Research Article A New Method of Moments for the...

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013, Article ID 840218, 6 pageshttp://dx.doi.org/10.1155/2013/840218

Research ArticleA New Method of Moments for the Bimodal ParticleSystem in the Stokes Regime

Yan-hua Liu1 and Zhao-qin Yin2

1 College of Mechanical and Electrical Engineering, Hohai University, Changzhou 213022, China2 China Jiliang University, Hangzhou 310018, China

Correspondence should be addressed to Yan-hua Liu; [email protected]

Received 22 September 2013; Accepted 30 October 2013

Academic Editor: Jianzhong Lin

Copyright © 2013 Y.-h. Liu and Z.-q. Yin. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

The current paper studied the particle system in the Stokes regime with a bimodal distribution. In such a system, the particles tendto congregate around two major sizes. In order to investigate this system, the conventional method of moments (MOM) should beextended to include the interaction between different particle clusters. The closure problem for MOM arises and can be solved by amultipoint Taylor-expansion technique.The exact expression is deduced to include the size effect between different particle clusters.The collision effects between different modals could also be modeled. The new model was simply tested and proved to be effectiveto treat the bimodal system.The results showed that, for single-modal particle system, the results from newmodel were the same asthose from TEMOM. However, for the bimodal particle system, there was a distinct difference between the two models, especiallyfor the zero-order moment. The current model generated fewer particles than TEMOM. The maximum deviation reached about15% for𝑚

0and 4% for𝑚

2. The detailed distribution of each submodal could also be investigated through current model.

1. Introduction

The particulate matter has become one of the most dan-gerous pollutants to the atmospheric environment and thehealth of human beings. It will reduce the visibility ofthe atmosphere and cause the traffic crowding and seriousaccidents. The fine particles (PM2.5) will also be breathedinto the bronchus of human beings, followed by severalkinds of respiratory diseases. The lungs will absorb the fineparticles and cardiovascular disease will come into being [1].However, the mechanism of the generation and evolution ofthe particulate matter still remains to be clarified. Hence, ithas both theoretical and realistic senses to study the dynamicsof the particulate matter.

Previous study on the aerosol dynamics usually supposesthat the particle system is monodispersed (i.e., the system hasonly one scale) or multidispersed (i.e., the system has multi-scales) but is in a log-normal distribution in size [2]. Suchkinds of assumptions will greatly simplify the problems, anda series of approximate or precise solutions will be obtained.However, these assumptions are based on the experimental

measurement and cannot be applied to all the cases. There isanother type of particle size distribution, namely, bimodal ormultimodal distribution. For example, the newborn particlestogether with the background particles compose the bimodaldistribution system. Furthermore, the newborn particlesmayalso exhibit a multimodal or bimodal distribution [3]. Pugat-shova et al. [4] and Lonati et al. [5] measured the particulatematter in the urban on-road atmosphere in different citiesand times.Themultimodal distribution was observed. At thistime, unacceptable error may appear using mono-dispersedor log-normal assumption.

Take the bimodal system, for example: the particles gatheraround two independent particle sizes. In order to study sucha system, the particle size distribution should be separatedinto two sub-PSDs [6]. The dynamics of the system may beobtained according to the two subparticle clusters. Under thisdescription, the governing equations of the particle systemshould be modified to represent the additional coagulationeffect [7]; that is, the collision of particles is artificiallyseparated into two kinds: internal coagulation and external

2 Abstract and Applied Analysis

coagulation. Because the typical particle diameter of thebimodal system is 5 nm to 2.5 𝜇m, whichmeans that particleslie in different dynamic regimes (free molecular regime,transition regime and continuum regime), the coagulationin such a wide range should also be treated separately. Thecurrent study will focus on the continuum (Stokes) regime.

