Quadrature-based moment methods for the population balance...

Quadrature-based moment methods for thepopulation balance equation: an algorithm review

Dongyue Li, Zhipeng Li, Zhengming Gao∗State Key Laboratory of Chemical Resource Engineering, School of Chemical Engineering,

Beijing University of Chemical Technology, Beijing, ChinaEmail: [email protected] (Z. Gao), [email protected] (D. Li)

Abstract—The dispersed phase in multiphase flows can bemodeled by the population balance model (PBM). A typicalpopulation balance equation (PBE) contains terms for spatialtransport, loss/growth and breakage/coalescence source terms.The equation is therefore quite complex and difficult to solveanalytically or numerically. The quadrature-based moment meth-ods (QBMMs) are a class of methods that solve the PBE byconverting the transport equation of the number density function(NDF) into moment transport equations. The unknown sourceterms are closed by Gaussian quadrature. Over the years, manyQBMMs have been developed for different problems, such asthe quadrature method of moments (QMOM), direct quadraturemethod of moments (DQMOM), extended quadrature method ofmoments (EQMOM), conditional quadrature method of moments(CQMOM), extended conditional quadrature method of moments(ECQMOM) and hyperbolic quadrature method of moments(HyQMOM). In this paper, we present a comprehensive algo-rithm review of these QBMMs. The mathematical equations forspatially homogeneous systems with first-order point processesand second-order point processes are derived in detail. Thealgorithms are further extended to the inhomogeneous systemfor multiphase flows, in which the computational fluid dynamics(CFD) can be coupled with the PBE. The physical limitationsand the challenging numerical problems of these QBMMs arediscussed. Possible solutions are also summarized.

Index Terms—PBE; QBMM; Multiphase flow; CFD

I. INTRODUCTION

The population balance model (PBM) can be used to de-scribe the evolution of a population of particles by the numberdensity function (NDF) for many different industrial pro-cesses, such as gas-liquid dispersions [1]–[14], liquid-liquiddispersions [15]–[28], gas-particle flows [29]–[37], [37]–[41],aerosol engineering [42]–[45], crystallization [46]–[54], react-ing flows and combustions [55]–[62], soot formation [63]–[65], and sprays [66]–[72], to cite just a few. Analysis ofthe particulate system seeks to synthesize the behavior of thepopulation of particles and its environment from the behaviorof single particles in their local environments. Readers who areinterested in the numerical aspects of the industrial processesthat are listed above are referred to the latest works [73]–[76].Among researchers, chemical engineers have put populationbalances to the most diverse use for multiphase flows [77].

However, due to the complex characteristics of the pop-ulation balance equation (PBE), the analytical solution canbe acquired only under a rigid assumption [78]–[81]. Differ-ent numerical methods were developed and applied to solvethe PBE, such as the class method (CM) [54], [82]–[84],

method of moments (MOM) [85]–[87], method of weightedresiduals (MWR) [88]–[91], Monte-Carlo method [92]–[101],direct quadrature spanning tree (DQST) method [102], [103],Lattice Boltzmann method [50], and method of manufacturedsolutions (MMS) [80], [104], [105] and others [106]–[108].Each can be further divided into sub-methods, such as thehigher-order moment conserving method of classes (HMMC)[109] and cell averaged technique (CAT) method [110], [111]in the CM, and the Taylor-series expansion method of moment(TEMOM) [112]–[115], finite-size-domain complete set oftrial functions method of moments (FCMOM) [116], [117] andmoment projection method (MPM) [118], [119] in the MOM.Each method has disadvantages and advantages over the oth-ers. For example, the CM can provide information on the shapeof the NDF. However, it requires many discretized sectionsto achieve desirable accuracy, which is highly computational-resource demanding [77]. The multivariate CM may requiremore than 10 thousand equations to be solved in certain casesand the algorithm is cumbersome [6], [120]. The MOM canpredict the mean Sauter diameter as accurately as the CM,but the main drawback is that the NDF can not be predicteddirectly. However, the derivative methods that are based onthe MOM are very popular due to their balance betweenaccuracy and computational resource requirements, especiallywhen the PBE is coupled with computational fluid dynamics(CFD). The derivative methods that are based on the MOMare shown in Table. 1. The algorithms that employ Gaussianquadrature to close the source terms are grouped into the so-called quadrature-based moment methods (QBMMs), whichdistinguish themselves from others by using the quadratureweights and abscissas to calculate the unclosed terms.

Hulburt and Kats [85] pioneered the work of developingthe MOM to solve the PBE. The MOM solves the PBE bytracking the time dependence of just the lower-order momentsof the NDF, as the lower-order moments are sufficient for theinvestigation of the industrial processes by engineers. This isthe point where it differentiates from the CM, in which theNDF is discretized as sections and each section is transportedaccording to it’s own transport equation. The source terms inPBE are closed by expressing them in terms of the lower-ordermoments. Therefore, the moment equation system is closedand the computational resource requirement is dramaticallydecreased. However, moment closure in the MOM requires arigid restriction on the mathematical form of the source term,

which becomes an obstacle for applying the MOM to realisticprocesses.

To eliminate the rigid restriction in the MOM, McGraw[121] developed the quadrature method of moments (QMOM)to solve the PBE for growth problems. In the QMOM, the mo-ments of the NDF are approximated by the n-point Gaussianquadrature, and the NDF can be approximated by the abscissaswith a delta function. The abscissas (nodes) and weightscan be calculated from the moments by a moment inversionalgorithm, e.g., the product-difference (PD) algorithm [122]or the wheeler algorithm [123]. It was shown by McGraw[121] that the growth term can be easily approximated byGaussian quadrature and it does not need to comply with aspecific mathematical form. Meanwhile, the prediction of thetime evolution of m0 to m5 agrees well with the analyticalsolution, with a relative error of less than 0.2% [121]. TheQMOM was then applied to the coagulation problem [124],and the breakage and coalescence problems [125], [126]. Dueto the simplicity of the QMOM, it can be easily coupled witha CFD solver.

However, the QMOM is only applicable to univariate dis-tributions due to the limitations of Gaussian quadrature. Toovercome this problem, Wright et al. [140] proposed a direct-inversion method in the context of particle coalescence andsintering. This method is usually referred to as the multivariateQMOM or the Brute-force QMOM [73]. Yoon and McGraw[141] developed the Tensor-product QMOM in the context ofaerosol modeling. Baldyga et al. [142] developed the GaussianCubature (GC) technique for crystallization, aggregation andsintering. Another method is the direct quadrature methodof moments (DQMOM) which was developed by Marchisioand Fox [127]. The main feature of the DQMOM is thatthe primitive variables (weights and abscissas) are transportedinstead of the conservative variables (moments). Therefore,application of the moment inversion algorithm is not necessaryafter the weights and abscissas have been found. Moreover, ex-tending the univariate DQMOM to the multivariate DQMOMis straightforward and it has been applied to different industrialprocesses [143]–[147].

One of the biggest drawbacks of the DQMOM is that it isaffected by problems that are related to the proper conservationof some moments of the NDF when it is coupled with CFD.These problems were prevented by the fully conservative DQ-MOM algorithm [6]. When the DQMOM is applied to purelyhyperbolic transport equations, it will fail if there are shocksin the abscissas due to the discontinuity. This situation occursmost commonly when the velocity is the internal coordinate.For such cases, only the multivariate moment method appearsto be able to overcome these problems. Thus, the CQMOM[120], [128] was proposed for solving the multivariate systemefficiently and accurately. It employs conditional probabilityfunction theory to approximate the multivariate NDF. Theconservative variables (the mixed moments) are transportedand it is more robust than the DQMOM for problems in whichshocks may occur (e.g., the gas velocity can be discontinuousin the Euler equation).

All the QBMMs that were discussed above only predicta discontinuous NDF, which will lead to problems in somecases. For example, if the PBE is used to simulate evaporationproblems, a term that explains the loss of particles of zerosize appears in the zero-order moment equation. To evaluatethis term, the value of the NDF for a zero-size particle isrequired. If the QBMM was employed in large eddy simu-lation (LES) framework, the velocity dispersion around thevelocity abscissa needs to be captured [148]. One possiblesolution is to increase substantially the number of abscissasso that the phase space is adequately discretized. However,the moment inversion algorithm is not accurate for N largerthan approximately 10 [73]. The QBMMs that were discussedabove will fall due to the discontinuous NDF reconstruction.To predict the continuous NDF, the EQMOM [129], [148] wasdeveloped. It can be viewed as an enhanced version of theQMOM, which employs an existing kernel density function(KDF) to approximate the NDF. After the moment transportequations have been solved, the parameters of the KDF canbe calculated and a continuous NDF can be reconstructed. Tocalculate the additional unknown variance σ in the KDF, oneadditional moment transport equation is necessary. At last, itshould be noted here that the hyperbolic quadrature method ofmoments (HyQMOM) was developed recently to circumventthe known disadvantages of the CQMOM when it was appliedto KE [132] to prevent weak-hyperbolicity. It can be alsocombined with the CQMOM for problems with multivariateNDF and a CHyQMOM was constructed.

It can be seen that a large variety of QBMMs have beendeveloped in past works to solve the PBE with univariateNDF or multivariate NDF. When the method is coupledwith CFD for inhomogeneous multiphase systems, specialnumerical treatments need to be applied to ensure robustnessand stability. The focus of this paper is to present an algorithmreview for these methods. The advantages and drawbacks ofeach QBMM are reviewed in depth and summarized. Theremainder of the work is organized as follows. In section 2 thealgorithm of the QMOM, DQMOM, EQMOM, CQMOM andECQMOM for spatially homogeneous systems is described. Insection 3 the CFD-PBE coupling algorithm for spatially inho-mogeneous multiphase systems is discussed. In section 4 thenumerical aspects of the spatially transported moments, suchas the boundedness problem, higher-order advection schemes,realizability of the moments, and reconstructed continuousNDF are investigated. Finally, conclusions are drawn in section5.

II. ALGORITHMS FOR SPATIALLY HOMOGENEOUSSYSTEMS

This NDF transport equation in the PBM can be given bydifferent names. The population balance equation (PBE) isusually used to describe the evolution of the NDF, namely,n(t,x, ξ), which is independent of the velocity but dependenton an internal coordinate vector ξ, which can be the particlesize d or the composition φ. The kinetic equation (KE) isusually used to describe the evolution of the velcocity number

TABLE I: Summary of the moment-based algorithms in the literature.

