lattice boltzmann method for multi-dimensional population balance models in crystallization
TRANSCRIPT
Chemical Engineering Science 70 (2012) 121–134
Contents lists available at ScienceDirect
Chemical Engineering Science
0009-25
doi:10.1
� Corr
E-m
journal homepage: www.elsevier.com/locate/ces
Lattice Boltzmann method for multi-dimensional populationbalance models in crystallization
Aniruddha Majumder a, Vinay Kariwala a,�, Santosh Ansumali b, Arvind Rajendran a
a School of Chemical and Biomedical 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 12 October 2010
Received in revised form
29 March 2011
Accepted 22 April 2011Available online 2 June 2011
Keywords:
Crystallization
Dynamic simulation
High resolution method
Lattice Boltzmann method
Particulate process
Population balance
09/$ - see front matter & 2011 Elsevier Ltd. A
016/j.ces.2011.04.041
esponding author. Tel.: þ65 6316 8746; fax:
ail address: [email protected] (V. Kariwala).
a b s t r a c t
In this work, lattice Boltzmann method (LBM) is developed for efficient and accurate solution of multi-
dimensional population balance equations (PBEs) used to model crystallization processes with growth
and nucleation. Detailed derivation of LBM for multi-dimensional advection equation is presented,
where the velocity is a function of space coordinates. The developed scheme is subsequently applied to
solve multi-dimensional PBEs with size-dependent growth rate by drawing an analogy between the
advection equation and the PBE. The computational advantage of LBM is shown by solving several
benchmark examples taken from the literature and comparing the results with those obtained using
well-established high resolution (HR) method. It is found that LBM provides at least as accurate
solution as HR method, while requiring lower computation time.
& 2011 Elsevier Ltd. All rights reserved.
1. Introduction
In recent years, the advancement of computational capabilityand the improvement in the particle size measurement techni-ques (Ruf et al., 2000; Larsen and Rawlings, 2009; Kempkes et al.,2009; Darakis et al., 2010) have greatly increased the interest inpopulation balance equations (PBEs) for modeling crystallizationprocesses (Nagy et al., 2008; Wang et al., 2008; Borchert et al.,2009). Many of the crystalline entities encountered in pharma-ceutical, photographic and other industries show anisotropicmorphologies (Ma et al., 2002b). The PBEs describing suchcrystallization processes are multi-dimensional, i.e., the growthof the crystals is associated with the change in multiple char-acteristic lengths. The numerical solution of multi-dimensionalPBEs is computationally challenging by construction, which is oneof the main difficulties in understanding as well as model basedcontrol of multi-dimensional crystallization processes. With theavailability of high speed computers, studies on multi-dimen-sional PBEs, however, have started appearing in the last few years(Ma et al., 2002a; Puel et al., 2003; Gunawan et al., 2004; Qamaret al., 2007; Ma et al., 2007; Pinto et al., 2008; Ma and Wang,2008; Borchert et al., 2009).
In this work, we consider multi-dimensional crystallizationprocesses with growth and nucleation. In industrial applications,
ll rights reserved.
þ65 6794 7553.
the crystallizer is often assumed to be well mixed, i.e., the crystalsize distribution (CSD) is considered to be independent of thespatial coordinates. Then, the PBE can be written as
@
@tnðx,tÞþrx � ðGðx,C,TÞnðx,tÞÞ ¼ B0ðC,TÞdðx�x0Þ, ð1Þ
where n(x,t) is the CSD, x denotes the characteristic lengths of thecrystal, x0 is the nuclei size, G is the crystal growth rate, B0 is thenucleation rate, d is the Dirac delta distribution, C is the soluteconcentration in the solution and T is the solution temperature.As the crystals nucleate and grow by consuming solute moleculesfrom the solution, the concentration of the solute and thus therates of the nucleation and growth processes decrease with time.Thus, appropriate mass balance equation has to be coupled withEq. (1) to describe the dynamic evolution of CSD.
The available numerical methods for solving PBEs can bebroadly divided into the following five classes: (1) the methodof moments; (2) the method of characteristics; (3) the method ofweighted residuals; (4) the Monte Carlo method and (5) thediscretized methods. In the method of moments (MOM), insteadof CSD, its moments are calculated. The PBE is reduced to a set ofmoment equations, which are solved using ordinary differentialequation (ODE) solver (Hulburt and Katz, 1964). Although simple,the use of this method, requires that the moment equations areclosed, which is often violated for complex systems, e.g., whengrowth rate is a non-linear function of crystal size. The issue ofclosure can be handled using quadrature method of moment(QMOM), which uses a quadrature approximation of the moments
A. Majumder et al. / Chemical Engineering Science 70 (2012) 121–134122
(McGraw, 1997; Wright et al., 2001; Gimbun et al., 2009;Kariwala et al., in press). This technique, however, loses itssimplicity for multi-dimensional problems (Su et al., 2007). Sinceonly moments are tracked in this method, the CSD is recon-structed using linear and non-linear inversion approaches, orapproximated with a gamma, beta or log-normal distribution(Aamir et al., 2009). Thus, the numerical errors in the fitteddistribution can be large, if the assumed distribution does notaccurately parameterize the actual distribution.
The method of characteristics (Kumar and Ramkrishna, 1997;Qamar et al., 2008) solves the PBE by reducing the PBE to a systemof ODEs. This method is useful for solution of PBEs with pure growthprocesses. In simultaneous processes, which include growth, aggre-gation and breakage, this method has been used to solve the growthpart only and combined with other discretized techniques whichsolve the aggregation and breakage parts (Kumar and Ramkrishna,1997; Qamar et al., 2008). Implementation of nucleation in thistechnique requires special efforts, where new cells need to be addedin the computational domain to accommodate the nuclei.
The method of weighted residuals (Singh and Ramkrishna, 1977)also reduces the PBE to a system of ODEs by approximating the CSDwith a linear combination of basis functions. The basis functionsused can be global or piecewise low-order polynomials, which arelocally nonzero. The latter method, known as finite element method(FEM), has been used for solving PBEs including processes withsimultaneous growth and aggregation (Nicmanis and Hounslow,1998; Mahoney and Ramkrishna, 2002; Alexopoulos et al., 2009).The successful application of this method, however, often requiresknowledge of the resulting CSD a priori. The drawbacks of thesemethods include the inability to capture the discontinuity that mayarise along the separatrix (curve that divides states arising frominitial conditions from those arising from boundary conditions) andthe computational overhead resulting due to evaluation of doubleintegrals in the Galerkin formulation (Costa et al., 2007).