Generally, the particle balance equation (PBE) governsthe detailed evolution process of PSD and can be numericallysolved. However, because of its huge computation resourceto solve the PBE directly, the method of moment (MOM)[2, 8, 9] is often taken into account as an alternation. Ittakes several moments of PSD in particle volume spaceand converts PBE into moment equations. Each momenthas its physical meaning: zero-order moment represents thenumber concentration, first-order moment represents thevolume concentration, and second-order moment is relatedto the polydispersity. Although MOM cannot directly giveout the evolution of specific PSD, it can obtain the statisticalcharacteristics of particle system and the calculation duringthis procedure reduces to an acceptable level. As a matter offact, MOM is widely used in the research of aerosol dynamicsfor its simplicity and low computational cost.

One limitation of MOM is the closure problem dueto the coagulation term in PBE. When PBE is convertedinto moment equations, the coagulation term will be trans-formed into fractional moments, which cannot be explicitlyexpressed and mathematical models should be introducedinto MOM to solve this problem, the so-called closureproblem. There are typically three kinds of methods: prede-termined PSD [2], quadrature method of moment (QMOM)[10], and Taylor-expansion method of moment (TEMOM)[11].Thefirst class often supposes that the PSD is a log-normaldistribution, and the coagulation term can be directly deter-mined. It can only be applied to the log-normal distributedparticle system. QMOM utilizes the Gaussian quadraturemethod to evaluate the coagulation term in the momentequations. The pre-determined PSD is not necessary, butthe computation is easy to diverge. TEMOM expands thenonlinear term in the collision kernel using the Taylorexpansion. Finally, the coagulation term can be expressed asa linear combination of different moments. TEMOM has itssuperiority on its easy expression, high precision, and lowcomputational cost. It is widely used in the research of theaerosol dynamics [12–15].

However, using TEMOM to study the bimodal systemhas some problems. TEMOM expands the collision kernelfunction at the average diameter 𝑢

0. For the internal collision,

there is no problem, but, for the external collision in bimodalsystem, this expansion should be extended. For a typicalbimodal system, there are two clusters of particles withdifferent diameters, and the total numbers of particles in eachcluster are also different. This fact will contribute to the factthat the average diameter of the whole systemmay lie aroundthe first mode or the second mode or even the mid place ofthe two modes. If the external collision term also expandsat the average diameter of the system, additional error willdecrease the accuracy of the simulation. In TEMOM, theconvergent region is (0, 2𝑢

0) [16], while, for bimodal system,

one mode may lie outside this region if both modes expand

at the same point. This possibility may lead to the divergenceof the calculation.

Hence, the Taylor-expansion method of moments shouldbe developed to be applied to the bimodal particle system toimprove the accuracy and the stability. The current researchwill focus on the multipoint Taylor-expansion method ofmoments, and the Stokes regime is preferred for ease.

2. Mathematical Theories

Considering the typical system with Brownian coagulationonly, PSD satisfies the PBE as [2]

𝜕𝑁 (V)𝜕𝑡

=

1

2

∫

V

0

𝛽 (𝑢, V − 𝑢)𝑁 (𝑢)𝑁 (V − 𝑢) 𝑑𝑢

− 𝑁 (V, 𝑡) ∫∞

0

𝛽 (V, 𝑢)𝑁 (𝑢) 𝑑𝑢,

(1)

where𝑁(V) is the size distribution function, whichmeans thenumber of particles with a volume V, 𝑢 and V are the particlevolumes, and 𝛽 is the Brownian coagulation coefficient.

In order to convert PBE into moment equations, thedefinition of moments is introduced

𝑚𝑘 (

𝑡) = ∫

∞

0

𝑁(V, 𝑡) V𝑘𝑑V𝑎. (2)

Applying (2) to (1), the moment equations are obtained:

𝜕𝑚𝑘

𝜕𝑡

=

1

2

∬

∞

0

{[(V1+ V2)𝑘− V𝑘1− V𝑘2]

× 𝛽 (V1, V2)𝑁 (V

1)𝑁 (V

2)}𝑑V1𝑑V2.

