticequations (Succi, 2001). In the discrete form, the simplest possible
model has three types of fictitious particles with velocitiesci¼{0,c,�c}, i.e., stationary (0), right moving (+), left moving(�) (Karlin et al., 2006), as illustrated in Fig. 1.
The simplest set of kinetic equations for the particles are givenby the discrete Boltzmann equation (Karlin et al., 2006)
@tfiþci@xfi ¼�1
t ðfi�f eqi ðrðf Þ,vÞÞ, i¼ 0,1,2 ð3Þ
where t40 is some relaxation time related to diffusioncoefficient, fi are the discrete Boltzmann distribution functionsand fi
eq are the equilibrium discrete Boltzmann distributionfunctions chosen appropriately to recover diffusion equation aslong-time long-wavelength limit of the kinetic equation. InEq. (3), the subscripts 0, 1 and 2 refer to stationary, right movingand left moving particles, respectively. The left hand side of Eq. (3)denotes the free flight, while right hand side represents therelaxation of the particles to equilibrium (Bhatnagar–Gross–Krook(BGK) approximation for collision) (Bhatnagar et al., 1954). Theequilibrium distributions can be found by minimizing appropriateentropy function, which is discussed next.
2.2. H-function and equilibrium distributions
In order to formulate the discrete velocity model consistentwith the H-theorem (analogue of second law of thermodynamics),the idea of entropic LBM was proposed (Karlin and Gorban, 1998;Karlin et al., 1999; Ansumali and Karlin, 2002a, b; Ansumali et al.,2003). The goal of such approaches was to construct discretekinetic models based on discrete H-functions relevant to thedynamics considered. The resulting models, by construction, areunconditionally stable, a highly desirable property in anynumerical scheme (Karlin et al., 1999; Ansumali and Karlin,2000). As the usual formulation of the LBM can be recovered asTaylor series approximation of the entropic formulation, we willnot distinguish between the two formulations and discuss onlyentropic approach in the present work.
In the entropic formulation of LBM, similar to the Boltzmannequation, the dynamics are chosen such that at equilibrium anunderlying entropic function is minimized. The entropy functionin LBM relevant to hydrodynamics is typically a discrete Kullback(1997) form of entropy. The discrete form of H-function can bewritten as
H¼X2
i ¼ 0
fi lnfi
wi�1
� �ð4Þ
where wi are the weights. For this 1D case, the weights can beselected as wi¼{4/6,1/6,1/6} (Karlin et al., 1999; Ansumali andKarlin, 2002b). The equilibrium distribution of the particlepopulation is obtained by minimizing the H-function under thefollowing constraints:
X2
i ¼ 0
fi ¼ f0þ f1þ f2 ¼ r ð5Þ
X2
i ¼ 0
cifi ¼ cðf1�f2Þ ¼ ru ð6Þ
where u is the average velocity. Eqs. (5) and (6) represent the localconservation of mass and momentum, respectively. By solving theminimization problem, we get the equilibrium values of the
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discrete Boltzmann distribution function fieq, parameterized by
two variables, r and u, as
f eq0 ðr,uÞ ¼
2r3½2�
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þu2=c2
s
q� ð7Þ
f eq1 ðr,uÞ ¼
r3½ðuc�c2
s Þ=2c2s þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þu2=c2
s
q� ð8Þ
f eq2 ðr,uÞ ¼
r3½�ðucþc2
s Þ=2c2s þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þu2=c2
s
q� ð9Þ
where cs ¼ c=ffiffiffi3p
is analogous to the speed of sound in the system.An equivalent moment representation of the kinetic equationsgiven by Eq. (3) with equilibria given by Eqs. (7)–(9) is
@trþ@xðruÞ ¼ 0 ð10Þ
@tðruÞþ@xP¼1
tðrv�ruÞ ð11Þ
@tPþc2@xðruÞ ¼1
t ðPeq�PÞ ð12Þ
where P¼(f1+ f2)c2 and Peq ¼ rc2s ð2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þMa2
p�1Þ. Here, Ma denotes
the Mach number defined as v/cs. Note that unlike hydrodynamics,where both mass and momentum are conserved, in the presentcase, collision in Eq. (3) only conserves the density r, becauseequilibrium is computed at a fixed velocity v (Karlin et al., 2006).Later in Section 2.3, we will show that this breaking of momentumconservation in long-time and long-wavelength limit impliesadvection–diffusion equation for the density r.
To show that the H-function is a non-increasing function, wemultiply both sides of Eq. (3) with lnðfi=f eq
i ð1,vÞÞ, and sum over alldiscrete velocities, which yields
@tH1þ@xJH ¼�s ð13Þ
where H1, JH and s are the relevant thermodynamic entropyfunction, flux term and entropy production terms, respectively,and are given as
H1 ¼X2
i ¼ 0
fi lnfi
f eqi ð1,vÞ
�1
!ð14Þ
JH ¼X2
i ¼ 0
cifi lnfi
f eqi ð1,vÞ
�1
!ð15Þ
s¼ 1
tX2
i ¼ 0
ðfi�f eqi ð1,vÞÞ ln
fi
f eqi ð1,vÞ
!ð16Þ
For any positive, real and nonequal values of fi and fieq, s40.