1st level derivatives 2nd level derivativesQBMM QMOM [121], DQMOM [127], DQMOM-FC [6], CQMOM [128], EQMOM [129],

ECQMOM [73], Cumulative QMOM [130], ADQMOM [131], HQMOM [132], CHyQ-MOM [132]

SQMOM [133] OPOSPM [134], MPSPM [135], SM-SQMOM [25]FCMOM [116]TEMOM [112] DEMM [136], GTEMOM [137]MOMIC [138]S − γ model [139]MPM [118]

function (VDF), namely, n(t,x,Ud), in which different veloc-ities of the particles are considered. The KE is also called theWilliams-Boltzmann equation in the field of sprays [149] andthe general particle-dynamic equation in the field of aerosols[150]. When the NDF is dependent on both the velocity andother mesoscale variables (e.g., particle size), the governingequation for the NDF (e.g., n(t,x, ξ,Ud)) is usually calledthe generalized population balance equation (GPBE). In thiswork, we focus on the algorithms of the QBMMs for the PBEin spatially homogeneous and inhomogeneous systems. Unlessotherwise stated, the internal coordinate is the particle size.

Consider a single computational volume, as shown in Fig.1.The NDF describes the number concentration of particleswith sizes between d and d + dd. Particles can break upinto smaller particles or coalesce with other particles to formlarger particles due to movement or external forces. The PBEdescribes the time evolution of the NDF. According to Fig.1,the unknown NDF is a distribution function instead of a vectorvalue or a scalar value (e.g., U or p). The following PBEgoverns the evolution of the NDF:

∂n(t,x, ξ)

∂t+∇x ·(Udn(t,x, ξ))+∇ξ ·(ξn(t,x, ξ)) = S, (1)

where n(t,x, ξ) is the NDF; x is the location vector, which isoften referred to as the external coordinate; ξ is the internalcoordinate vector; Ud is the particle’s velocity, which isassumed to be known in the PBE; S is the possible sourceterm which is often called point processes in the field of PBM[73]; ∇x and ∇ξ are divergence operators in the physical andphase spaces, respectively. Eq. (2) can be simplified furtherfor different problems. For example, if only the particle sizeis included in the internal coordinate and continuous changesof particle size (the zero-point processes) can be neglected,Eq. (2) can be written as

∂n(d)

∂t= S (2)

for a homogeneous system and

∂n(d)

∂t+∇ · (Udn(d)) = S (3)

for an inhomogeneous system (the dependences of n(t,x, ξ)on t and x are omitted for simplicity). In Eq. (3), Ud is oftencalculated by a CFD model. Nevertheless, after the simpli-fication, the exact solution of Eq. (2) can be only acquired

with rigid assumptions on the non-linear source terms. Inthe following, different solving methods for a homogeneoussystem are studied in details.

Fig. 1: An example of the NDF in one computational volume.

A. MOM

Although the MOM does not belong to the QBMMs sinceit is not based on the Gaussian quadrature closure, it is stillnecessary to discuss it here because it was the work of Hulburtand Katz [85] in which the NDF transport equation was firsttransformed into the moment transport equations. In the MOM,the k-th order moment of the NDF is defined by the followingequations:

mk =

∫ ∞0

dkn(d)dd. (4)

The lower-order moments may have specific physical mean-ings. For example, m0 represents the total number concentra-tion, m2 represents the total particle area, and m3 representsthe total particle volume. The mean particle size can be definedas the ratio mk+1/mk for any value of k. Usually, the meanSauter diameter is used, which is defined by m3/m2. If wemultiply the L.H.S. side of Eq. (2) by dk and integrate it overd, the NDF transport equation is transformed into momenttransport equations:∫ ∞

0

dk∂n

∂tdd =

∂mk∂t

=

∫ ∞0

Sdd. (5)

Eq. (5) is not closed since the source terms include theunknown functions. The MOM imposes rigid constraint on thesource terms to approximate them by the moments. Consider

a typical zero-order-process, the growth source term can bewritten as the following equation:

S = −∂Gn∂d

, (6)

where G is the growth function. If we multiply the R.H.S. ofEq. (6) by dk and integrate over d, it can be simplified by thefollowing relationship:

−∫ ∞

0

∂Gn

∂ddkdd = −

∫ ∞0

dkd (Gn)

= dkGn |∞0 +k∫ ∞

0

Gndk−1dd

= k

∫ ∞0

Gndk−1dd.

(7)

In Eq. (7), dkGn |∞0 equals zero since the tails of the NDFfunction tend to zero. By substituting Eq. (7) into Eq. (5), themoment transport equations with the growth source term canbe written as:

∂mk∂t

= k

∫ ∞0

Gndk−1dd. (8)

Eq. (8) is not closed either because its R.H.S. depends on theunknown NDF. However, as reported in the work of Hulburtand Katz [85], if the growth term has a special shape, such asφ = G/d, where G is a constant value, Eq. (8) can be writtenas follows:

∂mk∂t

= kG

∫ ∞0

ndk−2dd = kGmk−2. (9)

For simplicity, but without loss of generality, Eq. (9) can beexpanded to the following set of moment equations:

∂m0∂t

= 0,

∂m1∂t

= Gm−1,

∂m2∂t

= Gm0,

∂m3∂t

= Gm1.

(10)

The systems in Eq. (10) has four equations with five unknownvariables. Hulburt and Katz [85] further assumed that the NDFhad a special mathematical form that was parametrized by itsmoments. For example, based on the Gamma distribution, theNDF can be expressed by the leading terms of the Laguerreseries expansion. By this assumption, m−1 can be calculatedby [85]

m−1 =m20m1

2m21 −m0m2. (11)

By substituting Eq. (11) into Eq. (10), the equation system isclosed.

In the MOM, the moment equation system is self-closedby imposing a rigid restriction on the source term and theNDF. However, in contrast to the CM, only four momenttransport equations need to be solved. One may ask whetherother possible source terms, such as breakage and coalescence

terms, can be treated in a similar way to form a self-closedequation system. Unfortunately, to the best of the author’sknowledge, this is impossible due to their highly non-linearcharacteristics. This limits the ability to apply the MOM torealistic industrial processes. However, the MOM successfullytransforms the NDF transport equation to moment transportequations, which constitute the fundamental basis of otherQBMMs.

B. QMOM

The main strategy of the QMOM [121] is to approximate thetedious integrals that appear in the source terms by the n-pointGaussian quadrature. Mathematically, the Gaussian quadratureseeks to solve the integral numerically. For any given integral,it can be approximated by the following equation:∫

Ω

f(x)dx ≈N∑i=1

wif(xi), (12)

where wi are the weights, xi are the abscissas, and N isthe number of weights (which is equal to the number ofabscissas). The Gaussian quadrature calculates the integral byN ’s weights and abscissas of order 2N−1. The essence of thequadrature-based closure is that the abscissas and weights canbe completely specified in terms of the lower-order momentsof the unknown NDF. For example, the k-th order momentscan be computed by

mk =

∫ ∞0

dkn(d)dd =

N∑i=1

dkiwi. (13)

Eq. (13) can be written in the following expanded form (N =2):

w1 + w2 = m0,

d1w1 + d2w2 = m1,

d21w1 + d22w2 = m2,

d31w1 + d32w2 = m3,

(14)

For any given initial moments mk that correspond to a realNDF, Eq. (14) can be used to calculate the abscissas andweights. The direct solution to Eq. (14) requires a non-linearsolver, such as the Newton-Raphson method [151]. McGraw[121] proposed to use the product-difference (PD) algorithm[122] to find the abscissas and weights, where abscissa i andweight i correspond to the eigenvalue and the first componentof the ith eigenvector of the constructed tridiagonal matrix,respectively. The PD algorithm, or another similar algorithm(the Wheeler algorithm [123]), is usually called the momentinversion algorithm in the field of PBM. After the abscissasand weights have been calculated by the moment inversionalgorithm, the source terms can be calculated and the equationsystem is closed. In the following, the breakage (first-orderpoint process) and coalescence (second-order point process)source terms are included in the moment transported equationsas an example to demonstrate how the moment transportequations are closed in the QMOM.

The breakage and coalescence source terms are defined by[77]

S(d)

=d2

2

d∫0

a(

(d3 − d′3) 13 , d′)

(d3 − d′3) 23n(

(d3 − d′3) 13)n(d′) dd′

− n(d)∞∫

0

a(d, d′)n(d′) dd′ +

∞∫d

g (d′)β (d|d′)n (d′) dd′

− g (d)n (d) , (15)

where a(d, d′) is the coalescence kernel, which quantifies therate of coalescence of particles of size d (daughter particle)and d′ (mother particle); g(d) is the breakage kernel, whichquantifies the frequency of breakage of particles of diameter d;and β (d|d′) is the daughter size distribution function, whichdescribes the number and size of particles that are formed bya breakage event. The first and second terms in the R.H.S.of Eq. (15) represent birth and death due to coalescence; thethird and fourth terms in the L.H.S. of Eq. (15) represent birthand death due to breakage.

With the Gaussian quadrature, the death term and the birthterm of the breakage source term can be calculated by∫ ∞

0

dkg (d)n (d) dd =

N∑i=1

dki g(di)wi (16)

and

∞∫0

dk

∞∫d

g (d′)β (d|d′)n (d′) dd′ dd

=

∞∫0

n (d′) g (d′)

d′∫0

β (d|d′)dk dd

dd′=

N∑i=1

wig(di)

di∫0

β (d|di)dk dd

. (17)For a given shape of the daughter size distribution function,e.g., β (d|di) = 2/di, an solution can be derived:

di∫0

β (d|di) dkdd =di∫

0

2

didkdd =

2

k + 1dki . (18)

By substituting Eq. (18) into Eq. (17), the birth term of thebreakage becomes

∞∫0

dk

∞∫d

g (d′)β (d|d′)n (d′) dd′ dd

=2

k + 1

N∑i=1

wig(di)dki . (19)

Similarly, the death term and the birth term of the coalescencecan be calculated by

∞∫0

dk

n(d) ∞∫0

a(d, d′)n(d′) dd′

dd=

∞∫0

dkn(d)

∞∫0

a(d, d′)n(d′) dd′

dd=

N∑i=1

widki

∞∫0

a(di, d′)n(d′) dd′

=

N∑i=1

widki

N∑j=1

a(di, dj)wj

(20)and

∞∫0

dk

d22

d∫0

a(

(d3 − d′3) 13 , d′)

(d3 − d′3) 23

n(

(d3 − d′3) 13)n(d′) dd′

)dd

=

∞∫0

(u3 + d′3

) k3

d22

∞∫0

a (u, d′)

u2n (u)n(d′) dd′

u2d2

du

=1

2

∞∫0

(u3 + d′3

) k3

∞∫0

a (u, d′)n (u)n(d′) dd′

du=

1

2

∞∫0

n(u)

∞∫0

a (u, d′)(u3 + d′3

) k3 n(d′) dd′

du=

1

2

N∑i=1

wi

N∑j=1

a(di, dj)(d3i + d

3j

) k3 wj

. (21)where u3 = d3 − d′3. By substituting Eq. (16), Eq. (19),Eq. (20), Eq. (21) and Eq. (15) into Eq. (5), the moment trans-port equations for the QMOM with breakage and coalescencesource terms can be written as follows:

∂mk∂t

=1

2

N∑i=1

wi

N∑j=1

a(di, dj)(d3i + d

3j

) k3 wj

−

N∑i=1

widki

N∑j=1

a(di, dj)wj

+

N∑i=1

wig(di)

di∫0

β (d|di)dk dd

− N∑i=1

dki g(di)wi. (22)

After the weights and abscissas are calculated by the momentinversion algorithm from the given initial moments, Eq. (22)can be solved for the moments for the next time step. Inthe QMOM, the integrals in the source terms are transformedto summation operations by Gaussian quadrature. If N = 3,

only 6 moment transport equations need to be solved, whichrequires a total of 9 + 9 + 9 + 3 = 32 summation operations.

