Research ArticleAnalytical Solutions for Composition-Dependent Coagulation

Manli Yang12 Zhiming Lu1 and Jie Shen1

1Shanghai Institute of Applied Mathematics and Mechanics Shanghai University Shanghai 200072 China2Jiyang College Zhejiang Agriculture and Forestry University Zhuji 311800 China

Correspondence should be addressed to Zhiming Lu zmlushueducn

Received 6 January 2016 Accepted 4 April 2016

Academic Editor Babak Shotorban

Copyright copy 2016 Manli Yang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Exact solutions of the bicomponent Smoluchowskirsquos equation with a composition-dependent additive kernel 119870(V119886 V119887 V1015840119886 V1015840119887) =

120572(V119886+ V1015840119886) + (V

119887+ V1015840119887) are derived by using the Laplace transform for any initial particle size distribution The exact solution for an

exponential initial distribution is then used to analyse the effects of parameter 120572 on mixing degree of such bicomponent mixturesand the conditional distribution of the first component for particles with given mass The main finding is that the conditionaldistribution of large particles at larger time is a Gaussian function which is independent of the parameter 120572

1 Introduction

Modelling of a number of industrially important processessuch as coagulation and growth of aerosols [1] granulationof powders [2] crystallization [3] crystal shape engineering[4] and synthesis of nanoparticles [5] requires particles to beidentified with two or more of their attributes such as massfor two or more different compositions mass and surfacearea mass of primary particles and binder volume particlevolume and uncapped surface area In general cases thecoagulation kernel is a function of both size and compositionof the particles In granulation for instance the surfaceproperties (surface energy roughness) of granules determinethe efficiency by which granules are coated by the binder[6] Therefore components with different wetting propertiesmay exhibit markedly different behavior during coagulationCompositional effects introduce yet another dimension in theinteraction between particles in coagulation

This multicomponent coagulation problem was broughtto focus by Lushnikov [7] and later by Krapivsky and Ben-Naim [8] for systems in which the coagulation kernel isindependent of composition Vigil and Ziff [9] summarizedthe solutions of Lushnikov and showed that in these casesthe compositional distribution is a Gaussian function Morerecently Matsoukas et al formulated the bicomponent prob-lem in terms of one population balance equation for the size

distribution and another for the distribution of componentsand provided solutions for kernels that are independent ofcomposition [10ndash12] These solutions have shown that forsuch kernels the distribution of components followsGaussianscaling that is independent of the details of the kernel

Without loss of generality only two-component coagula-tion problem is considered and two components are given bytheir mass (or volume) (V

119886 V119887)

The governing equation for this coagulation problem isthe following population balance equation (PBE)

120597119873 (V119886 V119887 119905)

120597119905=1

2int

V119886

0

119889V1015840119886

sdot int

V119887

0

119870(V119886minus V1015840119886 V119887minus V1015840119887 V1015840119886 V1015840119887)119873 (V

119886minus V1015840119886 V119887minus V1015840119887 119905)

sdot 119873 (V1015840119886 V1015840119887 119905) 119889V1015840

119887minus 119873 (V

119886 V119887 119905) int

infin

0

119889V1015840119886

sdot int

infin

0

119870(V119886 V119887 V1015840119886 V1015840119887)119873 (V1015840

119886 V1015840119887 119905) 119889V1015840

119887

(1)

which is an extension of Smoluchowskirsquos equation for one-component coagulation where 119873(V

119886 V119887 119905) is the number

of density functions at time 119905 such that 119873(V119886 V119887 119905)119889V119886119889V119887

represents the number concentration of particles in the size

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 1735897 8 pageshttpdxdoiorg10115520161735897

2 Mathematical Problems in Engineering

range of 119886-component V119886to V119886+ 119889V119886 and the size range of

119887-component V119887to V119887+ 119889V119887 119870(V119886 V119887 V1015840119886 V1015840119887) is the coagula-

tion rate coefficient Recently Fernandez-Dıaz and Gomez-Garcıa by using Laplace transform obtained an exact ana-lytical solution for (1) with the additive kernel (which isindependent of composition) for any initial particle sizedistribution (PSD) [13] They further analysed the behaviorof the solution for larger sizes and time and found that thescaling solution cannot be used to describe the behavior ofthe number of the particle size distributions In this studywe extend Fernandez-Dıaz and Gomez-Garcıarsquos procedure tosolve (1) with a composition-dependent kernel (see (2) innext section) and analyse the effects of parameter 120572 on theproperties of bicomponent coagulation