Hence, H1 monotonically decreases when the system is away fromequilibrium and is minimum at equilibrium.
2.3. Chapman–Enskog analysis
From the moment representation in Eqs. (10)–(12) of thekinetic equation given by Eq. (3), it is clear that in the limit oft-0, both u and P relax quickly to their equilibrium values. Thisimplies that when t-0, the continuity equation reduces to thedesired advection equation. Here, the role of small but finite t is tomake the system singularly perturbed and one would physicallyexpect a small correction from the equilibrium values. TheChapman–Enskog method is one of the most widely usedmethods to find such a correction. In Chapman–Enskog analysis,one expands the fast variables in powers of a smallnessparameter. Thus, in the present case, u and P can be expandedin small parameter t as
u¼ ueqþtuð1Þ þt2uð2Þ þ � � � ð17Þ
P¼ PeqþtPð1Þ þt2Pð2Þ þ � � � ð18Þ
Furthermore, the time derivative operator is also expanded inthe powers of smallness parameter. In the present case, for anyvariable f
@tf� f@ð0Þt þt@ð1Þt þt2@ð2Þt þ � � �gf ð19Þ
where @ð0Þt represents slow dynamics and the rest of the terms onthe right hand side of Eq. (19) represent fast dynamics. Bysubstituting the expansion values in Eqs. (10)–(12), we get
½@ð0Þt þt@ð1Þt þt2@ð2Þt þ � � ��r¼�@x½rvþtruð1Þ þt2ruð2Þ þ � � �� ð20Þ
The time derivatives operators such as @ð0Þt and @ð1Þt are definedfrom the consistency relations on slow variables. In present case,it would mean
@ð0Þt r¼�@xðrvÞ ð21Þ
@ð1Þt r¼�@xðruð1ÞÞ ð22Þ
Finally, upon substitution of Eqs. (17)–(19) into Eqs. (10)–(12),at O(1) we obtain
v@ð0Þt rþ@xPeq ¼�ruð1Þ ð23Þ
Using Eq. (21) and substituting the value of Peq in Eq. (23), theexpression for ru up to linear order in t is
ru� rv�tc2s @xr�tðrvÞ@xv ð24Þ
where terms of order Ma4 are neglected. Substituting this value ofru in Eqs. (10)–(12), we get
@trþ@xðrvÞ�@xðD@xrÞ ¼ tcs@x½Ma @xðrvÞ�|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}A
ð25Þ
where diffusivity D¼ tc2s ð1�Ma2
Þ. The term on the right hand sideof Eq. (25) is the error term, which is of the order OðMa tÞ and isvery small. When v is independent of spatial coordinate x,A¼ tc2
s Ma2 and Eq. (25) reduces to
@trþv@xr�D@2xr¼ 0 ð26Þ
where D¼ tc2s . The solution of this advection–diffusion equation
in the limit of t-0 approaches the viscous solution of the desiredadvection equation. In order to achieve the desired viscoussolution, we have to keep the diffusion coefficient as small aspossible. This can be done by choosing sufficiently small values ofc and t.
2.4. Discretization of the kinetic equation
In order to discretize the kinetic equation in time andspace, we integrate Eq. (3) over time dt using trapezoidal rule toobtain
fiðxþcidt,tþdtÞ � fiðx,tÞ�dt
2t½ðfiðx,tÞ�f eq
i ðx,tÞÞþðfiðxþcidt,tþdtÞ
�f eqi ðxþcidt,tþdtÞÞ� ð27Þ
It should be noted that no space discretization is done so far. InEq. (27), if dt is chosen such that dx¼ cdt, after every time step theparticles are moved to the next lattice node and space discretiza-tion is exact. We note that fiðxþcidt,tþdtÞ appears on both sidesof Eq. (27). In order to create an efficient explicit numericalscheme, we define a new set of functions as
giðx,tÞ ¼ fiðx,tÞþdt
2tðfi�f eq
i Þ ð28Þ
such that local conservation is the same for both variables, i.e.
gieq(g(x,t)) ¼ fi
eq(f(x,t)). Now, substituting for fi and fieq in terms of gi
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and gieq in Eq. (27), we get
giðxþcidt,tþdtÞ ¼ ð1�abÞgiðx,tÞþabgeqi ðgðx,tÞÞ ð29Þ
where a¼ 2 and b¼ dt=ð2tþdtÞ. In this way, we arrive at anexplicit expression in terms of gi, which are not exactly particlepopulations but can serve our purpose to solve the advectionequation. The expression for giðxþcidt,tþdtÞ in Eq. (29) is aconvex linear combination of gi(x,t) and gi
eq(g(x,t)), if 0rbr1=2.Thus, non-linear stability can be ensured by choosing b in thisrange provided gi are positive at t¼0. However, such a choiceresults in a very expensive numerical scheme, as it requires takingtime steps smaller than 2t. Fortunately, the method is linearlystable over a much wider range of b, i.e. 0rbr1 when a¼ 2(Karlin et al., 2006). For numerical simulations, we are moreinterested in the limit of b-1 (where t can be set to zero). In thislimit, if the grid resolution is not sufficient, gi can becomenegative and non-linear stability of the method is not guaranteed.The non-linear stability can be guaranteed by using entropicintegrator (Karlin and Gorban, 1998; Karlin et al., 1999). The basicidea behind entropic integrator is to choose a such that within asingle time step, increase of entropy can be guaranteed; see Karlinet al. (2006) for more details. It can be shown that when gridresolution is sufficient, the value of a, which ensures this,approaches 2. In the present work, we are only interested in fullyresolved simulations and thus fix a¼ 2. Here, it needs to bementioned that the monotonicity of the solution is not guaran-teed by the entropic method. A limiter based extension of LBMcan cure monotonicity issue (Brownlee et al., 2007).