The solution procedure of the QMOM is summarized asfollows:

1) From the given initial moments, calculate the weightsand abscissas by the moment inversion algorithm (e.g.,PD algorithm).

2) Calculate the source term as reported in Eq. (22).3) Calculate the unknown time derivatives of the moments

in Eq. (22).4) The iterative procedure is repeated from step 1.The QMOM does not provide the information of the NDF

directly since it only tracks its moments. The abscissas andweights have no exact physical meaning because they are onlyparts of the quadrature approximation. Nevertheless, they canbe viewed as an approximation of the real NDF, as reported inFig.2. Under such assumption, the NDF can be reconstructedby a sum of N delta functions, where each delta functionrepresents a class of particles. Therefore, the NDF can beapproximated by

n(d) =

N∑i=1

wiδ(d− di). (23)

The abscissas represent the particle sizes and the weights aretheir relative volume fractions. Eq. (23) is quite similar to themathematical definition in the CM in which the NDF is definedby [152]

n(d) =

N∑i=1

Niδ(d− di). (24)

where Ni is the number of particles with diameter di. Themain difference between the QMOM and the CM is that in theCM, the discrete particles need to be defined a-priori, whichmay introduce some problems when the engineer does not haveinformation on the particle size interval. The only solution isto discretize the particle size over a large enough range toinclude all possible sizes. However, it forms many transportequations. In contrast, only few abscissas (e.g., N = 3) arenecessary in the QMOM and they move along the phase spaceaxis flexibly. Meanwhile, they change their volume fractions(weights) to minimize the committed error.

Compared with other QBMMs, the QMOM is the simplest,and all the other algorithms that are discussed in the followingare derived from the main concept of the QMOM. Due toits simplicity and robustness, the QMOM was employed tosimulate a large variety of applications, such as gas-liquid dis-persions [5], [153], liquid-liquid dispersions [21], gas-particleflows [32], pipe flows [154], aerosol dynamics [155], andsprays [156], to cite a few. In 2016, the original paper [121]in which the QMOM was developed was awarded the AS&TOutstanding Publication Award by the American Associationfor Aerosol Research.

C. DQMOM

The original QMOM can be only applied to problems witha univariate NDF due to the limitations of the Gaussian

quadrature. One of the motivations for the DQMOM [127]was to develop an algorithm for solving the multivariate NDFtransport equation. Another motivation was to achieve strongcoupling between the abscissas and phase velocities, whichwill be discussed in section 3.2. The main difference of theDQMOM with other QBMMs is that it tracks directly theprimary variables that appear in the quadrature approximation,rather than tracking the moments. In this section, we focus onthe algorithm of the DQMOM for the univariate NDF transportequation, in which the difference between the DQMOM andthe QMOM can be emphasized. It is straightforward to extendthe DQMOM to problems with a multivariate NDF.

By substituting Eq. (23) into Eq. (2), the NDF transportequation can be written as follows:

N∑i=0

∂wiδ(d− di)∂t

=

N∑i=0

δ(d− di)∂wi∂t−

N∑i=0

wiδ′(d− di)

∂di∂t

= S. (25)

Eq. (25) can be reformulated as

N∑i=0

δ(d− di)∂wi∂t

−N∑i=0

(δ′(d− di)

∂widi∂t

− δ′(d− di)di∂wi∂t

)

=

N∑i=0

(δ(d− di) + δ′(d− di)di

∂wi∂t

)

−N∑i=0

(δ′(d− di)

∂widi∂t

)= S. (26)

Applying the moment transform to Eq. (26) yields

∞∫0

(N∑i=0

(δ(d− di) + δ′(d− di)di

∂wi∂t

)

−N∑i=0

(δ′(d− di)

∂widi∂t

))dkdd = Sk. (27)

Given that∫dkδ(d − di)dd = dki and

∫dkδ′(d − di)dd =

−kdk−1i , Eq. (27) can be simplified to the following:

(1− k)N∑i=0

dki∂wi∂t

+ k

N∑i=0

dk−1i∂widi∂t

= Sk, (28)

which can be written in expanded form (e.g., k = 4 and N =

2):

∂w1∂t

+∂w2∂t

= S0,

∂w1d1∂t

+∂w2d2∂t

= S1,

−d21(∂w1∂t

+∂w2∂t

)+ 2d2

(∂w1d1∂t

+∂w2d2∂t

)= S2,

−2d31(∂w1∂t

+∂w2∂t

)+ 3d22

(∂w1d1∂t

+∂w2d2∂t

)= S3.

(29)

Fig. 2: Approximations of the real NDF (red solid lines) withtwo shapes (top and bottom) by the fixed-pivot CM (left) andthe QMOM (right) with two abscissas (black solid lines) andthree abscissas (green solid lines). The real NDF evolves withrespect to time and its shape changes. In the QMOM, not onlydo the weights evolve with time but also the abscissas are freeto move in the phase space. In the fixed-pivot CM, the particlesize interval is defined in advance and particle sizes are fixed;only the weights of the particles evolve with respect to time.

Eq. (29) can be written in matrix form:1 1 0 00 0 1 1−d21 −d21 2d2 2d2−2d31 −2d31 3d22 3d22

∂w1∂t∂w2∂t

∂w1d1∂t

∂w2d2∂t

=S0S1S2S3

. (30)Once the weights and the abscissas have been computedfrom the given initial moments, the source terms in Eq. (30)can be calculated in the same way as in the QMOM (seeEq. (22)). The matrix that is defined in the L.H.S. of Eq. (30),which is denoted as A, depends only on the abscissas. Ifthe abscissas are unique, A will be of full rank for arbitraryN . Subsequently, the unknown time derivative can be simplycalculated by A−1S:

∂w1∂t∂w2∂t

∂w1d1∂t

∂w2d2∂t

=

1 1 0 00 0 1 1−d21 −d21 2d2 2d2−2d31 −2d31 3d22 3d22

−1

S0S1S2S3

=a0a1b0b1

.(31)

The solution procedure of the DQMOM is summarized asfollows:

1) From the given initial moments, calculate the weightsand abscissas by the moment inversion algorithm (e.g.,the PD algorithm).

2) Calculate the source terms by Eq. (22) and construct thematrix A by Eq. (30).

3) Calculate the weights and abscissas by Eq. (31).4) The iterative procedure is repeated from step 2.The essence of the DQMOM is that the weights and

abscissas are calculated and transported directly. Therefore, themoment inversion algorithm is only necessary for determiningthe weights and abscissas in step 1. This may be an advantageover the QMOM because the convergence problem in themoment inversion algorithm may be prevented. However, thesuccessful computation of DQMOM is based on the assump-tion that the matrix A is full rank. When the abscissas arenon-distinct, A is not full rank; it is singular. In such cases,Eq. (30) does not have a unique solution. In the context ofQBMM, this means that not all the delta functions are neededto represent the NDF [127]. To overcome this problem, whichis due to the identical node’s values, it often suffices to addsmall perturbations to the non-distinct abscissas, so that Abecomes full rank [127]. Due to its easy extension to problemswith multivariate NDF, the DQMOM has been employed inmany different applications [157]–[164].

D. EQMOM

The QMOM and the DQMOM employ the delta functionto approximate the real NDF. In such cases, the continuousNDF is represented discontinuously. From a mathematicalpoint of view, the real NDF is just a non-negative probabilitydensity function. Instead of using the Dirac delta function toapproximate the NDF, the EQMOM approximates the NDFby a non-negative distribution function, such as the Gammadistribution function [129], Beta distribution function [129],Gaussian distribution function [148] or log-normal distributionfunction [165]. The distribution function is called the kerneldensity function (KDF) in the context of the EQMOM. Dueto the capability of reconstructing a continuous NDF, theEQMOM has been applied extensively in problems in whichthe reconstruction of a continuous NDF is necessary [166]–[172]. In the following, the EQMOM with a log-normal KDFis presented as an example to investigate the algorithm. It isstraightforward to employ other existing distribution functionsas the KDF.

In the log-normal EQMOM, the NDF is approximated by

n(d) ≈N∑i=0

wiδσ(d, di), (32)

where δσ(d, di) is the log-normal distribution function, whichis expressed as follows:

δσ(d, di) =1

dσ√

2πexp

(− (lnd− di)

2

2σ2

), (33)

where σ is the variance of the log-normal distribution function;di and wi are the abscissas and weights, which are usuallycalled the primary abscissas and primary weights in the contextof the EQMOM. The parameter σ is assumed to be the samefor all the KDFs to obtain a single non-linear equation. Theconcept of the reconstructed continuous NDF of the EQMOMis presented in Fig.3. It can be seen that the continuousNDF that is reconstructed from the EQMOM overlaps thereal NDF if the original real NDF follows a log-normaldistribution function, whereas the QMOM only predicts threediscontinuous abscissas.

Fig. 3: Approximations for the real log-normal NDF (red solidline) by three nodes QMOM (left) and one primary node log-normal EQMOM (right). Black solid line: the abscissas of theQMOM. Black dashed line: reconstructed continuous NDF bythe log-normal EQMOM.

The moment transport equations in the EQMOM can bederived in the same way as in the MOM or QMOM so theyare identical to the equations that are reported in Eq. (5).Moreover, the source terms that are calculated by the EQMOMshould be identical to them as well. One may, therefore, expectthe mean particle size that are predicted by the EQMOMand the QMOM to be identical. However, because anotherunknown parameter σ in the EQMOM needs to be calculated,an additional moment equation is necessary to close theequation system. For simplicity, but without loss of generality,the algorithm of the log-normal EQMOM with two primaryabscissas is discussed in the following section.