The precise mathematical connection between population bal-ances and the Monte Carlo (MC) approach (Landau and Binder,2005) was shown by Ramkrishna (1981). The MC method is basedon the principle that the dynamic evolution of an extremely largepopulation of particles can be followed by tracking the correspond-ing changes or events (e.g., growth and aggregation) occurringin a smaller number of sample particles (Meimaroglou andKiparissides, 2007). This method has been used to solve PBEs withgrowth and aggregation (Smith and Matsoukas, 1998; Tandon andRosner, 1999). As the complexity and the dimension of the problemincrease, the MC method becomes more suitable for solving theproblem. The major drawback of this method is that the accuracylargely depends on number of sample particles considered(Meimaroglou and Kiparissides, 2007). A comparative study ofthe different MC methods for particulate processes has beenreported by Zhao et al. (2007).
Numerous discretization methods for solution of multi-dimen-sional PBE with different orders of accuracy have been investi-gated, e.g., method of classes (Puel et al., 2003; Alopaeus et al.,2007; Ma and Wang, 2008), weighted essentially nonoscillatory(WENO) method (Hermanto et al., 2009; Bouaswaig and Engell,2009) and two level discretization algorithm (Pinto et al., 2008).Some methods, such as the fixed pivot (FP) method (Kumar andRamkrishna, 1996; Nandanwar and Kumar, 2008) and the cellaverage technique (Kumar et al., 2008), which is a modification ofFP method, are specially developed for processes with aggrega-tion and breakage. However, the extension of cell averagetechnique to three or higher dimensional problems is computa-tionally very expensive and special treatment is required to reducethe computational cost (Kumar et al., 2008). In recent years, adiscretized method known as high resolution (HR) finite volumemethod has become popular for solving PBEs (Ma et al., 2002a;
Gunawan et al., 2004; Qamar et al., 2007; Gunawan et al., 2008;Majumder et al., 2010b). This method incorporates flux limiterfunctions with the traditional finite volume method, which enablesit to capture propagation of sharp fronts without excessive numer-ical dispersion (LeVeque, 2002). A drawback of this method is thatdimensional splitting is used to include flux limiters in multi-dimensional problems, which introduces an additional splittingerror, except for simple cases (LeVeque, 2002).
Recently, we introduced lattice Boltzmann method (LBM)(Benzi et al., 1992; Succi, 2001) for solution of 1D PBEs usedto model crystallization processes with growth and nucleation(Majumder et al., 2010a). It was found that LBM provides moreaccurate and efficient solution compared to the HR method. Theobjective of this paper is to develop LBM for solution of multi-dimensional PBEs with growth and nucleation, and show theadvantages of the proposed method using benchmark examples.We recall that in LBM, instead of solving the PBE by discretization,particle kinetic equations are solved. Through multiscale analysis(Chapman–Enskog expansion), it is shown that in the long-time,long-wavelength limit, the particle kinetic equations approximatethe PBE with an advection-diffusion equation. The performance ofthe proposed scheme is analyzed for both smooth and non-smoothdistributions. It is shown that LBM has second order convergencefor smooth distribution and lower order convergence for non-smooth distribution. Finally, some case studies for multi-dimen-sional PBEs are presented, which demonstrate that LBM provides atleast the same level of accuracy as the well-established HR method,while the computational time for LBM is lower.
2. Lattice Boltzmann method
Lattice based methods for simulation of hydrodynamics havereceived significant attention in the past two decades. In thesemethods, instead of solving the hydrodynamic equations bydiscretization, pseudo-particle kinetics is used in such a way thatthe hydrodynamic equations are recovered at the appropriatescale. LBM was initially developed as a spin-off of one suchmethod known as lattice gas cellular automata (LGCA) (Frischet al., 1986; Succi, 2001). The key distinguishing feature of LBM isthe introduction of the Boltzmann equation to overcome theproblem of statistical noise encountered in LGCA.
With an appropriate choice of the mesoscale equilibrium dis-tribution, various macroscale dynamics can be simulated by latticeBoltzmann type method. In its typical implementation, velocitydistribution functions corresponding to a set of fictitious particlesare introduced. These particles reside and interact only at the nodesof a predefined lattice. System dynamics and complexity emerge bythe repeated application of local rules for the motion, collision andre-distribution of these coarse-grained particles. One of the mainadvantages of LBM is its ease of implementation. In addition, due tothe local dynamics involving interaction of each lattice node withthe nearest neighboring nodes only, LBM allows for efficientparallelization (Amati et al., 1997; Kandhai et al., 1998; Mazzeoand Coveney, 2008). LBM has been applied to various problems; seefor example Benzi et al. (1992), Chen and Doolen (1998), Chen et al.(2003), Mantle et al. (2001), Ansumali et al. (2007), Kim et al. (2008),Yudistiawan et al. (2010) and Majumder et al. (2010a).
LBM has also been used previously to model growth of a singlecrystal, see e.g., De Fabritiis et al. (1998), Miller et al. (2001), Kanget al. (2004) and Rasin et al. (2005). In comparison, this work dealswith the evolution of the properties of a population of crystals,which requires solving the PBE. The mathematical structure of theadvection equation is the same as the PBE with growth process.Thus, we first present the derivation of the LBM scheme for multi-dimensional advection equation with space dependent velocity in
13
6
4 87
2 5
0
Fig. 1. The lattice velocities of D2Q9 model.
Table 2Velocity index and weights for 3D lattice models.
Velocities Velocity index Weights
D3Q27 D3Q19 D3Q15 D3Q27 D3Q19 D3Q15
0 0 0 0 8
27
1
9
2
9(7c,0,0), (0,7c,0),
(0,0,7c)
1–6 1–6 1–6 2
27
1
18
1
9
(7c,7c, 0), (7c,0, 7c),
(0,7c, 7c)
7–18 7–18 – 1
54
1
36
–
(7c,7c, 7c) 19–26 – 7–14 1
216
– 1
72
A. Majumder et al. / Chemical Engineering Science 70 (2012) 121–134 123
the subsequent section. The developed LBM scheme is then appliedto solve PBE with size-dependent growth by showing analogybetween the advection equation and the PBE in Section 3.
2.1. Principle
We consider the following advection equation:
@trþrr � ðvrÞ ¼ 0, ð2Þ
where r is the concentration of the species being transported(passive scalar), r is the space coordinate and v is the advection orbulk fluid velocity. LBM uses a mesoscopic representation of theadvection process obtained by considering collection of somefictitious particles, whose average dynamics resembles advectionof macroscopic density. Unlike an individual molecule, which movesrandomly, these particles are allowed to assume only certain chosenvelocities. These velocities are chosen such that they are consistentwith the symmetry and isotropy requirement of the macroscopicdynamics. The kinetic equations for the particles are modeled by thefollowing discrete Boltzmann equation with Bhatnagar–Gross–Krook(BGK) approximation (Karlin et al., 2006)
@t fiþci � rrfi ¼�1
t ðfi�f eqi ðrðf Þ,vÞÞ, i¼ 0,1, . . . ,m, ð3Þ
where fi is the discrete Boltzmann distribution function, ci is thediscrete velocity, fi
eq is the discrete equilibrium distribution function(discrete analog of the Maxwell–Boltzmann distribution of conti-nuum kinetic theory), obtained by minimizing an appropriateentropy function subject to constraints of mass and momentumconservation, and m is the number of nonzero velocities.