L1 L4L2 L3Lbuffer LN
ΔL
Fig. 2. Buffer point for nucleation.
3. Lattice Boltzmann method for population balanceequations
In this section, we describe the application of LBM for solving1D PBEs. For processes with no nucleation and constant growthrate, i.e. where G is independent of L and t, the 1D PBE becomes
@tnþG@Ln¼ 0 ð30Þ
In Eq. (30), n, G and L are analogous to r, v and x in the advectionequation, respectively. Thus, the LBM scheme for the advectionequation can be applied to solve the 1D PBE in Eq. (30). When thegrowth rate depends on the size of the crystals or varies withtime, some changes are needed. Similarly, a strategy is needed toincorporate nucleation. These issues are discussed next.
3.1. Size-dependent growth rate
For size dependent growth, the error term A in Eq. (25) is oforder OðMa tÞ. Thus, for fixed t, when the Mach number issufficiently small, the right hand side of Eq. (25) can be neglectedand the current scheme can be applied to processes with sizedependent growth rate. Smaller Mach number, however, leads tosmaller time steps, which in turn increases the solution time. Toovercome this difficulty, we present an alternate method invol-ving coordinate transformation such that in terms of transformedvariable, the PBE correspond to a process with size independentgrowth rate. This allows us to use larger time steps and thusreduce the simulation time. A similar method has been used byMatsoukas and Lin (2006) in the context of Fokker–Planckequation.
Let us define z ¼ z(L). As the number of crystals in a specifiedsize interval in both of the coordinate systems are the same, theCSDs in terms of z and L are related as
hðz,tÞ ¼ nðL,tÞdL
dzð31Þ
where h(z,t) is the number density in the transformed coordinatesystem. Substituting for n(L,t) in Eq. (1), we get
@t hdL
dz
� ��1 !
þ@L GhdL
dz
� ��1 !
¼ 0 ð32Þ
Without loss of generality, we may define
dL
dz¼ GðLÞ ð33Þ
3z¼
ZdL
GðLÞþk ð34Þ
where k is the constant of integration. Thus, in the transformedcoordinate z, the PBE becomes
@thþ@LhdL
dz
� �¼ 0 ð35Þ
3@thþ@zh¼ 0 ð36Þ
which is equivalent to a PBE with size independent growth withgrowth rate being unity. Thus the LBM scheme developed earlierin this paper can be used directly.
3.2. Time-dependent growth rate
Next, we consider the case, where in addition to crystal size,the growth rate also varies with time, i.e. G¼G(L,t). We assumethat the growth rate can be written as
GðL,tÞ ¼ G1ðLÞG2ðtÞ ð37Þ
which is often satisfied for the commonly used models (Lim et al.,2002; Gunawan et al., 2004). By defining
dL
dz¼ G1ðLÞ ð38Þ
and following the analysis in Section 3.1, the PBE in terms of thetransformed variable z becomes
@thþG2@zh¼ 0 ð39Þ
Thus, the dependence of growth rate on CSD can be handled asbefore.
3.3. Nucleation
Finally, we consider the general case of a batch crystallizationprocess with varying growth rate and nucleation. With similaranalysis as done in Section 3.2, the PBE in terms of transformedvariable is
@thþG2@zh¼ B0ðtÞdð0ÞdL
dzð40Þ
In the following discussion, we interpret the nucleation termas a boundary condition for the PBE in Eq. (40) (Ramkrishna,2000),
nð0,tÞGð0Þ ¼ B0ðtÞ ð41Þ
3hð0,tÞG2ð0Þ ¼ B0ðtÞdL
dz
����z ¼ 0
ð42Þ
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Nucleation as boundary condition can be treated in the sameway as open boundaries, where fluid inlets are present. One wayto handle nucleation is to consider an extra grid point as bufferbefore the first grid point of the computational domain (see Fig. 2)and constantly refilling the buffer with the equilibriumpopulation corresponding to the desired value of the numberdensity at size zero as (Succi, 2001)
gi,buff ¼ geqi ðhð0,tÞÞ ð43Þ
3.4. Implementation
The numerical implementation of LBM involves coding Eq.(29), where right hand side represents collision/relaxation and lefthand side represents streaming of the particles on the latticenodes. During the implementation, these collision and propaga-tion phenomena are split into two consecutive steps. Thenumerical implementation of propagation step is shown inFig. 3, where only a portion of the computational domain isconsidered and streaming of the particle populations areperformed based on their velocities.