(3)

In current paper, the Stokes regime is studied, and thecollision kernel 𝛽may be rewritten as

𝛽c (V1, V2) = 𝐵[2 + (

V2

V1

)

1/3

+ (

V1

V2

)

1/3

] , (4)

where 𝐵 = 2𝑘𝑏𝑇/𝜇, 𝑘

𝑏is the Boltzmann constant, 𝑇 is the

environment temperature, and 𝜇 is the molecular viscosity ofgas.

When investigating the bimodal system, PSD can beexpressed as 𝑁(V, 𝑡) = 𝑁

𝑖(V, 𝑡) + 𝑁

𝑗(V, 𝑡). PBE for each sub-

PSD can be established. Apply the definition equation (2) tothe PBEs. The moment equations can be attained for bothcluster 𝑖 and cluster𝑗 listed as follows:

𝜕𝑚𝑖

𝑘

𝜕𝑡

= 𝐶𝑖𝑖

𝑘+ 𝐷𝑖𝑗

𝑘,

𝜕𝑚𝑗

𝑘

𝜕𝑡

= 𝐶𝑗𝑗

𝑘+ 𝐸𝑖𝑗

𝑘,

(5)

Abstract and Applied Analysis 3

where

𝐶𝑖𝑖

𝑘=

1

2

∬

∞

0

{[(𝑢 + V)𝑘 − 𝑢𝑘− V𝑘]

× 𝛽 (𝑢, V)𝑁𝑖(𝑢)𝑁𝑖(V)} 𝑑𝑢 𝑑V,

(6)

𝐷𝑖𝑗

𝑘= − ∬

∞

0

𝑢𝑘𝛽 (𝑢, V)𝑁𝑖 (𝑢)𝑁𝑗 (V) 𝑑𝑢 𝑑V, (7)

𝐸𝑖𝑗

𝑘= ∬

∞

0

{[(𝑢 + V)𝑘 − V𝑘]

× 𝛽 (𝑢, V)𝑁𝑖 (𝑢)𝑁𝑗 (V)} 𝑑𝑢 𝑑V.

(8)

Note that 𝐶𝑖𝑖𝑘and 𝐶

𝑗𝑗

𝑘are only related to 𝑁

𝑖or 𝑁𝑗. These

two terms represent the internal coagulation effect in thecluster 𝑖 or 𝑗. As a result, the single point binary Taylorexpansion is used to deal with these two terms (at 𝑢

1or 𝑢2).

The results from the typical TEMOM can be directly used. In(9)𝑚𝑘represents 𝑘th moment of PSD𝑁

𝑖or𝑁𝑗. Consider

𝐶𝑖𝑖

0=

𝐵𝑚𝑖2

0

81𝑚𝑖2

1

(−151𝑚𝑖4

1− 2𝑚𝑖2

2𝑚𝑖2

0− 13𝑚

𝑖

2𝑚𝑖2

1𝑚𝑖

0) ,

𝐶𝑖𝑖

1= 0,

𝐶𝑖𝑖

2=

−2𝐵1

81𝑚𝑖2

1

(2𝑚𝑖2

2− 13𝑚

𝑖

2𝑚𝑖2

1𝑚𝑖

0− 151𝑚

𝑖4

1) .

(9)

The approximation of 𝐷𝑖𝑗

𝑘and 𝐸

𝑖𝑗

𝑘will be deduced in

the following part. Substitute (4) into (7) and (8). A lot offractional moments will appear in the expression of 𝐷𝑖𝑗

𝑘and

𝐸𝑖𝑗

𝑘, which can be approximated through the Taylor expansion

of V𝑝 (𝑝 is fraction) at 𝑢1or 𝑢2. Consider

𝑚𝑝≈

𝑢𝑝−2

(𝑝2− 𝑝)

2

𝑚2

− 𝑢𝑝−1

(𝑝2− 2𝑝)𝑚

1+

𝑢𝑝

2

(𝑝2− 3𝑝 + 2)𝑚

0.