The lattice models available for 2D problems are D2Q5, D2Q7and D2Q9, where D stands for dimensions and Q stands for thenumber of discrete velocities used in the model (Succi, 2001).Lattice velocities and weights for D2Q9 and D2Q5 models arepresented in Table 1, where c denotes nonzero velocity compo-nent of the particles along the coordinate axes. Lattice velocitiesfor D2Q9 model are shown in Fig. 1, where the lattice velocitiesfor D2Q5 differ from D2Q9 in the sense that it does not includethe diagonal velocities. The D2Q7 model is not considered here, asit involves a hexagonal lattice, which is more difficult to imple-ment than the rectangular lattices used in D2Q5 and D2Q9models. The lattice models available for 3D problems areD3Q15, D3Q19 and D3Q27 (Succi, 2001). The lattice velocitiesfor these models and associated weights are presented in Table 2.Fig. 2 shows the lattice velocities of D3Q27 model with theindexing used in Table 2. Note that D3Q19 does not include thevelocities with magnitude
ffiffiffi3p
c, while D3Q15 does not includevelocities with magnitude
ffiffiffi2p
c. Thus, the various models differmainly in terms of the discrete velocities and associated weights.
2.2. Equilibrium distribution
The equilibrium distributions are chosen such that they minimizean underlying entropy function and thus are in accordance with the
Table 1Velocity index and weights for 2D lattice models.
Velocities Velocity index Weights
D2Q9 D2Q5 D2Q9 D2Q5
0 0 0 4
9
2
6(7c,0), (0,7c) 1–4 1–4 1
9
1
6(7c,7c) 5–8 – 1
36
–
second law of thermodynamics. The discrete form of the H-function(negative of entropy functional) is given as (Karlin et al., 1999)
H¼Xm
i ¼ 0
fi lnfi
wi
� �: ð4Þ
The H-function is non-increasing and serves as a Lyapunovfunction for the dynamics of the system. Thus, the advantage ofthis entropic approach is that the method obtained is uncondi-tionally stable by construction, which is regarded as a highlydesirable property of numerical schemes (Karlin et al., 1999;Ansumali and Karlin, 2000).
The equilibrium distributions can be found by minimizing theH-function subject to the constraints of conservation of mass andmomentum, i.e.,
Xmi ¼ 0
fi ¼ r, ð5Þ
Xmi ¼ 0
cifi ¼ ru, ð6Þ
where u is the local average velocity. The formal solution of thisminimization problem is of the Boltzmann type
f eqi ¼ expðAþByciyÞ, ð7Þ
whereA and By are the Lagrange multipliers and can be obtained bysubstituting equilibrium expression into Eq. (5). The exact solutionfor the equilibrium distribution can be obtained for few importantcases, e.g., D2Q9 model, and is given as (Ansumali et al., 2003)
f eqi ðr,ugÞ ¼ rwi
YDg ¼ 1
2�
ffiffiffiffiffiffiffiffiffiffiffiffi1þ
u2g
c2s
r ! 2ffiffi3p
ugcsþ
ffiffiffiffiffiffiffiffiffiffiffiffi1þ
u2g
c2s
r1�
ugffiffiffiffiffið3Þp
cs
� �0BB@
1CCA
cig=ffiffiffiffiffið3Þp
cs
� �, ð8Þ
12
5
6
3
4
13
10
16
8
18
12
117
14
9
15
21
22 23
24
26
25
20
1917
0
Fig. 2. The lattice velocities of D3Q27 model, (a) stationary velocity, (b) velocities with magnitude c, (c) velocities with magnitudeffiffiffi2p
c, (d) velocities with magnitudeffiffiffi3p
c.
A. Majumder et al. / Chemical Engineering Science 70 (2012) 121–134124
where cs ¼ c=ffiffiffi3p
is the speed of sound in the system. A seriesexpansion of the equilibrium distributions can be performed whichis accurate up to O(u2) to obtain the following expression valid forall lattices (Karlin et al., 1999; Boghosian et al., 2003; Ansumali et al.,2003)
f eqi ðr,uÞ ¼ rwi 1þ
ci � u
c2s
þ1
2
ðci � uÞ2
c4s
�1
2
u � u
c2s
" #: ð9Þ
It should be noted that conserving both mass and momentumduring collision will recover Navier–Stokes equation in the long-time long-wavelength limit. Since we are interested in the advectionequation, where the velocity field is given and not disturbed by theadvected species, equilibrium distribution f eq
i ðrðf Þ,uÞ is calculated atthe given velocity u¼v (Karlin et al., 2006). It is shown in Section 2.3that breaking this momentum conservation leads to advection-diffusion equation for r in the long-time and long-wavelength limit.
2.3. Recovering macroscopic description from kinetic equations
The Chapman–Enskog expansion is one of the most widelyused multiscale analysis method to find long-time, long-wave-length limit of the kinetic equation. As shown in the Appendix,solving the kinetic equation recovers the following advection-diffusion equation at the macroscopic scale, when the advectionvelocity v is independent of space coordinate r
@r@tþrr � ðrvÞ�Dr2
rr¼ 0, ð10Þ
where the diffusion coefficient D¼ tc2s . However, when advection
velocity is a function of space coordinate r, the equation recov-ered at the macroscopic level is
@r@tþrr � ðrvÞ�Dr2
rr¼ trr � ðrv � rrvÞ: ð11Þ
The term on the right-hand side of Eq. (11) is the anomaly or errorterm and is OðtMa2
Þ, where Ma¼ jvj=cs is the Mach number.
When Ma is sufficiently small ðMa� 0:01Þ, the error due to thisanomaly term is negligible. In the limit t-0, Eqs. (10) and (11)reduce to the desired advection equation, i.e., Eq. (2). Hence, LBMmodels the advection equation with an advection-diffusion equa-tion. Thus, we see that solution of the kinetic equations at themesoscopic level is equivalent to obtaining the viscous solution ofthe advection equation. In the next section, the numerical solu-tion of the kinetic equation is discussed.