4. Case studies
In this section, LBM is applied to solve a few benchmarkproblems and the accuracy is compared with finite volume HRmethod. The HR method, which is considered the state of the artfor simulation of compressible gas dynamics, has been success-fully applied to crystallization processes by several researchers(Ma et al., 2002; Gunawan et al., 2004; Qamar et al., 2008). The HRmethod used in this paper is taken from Gunawan et al. (2004). Inthis method, the Lax–Wendroff scheme is used with van Leer fluxlimiter. The accuracy of the schemes is compared by computingthe L1- and L2�norms of the error defined as follows:
L1�norm¼1
N
XN
i ¼ 1
jnðLi,tf Þ�nexactðLi,tf Þj ð44Þ
f0(3)
f1(4) f1(5)
f2(3) f2(5)
f0(4)
f1(3)
f0(
Left moving
f0(3) f0(4) f
f2(4)
f1(2) f1(3) f1(4)
f2(4) f2(5) f2(6)
Fig. 3. Advection of the particle populations in a 1D lattice, where f0, f1 and f2 denote the
(a) Before advection, (b) after advection.
L2�norm¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
N
XN
i ¼ 1
ðnðLi,tf Þ�nexactðLi,tf ÞÞ2
vuut ð45Þ
where Li are the discrete points of the chosen grid, tf is the finalsimulation time and N is the number of grid points.
The stability requirement for HR method is that the Courantnumber jðGDt=DLÞjr1 (LeVeque, 1992). For a given grid size ðDLÞ,we consider that the time interval ðDtÞ is chosen such that themaximum Courant number is approximately 0.1. For LBM, theparticle velocity c needs to be chosen such that the Mach numberis sufficiently low. Similar to Karlin et al. (2006), we choose c to be10 times of the growth rate and for that the Mach number for agiven grid size is approximately 0.173. Selecting the relaxationparameter b closer to 1 provides smaller diffusion coefficient butit may introduce oscillation, especially for nonsmooth distribu-tions. Thus b is chosen such that the diffusion coefficient issufficiently small and oscillation is minimal. In the first twoexamples, accuracy and convergence of LBM are comparedwith the HR method. The third example is more realistic, wherea model of batch crystallization process is considered. Allcomputations are performed using a Windows Vista PC with anIntel CoreTM2 Duo Processor 6600 (2.40 GHz, 2 GB RAM) usingMatlab s 2007a.
4.1. Size independent growth rate (smooth distribution)
We first consider a crystallization process with constantgrowth rate. The initial distribution is given by the followingsteep Gaussian profile (Karlin et al., 2006):
nðL,0Þ ¼ 1:0þ0:5expð�5000ðL=N�0:25Þ2Þ ð46Þ
The crystal growth rate is G¼ 0:1mm s�1 and the size domain ofinterest is 02800mm. The dynamics of this process can berepresented by Eq. (30), which is solved using LBM and HRmethods. The HR method is used with DL¼ 0:1mm and Dt¼ 0:1 s.The LBM parameters used for simulation are b¼ 0:99999993 andc¼ 1mm s�1. Further increase in b does not reduce the diffusioncoefficient significantly. To explore the convergence of thesemethods, a periodic boundary condition is used, i.e. crystalsexiting the domain are reintroduced in the domain with size zero.
f1(6) f1(7)
f0(6) f0(7)5)
Rightmoving
0(5)
f2(6) f2(7)
f1(5) f1(6)
f0(7)f0(6)
f2(7) f2(8)
stationary, right moving and left moving populations of the particles, respectively.
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We consider that the final time is 8000 s, which is the time takenby the distribution to complete 1 cycle. In other words, theanalytical solution at final time is the same as the initialdistribution.
The initial and final distributions obtained using LBM and HRmethod are shown in Fig. 4(a). From the zoomed view of the finaldistribution shown in Fig. 4(b), it can be noted that LBM is able tocapture the sharp peak of the gaussian distribution, whereas HRmethod shows relatively more diffusion. The diffusion is also notsymmetric for HR method as can be seen from Fig. 4(b).
The L1- and L2�norms of error of the numerical solutions areplotted against different size intervals ðDLÞ in Fig. 5(a) and (b),respectively. We see that the curves for LBM are situated underthe curves for HR method. This means that for a particular gridsize, solution provided by LBM is more accurate than HR method.Slope of these curves are indicative of the convergence of themethods. LBM shows second order convergence, whereas HRmethod has a lesser order convergence for both error norms. Thiscould be due to the fact that HR method uses flux limiters whichare designed to work well for discontinuities, but takes conser-vative approach for smooth distribution. The computation timerequired for the simulation results shown in Fig. 4 is 23 s for LBMand 54 s for HR method, which indicates the efficiency of theproposed scheme.
4.2. Size independent growth rate (non-smooth distribution)
Here, we consider another process with size independentgrowth rate but with non-smooth initial distribution. This case
0 200 400 600 8000.9
1
1.1
1.2
1.3
1.4
1.5
Num
ber
dens
ity, n
(#/
μm)
Initial and exactHR (ΔL = 0.1, Δt = 0.1)LBM
Crystal size, L (μm)
Fig. 4. Size distribution for LBM and HR method (1D smooth distribution). (a) Initial an
peak of the Gaussian.