The moments of the NDF can be expressed by the analyticalsolution of the integral of the log-normal distribution function:

mk =

∫ ∞0

w1

dσ√

2πexp

(− (lnd− d1)

2

2σ2

)dd

+

∫ ∞0

w2

dσ√

2πexp

(− (lnd− d2)

2

2σ2

)dd

= w1 exp

(kd1 +

k2σ2

2

)+ w2 exp

(kd2 +

k2σ2

2

), (34)

which can be written in the following expanded form (e.g.,

N = 5):

m0 = w1 + w2

m1 = w1e

(d1+

σ2

2

)+ w2e

(d2+

σ2

2

)m2 = w1e

(2d1+2σ2) + w2e(2d2+2σ2)

m3 = w1e

(3d1+

9σ2

2

)+ w2e

(3d2+

9σ2

2

)

m4 = w1e

(4d1+

16σ2

2

)+ w2e

(4d2+

16σ2

2

)(35)

Eq. (35) is closed for five unknown parameters with fiveequations. Directly solving Eq. (35) requires a non-linearsolver, which is computational-resource demanding. However,if we eliminate w1, w2, d1, and d2 from Eq. (35), a polynomialequation of σ can be found. Unfortunately, it is not convenientto obtain the analytical solution of the polynomial equationdue to the highly non-linear dependence on σ. An iterativeroot finding procedure, such as Ridder’s method [129], [151],can be employed to update σ. Setting

z = eσ2

2 , ξ1 = ed1 , ξ2 = e

d2 , (36)

Eq. (35) leads to

m0 = w1 + w2 = m∗0,

m1 = z (w1ξ1 + w2ξ2) = zm∗1,

m2 = z4(w1ξ

21 + w2ξ

22

)= z4m∗2,

m3 = z9(w1ξ

31 + w2ξ

32

)= z9m∗3,

m4 = z16(w1ξ

41 + w2ξ

42

)= z16m∗4.

(37)

where m∗k is mathematically equivalent to the moments thatare defined in the QMOM in which the NDF is approximatedby a delta function. The σ finding procedure consists of thefollowing main steps: Calculate the corresponding mk andm∗k from a given initial σ. Update the primary weights andabscissas from the first four m∗k by the moment inversionalgorithm. Determine whether the given initial value of σsatisfies the fifth equation in Eq. (37), namely

f(σ) = m4 − e8σ2

m∗4?= 0, (38)

where m4 is the given initial moment and m∗4 equals∑5i=1 wiξ1. If f(σ) is smaller than the machine tolerance,

the guessed value of σ satisfies Eq. (37). Ridder’s method canbe used to find the root of Eq. (38), which is σ. The detailsof Ridder’s method can be found in the literature [151]. Here,we only summarise the iterative procedure that was applied tothe EQMOM:

1) Calculate the midpoint σ3 = 0.5(σ1 + σ2) from twogiven initial roots, σ1 and σ2.

2) Calculate σ4 by

σ4 = σ3 + (σ3−σ1)sign (f(σ1)− f(σ2)) f(σ3)√

f(σ3)2 − f(σ1)f(σ2). (39)

3) Check whether f(σ4) = 0. If yes, the iterative proce-dure converges. Otherwise, repeat the iterative procedurefrom step 1.

After σ has been found, the continuous NDF can beconstructed by Eq. (32). To calculate the moments for the nexttime step, the source terms need to be calculated. One maywant to calculate the source terms in the EMOM following theprocedure that was employed in the QMOM. However, eventhe moment transport equations for the EQMOM are identicalwith the equations in the QMOM, the closure in the EQMOMis much more complicated than in the QMOM due to thecontinuous reconstructed NDF. In the following, the breakagesource terms were employed as an example to demonstratethat secondary abscissas and weights need to be introduced tocalculate the source terms.

The death term of the breakage can be calculated by∫ ∞0

dkg (d)n (d) dd =

∫ ∞0

dkg (d)

(N∑i=0

wiδσ(d, di)

)dd

=

N∑i=1

wi

∫ ∞0

dkg (d) δσ(d, di)dd

=

N∑i=1

wi

∫ ∞0

dkg (d)1

dσ√

2πexp

(− (lnd− di)

2

2σ2

)dd.

(40)

Define

s2 =(lnd− di)2

2σ2→ d = exp(σs

√2 + di)

→ dd =√

2σ exp(σs√

2 + di)ds. (41)

Substitute Eq. (41) into Eq. (40):

N∑i=1

wi

∫ ∞0

dkg (d)1

dσ√

2πexp

(− (lnd− di)

2

2σ2

)dd

=

N∑i=1

wi

∞∫−∞

(exp(σs

√2 + di)

)kg(exp(σs

√2 + di)

)exp(σs

√2 + di)σ

√2π

e−s2√

2 exp(σs√

2 + di)ds

=1√π

N∑i=1

wi

∞∫−∞

(exp(σs

√2 + di)

)kg(

exp(σs√

2 + di))e−s

2

ds. (42)

Eq. (42) can be calculated by the Gauss-Hermite formula:

1√π

N∑i=1

wi

∞∫−∞

(exp(σs

√2 + di)

)kg(

exp(σs√

2 + di))e−s

2

ds

=1√π

N∑i=1

wi

Nj∑j=1

(exp(σsj

√2 + di)

)kg(

exp(σsj√

2 + di))wj

), (43)

TABLE II: The numbers of the summation operations for theQMOM and the EQMOM with different numbers of abscissas.

Number ofnodes and weights

Number ofsummation operation

QMOM N = 2 4N = 3 9

EQMOM N = 2, N′ = 10 400

N = 2, N ′ = 20 1600

where Nj is the number of the secondary abscissas sj andweights wj . Eq. (43) is the final form of the death term ofthe breakage in the EQMOM. It can be seen that sj and wjare introduced in order to calculated the integral in Eq. (43).We can choose a large value of Nj to improve the accuracyof Eq. (43). The calculation of other terms is neglected herefor brevity. The final breakage and coalescence source termsare reported as follows:

Sk ≈1

2π

N∑i1=1

N∑i2=1

wi1wi2

Nj∑j1=1

Nj∑j2=1

wj1wj2[(�3i1,j1 + �

3i2,j2

)k/3 − �ki1,j1 − �ki2,j2] ai1,j1,i2,j2+

1√π

N∑i1=1

N∗∑j1=1

wi1w∗j1gi1,j1

(bk

i1,j1 − �ki1,j1

), (44)

where ai1,j1,i2,j2 = a(�i1,j1 , �i2,j2), gi1,j1 = g(�i1,j1), andwhere:

bk

i1,j1 =

∞∫0

dkβ (d|�i1,j1) dd, �i1,j1 = �(sj , di) (45)

The solution procedure of the EQMOM is summarized asfollows:

1) Guess σ, calculate the corresponding m∗k by Eq. (37)from the given first even initial moments. Calculate theprimary weights and abscissas by the moment inversionalgorithm (e.g., the PD algorithm).

2) Determine whether the convergence criterion is satisfied,which is given in Eq. (38). If yes, calculate the sourceterms. Otherwise, apply Ridder’s iterative procedureuntil the convergence criterion is satisfied.

3) Calculate the source terms and the unknown time deriva-tives of the moments in Eq. (22).

4) The iterative procedure is repeated from step 1.In contrast to the QMOM, the EQMOM employs 2N + 1

moments to calculate the N primary weights and abscis-sas, and an additional parameter σ, which can be usedto reconstruct a continuous NDF. Obviously, the additionalmoment transport equation imposes increased computationalresource requirement. In addition, an iterative procedure withthe Ridder’s method is carried out in each time-step loop.This dual-iterative feature of the EQMOM leads to slowcalculation. Sometimes the calculations diverge, depending onthe initial guess for σ. However, it was found that most of thecomputational resources were spent on the calculation of the

source terms [24] and the EQMOM can be much slower thanother QBMMs, such as the DQMOM or even the CM [173].As reported in Table. II, there are only N × N summationoperations for the coalescence source terms for the QMOM.However, for the EQMOM with N primary abscissas and N ′

secondary abscissas, there are (N×N ′)×(N×N ′) summationoperations for the coalescence source terms.

E. CQMOM

The CQMOM was first introduced and validated for passivescalars transportation [120] for Flash Nanoprecipitation. Oneyear later, the comprehensive theory of the CQMOM waspresented by Yuan et al. [128], and it was applied to activescalars (e.g., velocities). Compared with the QMOM and theEQMOM and other methods (e.g., the multivariate QMOM[140], the tensor-product QMOM [141] or the DQMOM),the CQMOM is a more popular method for problems withmultivariate NDF. It is based on conditional density function(CDF) theory. The CDF represents the probability of havingone internal coordinate within an infinitesimal limit whenone or more of the other internal coordinates are fixed andequal to a specific value [73], [128]. For clarity, we willonly discuss the algorithm of the CQMOM with two internalcoordinates, namely, particle size d and particle composition φ.Moreover, two abscissas for the particle size and compositionare employed as an example to demonstrate the algorithm ofthe CQMOM. It is straightforward to increase the number ofinternal coordinates for other problems.

The bivariate NDF, which is denoted as n(d, φ), can bewritten as

n(d, φ) = n(φ|d)n(d), (46)

where n(φ|d) corresponds to the CDF for φ given a fixed valueof d. Therefore, the conditional moments for φ are defined by

〈φ〉j (dα) =∫φjn(φ|dα)dφ =

Nβ∑β=1

wα,βφjα,β , (47)

where wα,β and φα,β are the conditional weights and abscissasfor φ, respectively; Nβ is the number of the abscissas for φ.Eq. (47) implies that any given value of dα corresponds toa conditional moment set 〈φ〉j (dα). The conditional momentset can be used to find the conditional weights and abscissasfor the conditioned internal coordinate. It is straightforward toshow that

mi,j =

∫ ∫diφjn(d, φ)dddφ

=

∫di(∫

φjn(φ|d)dφ)n(d)dd

=

Nα∑α=1

wαdiα

(∫φjn(φ|dα)dφ

)=

Nα∑α=1

wαdiα 〈φ〉

j(dα)

=

Nα∑α=1

wαdiα

Nβ∑β=1

wα,βφjα,β

, (48)

where Nα is the number of abscissas for d. Eq. (48) canbe used to find the conditional weights and abscissas forthe conditional (secondary) internal coordinate from the puremoments (mi,0). For example, if j = 0, the weights andabscissas can be calculated by the moment inversion algorithmas discussed in the previous sections:

m0,0m1,0m2,0m3,0

PD→w1w2d1d2

. (49)

After the weights and abscissas have been found, the condi-tional weights and abscissas for the second internal coordinatecan be calculated from the conditional moments by Eq. (47).These unknown conditional moments can be calculated byEq. (48). For example, after d1 and d2 have been calcu-lated, the conditional moments 〈φ〉j (dα) can be calculatedby Eq. (48), which can be written as the following expandedform (e.g., Nα = Nβ = 2):

m0,1 = w1 〈φ〉1 (d1) + w2 〈φ〉1 (d2),m1,1 = w1d1 〈φ〉1 (d1) + w2d2 〈φ〉1 (d2),m0,2 = w1 〈φ〉2 (d1) + w2 〈φ〉2 (d2),m1,2 = w1d1 〈φ〉2 (d1) + w2d2 〈φ〉2 (d2),m0,3 = w1 〈φ〉3 (d1) + w2 〈φ〉3 (d2),m1,3 = w1d1 〈φ〉3 (d1) + w2d2 〈φ〉3 (d2).