(10)

Making use of (10), 𝐷𝑖𝑗𝑘and 𝐸

𝑖𝑗

𝑘can be expressed as a linear

combination of𝑚𝑖𝑘and𝑚

𝑗

𝑘. Moreover

𝐷0= −

∑𝑎𝑚𝑛

𝑚𝑖

𝑚𝑚𝑗

𝑛

81

, 𝐷1=

∑𝑏𝑚𝑛

𝑚𝑖

𝑚𝑚𝑗

𝑛

81

,

𝐷2=

∑ 𝑐𝑚𝑛

𝑚𝑖

𝑚𝑚𝑗

𝑛

81

, 𝐸0= 0,

𝐸1=

∑𝑑𝑚𝑛

𝑚𝑖

𝑚𝑚𝑗

𝑛

81

, 𝐸2= −

∑ 𝑒𝑚𝑛

𝑚𝑖

𝑚𝑚𝑗

𝑛

81

.

(11)

The exact expressions of the coefficients in𝐷0,𝐷1,𝐷2,𝐸1,

and 𝐸2are listed in the appendix.

3. Tests and Discussion

In order to verify the deduction, both theoretical and numer-ical validations are performed, respectively.

𝜏 = Bt

102

101

100

10−1

0 2 4 6 8

m0 1-point TEMOMm1 1-point TEMOMm2 1-point TEMOM


m0,m

1, a

ndm

2Figure 1: The evolution of moments for Case I using differentexpansion schemes.

Note that, if 𝑁𝑖

= 𝑁𝑗

= 𝑁/2, (5) turns into two setsof moment equations with monomodal distribution. If (5)and set 𝑚

𝑘= 𝑚𝑖

𝑘+ 𝑚𝑗

𝑘, the theoretical systematic moment

equations are attained:

𝜕𝑚𝑘

𝜕𝑡

= 4𝐶𝑖𝑖

𝑘. (12)

Substitute 𝐷0, 𝐷1, 𝐷2, 𝐸1, and 𝐸

2into (5), and set 𝑢

1=

𝑢2

= 𝑚1/𝑚0, 𝑟 = 1. The right side of new equation just

equals 4 times of (9), which is consistent with the theoreticalequation (12).

Two simple bimodal systems are simulated to validate thecurrent model. The single point and multipoint expansionmethods are both taken into account and the results arecompared with each other to show the validity and accuracy.The initial size distributions both satisfy the log-normaldistribution as follows:

𝑁(V, 𝑡) = 𝑁0exp

(− (ln2 (V/V𝑔)) / (2𝑤

2

𝑔))

(√2𝜋V𝑤𝑔)

. (13)

For Case I, 𝑁𝑖0= 𝑁𝑗

0= 1.0, V𝑖

𝑔= V𝑗𝑔

= √3/2, and𝑤𝑖𝑔

=

𝑤𝑗

𝑔= √ln(4/3) [8], which represents a monomodal system

and the PSD is separated into two equal sub-PSDs. For CaseII,𝑁𝑖0= 1.0, V𝑖

𝑔= √3/2 and𝑤

𝑖

𝑔= √ln(4/3) and𝑁

𝑗

0= 0.1𝑁

𝑖

0,

V𝑗𝑔

= 1000V𝑖𝑔, and 𝑤

𝑗

𝑔= 0.1𝑤

𝑖

𝑔, which represents a bimodal

system consisting of two log-normal sub-PSDs.Figure 1 shows the results of Case I for both single point

TEMOM and multipoint TEMOM. From the figure, a goodagreement is obtained. This is because the particle system is,in the final analysis, a mono-modal system. The consistency


𝜏 = Bt

102

103

104

105

101

100

10−1

0 2 4 6 8



m0,m

1, a

ndm

2

Figure 2: The evolution of moments for Case II using differentexpansion schemes.

between two methods is just as the same as the theoreticalanalysis at the beginning of this paragraph.