ln (
-
norm
)
−3 −2.5 −2 −1.5 –1
−15
−14
−13
−12
−11
−10
−9
−8
−7
−6
ln(ΔL)
Slope = 2
Slope = 1.2
HR (Courant no.= 0.1)LBM (Ma no.= 0.173)
Fig. 5. Error norms for LBM and HR method (1D smooth dis
study provides us with useful information about the ability of thenumerical schemes to deal with distributions where disconti-nuities are present. Discontinuity may occur in a seeded batchcrystallization due to secondary nucleation which causes suddenrise in the number of the crystals of nearly zero size (Qamar et al.,2006). The initial distribution is taken as (Motz et al., 2002)
nðL,0Þ ¼1010# m�1 if 2:0mmo L r10mm
0:0 elsewhere
(
The crystals are considered to be growing at the rate of0:1mm s�1. This problem is solved for 1 cycle (1000 s) whichmeans that the final distribution is the same as initial distribution.The HR method is used with DL¼ 0:1mm and Dt¼ 0:1 s. The LBMparameters used for simulation are b¼ 0:995 and c¼ 1mm s�1.Final distribution is shown in Fig. 6(a) and the zoomed view of thedistribution is shown in Fig. 6(b). We see that both methods sufferfrom similar amount of diffusion while diffusion for HR method isagain not symmetric.
The L1- and L2�norms of error are also shown in Fig. 7(a) and(b), respectively. It is seen from these plots that both methodshave similar convergence. However, the convergence rates aremuch lower than the convergence rate seen for smoothdistributions. It is noted that curves for LBM and HR methodintersect at a point, beyond which the HR method provides moreaccurate solution for a given grid size. However, for the same levelof accuracy, although LBM requires finer grid, it still requires lesssolution time than HR method. For example, for the L1�norm tobe 4.6�107, the computation time required for LBM is 1 s,whereas HR method requires 4 s.
198 199 200 201 202 203 204
1.485
1.49
1.495
1.5
Num
ber
dens
ity, n
(#/
μm)
Initial and exactHR (Δx = 0.1, Δt = 0.1)LBM (ΔL = 0.1)
Crystal Size, L (μm)
d final distribution after 1 cycle, (b) zoomed view of the distribution focusing the
−3 −2.5 −2 −1.5 −1−14
−12
−10
−8
−6
−4
ln (
-
norm
)
ln (ΔL)
Slope = 2
Slope= 0.51
HR (Courant no.= 0.1)LBM (Ma no.= 0.173)
tribution). (a) L1�norm of error, (b) L2�norm of error.
ARTICLE IN PRESS
0 20 40 60 80 1000
2
4
6
8
10
Num
ber
dens
ity, n
#/μ
m Initial and exactHR (ΔL = 0.1, Δt = 0.1)LBM (ΔL = 0.1)
2 4 6 8 10
9.75
9.8
9.85
9.9
9.95
10
10.05
10.1
Num
ber
dens
ity, n
#/μ
m
x 109 x 109
Initial and exactHR (ΔL = 0.1, Δt = 0.1)LBM (ΔL = 0.1)
Crystal size, L μm Crystal size, L μm
Fig. 6. Size distribution for LBM and HR method (1D non-smooth distribution). (a) Initial and final distribution after 1 cycle, (b) zoomed view of the distribution focusing
the peak of the pulse-type distribution.
−4 −3.5 −3 −2.5 −2 −1.5 −1 −0.517
17.5
18
18.5
19
19.5
−4 −3.5 −3 −2.5 −2 −1.5 −1 −0.519.4
19.6
19.8
20
20.2
20.4
ln (ΔL) ln (ΔL)
ln (
-
norm
)
ln (
-
norm
)
slope = 0.56
slope = 0.67
HR (Courant no.=0.1)LBM (Ma no. =0.173)
slope=0.39
slope= 0.27
HR (Courant no.=0.1)LBM (Ma no.=0.173)
Fig. 7. Error norms for LBM and HR method (1D non-smooth distribution). (a) L1�norm of error, (b) L2�norm of error.
A. Majumder et al. / Chemical Engineering Science 65 (2010) 3928–39363934
4.3. Batch crystallization with nucleation and size dependent growth
This test problem is taken from Gunawan et al. (2004) where1D crystallization with nucleation and growth is considered. Thegrowth rate is a function of crystal size, solution concentrationand temperature. The governing equation for the system iswritten as
@tnðL,tÞþ@L½GðL,CðtÞ,TðtÞÞnðL,tÞ� ¼ B0ðCðtÞ,TðtÞÞdð0Þ ð47Þ
where C is the solution concentration. Dominant mechanism forcrystal birth in this seeded batch crystallizer is the secondarynucleation given as (Randolph and Larson, 1988)
B0ðC,TÞ ¼ kbVC�CsatðTÞ
CsatðTÞ
� �b
ð48Þ
where kb and b are kinetic parameters, Csat denotes the saturatedsolution concentration and V is the total volume of the crystals inthe system given as
VðtÞ ¼
Z 10
L3nðL,tÞdL ð49Þ
The growth rate is given by the power law with lineardependence on size
GðL,C,TÞ ¼ kgC�CsatðTÞ
CsatðTÞ
� �g
ð1þ0:1LÞ ð50Þ
where kg and g are the kinetic parameters. Similar to Gunawanet al. (2004), we consider that the kinetic parameters correspondto crystallization of potassium nitrate (KNO3).