(50)

Eq. (50) can be written in the following matrix form:

[〈φ〉j (d1)〈φ〉j (d2)

]=

[w1 w2w1d1 w2d2

]−1 [m0,jm1,j

](51)

As long as the abscissas di are distinct, the Vandermondelinear system in Eq. (51) is of full rank and the necessary con-ditional moments 〈φ〉j (d1) and 〈φ〉j (d2) can be calculated.By virtue of Eq. (47), the conditional weights and abscissascan be calculated by the following relationship:

〈φ〉0 (d1)〈φ〉1 (d1)〈φ〉2 (d1)〈φ〉3 (d1)

PD→w1,1w1,2φ1,1φ1,2

,〈φ〉0 (d2)〈φ〉1 (d2)〈φ〉2 (d2)〈φ〉3 (d2)

PD→w2,1w2,2φ2,1φ2,2

. (52)

The calculation of the source terms in the CQMOM needsto take into account the multivariate NDF. For example, thesource terms for the breakage and coalescence processes are

expressed as follows [6], [174]:

S(d, φ) =d2

2

d∫0

φ∫0

a((d3 − d′3)1/3, d′

)(d3 − d′3)2/3

n(

(d3 − d′3)1/3, (φ− φd′))n(d′, φd′) dd

′ dφd′

− n(d, φ)∞∫

0

a(d, d′)n(d′, φd′) dd′ dφd′

+

∞∫d

∞∫φ

g (d′)β (d, φ|d′, φd′)n (d′, φd′) dd′ dφd′

− g (d)n (d, φ) , (53)

Eq. (53) is different from Eq. (15) due to the existence ofthe multivariate NDF, and the closure is different. Moreover,

the DSD function needs to be extended. Readers who areinterested in the theory are referred to the latest works [6],[73] for comprehensive information. Here, we employ thedeath term and the birth term of the breakage source termas an example, to demonstrate the calculation procedure. Thebreakage sink term can be calculated as follows:

∞∫0

∞∫0

diφjg (d)n (d, φ) dddφ

=

∞∫0

dig (d)

∞∫0

φjn (φ|d) dφ

n(d)dd=

Nα∑α=1

diαg(dα)wα

Nβ∑β=1

φjα,βwα,β

. (54)The breakage source term can be calculated as follows:

∞∫0

∞∫0

diφj∞∫d

∞∫φ

g (d′)β (d, φ|d′, φd′)n (d′, φd′) dd′ dφd′ dddφ

=

∞∫0

n(d′)g(d′)

∞∫0

n(φd′ |d′))

d′∫0

φd′∫0

diφjβ (d, φ|d′, φd′) dddφ

dφd′ dd′

=

Nα∑α=1

wαg(dα)

Nβ∑β=1

wα,β

dα∫0

φα,β∫0

diφjβ (d, φ|dα, φα,β) dddφ

. (55)

For a given bivariate DSD function [6]:

β (d, φ|dα, φα,β)

= 180d2

d3α

(d3

d3α

)2(1− d

3

d3α

)2δ

(φ− d

3

d3αφα,β

), (56)

the analytical solution can be calculated by

dα∫0

φα,β∫0

diφjβ (d, φ|dα, φα,β) dddφ

= 3240diαφ

jα,β

(i+ 3j + 9)(i+ 3j + 12)(i+ 3j + 15). (57)

By grouping Eq. (57), Eq. (55) and Eq. (54), the breakage

source terms can be written as

Sbri,j =

Nα∑α=1

wαg(dα)

Nβ∑β=1

3240wα,βdiαφ

jα,β

(i+ 3j + 9)(i+ 3j + 12)(i+ 3j + 15)

−Nα∑α=1

diαg(dα)wα

Nβ∑β=1

φjα,βwα,β . (58)

As illustrated in Fig. 4, the solution procedure of theCQMOM is summarized as follows:

1) Calculate the abscissas and weights for the first internalcoordinate from the given initial pure moments.

2) Construct the Vandermonde matrix from the abscissasand the weights.

3) From the Vandermonde matrix and the given initialmixed moments on the second internal coordinate, cal-culate the conditional moments 〈φ〉j (dα) by Eq. (51).

4) From the calculated conditional moments, calculate theconditional abscissas and the weights by the momentinversion algorithm. Calculate the source terms byEq. (58).

5) The iterative procedure is repeated from step 1.As an algorithm for problems with a multivariate NDF, the

CQMOM is suitable and applicable for simulating the multi-phase system with mass transfer [174], [175] and chemical re-actions [176], in which the internal coordinates include particlesize and composition. In addition, the CQMOM is extensivelyapplied to the multi-dimensional KE for gas-particle flows, inwhich the velocity components (e.g., u and v) can be treated asdifferent internal coordinates. For such flows, the most obviousmanifestation of non-equilibrium behavior is particle trajectorycrossing (PTC), which can be only predicted with multiplevelocity nodes by the CQMOM. In such cases, particles withthe same size can be transported with different velocities(abscissas). Readers who are interested in this subject arereferred to other works [128], [148], [171], [177].

F. ECQMOM

The CQMOM can be considered as an improved QMOM-based method for problems with a multivariate NDF. In suchcases, the NDF is discontinuously approximated. Is it possibleto merge the CQMOM with the EQMOM to predict the con-tinuous multivariate NDF? The answer to this question seemsto be yes. The ECQMOM concept was first developed andapplied for gas-particle flows in the 2D large eddy simulation(LES) framework by [148] and was discussed in detail by[73]. In the work of Chalons et al. [148], the ECQMOM withGaussian KDF was called multi-Gaussian (MG) quadraturemethod. Because they aimed at LES, only the ECQMOM cancapture the velocity dispersion around each quadrature point[166], [171], [178]. Similar to the EQMOM, the multivariateNDF in the ECQMOM is approximated by

n(d, φ) ≈Nα∑α=1

wαδσ (d, dα)

Nβ∑β=1

wα,βδσα (φ, φα,β)

,(59)

where σ and σα are the variances for the first internalcoordinate and the second internal coordinate, respectively.To solve for these two additional unknown variances in theunclosed equation system, two additional moment equationsare necessary. The solution procedure of the ECQMOM isillustrated in Fig. 4. The calculations of the source terms andthe moment transport equations are similar to the CQMOMand the EQMOM. Therefore, the calculation procedure of theECQMOM is omitted here for simplicity.

III. CFD-PBE COUPLING FOR INHOMOGENEOUSMULTIPHASE SYSTEMS

All the algorithms that were discussed above can be ex-tended for spatially inhomogeneous multiphase systems. Insuch cases, the flow field information (e.g., turbulent energydissipation rate) needs to be specified. The information of theflow field is usually provided by the CFD simulation. Then,a CFD-PBE coupling procedure should be formulated. TheCFD-PBE coupling can be implemented in any CFD code,such as ANSYS Fluent or OpenFOAM. Recently, an open-source OpenFOAM-based code which is called OpenQBMM

was released. The QMOM, EQMOM and CQMOM wereimplemented with multiple sub-models [179]. In the coupledCFD-PBE system, a corresponding spatial advection term anda diffusion term for fine particles [73], [180] need to beincluded, which will be discussed in the following sections.

A. CFD coupling with the QMOM and the EQMOM

In the QMOM and EQMOM in which the moments aretransported, the moment advection terms and diffusion terms(for fine particles) should be included. Take the advectionterm as an example. If we apply the moment transformationin Eq. (3), the moment transport equations can be written asfollows:

∂mk∂t

+∇ · (Udmk) = Sk, (60)

where Sk is the integral form of the source term, Ud isthe dispersed phase velocity, which can be predicted by theCFD. Depending on the specific problem, Eq. (60) needs tobe addressed separately. If it is coupled with a single-phaseCFD solver, it is common to implement the moment transportequations as passive scalar transport equations, which can beviewed as a type of one-way coupling. In such cases, thesolution of the PBE has no effect on the flow field and theimplementation is straightforward. If it is coupled with a two-phase CFD solver, in which the PBE is used to describethe dispersed phase’s movement and evolution, Ud can becalculated by the two-phase CFD solver (e.g., two-fluid model)and the effect of the particle size on the interfacial forceexchange term needs to be considered. In the incompressibletwo-fluid model (TFM) without mass and heat transfer, thegoverning equations are reported by [181]–[183]

∂αd∂t

+∇ · (αdUd) = 0, (61)

∂ (αdUd)

∂t+∇·(αd (Ud ⊗Ud))+∇·(αdτd)+∇·(αdRd)

= −αd∇p+ αdg + Md, (62)

∂ (αcUc)

∂t+∇ · (αc (Uc ⊗Uc)) +∇ · (αcτc) +∇ · (αcRc)

= −αc∇p+ αcg −Md, (63)

where αd is the volume fraction of the dispersed phase, ρdis its density and Ud is its average velocity, p is the pressureshared by the two phases, τd and τc are the viscous-stresstensors, Rd and Rc are the Reynolds-stress tensors, g isthe gravitational acceleration vector and Md is the interfacialforce term. The CFD-PBE coupling lies in the calculationof the Md. In the TFM, the particle size is assumed tobe constant. However, in the CFD-PBE coupling, the meanSauter diameter for each computational grid is calculatedfrom the predicted abscissas and weights, and the growth andbreakage/coalescence can be included. Moreover, it is commonfor strong breakage and coalescence to occur in the turbulentmixing processes. The TFM will fail in predicting such aphenomenon.