2.4. Discretization of the kinetic equation
In order to solve the kinetic equation numerically, we dis-cretize it in time and integrate it over Dt using the trapezoidalrule to obtain
fiðrþciDt,tþDtÞ � fiðr,tÞ�Dt
2t½ðfiðr,tÞ�f eq
i ðr,tÞÞþðfiðrþciDt,tþDtÞ
�f eqi ðrþciDt,tþDtÞÞ�: ð12Þ
If Dt is chosen such that Dr¼ ciDt, after every time step, theparticles move to the next lattice node and space discretization isexact. It is worth noting that fiðrþciDt,tþDtÞ appears on bothsides of Eq. (12). Thus, in order to create an efficient explicitnumerical scheme, a new set of functions are defined as
giðr,tÞ ¼ fiðr,tÞþDt
2tðfi�f eq
i Þ: ð13Þ
It should be noted from Eq. (13) that local conservation is the
same for fi and gi, e.g., r¼Pm
i ¼ 0 gi ¼Pm
i ¼ 0 fi. Moreover, from
Eq. (9), fieq is a functional of fi and depends on particle population
only via its lower order moments r and u. Since the distributionsfi and gi have identical r and u, they also have the same
equilibrium, i.e., f eqi ¼ geq
i . Now, substituting for fi and fieq in terms
of gi and gieq in Eq. (12), we get
giðrþciDt,tþDtÞ ¼ ð1�abÞgiðr,tÞþabgeqi ðgðr,tÞÞ, ð14Þ
A. Majumder et al. / Chemical Engineering Science 70 (2012) 121–134 125
where
a¼ 2, b¼Dt
2tþDt: ð15Þ
Thus, we arrive at an explicit expression in terms of gi. Although gi
are not exactly particle populations, since these functions aredefined such that local conservation is the same for both vari-ables, the average quantity rðx,tÞ can be recovered from gi.
The expression for giðrþciDt,tþDtÞ in Eq. (14) is a convex linearcombination of giðr,tÞ and geq
i ðgðr,tÞÞ, if 0rbr1=2. Thus, non-linear stability can be ensured by choosing b in this range providedgi are positive at t¼0. However, in practical problems, one mightbe interested in the limit of b-1 (i.e., in limit of vanishingdiffusion, D-0), then it is possible to ensure non-linear stabilityby using entropic integrator where a is taken as solution of
Hðf Þ ¼Hðf þaðf eq�f ÞÞ: ð16Þ
Further discussion on the stability of this method has beenprovided by Karlin et al. (2006).
Numerical implementation of LBM scheme developed foradvection equation requires solving Eq. (14). In practice, thisequation is split into two consecutive steps: (1) collision and(2) streaming. The collision step involves evaluation of the right-hand side of Eq. (14), while streaming step involves shifting thevalues of the distribution functions to the next lattice node basedon their discrete velocities.
3. Lattice Boltzmann method for populationbalance equations
In this section, solution of PBEs using LBM is discussed byidentifying the analogy between the advection equation and thePBE. If we compare the advection equation given by Eq. (2) andthe PBE given by Eq. (1), we see that both equations have thesame mathematical structure, where CSD n is analogous to fluiddensity r, growth rate G is analogous to the local fluid velocity v,and space coordinate r is analogous to the characteristic length x.Thus, the LBM scheme developed for the advection equation canalso be applied for solving the PBE with size-independent or size-dependent growth rate (with Ma� 0:01). In the ensuing discussion,we present a more efficient scheme for handling size-dependentgrowth problem and implementation of nucleation.
3.1. Size-dependent growth
The growth rate of the crystals in a crystallizer often dependson the size of the crystals, e.g., the larger crystals grow faster thanthe smaller ones (Gunawan et al., 2004). In such cases, thedeveloped LBM scheme can be used with smaller Ma, so thatthe anomaly term on the right-hand side of Eq. (11) is smallenough to be neglected. A smaller Ma is maintained by choosinglarge particle velocity c. This results in smaller time step asDt¼miniDxi=c, which ultimately leads to longer computationaltime. In order to improve the computational efficiency, a coordi-nate transformation is employed so that in the transformedcoordinate system, the size-dependent PBE becomes a size-independent one and thus larger Ma can be used. In the followingdiscussion, a brief description of the coordinate transformationmethod is presented; see Majumder et al. (2010b) for details.
The size-dependent growth rate of the crystals is often modeledas (Abegg et al., 1968; Lim et al., 2002; Gunawan et al., 2004)
Gi ¼FiðtÞOiðxiÞ, where i¼ 1,2, . . . ,D, ð17Þ
where FiðtÞ and OiðxiÞ characterize the size and time dependenceof the growth rate. To avoid singularity, we assume that the sign ofOiðxiÞ does not change with OiðxiÞa0 in the computational
domain. Now, we define the transformed coordinate as z¼fz1,z2, � � � ,zDg, where zi ¼ ziðxÞ. Then, the Jacobian matrix for thistransformation is given as
Tij ¼@xi
@zj, i,j¼ 1,2, . . . ,D: ð18Þ
The CSD in the original coordinate system n(x,t) is related to theCSD in the transformed coordinate system h(z,t) as
nðx,tÞ ¼ hðz,tÞjdetðTÞj: ð19Þ
We consider that zi are chosen such that Tii40 over the domainðLmin,LmaxÞ and Tij¼0, ia j. Then, zi ¼ ziðxiÞ and
jdetðTÞj ¼Ymi ¼ 1
dxi
dzi: ð20Þ
Substituting for n in Eq. (1) using Eq. (19) and defining thetransformation as
Oi ¼dxi
dzi, ð21Þ
we have
@h
@tþXm
i ¼ 1
Fi@h
@zi¼ B0dðx�x0Þ
Ymi ¼ 1
OiðxÞ, ð22Þ
which is a PBE with size-independent growth rate (Majumderet al., 2010b). This transformed PBE can be solved with larger Maand the original CSD n(x,t) can be recovered at any time instantfrom Eq. (19).
3.2. Nucleation
Nucleation is the formation of the rudimentary particlesduring crystallization. The size of the nuclei is often assumed tocorrespond to the smallest cell considered in numerical simula-tion and nucleation is treated as boundary condition. One simpleway to implement nucleation as boundary condition is to addthe equilibrium population of the particles corresponding to thenumber of nuclei formed in the first cell during that time step as
giðx0,tþDtÞ ¼ giðx0,tÞþwiB0DtQ
iDxi,1, ð23Þ
where Dxi,1 is the size of the first grid in the ith dimension andx0 ¼ ½x1,0,x2,0, . . . ,xD,0� is the first grid point.
4. Analysis of the proposed scheme
In this section, the performance of the proposed scheme isanalyzed for smooth and non-smooth distributions in terms oferror norms defined as
L1�norm¼X
i
Xj
Xk
jEijkjDx1,iDx2,jDx3,k, ð24Þ
L2�norm¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX
i
Xj
Xk
E2ijkDx1,iDx2,jDx3,k
s, ð25Þ
where the error matrix Eijk ¼ neijk�nijk with nijk
e being the exactsolution. The error norms for 2D problems are defined similarly.For the 2D case studies, the D2Q9 model of LBM is used as it ismore accurate and stable than the D2Q5 model. For the 3D casestudy, the D3Q19 model is used. The reason behind choosingD3Q19 model is that it has been shown to provide a balancebetween the stability and the computational cost for hydrody-namic simulations (Mei et al., 2000).