Here we assume that the nuclei form at negligible size andconsumption of material due nucleation is negligible. Then themass balance for the liquid phase is given as
dC
dt¼�3rc
Z 10
GnL2 dL ð51Þ
where rc is the crystal density. The saturated solution concentra-tion is
Csatðg=g of waterÞ ¼ 1:721� 10�4T2þ5:88� 10�3Tþ0:1286 ð52Þ
The following temperature profile is used:
TðtÞ ð3CÞ ¼ 32�4ð1�e�t=18600Þ ð53Þ
The initial distribution of the crystal is given as
nðL,0Þ ¼�3:48� 10�4L2þ0:136L�13:3 if 180:5mmrLr210:5mm
0 elsewhere
(
ð54Þ
For HR method, we use DL¼ 0:5mm and Dt¼ 0:01 s. Theapplication of LBM requires an appropriate coordinate transfor-mation. Based on Eq. (50), we note that
dL
dz¼ 1þ0:1L ð55Þ
By choosing the integration constant to be zero,
z¼ 10 lnð1þ0:1LÞ ð56Þ
In terms of transformed variable z,
@thðz,tÞþkgC�CsatðTÞ
CsatðTÞ
� �g
@zhðz,tÞ ¼ B0ðCðtÞ,TðtÞÞdð0Þ ð57Þ
ARTICLE IN PRESS
−0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Num
ber
dens
ity, n
(#/
μm)
InitialHR (Δx = 0.5, Δt = 10−2)LBM (Δz = 0.016)
900 950 1000 1050
2
4
6
8
10
12
14
16
18
x 10−3
Num
ber
dens
ity, n
(#/
μm)
HR (Δx = 0.5,Δ t = 10−2)LBM (Δz = 0.016)
Size, L (μm) Size, L (μm)
103102101100
Fig. 8. Size distribution for LBM and HR method (1D batch process). (a) Initial and final distribution after 1000 s, (b) zoomed view of the final distribution focusing the
larger crystals. Kinetic parameters used are b¼ 1:78,kb ¼ 4:64� 10�7 , g ¼ 1:32, kg ¼ 1:16� 102 , rc ¼ 2:11� 10�12.
A. Majumder et al. / Chemical Engineering Science 65 (2010) 3928–3936 3935
as ðdL=dzÞð0Þ ¼ 1. The initial distribution becomes
hðz,0Þ ¼ nðL,0Þexpð0:1zÞ ð58Þ
The volume V is computed as
VðtÞ ¼ 1000
Z 10ðexpð0:1zÞ�1Þ3hðz,tÞdz ð59Þ
where the rate of change of solution concentration becomes
dC
dt¼�3rc
Z 10
G1ðtÞG2ðLÞnðL,tÞL2 dL ð60Þ
¼�300rcG1ðtÞ
Z 10
expð0:1zÞhðz,tÞðexpð0:1zÞ�1Þ2 dz ð61Þ
LBM is used with the parameters b¼ 0:995, c¼ 0:7mm s�1 and auniform grid in z with 3200 points (same as HR method), whichresults in a logarithmic grid in L. The final distribution seen after1000 s are shown in Fig. 8(a). It can be seen that except for smallvalues of L, the final distribution obtained using the two methodsare indistinguishable. However, from the zoomed view of thedistribution shown in Fig. 8(b), we see that LBM is able tomaintain the shape of the distribution originating from seedbetter than the HR method.
The analytical solution of Eqs. (47)–(53) is not available. To getan estimate of the error, the distribution obtained using HRmethod is compared with the corresponding distribution with HRmethod itself with N¼35 200 and Dt¼ 7:57� 10�4 s.Similarly, the solution from LBM method is compared withthe corresponding distribution obtained by LBM with N¼35 200and c¼ 0:7mm s�1. For ‘‘exact’’ solutions, N is chosen to be 35 200,as with this choice all the grid points for N¼3200 coincide with asubset of grid points for N¼35 200 avoiding the need forinterpolation during the calculation of error. We find that the L1
and L2�norms of error are 4.07�10�5 and 2.07�10�4, respec-tively, for HR method and 6.32�10�5 and 5.34�10�4, respec-tively, for LBM. The computation times required by the HRmethod and LBM are 60 and 6 s, respectively. Thus with samenumber of grid points LBM is much faster than HR method.
5. Conclusions
In this article, we show the applicability of lattice Boltzmannmethod (LBM) to solve 1D population balance equations (PBEs)representing crystallization process with nucleation and growth.Traditional LBM is applicable to PBEs with size dependent growthrate with the use of small Mach number, which translates into
larger computational time. To improve the efficiency, a coordinatetransformation technique is proposed, which converts the PBEwith size dependent growth to a PBE with size independentgrowth. Benchmark examples drawn from literature are used toshow that for smooth distribution, the proposed method providessecond order convergence whereas convergence rate for highresolution (HR) method is much lower. For non-smooth distribu-tion and batch crystallization example, both of these methodshave accuracies of the same order. However, for the same level ofaccuracy, the computation time for LBM is lower than HR methodby factors of 2–10.