Fig. 4: Solution procedure for CQMOM (left) and ECQMOM (right) with two internal coordinates (d and φ). The elementsin gray were given as the initial mixed moments (mi,j).

The coupling of CFD with the moment transport equationsis summarized as follows:

1) Calculate the interfacial force exchange term (e.g., dragforce) from the mean Sauter diameter, which is com-puted from the initial moments (e.g., m3/m2).

2) Calculate the information of the flow field by the TFMwith the updated interfacial force exchange term.

3) Feed the calculated advection velocity into the momenttransport equations and update the moments for the nexttime step.

4) The iterative procedure is repeated from step 1.The coupling of CFD with the moment transport equationsis suitable for a large variety of applications. However, itcan be seen that all moments are transported with the samedispersed velocity, as reported in Eq. (60). Thus, all particlesare moving with the same velocity: Ud. This approximation isonly applicable when the Stokes number is sufficiently smallthat the shape of the NDF is generally narrow enough not tohave a significant size-segregation effect.

B. CFD coupling with the DQMOM

If the DQMOM is employed to solve the PBE, anothercoupling procedure is required because the moment transportequations are transformed to the weights/abscissas transportequations. The weights and abscissas can be transported withdifferent velocities, which is another important feature ofthe DQMOM. The coupling is therefore applicable for theproblems with a wide NDF shape. Extending the homoge-neous DQMOM algorithm to the inhomogeneous DQMOM isstraightforward; only the advection term needs to be includedin Eq. (31), which can be written as follows:

∂w1∂t +∇ · (w1U1)∂w2∂t +∇ · (w2U2)

∂w1d1∂t +∇ · (w1d1U1)

∂w2d2∂t +∇ · (w2d2U2)

=a0a1b0b1

(64)where U1 and U2 are the advection velocities for the firstabscissa and second abscissa, respectively. If we assume

U1 = U2 = Ud, (65)

the predicted abscissas and weights by the DQMOM willbe identical to the results that are predicted by the momenttransport equations as reported in Eq. (60). However, Eq. (65)goes against the motivation of developing the DQMOM tostrongly couple the internal coordinates with the phase ve-locities. To predict the different phase velocities for differentinternal coordinates, the DQMOM is usually coupled with themulti-fluid model (MFM) instead of the TFM, where only onedispersed phase velocity is predicted.

C. CFD coupling with the CQMOM and ECQMOM

In the moment transport equations in the CQMOM and theECQMOM, special treatments need to be included, dependingon the specific problems. If the phase-space diffusion isneglected, it is enough to add the advection terms into themixed moment transport equations as reported in Fig. 4. IfN1 = 2 and N2 = 2, ten and thirteen moment transportequations need to be solved for the CQMOM and the ECQ-MOM, respectively. This coupling procedure is similar to thecoupling of CFD with QMOM/EQMOM. The CQMOM andECQMOM was applied to simulate Flash Nanoprecipitationand droplet evaporation [120], [156]. However, for the masstransfer of species when chemical reactions occur, the phase-space diffusion terms should be included [174], [175]. Forsuch problems, the drift terms should be added into the NDFtransport equation reported in Eq. (1):

∂n(d, φ)

∂t+∇ · (Udn(d, φ)) +∇φ

(φ̇n(d, φ)

)= S (66)

where φ̇ is the rate of continuous change of particle compo-sition due to molecular phenomena. Transforming the thirdterm in the L.H.S. of Eq. (66) to the moment terms followsthe same procedure as that reported in Eq. (7). Anotherimportant application of the CQMOM is to solve the transportequation of the velocity distribution function (VDF) to capturethe PTC, which can be only predicted by multiple valuesof the advection velocities. However, it should be stressedhere that the PBE is not suitable for such processes, and thecoupling procedure that was discussed here is not suitablefor predicting PTC because the velocity is excluded from the

internal coordinate. As the discussion of the CFD couplingwith the KE is beyond of the scope of this work, readers whoare interested in this subject are referred to other works [128].

IV. NUMERICAL PROBLEMS AND LIMITATIONSA. Boundedness of the phase fraction and moments

The moments of the NDF represent some important physicalproperties of the population of particles and they have to obeyphysical rules. For example, when the internal coordinate isthe particle diameter, m3 is linearly dependent on the phasefraction:

m3 = kvα, (67)

where kv is the shape coefficient which equals π/6 for a sphereparticle. Under such assumption, the transport equation of m3(without source terms) is equivalent to the dispersed phasefraction transport equation, which implies

∂m3∂t

+∇ · (Udm3) = 0 (68)

equals∂αd∂t

+∇ · (Udαd) = 0. (69)

When Eq. (68) and Eq. (69) are solved by the finite volumemethod, the key issue is to ensure the boundedness of m3and the phase fraction, as unbounded solutions may lookreasonable but are completely erroneous. By close examinationof Eq. (69), the use of the upwind approximation can guaranteethat the phase fraction is bounded by zero, but it cannotguarantee that it is bounded by one. If the phase equations forboth phases are solved together and an appropriate boundeddiscretization scheme is applied, α > 0 and the constraintα 6 1 will be obeyed [184]. However, this method does notguarantee conservation, which is a critical issue in multiphaseflows with high density ratios because small phase fractionerrors may correspond to large mass-fraction errors. Weller[185] proposed using the following equation instead of theoriginal one:

∂αd∂t

+∇ · (Uαd)−∇ · (Ur(1− αd)αd) = 0, (70)

where Ur is the relative phase velocity and U is the averagephase velocity, which equals αdUd +αcUc. Eq. (70) ensuresthe boundedness of the phase fraction in theory. In the contextof the moments, m3 should be bounded roughly in [0, 1.91],as reported in Eq. (67). Unfortunately, the moment transportequations have the same boundedness problem as of the∇ ·Ud 6= 0 in Eq. (60). Buffo et al. [186] were inspired byWeller [185] to reformulate the moment transport equationsas follows:

∂mk∂t

+∇ · (Umk)−∇ · (Ur (1− αd)mk) = 0. (71)

It was proven that Eq. (71) ensure the boundedness of themoments [186]. However, the practical numerical procedurehas limitations as well:

1) Eq. (71) can only utilize the first-order scheme becauseof the realizability problems which will be discussed in

the next section. If a higher-order scheme was employedfor Eq. (70), and a first order scheme was employed forEq. (71) separately, the predict αd and π6m3 will bedifferent.

2) The strictly boundedness between 0 and 1 still can not beensured by Eq. (70) due to its non-linear dependence onαd as shown in the third term. A slightly negative valueof αd introduces divergence of the simulations. One pos-sible solution is to solve the equation for αd and αc it-eratively in each time step. However, it is computationaldemanding. Another practical solution is to employ theexplicit Flux-correct transport (FCT) algorithm to solveEq. (70), the variation of which was firstly implementedby OpenFOAM foundation in OpenFOAM-1.4.1 for thephase fraction equation. Therefore, it is reasonable toemploy the FCT algorithm for the moment transportequations as reported in Eq. (71). However, due to theanti-diffusion procedure in FCT [187], it might introducethe realizability problems. Such numerical procedureneeds further investigation.

B. Higher-order scheme and realizability

When a higher-order scheme is employed for the advectionterm, Eq. (71) tends to diverge. This problem is called the mo-ment realizability problem or the moment corruption problemin the field of PBM. From the aspect of CFD, the realizabilityrefers to the solution of a problem being physically realistic[188]. Algorithms that are not realizable may result in non-physical solutions or cause numerical methods to diverge. Inthe context of the QBMM, whenever the moment set corre-sponds to a non-negative NDF, the moment set is realizable.Unfortunately, the realizability of the moment set has beenproven to be difficult to maintain by many works when thePBE is coupled with CFD for inhomogeneous systems [116],[189]–[192]. This non-realizable moments set is called the“invalid moment sets” [189]. If the invalid moment set werefed into the moment inversion algorithm, unrealizable abscis-sas would be calculated (e.g., d < 0), thereby jeopardizingthe stability of the simulation. The necessary and sufficientcondition for the existence of a non-negative and unique NDFfor a moment set is the non-negativity of the Hankel-Hadamarddeterminant [193]

∆k,l =

∣∣∣∣∣∣∣∣∣mk mk+1 ... mk+1mk+1 mk+2 ... mk+l+1

......

......

mk+l mk+l+1 ... mk+l+l

∣∣∣∣∣∣∣∣∣ ≥ 0. (72)When the Hankel-Hadamard determinant is negative, the mo-ment sets are invalid. However, it is not reasonable to computeall the Hankel-Hadamard determinants in each time step.Laurent and Nguyen [192] proposed to check the positivityof the ζk which are parameters of the PD algorithm. When ζkis positive, realizability criterion is satisfied.

Different algorithms were developed in order to overcomethe problem. The correction algorithm [189] employs thefirst three moments to calculate the three parameters of the

log-normal distribution function and calculate the entire setof the realizable moments. The adaptive Wheeler algorithmreturns the largest possible set of weights and abscissas [128].However, when the moment transport equations are coupledwith the CFD, it does not make sense to correct the invalidmoment sets in each time step when the higher-order schemeis employed, as this leads to moment corruption. Vikas et al.[194] developed a realizable higher-order finite volume schemefor the KE with an advection term, and it was lately extendedby Vikas et al. [191] to the PBE with a diffusion term. Thisscheme was based on the kinetics-based finite volume method(KBFVM) [73] since it was originally developed for the KE.However, the KBFVM is also applicable to the PBE. In thefollowing, the higher-order scheme proposed by Vikas et al.[194] is called Vikas scheme.

Fig. 5: The computational cell labelled by “owner” and “neigh-bour” in the unstructured computational mesh framework.

For the unstructured computational mesh, as reported inFig.5, the NDF transport equations with constant dispersedvelocity Ud can be written as

∂n

∂t+∇ · (Udn) = 0. (73)

The source terms are neglected since they do not create non-realizability problems. If an explicit Euler time scheme is usedwith the finite volume method, the updated NDF for gridlabelled by “own” for the time step t+ ∆t can be calculatedby

nt+∆town = ntown − λ

∑f

F tfntf , (74)

where nown and nnei is the NDF for the owner grid and theneighbour grid, respectively; λ equals ∆t∆Vown , in which ∆t isthe time step, Vown is the mesh volume; F tf and n

nf is the flow

flux and the NDF defined at the cell face “f” at time step t. Ifa first-order upwind scheme is applied, the ntf can be inferredfrom the upwind cell’s value:

ntf =

{ntown, F

tf > 0

ntnei, Ftf < 0

(75)

Substituting Eq. (75) into Eq. (74) yields

nt+∆town

= ntown − λ∑

max(F tf , 0

)ntown − λmin

(F tf , 0

)ntnei

=

1− λ∑f

max(F tf , 0

)ntown−λ∑f

min(F tf , 0

)ntnei

(76)

It can be seen that the last term in the R.H.S. of Eq. (76) isnon-negative. If

∆t 6 min

∆Vown∑f max

(F tf , 0

) (77)

and the given initial NDF is non-negative, the NDF calculatedby Eq. (76) will be always non-negative. This concludes theproof that when a first-order upwind scheme is employedfor the PBE with an advection term, the reliability problemdisappears when the CFL condition is satisfied, as reported inEq. (77). The grouping procedure for the first two terms in theR.H.S of Eq. (76) is essential to solve the realizability problemand it forms the basis of the Vikas scheme [73], [194].