Figure 2 shows the results of Case II for both single pointTEMOM and multi point TEMOM. From the figure, anobvious deviation is found. It shows that, for a typical bimodalsystem, the particle size difference between different modelscan not be neglected. The value for multipoint TEMOM isalways smaller than that for TEMOM especially for 𝑚

0. This

means that the original TEMOM model will underestimatethe coagulation effect for the particle number concentration(𝑚0). Another interesting phenomenon is that𝑚

1is the same

for both of the two models. The reason is that 𝑚1physically

represents the volume fraction of particles. The particlecollision (coagulation) will not change the total volume or themass of particles. Hence,𝑚

1is a constant from the beginning

to the end.Define the error function as

𝐸𝑘=

𝑚m𝑘

− 𝑚s𝑘

𝑚s𝑘

, (14)

Where 𝑚m𝑘represents the moments in multi-point TEMOM

and 𝑚s𝑘represents the moments in original TEMOM. The

exact tendency of 𝐸𝑘is shown in Figure 3. According to the

figure, the maximum deviation for 𝑚0will be about 15% and

4% for𝑚2. For𝑚

0, the error function𝐸

0will increase in a very

short time, reach the maximum, and then decrease slowly.This phenomenon indicates that the difference in particle sizewill lead to a relatively large deviation at the very beginning ofcoagulation for bimodal particle systemwhen the TEMOM isselected for the bimodal particle system.

Figure 4 shows the different moments in modes 𝑖 and 𝑗

using the technique proposed in current paper. According to

0 2 4 6 8 10

0

0.05

E0

E1

E2

−0.1

−0.15

−0.05

𝜏 = Bt

E0,E

1, a

ndE2

Figure 3: The variation of error function 𝐸𝑘versus dimensionless

time 𝜏.

𝜏 = Bt

102

103

104

105

101

100

10−1

0 2 4 6 8

10−2

m0, mode im1, mode im2, mode i

m0, mode jm1, mode jm2, mode j

m0,m

1, a

ndm

2

Figure 4: The evolution of moments for Case II with differentmodes.

the figure, an obvious reduction is found for each moment𝑚𝑘in mode 𝑖, which means that the coagulation will lead to

the decrease of 𝑚0(particle number concentration) and 𝑚

1

(particle volume fraction). Particularly the volume fractionof particles, 𝑚

1, no longer keeps a constant because of the

external collision with particles in mode 𝑗 and the new birthof bigger particles. For particles in mode 𝑗, the internal

Abstract and Applied Analysis 5

coagulation in mode 𝑗 will lead to the decrease of 𝑚0, while

the external coagulation betweenmode 𝑖 andmode 𝑗will takeno effect on 𝑚

0. As a result, the slope of curve is flatter than

that in Figure 2. However, 𝑚1and 𝑚

2are comparable with

those in Figure 2, because these two parameters are relatedto the particle volume tightly. The average volume of particlein mode 𝑗 is much bigger than that in mode 𝑖, according tothe initial condition. In general, such a result indicates theimportance of current technique, giving more accurate resultand more detail for the complex bimodal particle system.

4. Conclusions

The current research showed a multipoint Taylor-expansionmethod of moments for the bimodal particle system inthe Stokes regime. A theoretical deduction was performedand brief results are given. Both theoretical validation andnumerical tests are implemented. The results show that, fora single-modal system, there is no difference between thetwo methods. However, for a bimodal system, although theevolution ofmoments has the same tendency, there is obviousdeviation between the twomethods. For the case investigatedin current paper, the maximum deviation for𝑚

0is about 15%

and 4% for𝑚2. Eachmoment𝑚

𝑘in mode 𝑖will decrease.The

technique proposed in this paper will bring in the accuracyand details of particles. This method can be further extendedto the multi-modal system

Appendix

The coefficients in (11) are listed with the definition 𝑟 =

(𝑢1/𝑢2)1/3. Consider

𝑎00

= 70𝑟 + 70𝑟−1

+ 162, 𝑎01

= 35𝑢−1

1(2𝑟2− 𝑟4) ,

𝑎02

= 𝑢−2

1(10𝑟7− 14𝑟5) , 𝑎

10= 35𝑢

−1

1(2𝑟 − 𝑟

−1) ,

𝑎11

= −35𝑢−2

1(𝑟4+ 𝑟2) , 𝑎

12= 𝑢−3

1(10𝑟7+ 7𝑟5) ,

𝑎20

= 𝑢−2

1(10𝑟−1

− 14𝑟) , 𝑎21

= 𝑢−3

1(7𝑟4+ 10𝑟2) ,

𝑎22

= −2𝑢−4

1(𝑟7+ 𝑟5) ;