The scope of this paper includes solution of 1D PBEs and theextension of the current LBM scheme to multidimensionalproblems is currently being investigated by choosing appropriatelattice and corresponding particle kinetics. Our preliminarystudies show that LBM is much faster than HR method for 2DPBEs. Work is also in progress to extend LBM to PBEs withaggregation and breakage phenomena by modelling them asforcing terms in the LB equation. For simulating the effect of flowcondition or mixing on crystallization process, coupling of PBEsand computational fluid dynamics (CFD) is required (Kulikovet al., 2006; Woo et al., 2006). In such cases, LBM may prove to beadvantageous as it provides faster solution and handles hydro-dynamics naturally.
Notation
B0
rate of nucleation, # s�1c
velocity component of the quarks/particles alongcoordinate axes, mm s�1
cs
speed of sound, mm s�1C
concentration of the solution, g/g Csat saturation concentration of the solution, g/g D diffusivity when v is a function of x, mm2 s�1D
diffusivity when v is independent of x, mm2 s�1f
Boltzmann distribution function, # mm�1fi
discrete Boltzmann distribution function, # mm�1fieq
equilibrium discrete Boltzmann distribution, # mm�1
gi
re-defined discrete distribution function, # mm�1G
growth rate of crystals, mm s�1h
distribution function in transformed coordinate system,# mm�1
H
entropy function, # mm�1JH
entropy flux, # s�1
ARTICLE IN PRESS
A. Majumder et al. / Chemical Engineering Science 65 (2010) 3928–39363936
L
characteristic length of crystals, mm m mass of particles, g Ma Mach number, dimensionless n crystal size distribution, # mm�1N
number of grid points, dimensionless t time, s T temperature, 1C u average fluid velocity v advection velocity of the fluid, mm s�1x
coordinate in physical space, mm z size in the transformed coordinate system, mmGreek letters
b
relaxation parameter, dimensionlessr
density of fluid, g mm�3t
relaxation time, sAcknowledgement
The authors thank Wahyu Perdana Yudistiawan for discussionsand suggestions. The financial support from Nanyang Technolo-gical University, Singapore through grant no. RG25/07 is grate-fully acknowledged.
References
Alopaeus, V., Laakkonen, M., Aittamaa, J., 2007. Solution of population balanceswith growth and nucleation by high order moment-conserving method ofclasses. Chem. Eng. Sci. 62 (8), 2277–2289.
Ansumali, S., Karlin, I.V., 2000. Stabilization of the lattice Boltzmann method bythe H theorem: a numerical test. Phys. Rev. E 62 (6), 7999.
Ansumali, S., Karlin, I.V., 2002a. Entropy function approach to the latticeBoltzmann method. J. Statist. Phys. 107, 291–308.
Ansumali, S., Karlin, I.V., 2002b. Single relaxation time model for entropic latticeBoltzmann methods. Phys. Rev. E 65, 056312.
Ansumali, S., Karlin, I.V., Ottinger, H.C., 2003. Minimal entropic kinetic models forsimulating hydrodynamics. Europhys. Lett. 63, 798–804.
Bhatnagar, P.L., Gross, E.P., Krook, M., 1954. A model for collision processes ingases. I. Small amplitude processes in charged and neutral one-componentsystems. Phys. Rev. 94 (3), 511.
Brownlee, R., Gorban, A., Levesley, J., 2007. Nonequilibrium entropy limiters inlattice Boltzmann methods. Phys. A Statist. Mech. Appl. 387 (2–3), 385–406.
Chen, H., Chen, S., Matthaeus, W., 1992. Recovery of the Navier–Stokes equationusing a lattice-gas Boltzmann method. Phys. Rev. A 45, R5339–R5342.
Chen, H., Kandasamy, S., Orszag, S., Shock, R., Succi, S., Yakhot, V., 2003. ExtendedBoltzmann kinetic equation for turbulent flows. Science 301 (5633), 633–636.
Chen, S., Doolen, G.D., 1998. Lattice Boltzmann method for fluid flows. Annu. Rev.Fluid Mech. 30 (1), 329–364.
Freund, H., Zeiser, T., Huber, F., Klemm, E., Brenner, G., Durst, F., Emig, G., 2003.Numerical simulations of single phase reacting flows in randomly packedfixed-bed reactors and experimental validation. Chem. Eng. Sci. 58 (3–6),903–910.
Frisch, U., Hasslacher, B., Pomeau, Y., 1986. Lattice-gas automata for the Navier-Stokes equation. Phys. Rev. Lett. 56, 1505.
Gunawan, R., Fusman, I., Braatz, R.D., 2004. High resolution algorithms for multi-dimensional population balance equations. A.I.Ch.E. J. 50 (11), 2738–2749.
Gunawan, R., Fusman, I., Braatz, R.D., 2008. Parallel high-resolution finite volumesimulation of particulate processes. A.I.Ch.E. J 54 (6), 1449–1458.
Higuera, F., Succi, S., Benzi, R., 1989. Lattice gas-dynamics with enhancedcollisions. Europhys. Lett. 9, 345–349.
Hulburt, H.M., Katz, S., 1964. Some problems in particle technology: a statisticalmechanical formulation. Chem. Eng. Sci. 19 (8), 555–574.
Immanuel, C.D., Doyle, F.J., 2003. Computationally efficient solution of populationbalance models incorporating nucleation, growth and coagulation: applicationto emulsion polymerization. Chem. Eng. Sci. 58 (16), 3681–3698.