The time steps for LBM in all the simulations are chosen suchthat the particle velocity c¼ 10maxgðGgÞ for size-independent
A. Majumder et al. / Chemical Engineering Science 70 (2012) 121–134126
growth, where g¼ 1,2,3. On the other hand, the time step for HRmethod is chosen such that the Courant number, DtGg=Dxg is0.1 or less in all directions (LeVeque, 2002; Majumder et al.,2010b). The LBM parameter b is taken to be 0.99995 for smoothdistributions. For non-smooth distributions, b has to be chosencarefully in order to avoid oscillations and thus a lower value of bis used. We have taken b to be 0.994 and 0.99 for 2D and 3D non-smooth distributions, respectively. All simulations are performedon a PC with an Intel Xenon X5450 processor (3.00 GHz, 4 GBRAM) using Matlab 2007b.
4.1. Smooth distribution
We consider the following Gaussian as the initial distribution
nðx1,x2,x3,0Þ ¼ exp �1
1000ððx1�mÞ2þðx2�mÞ2þðx3�mÞ2Þ
� �, ð26Þ
Fig. 3. Final distribution for the 2D Gau
−2.5 −2.4 −2.3 −2.2 −2.1 −2−5
−4.5
−4
−3.5
−3
−2.5
log (Δx)
log
(L1−
norm
)
LBM (D2Q9)HR
Slope = 1.8
Slope = 2.1
−2.3 −2.2 −2.1 −2 −1.9 −1.8 −1.7
−4.5
−4
−3.5
−3
log (Δx)
log
(L1−
norm
)
LBM (D3Q19)HR
Slope = 1.6
Slope = 2.2
Fig. 4. Convergence plots for the Gaussian distribution. Solid curves represent the b
(d) 3D Gaussian.
where m¼ 0:5 mm. The computational domain is taken as½0 1 mm� � ½0 1 mm� � ½0 1 mm�. The crystals are assumed to begrowing at a constant rate of G1 ¼ G2 ¼ G3 ¼ 0:1 mm=s. Thus, theCSD will just shift along the diagonal with time. The analyticalsolution of this problem can be found easily and is given as
nðx1,x2,x3,tÞ ¼ nðx1�G1t,x2�G2t,x3�G3t,0Þ: ð27Þ
The same Gaussian distribution is used for 2D analysis with x1
and x2 components. The problem is solved using LBM and HRmethod with periodic boundary condition, i.e., the crystals leav-ing the domain are reintroduced at the beginning of the domain.
Different grid sizes are used to solve the problem and theperformance is compared in terms of accuracy and convergence.The final distributions for the 2D Gaussian after a full cycle, i.e., attf¼10 s, obtained with grid size Dx1 ¼Dx2 ¼ 0:005 mm for bothmethods are shown in Fig. 3. It can be seen from these plots thatLBM is less diffusive than HR method, as it is able to capture the
ssian. (a) LBM and (b) HR method.
−2.5 −2.4 −2.3 −2.2 −2.1 −2−4.5
−4
−3.5
−3
−2.5
−2
−1.5
log (Δx)
log
(L2−
norm
)
LBM (D2Q9)HR
Slope = 1.7
Slope = 2
−2.3 −2.2 −2.1 −2 −1.9 −1.8 −1.7
−4
−3.5
−3
−2.5
−2
log (Δx)
log
(L2−
norm
)
LBM (D3Q19)HR
Slope = 1.5
Slope = 2.3
est fitted straight lines. (a) 2D Gaussian, (b) 2D Gaussian, (c) 3D Gaussian and
A. Majumder et al. / Chemical Engineering Science 70 (2012) 121–134 127
peak of the Gaussian better. A similar result is found for the 3DGaussian distribution (not shown here).
Convergence of LBM and HR method in terms of L1- andL2-norms is shown in Fig. 4 for the 2D and 3D Gaussian initialdistributions. It is worth noting that the curves for LBM aresituated below the curves for the HR method, which indicatesthat LBM provides more accurate solution for a given grid size. Inaddition, the convergence slopes of curves for LBM, as shown inFig. 4, are greater than those of HR method for both 2D and 3Dexamples. Thus, it can be concluded that LBM has better con-vergence properties than HR method for a smooth initial dis-tribution. The trade-off between accuracy and computation timeis shown in Fig. 5. These plots provides the useful informationabout the computational cost required to obtain a solution withtolerable error. It can be noted from these plots that curves forLBM are situated around the bottom left corner, which means thatfor a given error norm, LBM will be much faster (more than oneorder of magnitude) than HR method.
100 101 10210−5
10−4
10−3
Computation time (s)
L 1−n
orm
LBM (D2Q9)HR
102 103 10410−5
10−4
10−3
Computation time (s)
L 1−n
orm
LBM (D3Q19)HR
Fig. 5. Trade-off plots for the Gaussian distribution. Solid curves represent the fitted cur
and (d) 3D Gaussian.
Fig. 6. Final distribution for 2D non-smooth d
4.2. Non-smooth distribution
Next, we analyze the performance of LBM and HR method fornon-smooth distribution. The initial distribution is taken as thefollowing square pulse:
nðx1,x2,x3,0Þ ¼1 if 0:4 mmrx1,x2,x3r0:6 mm,
0 elsewhere:
(ð28Þ
The problem is solved using different gride sizes in the samecomputational domain as the Gaussian distribution for a full cycle.A sample distribution for 2D case at tf ¼10 s obtained with grid sizeDx1 ¼Dx2 ¼ 0:005 mm is shown in Fig. 6. We see that both LBM andHR method capture the sharp front of the square pulse withoutexcessive diffusion and oscillation. This feature is very importantfor the simulation of crystallization processes, which often involvespropagation of sharp fronts occurring due to nucleation. Similarconclusion is found for the 3D case (details not shown).
100 101
10−4
10−3
10−2
Computation time (s)
L 2−n
orm
LBM (D2Q9)HR
102 103 104
10−4
10−3
10−2
Computation time (s)
L 2−n
orm
LBM (D3Q19)HR
ves assuming power law relation. (a) 2D Gaussian, (b) 2D Gaussian, (c) 3D Gaussian
istribution. (a) LBM and (b) HR method.
A. Majumder et al. / Chemical Engineering Science 70 (2012) 121–134128
Convergence of both the methods in terms of L1- and L2-normsfor the 2D and 3D cases are shown in Fig. 7. It can be seen fromthese figures that both methods have very similar convergence for
−2.5 −2.4 −2.3 −2.2
−2.3
−2.25
−2.2
−2.15
−2.1
−2.05
−2
log (Δx)
log
(L1−
norm
)LBM (D2Q9)HR
Slope = 0.66
Slope = 0.61
−2.3 −2.2 −2.1 −2 −1.9 −1.8 −1.7
−2.6
−2.5
−2.4
−2.3
−2.2
log (Δx)
log
(L1−
norm
)
LBM (D3Q19)HR
Slope = 0.69
Slope = 0.57
Fig. 7. Convergence plots for non-smooth distribution. Solid curves represent the best fi
(d) 3D non-smooth.