When a higher-order scheme is employed, Eq. (76) does nothold and it can not be simplified by grouping. However, if wesubstitute Eq. (23) into Eq. (73), it leads to

nt+∆town =

N∑α=1

wtα,ownδ(d− dα,own)t

− λ∑f

F tf

(N∑α=1

wtα,fδ(d− dα,f )t). (78)

By virtue of the KBFVM, the weights and abscissas on thecell face f are split as follows:

Ff

N∑α=1

wtα,fδ(d− dα,f )t

=

N∑α=1

max(Ffown , 0)wtα,fownδ(d− dα,fown)

t

+

N∑α=1

min(Ffnei , 0)wtα,fneiδ(d− dα,fnei)

t. (79)


nt+∆town =

N∑α=1

wtα,ownδ(d− dα,own)t

− λN∑α=1

∑f

max(Ffown , 0)wtα,fownδ(d− dα,fown)

t

− λN∑α=1

∑f


t. (80)

The essence of the Vikas scheme is to employ the first-orderspatial scheme for the abscissas and a higher-order scheme for

the weights. Therefore, the abscissas on the cell face can beapproximated by

dα.fown = dα,own, dα.fnei = dα,nei (81)


nt+∆town =

N∑α=1

δ(d−dα,own)twtα,own − λ∑

f

wtα,fown max(F tfown , 0

)− λ

N∑α=1

∑f


t. (82)

For any ∆t that satisfies the condition

∆t 6 minown,α

wtα,own∆Vown∑f w

tα,fown

max(F tfown , 0

) (83)

the NDF is non-negative. It can be seen that in the Vikasscheme, if the first-order scheme is applied to the abscissasand the discritized time step satisfies Eq. (83), a higher-order scheme for the weights can be employed to preventthe moment realizability problem in the Euler explicit timemarching scheme. It is straightforward to extend the Vikasscheme to other problems, such as the PBE with diffusionterms [191]. Therefore, the discussion is omitted here forbrevity.

Another explanation which is easier to understand the Vikasscheme is also briefly discussed here. The spirit of the Vikasscheme is to employed the monotonic upwind scheme forconservation laws (MUSCL) type of reconstruction on thecanonical moments flux at the grid interface, which is calcu-lated by the piecewise linear (or polynomial) reconstruction ofthe weights and the piecewise constant reconstruction of othervariables. As shown in Fig. 6, for a uniform grid (labelled byI) with spacing ∆x, we have{

wIβ(x) = wIβ + σ

Iβ(x− xI)

dIβ(x) = dIβ

, xI − ∆x2≤ x ≤ xI + ∆x

2,

(84)where σIβ is the slope on the Ith cell. Substituting Eq. (84)into Eq. (13), it leads to

mIk(x) =

N∑β=1

(wIβ + σ

Iβ(x− xI)

) (dIβ)k. (85)

It can be seen that mIk(x), at any location x, satisfy Eq. (13)itself for any slope σIβ and these moments values are ensuredto be realizable. In the following we use the 1-D equation

∂mk∂t

+ u∂mk∂t

= 0 (86)

and the piecewise linear reconstruction of weights as anexample to show the procedure of Vikas scheme. It can bedivided into two main steps:

moments

weights

abscissas

II-1 I+1

Fig. 6: Piecewise linear reconstruction of the moments,weights and constant reconstruction for the abscissas.

1) Discretize Eq. (86) by explicit time-marching scheme:

M I,t+∆tk = MIk −

∆t

∆x

(F I+1/2 − F I−1/2

), (87)

where M I,t+∆tk, is the average value of mk over the Ithgrid at t+∆t, F I+1/2 and F I−1/2 are the moments fluxfunctions defined at the interface I + 1/2 and I − 1/2for Ith grid, respectively.

2) Instead of interpolating the flux from the moments,Vikas scheme calculates from the weights and abscissasas the following:

F I+1/2 =

N∑β=1

max(uIβ , 0)∆S

(wIβ + σ

Iβ

∆x

2

)(dIβ)i

+

N∑β=1

min(uI+1β , 0)∆S

(wI+1β − σ

I+1β

∆x

2

)(dI+1β

)i,

(88)

where ∆S = ∆y∆z, which is the grid interface surfacenormal to x-plane. The flux calculation corresponds tothe flux-vector splitting procedure: the first term in theR.H.S of Eq. (88) denotes the particles moving fromthe Ith grid crossing the I + 1/2 interface, whereas thesecond denotes the particles moving from the I + 1thgrid crossing the I + 1/2 interface. The flux at I − 1/2can be calculated by similar way.

Readers may find the Vikas scheme is only 1st-order accuratein time. Following the spirit of total variation diminishing(TVD) time stepping, multi-stage explicit time-integrationschemes can be used in practice to obtain a higher-order timescheme (the subscript are omitted for brevity):

m∗ = m+ F (m),

m∗∗ = m∗ + F (m∗),

mt+∆t =1

2(m∗ +m∗∗) ,

(89)

where F is the flux function. It can be easily proven thatthe two-stage second-order Runge-Kutta method combined theflux calculation based on Vikas scheme, as reported equationsabove, can ensure moments realizability. However, numericalproblems still exist as discussed as follows:

1) The numerical treatment of the moments are importantwhen the dispersed phase disappears. This phenomenonis called phase segregation in multiphase flow field. Forexample, in the bubble columns simulations, the initialbubble column is often filled by water, which implies αdand the kth order moment is theoretically zero due to theabsence of bubbles. However, such numerical challeng-ing settings introduce numerical problems for the TFMand QBMMs. In TFM, it leads to the singular problemwhich can be prevented by numerical manipulation onthe source term. In such cases, the “disappeared” phase’svelocity equals the terminal velocity. In QBMMs, thephase segragation implies the weights are zero in thesecomputational cells. According to Eq. (83), it implies thetime step tends to zero. Otherwise the advected momentswill be un-realizable. One possible solution would besetting the initial moments to be a relative small value[158]. Such numerical treatment needs further investiga-tions and verifications.

2) The Vikas scheme is suitable for Eq. (73). It was provenby Buffo et al. [186] that it introduces unbounded solu-tions. In order to ensure the boundedness, one possiblesolution is to employ Vikas scheme to solve Eq. (71), inwhich two advection terms exist. Meanwhile, the predic-tion of the m3 should be identical with αd if the TFMwas employed to simulate two-phase flows. However,the agreement of m3 and αd can be only obtained byapplying identical numerical scheme to these variables.The possible solution is to combine the FCT algorithmand Vikas scheme for the moment transport equationsto ensure boundedness. Such numerical treatment needsfurther investigations and verifications.

It should be noted here that the higher-order Vikas schemewas called “quasi-second-order” scheme by Laurent andNguyen [192], as of the higher-order scheme can not beemployed for the weights. Other higher-order schemes wereproposed by them which were called ζ scheme, ζ simpli-fied scheme, QW scheme and QW simplified scheme. Theseschemes are based on the calculation of ζ which can becalculated by the moment inversion algorithm (e.g. PD algo-rithm). The simplified schemes were also developed under the

KBFVM framework, and the ζ simplified scheme was recentlyimplemented in the OpenQBMM [179]. The QW simplifiedscheme shares large similarities with the Vikas scheme, but theinterpolation of weights and abscissas to cell faces are omitted.The key difference lies in that the CFL number for ζ simplifiedscheme and QW simplified scheme should be smaller than1/3 and 1/2, respectively. Readers who are interested in thisscheme are referred to their latest work [192]. Benchmark testcases of ζ based schemes need to be investigated in the future.

C. Reconstructed NDF

All the QBMMs track the moments instead of the NDF,which implies that the information of the rigorous NDF cannot be directly provided by the algorithm. The algorithm workswell for breakage and coalescence dominated processes andother similar processes where only the information of themean Sauter diameter is required. However, in certain cases itintroduces problems. For example, in evaporation simulations,it leads to a zero-order moment equation that contains a termthat corresponds to the loss of particles of zero size. In orderto evaluate this term, the value of the NDF at d = 0 is requiredbut is not available in the QMOM or the DQMOM. When theNDF was employed in LES framework, the velocity dispersionaround the velocity abscissas are also necessary. A possiblesolution is to increase the number of abscissas to a largeenough value (e.g., N > 100), such that the smallest abscissa’sexpected value tends to zero. However, it was proven thatthe moment inversion algorithm is not accurate for N largerthan approximately 10 [123], [195]. In some works, it iseven proven that these methods suffer from ill conditioningif N > 3 [133], [190], [196]. Thus, the QMOM and theDQMOM can not be employed for certain problems.

The NDF can be calculated from the moments if the NDFis defined a-prior. For example, if the NDF is expressed as

n = exp

(N∑i=1

AiLi

), (90)

where Ai is the unknown coefficients. The correspondingmoments can be written as

mk =

∫dk exp

(N∑i=1

Aidi

). (91)

Given N moments, the coefficients Ai can be found byNewton-Raphson method to reconstruct the NDF. Such methodwas adapted for crystallization problems [197]. The EQMOMfollows similar idea. In theory, the reconstructed NDF thatis predicted by the EQMOM will overlap with the shapeof the real NDF since the parameters of the KDF can becalculated analytically. However, because the values of σ areoften assumed to be identical in Eq. (33) to avoid the non-linear solver, error will be introduced if two or more primarynodes are employed. When the NDF is reconstructed from themoments, possible problems are listed as follows:

1) If the real NDF is assumed by summation of the KDFwith different values of σ, the predicted reconstructed

NDF by the EQMOM will deviate from the originalNDF [165], [170].

2) When many primary nodes are employed, as reported inFig. 7, the reconstructed NDF show oscillations, whichoriginate from the summation of the KDF [165], [170].