2 Exact Solution fora Composition-Dependent Kernel

The kernel considered in this study is given as

119870(V119886 V119887 V1015840119886 V1015840119887) = 120572 (V

119886+ V1015840119886) + (V

119887+ V1015840119887) (2)

where parameter 120572 determines the relative contribution ofcomposition 119886 to the coagulation Evidently the additivekernel studied in [13] is recovered if we are letting 120572 = 1 ofthe kernel given in (2)

To seek the exact solution of (1) with kernel given by (2)we can from (1) obtain the equation for the total number ofparticles

12059711987200(119905)

120597119905= minus [120572119872

10+11987201]11987200(119905) (3)

where 11987200

is the total number of particles 11987210

is the massof 119886-component and 119872

01is the mass of 119887-component The

solution of (3) is easily obtained 11987200(119905) = 119873

0exp(minus120601119905) =

1198730(1 minus 120591) with 120601 = 120572119872

10+ 11987201 1198730= 11987200(0) and the

characteristic coagulation time 120591 = 1 minus exp(minus120601119905)Following [9] the number concentration distribution can

be given as below

119873(V119886 V119887 119905)

= 11987200(119905) exp(minus

120572V119886+ V119887

120572V1198860+ V1198870

120591)119892 (V119886 V119887 120591)

(4)

Substituting (4) into (1) we have

120597119892 (V119886 V119887 120591)

120597120591=

120572V119886+ V119887

2 (120572V1198860+ V1198870)

sdot int

V119886

0

int

V119887

0

119892 (V119886minus V1015840119886 V119887minus V1015840119887 120591) 119892 (V1015840

119886 V1015840119887 120591) 119889V1015840

119886119889V1015840119887

(5)

For (5) we can use two-dimensional Laplace transform

119871 [119892 (1199041 1199042)]

= ∬

infin

0

exp (minus1199041V119886minus 1199042V119887) 119892 (V119886 V119887) 119889V119886119889V119887

= 119866 (1199041 1199042)

(6)

Taking the derivative in (6) follows

120597119866 (1199041 1199042 120591)

120597120591

= minus120572

120572V1198860+ V1198870

119866 (1199041 1199042 120591)

120597119866 (1199041 1199042 120591)

1205971199041

minus1

120572V1198860+ V1198870

119866 (1199041 1199042 120591)

120597119866 (1199041 1199042 120591)

1205971199042

(7)

This is Burgersrsquo equation in multidimension without adiffusive term It can be solved in the transformed space bythe Lagrange-Charpit method [14]

1199081= 1199041minus

120572

1198980

119866 (1199041 1199042) 120591

1199082= 1199042minus

1

1198980

119866 (1199041 1199042) 120591

119866 (1199041 1199042 120591) = 119866 (119908

1 1199082 120591)

(8)

With multidimensional Lagrange inversion [15] we obtain

119866 (1199081 1199082 120591)

=

infin

sum

1198961=0

infin

sum

1198962=0

(minus1)1198961+1198962 (120572120591119898

0)1198961

(1205911198980)1198962

11989611198962

1205971198961+1198962

1205971198961119905112059711989621199052

[119865 (1199051 1199052)] 1199051=1199041

1199052=1199042

(9)

with

119865 (1199051 1199052) = 1198661198961+1198962+1

(1199051 1199052)

minus (1199051minus 1199041) 1198661198961+1198962 (1199051 1199052)120597119866 (1199051 1199052)

1205971199051

minus (1199052minus 1199042) 1198661198961+1198962 (1199051 1199052)120597119866 (1199051 1199052)

1205971199052

(10)

That is

119866 (1199081 1199082 120591) =

infin

sum

1198961=0

infin

sum

1198962=0

(minus1)1198961+1198962 (120572120591119898

0)1198961

(1205911198980)1198962

11989611198962

1

1198961+ 1198962+ 1

1205971198961+1198962

1205971198961119904112059711989621199042

1198661198961+1198962+1

(1199041 1199042 0) (11)

when naming 119866(1199041 1199042 0) = 119866

0(1199041 1199042)