101 102
0.005
0.007
0.01
Computation time (s)
L 1−n
orm
LBM (D2Q9)HR
102 103 104
10−2.6
10−2.5
10−2.4
10−2.3
10−2.2
Computation time (s)
L 1−n
orm
LBM (D3Q19)HR
Fig. 8. Trade-off plots for non-smooth distribution. Solid curves represent the fitted
(c) 3D non-smooth and (d) 3D non-smooth.
non-smooth distribution, although the convergence rate is muchlower as compared to the smooth distribution. The lower con-vergence for non-smooth distribution can be understood from the
−2.5 −2.4 −2.3 −2.2−1.42
−1.4
−1.38
−1.36
−1.34
−1.32
−1.3
log (Δx)
log
(L2−
norm
)
LBM (D2Q9)HR
Slope = 0.31
Slope = 0.28
−2.3 −2.2 −2.1 −2 −1.9 −1.8 −1.7−1.6
−1.55
−1.5
−1.45
−1.4
log (Δx)
log
(L2−
norm
)
LBM (D3Q19)HR
Slope = 0.28
Slope = 0.32
tted straight lines. (a) 2D non-smooth, (b) 2D non-smooth, (c) 3D non-smooth and
LBM (D2Q9)HR
101 1020.038
0.042
0.046
0.05
Computation time (s)
L 2−n
orm
102 103 10410−1.6
10−1.5
10−1.4
Computation time (s)
L 2−n
orm
LBM (D3Q19)HR
curves assuming power law relation. (a) 2D non-smooth, (b) 2D non-smooth,
A. Majumder et al. / Chemical Engineering Science 70 (2012) 121–134 129
fact that in the vicinity of a discontinuity, HR method switches tothe lower order method to avoid oscillations that occur for higherorder discretization (LeVeque, 2002; Gunawan et al., 2004). Onthe other hand, in case of LBM, a smaller value of b has to be usedfor non-smooth distribution to avoid oscillations. However, fromthe trade-off plots shown in Fig. 8, it can be concluded that LBMperforms better than HR method in terms of L1-norm, i.e., for thesame level of accuracy in L1-norm, LBM is more efficient. How-ever, no significant advantage is found for LBM over HR method interms of L2-norm for 3D non-smooth distribution.
5. Case studies
In this section, we consider a few multi-dimensional batchcrystallization examples taken from the literature, which arecloser to practical problems than the examples considered inSection 4.
5.1. 2D batch crystallization
First, we consider a 2D batch crystallization example pre-sented earlier by Ma et al. (2002a) and Gunawan et al. (2004),which involves batch cooling crystallization of potassium dihy-drogen-phosphate (KDP) with initial CSD given as
nðx1,x2,0Þ ¼
�3:48� 10�4ðx2
1þx22Þ if 18:05 mmrx1,x2r21:05 mm,
þ0:136ðx1þx2Þ�26:6
0 else where:
8><>:
ð29Þ
The growth rate in the ith dimension and nucleation rate aregiven as
Giðxi,tÞ ¼ 0:1kgiSdi ðtÞð1þ0:6xiÞ, i¼ 1,2, ð30Þ
B0ðtÞ ¼ kbVðtÞSbðtÞ, ð31Þ
where S is the relative supersaturation defined as (C/Csat�1), di isthe growth exponent, b is the nucleation exponent and the totalvolume of the crystals V is
VðtÞ ¼
Z 10
Z 10
1
3x3
1þðx2�x1Þx21
� �nðx1,x2,tÞ dx1 dx2: ð32Þ
The mass balance equation is
dCðtÞ
dt¼�rc
Z 10
Z 10ð2G1ðx1x2�x2
1ÞþG2x21Þnðx1,x2,tÞ dx1 dx2, ð33Þ
where rc is the crystal density. The parameters for this system areshown in Table 3. The expression for Csat(T) and the temperatureprofile are given as (Ma et al., 2002a; Gunawan et al., 2004)
CsatðTÞ ¼ 9:3027� 10�5T229:7629� 10�5Tþ0:2087, ð34Þ
TðtÞð1CÞ ¼ 32�4ð1�e�t=310Þ: ð35Þ
Since this is a size-dependent growth rate problem, the time steprequired by LBM is smaller compared to a size-independent
Table 3Simulation parameters for 2D batch crystallization.
Parameter Value Units
b 2.04 Dimensionless
kb 7.49�10�8particles=mm3=s
d1 1.48 Dimensionless
kg1 12.1 mm=s
d2 1.74 Dimensionless
kg2 100.75 mm=s
rc 2.338�10�12g=mm3
problem, which in turn results in large computation time; seeSection 3.1 for details. In order to improve the computation time,the following transformations are introduced:
dxi
dzi¼ 1þ0:6xi, ð36Þ
3xi ¼ 1:667ðexpð0:6ziÞ�1Þ: ð37Þ
In terms of the transformed variables z1 and z2, the CSDs inboth the coordinate systems are related as
hðz1,z2,tÞ ¼ nðx1,x2,tÞexpð0:6ðz1þz2ÞÞ: ð38Þ
Thus, at any time the original CSD nðx1,x2,tÞ can be recovered fromhðz1,z2,tÞ using Eq. (38).
With this transformation, the growth rate for the problembecomes a size-independent one and hence can be solved moreefficiently. The same transformation can be applied while solvingwith the HR method as well (Majumder et al., 2010b). Whencoordinate transformation is used, the corresponding LBM and HRmethod are referred to as ‘LBMT’ and ‘HRT’ method, respectively.This problem is solved by HR, HRT, LBM and LBMT for acomparison of these techniques. The final time is taken as tf ¼80 sand the number of grid points used is N1¼240, N2¼480. It isworth noting that the time step for LBM is chosen such thatc¼ 80maxgðGgÞ to keep the Ma small. The reason for choosingsmall Ma for size-dependent growth rate is to ensure that theanomaly term on the right-hand side of Eq. (11) is small enoughto be neglected.
The final distributions obtained using these methods areshown in Fig. 9. The distributions shown in these plots look verysimilar except for the fact that CSD due to nucleation look muchsharper when coordinate transformation is used. Such behavior isexpected for this example as the transformed coordinate systemis mapped to the normal coordinate system by a logarithmicrelation, which concentrates more grid points in the smaller sizeregion and thus captures the nucleation dominated CSD better.
Analytical solution for this problem is not available. To checkthe accuracy, we take the solutions obtained by applying eachmethod with N1¼720, N2¼1440 as the ‘exact’ solution. Wecompare the results obtained using each method with its corre-sponding ‘exact’ solution to calculate the error norms. Thenumerical results are summarized in Table 4. From these results,it is seen that although these methods have similar accuracy, bothLBMT and HRT have lower computational cost compared to HRand LBM, respectively, due to the use of coordinate transforma-tion. However, among the four methods, LBMT can be concludedto be the best choice for maintaining the balance betweenaccuracy and computational cost.