𝑏00

= 𝑢1(14𝑟 − 10𝑟

−1) , 𝑏

01= −7𝑟

4− 10𝑟2,

𝑏02

= 2𝑢−1

1(𝑟7+ 𝑟5) , 𝑏

10= −112𝑟 − 40𝑟

−1− 162,

𝑏11

= 𝑢−1

1(56𝑟4− 40𝑟2) , 𝑏

12= 8𝑢−2

1(𝑟5− 2𝑟7) ,

𝑏20

= 𝑢−1

1(5𝑟−1

− 28𝑟) , 𝑏21

= 𝑢−2

1(14𝑟4+ 5𝑟2) ,

𝑏22

= −𝑢−3

1(4𝑟7+ 𝑟5) ;

𝑐00

= 𝑢2

1(5𝑟−1

− 28 𝑟) , 𝑐01

= 𝑢1(14𝑟4+ 5 𝑟2) ,

𝑐02

= −4𝑟7− 𝑟5, 𝑐

10= 𝑢1(98𝑟 − 25𝑟

−1) ,

𝑐11

= −49𝑟4− 25𝑟2, 𝑐

12= 𝑢−1

1(14𝑟7+ 5𝑟5) ,

𝑐20

= −196𝑟 − 25𝑟−1

− 162, 𝑐21

= 𝑢−1

1(98𝑟4− 25𝑟2) ,

𝑐22

= 𝑢−2

1(5𝑟5− 28𝑟7) ;

𝑑00

= 𝑢1(10𝑟−1

− 14𝑟) , 𝑑01

= 7𝑟4+ 10𝑟2,

𝑑02

= −2𝑢−1

1(𝑟7

+ 𝑟5) , 𝑑

10= 112𝑟 + 40𝑟

−1+ 162,

𝑑11

= 𝑢−1

1(40𝑟2− 56𝑟4) , 𝑑

12= 8𝑢−2

1(2𝑟7− 𝑟5) ,

𝑑20

= 𝑢−1

1(28𝑟 − 5𝑟

−1) , 𝑑

21= −𝑢−2

1(14𝑟4+ 5𝑟2) ,

𝑑22

= 𝑢−3

1(4𝑟7+ 𝑟5) ;

𝑒00

= 𝑢2

1(−28𝑟 + 5𝑟

−1+ 4𝑟−2

+ 4𝑟−4) ,

𝑒01

= 𝑢1(14𝑟4+ 5𝑟2+ 16𝑟 − 32𝑟

−1) ,

𝑒02

= −4𝑟7− 𝑟5− 2𝑟4− 8𝑟2,

𝑒10

= 𝑢1(98𝑟 − 25𝑟

−1− 32𝑟−2

+ 16𝑟−4) ,

𝑒11

= −49𝑟4− 25𝑟2− 128𝑟 − 128𝑟

−1− 324,

𝑒12

= 𝑢−1

1(14𝑟7+ 5𝑟5+ 16𝑟4− 32𝑟2) ,

𝑒20

= −196𝑟 − 25𝑟−1

− 8𝑟−2

− 2𝑟−4

− 162,

𝑒21

= 𝑢−1

1(98𝑟4− 25𝑟2− 32𝑟 + 16𝑟

−1) ,

𝑒22

= 𝑢−2

1(−28𝑟

7+ 5𝑟5+ 4𝑟4+ 4𝑟2) .

(A.1)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

The authors gratefully acknowledges the financial supportfrom the National Natural Science Foundation of Chinaunder Grant no. 11302070, the National Basic Research Pro-gram of China (973 Program) under Grant no. 2010CB-227102.