Mathematical Problems in Engineering 3

Applying Laplace inverse transform we obtain

119892 (V119886 V119887 120591)

=

infin

sum

1198961=0

infin

sum

1198962=0

(minus1)1198961+1198962 1205721198961

11989611198962

(120591

120572V1198860+ V1198870

)

1198961+1198962

sdot1

1 + 1198961+ 1198962

119871minus1

[1205971198961+1198962

1205971199041198961

11205971199041198962

2

1198661+1198961+1198962

0(1199041 1199042)]

(12)

By rearranging the multiple series in one we arrive at

119892 (V119886 V119887 120591) =

infin

sum

119896=0

(((120572V119886+ V119887) (120572V

1198860+ V1198870)) 120591)119896

(119896 + 1)

sdot 119871minus1

[1198661+119896

0(1199041 1199042)]

(13)

With (4) we finally obtain the general solutions of (1)

119873(V119886 V119887 120591) = 119873

0(1 minus 120591) exp(minus

120572V119886+ V119887

120572V1198860+ V1198870

120591)

sdot

infin

sum

119896=0

(((120572V119886+ V119887) (120572V

1198860+ V1198870)) 120591)119896

(119896 + 1)

sdot 119871minus1

[1198661+119896

0(1199041 1199042)]

(14)

This solution can be applied for any initial distributionsby determining the multidimensional Laplace transform1198660

1+119896

(1199041 1199042) If exponential initial PSD is assumed

119873(V119886 V119887 0) =

1198730

1198721011987201

exp(minusV119886

11987210

minusV119887

11987201

) (15)

The Laplace transform for this function is

1198660(1199041 1199042) =

1198730

1198721011987201

1

1199041+ 1119872

10

1

1199042+ 1119872

01

(16)

and we can obtain the explicit solution as below

119873(V119886 V119887 120591) =

1198730

1198721011987201

(1 minus 120591)

sdot exp(minus120572V119886+ V119887

120572V1198860+ V1198870

120591 minusV119886

11987210

minusV119887

11987201

)

sdot

infin

sum

119896=0

((V119886V1198871198721011987201) ((120572V

119886+ V119887) (120572V

1198860+ V1198870)) 120591)119896

119896 (119896 + 1)

(17)

It is easily verified that the solution in [13] is recovered if weare letting 120572 = 1 We will further analyse some interestingproperties of solution (17) in the next section and attentionis paid particularly to the effects of parameter 120572 on thecoagulation properties of such systems

3 The Effects of 120572 on the Coagulation

31 The Total Number and Mass of Particles with the Concen-tration 119888 Mixing degree of the mixtures is one of the keyissues in bicomponent coagulation problems To this end wedefine two important magnitudes introduced in [7 13] Oneis the total number of particles having the concentration 119888 ofthe first component

119873(119888 119905) = ∬

infin

0

119873(V119886 V119887 120591) 120575 (119888 minus

V119886

V119886+ V119887

)119889V119886119889V119887

(18)

and the mass of these particles

119872(119888 119905) = ∬

infin

0

(V119886+ V119887)119873 (V

119886 V119887 120591)

sdot 120575 (119888 minusV119886

V119886+ V119887

)119889V119886119889V119887

(19)

With the compositional-dependent additive kernel (1) andthe initial condition (15) we obtain

119873(119888 120591) =1 minus 120591

11987210119872011198602

infin

sum

119896=0

(3119896 + 1)

(119896 + 1) (119896)2(119863

1198603)

119896

119872 (119888 120591) =1 minus 120591

11987210119872011198603

infin

sum

119896=0

(3119896 + 2)

(119896 + 1) (119896)2(119863

1198603)

119896

(20)

with

119860 =119888

11987210

+1 minus 119888

11987201

+120572119888 + (1 minus 119888)

12057211987201+ 1119872

10

120591

1198721011987201

119863 =120572119888 + (1 minus 119888)

12057211987201+ 1119872

10

120591119888 (1 minus 119888)

(1198721011987201)2

(21)

As suggested in [13] we observe the process at differentaverage particles size 120590 with 120591 = 1 minus 120590minus12