5.2. 3D batch crystallization
We consider batch crystallization of potash alum(KAlðSO4Þ2 � 12H2O) where the shape of the crystals is beingtracked. The PBEs, which model such problems, are called mor-phological PBEs (Ma and Wang, 2008; Borchert et al., 2009). Inmorphological PBEs, the properties of the crystals denote thenormal distance of the crystal faces from the geometric center ofthat crystal. For example, the morphology of potash alum crystalsrequires three symmetric faces to be considered, which aref1 1 1g,f1 0 0g and f1 1 0g. Distances from the origin to these facesare denoted by x1, x2 and x3, respectively. The initial distributionat t¼500 s is given as (Ma and Wang, 2008)
nðx1,x2,x3,0Þ ¼ 2200exp �2
42ððx1�9Þ2þðx2�8Þ2þðx3�8:5Þ2Þ
� �: ð39Þ
Fig. 9. Final distribution after 80 s for 2D batch crystallization (KDP-H2O system).
Table 4Simulation results for 2D batch crystallization (KDP–H2O system).
Parameter HR HRT LBM LBMT
L1-error norm 0.158 0.260 0.160 0.271
L2-error norm 0.023 0.032 0.021 0.031
Computation time (s) 520 123 78 45
A. Majumder et al. / Chemical Engineering Science 70 (2012) 121–134130
The size-independent growth rates of the crystal faces are takenas (Ma and Wang, 2008)
G1 ¼ 7:753� 10�7S1:5, ð40Þ
G2 ¼ 1:744� 10�6S1:5, ð41Þ
G3 ¼ 1:124� 10�6S1:5: ð42Þ
The solubility of the solute Csat is a function of temperature T
and is given as
lnðCsatÞ ¼ 12:1=Tþ10:47lnðTÞ�65:73: ð43Þ
The solute concentration is calculated by coupling the PBE withmass balance equation as
CðtÞ ¼Cð0Þ�CsðtÞ
1�Ms=rcCsðtÞ, ð44Þ
CsðtÞ ¼rc
Ms
Zx
Zy
Zz
Vcnðx1,x2,x3,tÞ dx dy dz, ð45Þ
where Cs is the solid concentration, rc ¼ 1760 kg=m3 is the crystaldensity, Ms¼0.47439 kg/mol is the molecular weight of the solutebeing crystallized and Vc is the volume of a crystal calculated as
Vc ¼ 4ffiffiffi3p
x31�
ffiffiffi3p
x1�x2
3�3
ffiffiffi6p
=2x1�x3
2�
� 2ffiffiffi2p
x2�ffiffiffi6p
x1
�4
ffiffiffi6p
=2x1�x3
3�: ð46Þ
The problem is solved using D3Q19 model of LBM and HRmethod with 60�80�80 mesh points. As the analytical solutionof this problem is not available, the results obtained using boththe methods with 120�160�160 mesh points are taken as thecorresponding ‘exact’ solution to calculate the error norms. Thefinal distributions obtained after tf ¼1100 s are shown in Fig. 10.Although these plots look visually similar, from the contour plotsshown in Fig. 11, it is clear that the distributions obtained usingLBM are sharper than those obtained using the HR method.Furthermore, LBM preserves the shape of the final distribution,while the HR method suffers from anisotropic diffusion. Thisfinding is also in accordance with the error norms summarized inTable 5. From these results, it can be concluded that LBM providesmore accurate results than the HR method, while requiring lowercomputation time. It is worth noting that the computationaladvantage of LBM is higher for the 2D batch crystallization thanthis 3D example. A more efficient numerical implementation ofLBM for 3D crystallization process is currently being investigated.
6. Conclusions
Lattice Boltzmann method (LBM) is developed to obtainefficient and accurate solution of multi-dimensional PBEs usedto model crystallization processes with growth and nucleation. Itis shown by multiscale analysis that LBM approximates thedesired PBE with an advection-diffusion equation, where thediffusion coefficient is a user-defined parameter. For processeswith size-dependent growth rate, the traditional LBM requireslarger computation time in comparison with processes with size-independent growth rate. In order to improve the computationtime for such processes, a coordinate transformation is per-formed, which converts the problem to a size-independent one.
010
2030
020
400
500
1000
1500
2000
CSD
, n (#
/μm
3 ) LBM
020
40
020
400
500100015002000
CSD
, n (#
/μm
3 )
LBM
010
2030
020
400
500
1000
1500
2000
x1 (μm)
x2 (μm)
x1 (μm)
x2 (μm)
x3 (μm)
x2 (μm)
x3 (μm)
x2 (μm)
CSD
, n (#
/μm
3 )
HR
020
40
020
400
500100015002000
CSD
, n (#
/μm
3 )
HR
Fig. 10. Final distributions obtained at tf¼1100 s for 3D batch crystallization (Potash alum-H2O system). (a) CSD along x1, x2 axis at x3 ¼ 22 mm, (b) CSD along x2, x3 axis at
x1 ¼ 17 mm, (c) CSD along x1, x2 axis at x3 ¼ 22 mm and CSD along x2, x3 axis at x1 ¼ 17 mm.
100
100
100
400
400
700
70010001300
1600
x1 (μm)
x 2 (μ
m)
12 14 16 18 20 22 2422
24
26
28
30
32
34
1700
LBM
100
100
100
400
400
700
70010
00
1300
x1 (μm)
x 2 (μ
m)
12 14 16 18 20 22 2422
24
26
28
30
32
34
1594
HR
Fig. 11. Contour plot of the final distribution obtained at tf¼1100 s for 3D batch crystallization (Potash alum-H2O system) showing x1, x2 axis at x3 ¼ 22 mm. (a) LBM and
(b) HR method.
Table 5Simulation results for 3D batch crystallization (Potash alum–H2O system).
Parameter HR LBM
L1-error norm 2.55�104 2.06�104
L2-error norm 1.48�103 1.05�103
Computation time (s) 118 72
A. Majumder et al. / Chemical Engineering Science 70 (2012) 121–134 131
A detailed analysis of the performance of LBM for smoothand non-smooth distribution is presented. It is found that forsmooth distribution, LBM performs significantly better than thewell-established HR method in terms of convergence, accuracyand computational cost. For non-smooth distribution, however,both methods have similar convergence, while the computation
time for LBM is lower than HR method for the same level ofaccuracy. Few examples taken from the literature show theapplicability of LBM for solving practical problems. It is foundthat LBM is more efficient than HR method while maintaining atleast the same level of accuracy. Thus, LBM has the potential to beused for online model based control of crystallization processes(Nagy et al., 2008), which require fast and repetitive solution ofthe model equations. Accurate and efficient simulation of theseprocesses is also useful for process optimization and parameterestimation (Worlitschek and Mazzotti, 2004).
The number of discrete velocities in the available LB modelsincreases rapidly with problem dimensions, which affects thememory requirement and efficiency of LBM. To improve theefficiency of LBM for solving high-dimensional PBEs, the potentialof parallel implementation and the use of LB models with fewer
A. Majumder et al. / Chemical Engineering Science 70 (2012) 121–134132
discrete velocities is being explored. Work is also in progress toextend the current LBM scheme to more complex processes,e.g., crystallization processes with combined growth, aggregationand breakage phenomena by modeling aggregation and breakageas forcing terms in the LBM formulation.