References

[1] T. L. Chan, Z. Ning, J. S. Wang, C. S. Cheung, C. W. Leung,and W. T. Hung, “Gaseous and particle emission factorsfrom the selected on-road petrol/gasoline, diesel, and liquefiedpetroleum gas vehicles,” Energy and Fuels, vol. 21, no. 5, pp.2710–2718, 2007.

[2] R. B. Diemer and J. H. Olson, “A moment methodology forcoagulation and breakage problems, part 2: moment modelsand distribution reconstruction,” Chemical Engineering Science,vol. 57, no. 12, pp. 2211–2228, 2002.


[3] J. I. Jeong and M. Choi, “A bimodal moment model for thesimulation of particle growth,” Journal of Aerosol Science, vol.35, no. 9, pp. 1071–1090, 2004.

[4] A. Pugatshova, A. Reinart, and E. Tamm, “Features of themultimodal aerosol size distribution depending on the air massorigin in the Baltic region,” Atmospheric Environment, vol. 41,no. 21, pp. 4408–4422, 2007.

[5] G. Lonati, M. Crippa, V. Gianelle, and R. van Dingenen, “Dailypatterns of the multi-modal structure of the particle numbersize distribution in Milan, Italy,” Atmospheric Environment, vol.45, no. 14, pp. 2434–2442, 2011.

[6] H. Tang and J. Lin, “Research on bimodal particle extinctioncoefficient during Brownian coagulation and condensation forthe entire particle size regime,” Journal of Nanoparticle Research,vol. 13, no. 12, pp. 7229–7245, 2011.

[7] A. S. Koziol and H. G. Leighton, “The moments methodfor multi-modal multi-component aerosols as applied to thecoagulation-type equation,” Quarterly Journal of the RoyalMeteorological Society, vol. 133, no. 625, pp. 1057–1070, 2007.

[8] J. C. Barrett and J. S. Jheeta, “Improving the accuracy of themoments method for solving the aerosol general dynamicequation,” Journal of Aerosol Science, vol. 27, no. 8, pp. 1135–1142,1996.

[9] C. H. Jung and Y. P. Kim, “Numerical estimation of the effectsof condensation and coagulation on visibility using themomentmethod,” Journal of Aerosol Science, vol. 37, no. 2, pp. 143–161,2006.

[10] R.McGraw, “Description of aerosol dynamics by the quadraturemethod of moments,” Aerosol Science and Technology, vol. 27,no. 2, pp. 255–265, 1997.

[11] M. Yu, J. Lin, and T. Chan, “A new moment method for solvingthe coagulation equation for particles in Brownian motion,”Aerosol Science and Technology, vol. 42, no. 9, pp. 705–713, 2008.

[12] M. Yu and J. Lin, “Taylor-expansion moment method foragglomerate coagulation due to Brownian motion in the entiresize regime,” Journal of Aerosol Science, vol. 40, no. 6, pp. 549–562, 2009.

[13] M. Yu and J. Lin, “Binary homogeneous nucleation and growthof water-sulfuric acid nanoparticles using a TEMOM model,”International Journal of Heat and Mass Transfer, vol. 53, no. 4,pp. 635–644, 2010.

[14] M. Yu, J. Lin, and T. Chan, “Numerical simulation for nucleatedvehicle exhaust particulate matters via the temom/les method,”International Journal ofModern Physics C, vol. 20, no. 3, pp. 399–421, 2009.

[15] J. Lin, P. Lin, and H. Chen, “Research on the transport anddeposition of nanoparticles in a rotating curved pipe,” Physicsof Fluids, vol. 21, no. 12, pp. 1–11, 2009.

[16] M. Yu, J. Lin, H. Jin, and Y. Jiang, “The verification of the Taylor-expansion moment method for the nanoparticle coagulationin the entire size regime due to Brownian motion,” Journal ofNanoparticle Research, vol. 13, no. 5, pp. 2007–2020, 2011.

Submit your manuscripts athttp://www.hindawi.com

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttp://www.hindawi.com

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com

Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Stochastic AnalysisInternational Journal of

Research Article A New Method of Moments for the...

Documents

Transcript of Research Article A New Method of Moments for the...