Figures 1ndash3 show the evolution of total number and totalmass with concentration 119888 for 120572 = 01 1 and 20 respectivelyIt is shown that the overall behavior of the evolution oftotal number and mass is similar for different 120572 while theevolution for total number ((a) in Figures 1ndash3) and totalmass ((b) in Figures 1ndash3) is obviously different The curvesof total number do not tend to a Dirac-120575 function whereasthe curves of total mass do and this result is consistent withthe result in [13] The effect of 120572 can be clearly seen fromcomparison between Figures 2(b) and 3(b) and it is shownthat the maxima for the total mass approach the overallfraction from the left end for 120572 = 1 but from the right endfor 120572 = 20This is due to the fact that the larger the parameter120572 the bigger the contribution of the 119886-component to thecoagulation which leads to the quicker growth of particleswith higher concentration

32 Conditional Distribution for Mass-Given Particles Con-ditional distribution is another important issue in bicom-ponent coagulation problems Vigil and Ziff [9] postulatedthe general scaling law for two-component aggregation with

4 Mathematical Problems in Engineering

15

10

05

00

00 02 04 06 08 10

120590 = 2

120590 = 20

120590 = 100

120590 = 500

N(c)

c

(a)

0

10

20

30

40

50

60

120590 = 2

120590 = 50

120590 = 100

120590 = 500

00 02 04 06 08 10

M(c)

c

(b)

Figure 1 Number (a) and mass (b) for initial exponential distribution with 120572 = 01 and1198730= 1 and119872

10= 13 and119872

01= 1

00 02 04 06 08 10

120590 = 2

120590 = 20

120590 = 100

120590 = 500

15

10

05

00

N(c)

c

(a)

00 02 04 06 08 10

120590 = 2

120590 = 20

120590 = 100

120590 = 500

0

10

20

30

40

50M(c)

c

(b)

Figure 2 Number (a) and mass (b) for initial exponential distribution with 120572 = 1 and1198730= 1 and119872

10= 13 and119872

01= 1

00 02 04 06 08 10

120590 = 2

120590 = 20

120590 = 100

120590 = 500

15

20

10

05

00

N(c)

c

(a)

00 02 04 06 08 10

120590 = 2

120590 = 20

120590 = 100

120590 = 500

0

10

20

30

40

50

M(c)

c

(b)

Figure 3 Number (a) and mass (b) for initial exponential distribution with 120572 = 20 and1198730= 1 and119872

10= 13 and119872

01= 1

Mathematical Problems in Engineering 5

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

035

030

025

020

015

010

005

000

N(c|

a+b=x)

c

(a)

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

035

030

025

020

015

010

005

000

N(c|

a+b=x)

c

(b)

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

035

030

025

020

015

010

005

000

N(c|

a+b=x)

c

(c)

Figure 4 The compositional distributions for 119909 = 1 with1198730= 1119872

10= 13119872

01= 1 and (a) 120572 = 01 (b) 120572 = 1 and (c) 120572 = 20

nongelling kernels that bivariate PSD function can be theproduct of a normal distribution in particle composition andthe one-parameter scaling function for the correspondinghomogeneous coagulation problem And for large time thecompositional distribution in large particles is a Gaussianfunction We explore the conditional distribution for acomposition-dependent kernel given in (2) and again theeffects of 120572 are analysed in this section To this aim weintroduce the conditional 119886-component number distribution119873(119888 | 119909 = V

119886+ V119887) where 119888 = V

119886(V119886+ V119887) is 119886-component

concentration for particles given mass 119909Figure 4 shows the conditional distribution for particles

given mass 119909 = 1 with three different 120572 values It is shownthat the general behavior is similar for all three 120572 values forexample the total number decreases with the coagulationgoing on while the curves become flat The effects ofparameter 120572 in these small particles are nearly negligibleFigure 5 shows the conditional distribution for particles givenmass 119909 = 4 with three different 120572 values The differenceof 120572 being smaller than 1 and bigger than 1 is clearly seenby comparison of Figures 5(a) and 5(b) with Figure 5(c)

The obvious difference is that there exist two extreme pointsof the curve 119873(119888 | 119909 = V