Nomenclature
B0
rate of nucleation, # s�1c
nonzero velocity component of the particles along thecoordinate axes, mm s�1
cs
speed of sound, mm s�1C
concentration of the solution, g/g Cs solid concentration in the suspension, mol m�3Csat
saturation concentration of the solution, g/g D diffusivity, mm2 s�1D
dimension of the problem E error matrix, # mm�Dfi
discrete Boltzmann distribution function, # mm�Dfieq
equilibrium discrete Boltzmann distribution function,# mm�D
gi
re-defined discrete Boltzmann distribution function,# mm�D
G
growth rate of crystals, mm s�1h
distribution function in transformed coordinate system,# mm�D
H
entropy function, # mm�DJH
entropy flux, # s�Dm
number of nonzero discrete velocities, dimensionless Ma Mach number, dimensionless Ms molecular weight of the crystal, kg/m3n
crystal size distribution, # mm�DNi
number of grid points in the ith dimension, dimensionless P pressure, mmD�2 s�2r
space coordinate, mm S relative super saturation, dimensionless T temperature, 1C t time, s u local average velocity, mm s�1v
advection velocity, analogous to G in PBE, mm s�1x
characteristic length of crystals, mm zi size in the transformed coordinate system in the ithdimension, mm
Greek letters
b
relaxation parameter, dimensionlessFi
time-dependent part of growth rate in the ith dimension,mm s�1
rc
crystal density, g mm�3t
relaxation time, s Oi size-dependent part of growth rate in the ith dimension,mm s�1
Acknowledgments
The financial support from Nanyang Technological University,Singapore through AcRF Tier 1 Grant no. RG25/07 and Depart-ment of Science and Technology, India through Grant no. SR/S2/RJN-42/2008 is gratefully acknowledged.
Appendix A
A.1. Chapman–Enskog expansion
We present the Chapman–Enskog expansion to obtain long-time, long-wavelength limit of the kinetic equation. We start withthe moments of the kinetic equation given in Eq. (3), where theEinstein notation for tensors is adopted:Zeroth moment:
@
@trþ @
@rgðrugÞ ¼ 0: ð47Þ
First moment:
@
@t
Xi
ficigþ@
@rg
Xi
cigciyfi ¼1
tX
i
cigf eqi �
Xi
cigfi
!ð48Þ
)@
@tðrugÞþ
@
@rgPgy ¼
1
t ðrvg�rugÞ, ð49Þ
where g,y represents property coordinates and Pgy ¼P
icigciyfi isthe pressure.
Now, the non-conserved variables (u, P) are expanded in aseries of a small parameter t. The time derivatives appearing inthe equations are also expanded. The physical rationale is to splitthe dynamics of the system into slow and fast components. Uponexpansion, we get
ug ¼ vgþtuð1Þg þt2uð2Þg þ � � � ð50Þ
Pgy ¼ PeqgyþtPð1Þgy þt
2Pð2Þgy þ � � � ð51Þ
@
@tf¼
@ð0Þ
@tþt @
ð1Þ
@tþt2 @
ð2Þ
@tþ � � �
� �f: ð52Þ
From Eq. (47), we have
@ð0Þ
@tþt @
ð1Þ
@tþ � � �
� �rþ @
@rg½rvgþtruð1Þg þ � � �� ¼ 0: ð53Þ
Taking O(1) and OðtÞ terms,
@ð0Þ
@tr¼� @
@rgðrvgÞ, ð54Þ
@ð1Þ
@tr¼� @
@rgðruð1Þg Þ: ð55Þ
Eq. (49) can now be written as
@ð0Þ
@tþt @
ð1Þ
@tþ � � �
� �½rvgþtruð1Þg þ � � ��þ
@
@rg½PeqgyþtPð1Þgy þ � � ��
¼1
t½rvg�rvg�truð1Þg � � � ��: ð56Þ
Taking O(1) terms,
ruð1Þg ¼�@ð0Þ
@tðrvgÞ�
@
@rgPeqgy: ð57Þ
Here, Peqgy can be found by taking second moment of the equili-
brium distributions as
Peqgy ¼
Xi
f eqi cigciy
¼ rX
i
wicigciyþrc2
s
ul
Xi
wicigciycil
þruluZ
2c4s
Xi
wicigciycilciZ�ru2
2c2s
Xi
wicigciy ð58Þ
To simplify the expression for Peqgy, the following symmetry
properties of the lattice are used:
A. Majumder et al. / Chemical Engineering Science 70 (2012) 121–134 133
Xi
wi ¼ 1 ð59Þ
Xi
wicig ¼ 0 ð60Þ
Xi
wicigciy ¼ c2s dgy ð61Þ
Xi
wicigciycil ¼ 0 ð62Þ
Xi
wicigciycilciZ ¼ c4s DgylZ ð63Þ
Finally, we have
Peqgy ¼ rc2
s dgyþruluZ
2c4s
ðc4s DgylZÞ�
ru2
2c2s
c2s dgy
¼ rc2s dgyþ
r2½uluZðdgydlZþdgldyZþdgZdylÞ��
ru2
2dgy
¼ rc2s dgyþruguy: ð64Þ
The expression for equilibrium pressure at a given velocity vbecome
Peqgy ¼ rc2
s dgyþrvgvy: ð65Þ
Substituting this relationship for the equilibrium pressure inEq. (57) and using Eq. (54), we get
ruð1Þg ¼�vg@ð0Þ
@tr�vgvy
@
@ryr�ðrvgÞ
@
@ryvy�ðrvyÞ
@
@ryvg�c2
s
@
@ryrdgy
¼ vg@
@ryðrvyÞ�vgvy
@
@ryr�ðrvgÞ
@
@ryvy�ðrvyÞ
@
@ryvg�c2
s
@
@ryrdgy
¼�ðrvyÞ@
@ryvg�c2
s
@
@rgrdgy: ð66Þ
Then the expression for rug up to linear order of t becomes
rug ¼ rvgþtruð1Þg ð67Þ
rug ¼ rvg�ðtrÞ vy@vg@ry
� ��tc2
s
@r@ry
dgy: ð68Þ
Substituting these expressions in Eq. (47), we have
@r@tþ
@
@rgðrvgÞ�tc2
s
@2r@rg@ry
dgy ¼ t@
@rgðrvyÞ
@
@ryvg
� �: ð69Þ
When v is independent of space coordinate r, then the right-handside of Eq. (69) vanishes and we have
@r@tþ
@
@rgðrvgÞ�tc2
s
@2r@rg@ry
dgy ¼ 0: ð70Þ
An alternate and more sophisticated derivation of the advec-tion-diffusion equation from the lattice Boltzmann equation canbe found in Latt (2007).
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