3) The reconstructed NDF that is predicted by the EQMOMis highly dependent on the shape of the KDF. It canbe seen in Fig. 7 that if the log-normal EQMOM isemployed to predict the real NDF with a beta distributionshape, it will fail because the left tail of the log-normaldistribution tends to zero, which is not the case forthe beta distribution function. Similar problems werereported in the works by [129], [165], [170], [198]

However, it should be stress there that although the recon-structed NDF that is calculated from the moments may deviatefrom the accurate NDF, this does not affect the stability of thenumerical simulation. Moreover, the predicted moments areconsistent.

Fig. 7: Left: comparison of the real NDF (solid red line) withthe log-normal shape and the reconstructed NDF by the log-normal EQMOM with one primary node (dashed black line).Middle: comparison of the real NDF (solid red line) withthe log-normal shape and the reconstructed NDF by the log-normal EQMOM with two primary nodes (dashed black line).The dashed green lines represent the reconstructed NDF byeach primary node. Right: comparison of the real NDF (solidred line) with the beta distribution shape and the reconstructedNDF by the log-normal EQMOM with one primary node(dashed black line).

V. CONCLUSIONS

The PBE can be employed to model the dispersed phasein multiphase flows. The analytical solution of the PBE canbe only obtained under rigid assumptions. A large variety ofalgorithms for solving the PBE have been developed. Amongthem, the QBMMs were proven to be efficient, especially whenthe PBE was coupled with CFD simulations for multiphaseflows. Over the years, the QMOM, DQMOM, CQMOM,EQMOM, ECQMOM, and HyQMOM were developed fordifferent problems, and each has advantages and drawbacks. Inthe present work, the algorithms of these QBMMs for spatiallyhomogeneous systems with first-order and second-order pointprocesses was comprehensively studied. The numerical aspectswhen the QBMM was coupled with CFD for inhomogeneousmultiphase systems were investigated. The limitations of eachalgorithm were also discussed. The boundedness problem and

the realizability problem of the moment sets when a higher-order scheme was employed were presented and possiblesolutions were summarized.

The numerical challenges and the limitations of the QBMMare clear and further studies on a more robust and efficientalgorithm are indispensable. Ideally, a quadrature-based mo-ment algorithm should be able to provide a non-negative NDFaccurately within the internal coordinate limits without therealizability problem. The spatial numerical scheme shouldensure the boundedness of the moments. In addition, thealgorithm should be able to capture the mathematical char-acteristics of the equations to be solved.

In the future, the algorithm of the coupling between the CFDand the generalized population balance equation (GPBE) willbe reviewed.

ACKNOWLEDGEMENTS

This work is the updated version of our previous work(CJChE, 2019:27,483-500). Dongyue Li wants to acknowledgeProf. Daniele Marchisio for shedding light on the QBMM.He also wants to acknowledge the small, but active, CFDcommunity in China (cfd-china.com) for all their continuedencouragement.

APPENDIX A. A CONCRETE EXAMPLE OF INITIALMOMENTS CALCULATION

QBMM starts the iteration from the initial moments. Thesemoments represent statistical features of the NDF. In orderto calculate realizable moments, an initial NDF should beassumed. For diameter-based NDF, it is common to assumea log-normal distribution function:

n(d) ≈ 1dσ√

2πexp

(− (lnd− µ)

2

2σ2

), (92)

where σ and µ are the log-normal distribution parameterswhich can be computed by

µ = ln

(m2√v +m2

), (93)

σ =

√ln( vm2

+ 1), (94)

where m is the mean value,√v is the dtandard deviation.

The initial raw moments based on the log-normal NDF canbe expressed as follows:

mk = N exp

(kµ+

k2σ2

2

), (95)

where N is the number of particles in per unit, which can becalculated from the given phase fraction as follows:

N =αd

kv exp(3µ+ 3

2σ2

2

)), (96)

where kv is the shape parameter equals π/6. In practice, mcan be seen as the mean particle diameter (e.g., mean Sauterdiameter),

√v can be set equal to a small value (e.g., 15

%). Since the moments are calculated from the NDF, theyare ensured to be realizable.

Another simpler approach can be used. One can calculatethe initial moments directly from the weights and abscissasas reported in Eq. (13). The summation of weights shouldbe equal to N . For example, if three nodes QMOM is used,one can assume w1 = 500, w2 = 600, w3 = 700, d1 =0.005, d2 = 0.006 and d3 = 0.007. The moments can becalculated by mk =

∑wid

ki . In this approach, the presumed

NDF’s shape can be hardly obtained. It should be stressedthat di cannot be identical, otherwise the moments cannot berealizable.

REFERENCES

[1] V. Buwa, V. Ranade, Dynamics of gas–liquid flow in a rectangularbubble column: experiments and single/multi-group CFD simulations,Chemical Engineering Science 57 (2002) 4715–4736.

[2] T. Wang, J. Wang, Y. Jin, Population balance model for gas- liquidflows: Influence of bubble coalescence and breakup models, Industrial& Engineering Chemistry Research 44 (2005) 7540–7549.

[3] M. Laakkonen, V. Alopaeus, J. Aittamaa, Validation of bubble break-age, coalescence and mass transfer models for gas–liquid dispersion inagitated vessel, Chemical Engineering Science 61 (2006) 218–228.

[4] M. Bhole, J. Joshi, D. Ramkrishna, CFD simulation of bubble columnsincorporating population balance modeling, Chemical Engineering Sci-ence 63 (2008) 2267–2282.

[5] A. Buffo, D. Marchisio, M. Vanni, P. Renze, Simulation of coalescence,break up and mass transfer in gas-liquid systems by using montecarlo and quadrature-based moment methods, in: Ninth InternationalConference on CFD in the Minerals and Process Industries, 2012.

[6] A. Buffo, M. Vanni, D. Marchisio, R. Fox, Multivariate quadrature-based moments methods for turbulent polydisperse gas–liquid systems,International Journal of Multiphase Flow 50 (2013) 41–57.

[7] Y. Liao, D. Lucas, E. Krepper, Application of new closure modelsfor bubble coalescence and breakup to steam–water vertical pipe flow,Nuclear Engineering and Design 279 (2014) 126–136.

[8] Y. Liao, D. Lucas, Poly-disperse simulation of condensing steam-waterflow inside a large vertical pipe, International Journal of ThermalSciences 104 (2016) 194–207.

[9] A. Buffo, M. Vanni, P. Renze, D. Marchisio, Empirical drag closurefor polydisperse gas–liquid systems in bubbly flow regime: Bubbleswarm and micro-scale turbulence, Chemical Engineering Research andDesign 113 (2016) 284–303.

[10] X. Liang, H. Pan, Y. Su, Z. Luo, CFD-PBM approach with modifieddrag model for the gas–liquid flow in a bubble column, ChemicalEngineering Research and Design 112 (2016) 88–102.

[11] H. Pan, X. Chen, X. Liang, L. Zhu, Z. Luo, CFD simulations of gas-liquid-solid flow in fluidized bed reactors-A review, Powder Technol-ogy 299 (2016) 235–258.

[12] D. Cheng, S. Wang, C. Yang, Z. Mao, Numerical simulation ofturbulent flow and mixing in Gas-Liquid-Liquid stirred tanks, Industrial& Engineering Chemistry Research 56 (2017) 13050—-13063.

[13] K. Guo, T. Wang, Y. Liu, J. Wang, CFD-PBM simulations of a bubblecolumn with different liquid properties, Chemical Engineering Journal329 (2017) 116–127.

[14] G. Yang, K. Guo, T. Wang, Numerical simulation of the bubble columnat elevated pressure with a CFD-PBM coupled model, ChemicalEngineering Science 170 (2017) 251–262.

[15] M. Jaradat, M. Attarakih, H. Bart, Effect of phase dispersion and masstransfer direction on steady state RDC performance using populationbalance modelling, Chemical Engineering Journal 165 (2010) 379–387.

[16] V. Alopaeus, Analysis of concentration polydispersity in mixed liquid–liquid systems, Chemical Engineering Research and Design 92 (2014)612–618.

[17] J. Mitre, P. Lage, M. Souza, E. Silva, L. Barca, A. Moraes, R. Coutinho,E. Fonseca, Droplet breakage and coalescence models for the flowof water-in-oil emulsions through a valve-like element, ChemicalEngineering Research and Design 92 (2014) 2493–2508.

[18] M. Attarakih, S. Alzyod, M. Hlawitschke, H. Bart, OPOSSIM: apopulation balance-SIMULINK module for modelling coupled hydro-dynamics and mass transfer in liquid extraction equipment, ComputerAided Chemical Engineering 37 (2015) 257–262.

[19] J. Favero, L. Silva, P. Lage, Modeling and simulation of mixingin water-in-oil emulsion flow through a valve-like element using apopulation balance model, Computers & Chemical Engineering 75(2015) 155–170.

[20] A. Buffo, V. Alopaeus, Solution of bivariate population balance equa-tions with high-order moment-conserving method of classes, Comput-ers & Chemical Engineering 87 (2016) 111–124.

[21] Z. Gao, D. Li, A. Buffo, W. Podgórska, D. Marchisio, Simulation ofdroplet breakage in turbulent liquid–liquid dispersions with CFD-PBM:Comparison of breakage kernels, Chemical Engineering Science 142(2016) 277–288.

[22] A. Bourdillon, P. Verdin, C. Thompson, Numerical simulations ofdrop size evolution in a horizontal pipeline, International Journal ofMultiphase Flow 78 (2016) 44–58.

[23] D. Li, A. Buffo, W. Podgórska, Z. Gao, D. Marchisio, Droplet breakageand coalescence in liquid-liquid dispersions: comparison of differentkernels with EQMOM and QMOM, AIChE Journal 63 (2017) 2293–2311.

[24] D. Li, A. Buffo, W. Podgórska, D. Marchisio, Z. Gao, Investigation ofdroplet breakup in liquid-liquid dispersions by CFD-PBM simulations:The influence of the surfactant type, Chinese Journal of ChemicalEngineering 25 (2017) 1369–1380.

[25] S. Alzyod, M. Attarakih, A. Hasseine, H. Bart, Steady state modelingof Kühni liquid extraction column using the Spatially Mixed SectionalQuadrature Method of Moments (SM-SQMOM), Chemical Engineer-ing Research and Design 117 (2017) 549–556.

[26] A. Misra, L. De Souza, M. Illner, L. Hohl, M. Kraume, J. Repke,D. Thévenin, Simulating separation of a multiphase liquid-liquid sys-tem in a horizontal settler by CFD, Chemical Engineering Science 167(2017) 242–250.

[27] C. Qin, C

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