Karlin, I.V., Ansumali, S., Frouzakis, C.E., Chikatamarla, S.S., 2006. Elements of thelattice Boltzmann method. I: linear advection equation. Commun. Comput.Phys. 1 (4), 616–655.
Karlin, I.V., Ferrante, A., Ottinger, H.C., 1999. Perfect entropy functions of the latticeBoltzmann method. Europhys. Lett. 47, 182–188.
Karlin, I.V., Gorban, A., 1998. Maximum entropy principle for lattice kineticequations. Phys. Rev. Lett. 81, 6–9.
Kulikov, V., Briesen, H., Marquardt, W., 2006. A framework for the simulation ofmass crystallization considering the effect of fluid dynamics. Chem. Eng.Process. 45 (10), 886–899.
Kullback, S., 1997. Information Theory and Statistics. Dover Publications,New York.
Kumar, J., Peglow, M., Warnecke, G., Heinrich, S., 2008. The cell average techniquefor solving multi-dimensional aggregation population balance equations.Comput. Chem. Eng. 32 (8), 1810–1830.
Kumar, S., Ramkrishna, D., 1996. On the solution of population balance equationsby discretization—I. A fixed pivot technique. Chem. Eng. Sci. 51 (8),1311–1332.
Kumar, S., Ramkrishna, D., 1997. On the solution of population balance equationsby discretization—III. Nucleation, growth and aggregation of particles. Chem.Eng. Sci. 52 (24), 4659–4679.
LeVeque, R.J., 1992. Numerical Methods for Conservation Laws. Birkhauser, Berlin,Germany.
Lim, Y., Le Lann, J.-M., Meyer, X.M., Joulia, X., Lee, G., Yoon, E.S., 2002. On thesolution of population balance equations (pbe) with accurate fronttracking methods in practical crystallization processes. Chem. Eng. Sci. 57(17), 3715–3732.
Ma, D.L., Tafti, D.K., Braatz, R.D., 2002. High-resolution simulation of multi-dimensional crystal growth. Ind. Eng. Chem. Res. 41 (25), 6217–6223.
Mantle, M., Sederman, A., Gladden, L., 2001. Single-and two-phase flow in fixed-bed reactors: MRI flow visualisation and lattice-Boltzmann simulations. Chem.Eng. Sci. 56 (2), 523–529.
Matsoukas, T., Lin, Y., 2006. Fokker–Planck equation for particle growth bymonomer attachment. Phys. Rev. E 74 (3), 031122.
Mahoney, A.W., Ramkrishna, D., 2002. Efficient solution of population balanceequations with discontinuities by finite elements. Chem. Eng. Sci. 57 (7),1107–1119.
Marchisio, D.L., Fox, R.O., 2005. Solution of population balance equations using thedirect quadrature method of moments. J. Aerosol Sci. 36 (1), 43–73.
McGraw, R., 1997. Description of aerosol dynamics by the quadrature method ofmoments. Aerosol Sci. Technol. 27 (2), 255–265.
Motz, S., Mitrovic, A., Gilles, E.D., 2002. Comparison of numerical methods for thesimulation of dispersed phase systems. Chem. Eng. Sci. 57 (20), 4329–4344.
Qamar, S., Ashfaq, A., Angelov, I., Elsner, M.P., Warnecke, G., Seidel-Morgenstern,A., 2008. Numerical solutions of population balance models in preferentialcrystallization. Chem. Eng. Sci. 63 (5), 1342–1352.
Qamar, S., Elsner, M.P., Angelov, I.A., Warnecke, G., Seidel-Morgenstern, A., 2006. Acomparative study of high resolution schemes for solving population balancesin crystallization. Comput. Chem. Eng. 30 (6-7), 1119–1131.
Qian, Y.H., d’Humieres, D., Lallemand, P., 1992. Lattice BGK models for Navier–Stokes equation. Europhys. Lett. 17, 479–484.
Ramkrishna, D., 2000. Population Balances: Theory and Applications to ParticulateSystems in Engineering. Acadamic Press, San Diego, CA.
Randolph, A., Larson, M.A., 1988. Theory of Particulate Processes, second ed.Academic Press, San Diego, CA.
Rasin, I., Succi, S., Miller, W., 2005. A multi-relaxation lattice kinetic method forpassive scalar diffusion. J. Comput. Phys. 206 (2), 453–462.
Succi, S., 2001. Lattice Boltzmann Equation for Fluid Dynamics and Beyond. OxfordUniversity Press, New York.
Sullivan, S., Sani, F., Johns, M., Gladden, L., 2005. Simulation of packed bed reactorsusing lattice Boltzmann methods. Chem. Eng. Sci. 60 (12), 3405–3418.
Theodoropoulos, C., Sankaranarayanan, K., Sundaresan, S., Kevrekidis, I., 2004.Coarse bifurcation studies of bubble flow lattice Boltzmann simulations. Chem.Eng. Sci. 59 (12), 2357–2362.
Wolf-Gladrow, D., 2000. Lattice-gas Cellular Automata and Lattice BoltzmannModels. Birkhauser Verlag, Berlin, Germany.
Woo, X.Y., Tan, R.B.H., Chow, P.S., Braatz, R.D., 2006. Simulation of mixing effects inantisolvent crystallization using a coupled cfd-pdf-pbe approach. Cryst.Growth Des. 6 (6), 1291–1303.