119886+ V119887) for the case 120572 = 20

while only one extreme point exists for 120572 le 1 which meansmixing degree is worse for 120572 gt 1 than 120572 le 1 Comparisonof Figures 4 and 5 shows that 119886-component is mixed better inlarger particles than inminute particlesThis tendency can befurther illustrated in Figure 6 which shows the distributionfor 119909 = 100 It is clearly shown in Figure 6 that the mixture iswell mixed when 120590 gt 10 irrespective of the value 120572 It is alsonoticed that the distribution curves for 3 different 120572 valuescollapse each other to a Gaussian function which can beverified analytically in fact since from the exact solutions (17)we can obtain for large particles at large time the conditionaldistribution as below119873(119888 | 119909)

sim1

radic4120587 (1198882

0(1 minus 1198880)2

119909)

exp(minus119909 (119888 minus 119888

0)2

41198882

0(1 minus 1198880)2)

(22)

Equation (22) is a Gaussian function and is independent ofthe value 120572

6 Mathematical Problems in Engineering

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

0000

0005

0010

0015

N(c|

a+b=x)

c

(a)

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

0000

0005

0010

0015

N(c|

a+b=x)

c

(b)

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

0000

0005

0010

0015

N(c|

a+b=x)

c

(c)

Figure 5 The compositional distributions for 119909 = 4 with1198730= 1119872

10= 13119872

01= 1 and (a) 120572 = 01 (b) 120572 = 1 and (c) 120572 = 20

From Figure 6 we can find an obvious difference betweenthe results for 120572 = 20 and the results for 120572 le 1 Thedistribution approaches to the asymptotic steady solution(22) from right end when 120572 = 20 while the curves approachthis steady state from left end In other words the equilib-rium concentration 119888 of the compositional distribution isbigger than overall 119886-component mass concentration 119888

0=

11987210(11987210+ 11987201) when 120572 = 20 and smaller than overall

concentration when 120572 le 1 The mechanism is the same asexplained in Section 31 In fact it can be proved that when120572 gt 43 the equilibrium concentration is larger than theoverall fraction 119888

0at a finite time

4 Conclusion

In this paper we have obtained the exact solution ofSmoluchowskirsquos continuous two-component equation witha composition-dependent additive kernel which is given as119870(V119886 V119887 V1015840119886 V1015840119887) = 120572(V

119886+ V1015840119886) + (V119887+ V1015840119887) for any initial particle

size distributions The main characteristics of the solution

and the effects of parameter 120572 on the bicomponent coagu-lation have been analysed in detail for an exponential initialdistribution

The effects of 120572 on the total number and mass of particleswith concentration 119888 were first investigated and the resultsshow that the curves for the total number do not reduceto a Dirac-120575 function for any given 120572rsquos while the curves forthe total mass tend to a Dirac-120575 function at large time Theapproaching process is slightly different which depends onthe parameter 120572

Then the effects of 120572 on the conditional distribution ofthe first component in mass-given particles were discussedand it is found that the conditional distribution is a Gaussianfunction at large time which is independent of the valueof 120572 Besides the conditional distribution in particles withmoderate mass is quite different which strongly depends onthe parameter 120572

The exact solutions obtained in this paper are of potentialhelp to understanding the coagulation in a bicomponentsystem The solutions can also be used to test the validity of

Mathematical Problems in Engineering 7

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

5 times 10minus6

4 times 10minus6

3 times 10minus6

2 times 10minus6

1 times 10minus6

0

N(c|

a+b=x)

c

(a)

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

5 times 10minus6

4 times 10minus6

3 times 10minus6

2 times 10minus6

1 times 10minus6

0

N(c|

a+b=x)

c

(b)

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

5 times 10minus6

4 times 10minus6

3 times 10minus6

2 times 10minus6

1 times 10minus6

0

N(c|

a+b=x)

c

(c)

Figure 6The compositional distributions for 119909 = 100with1198730= 1119872

10= 13119872

01= 1 and (a) 120572 = 01 (b) 120572 = 1 and (c) 120572 = 20 Compared

with overall 119886-component mass concentration 1198880= 11987210(11987210+11987201) (here 119888

0= 025)

the numerical methods developed to solve multicomponentSmoluchowskirsquos equation

Symbols

V119886 V119887 The size of particles 119886 119887

11987200(119905) Particle number at time 119905

11987210 Mass of 119886-component

11987201 Mass of 119887-component

1198730 Initial number of particles

120601 Equivalent total mass of particles120591 Characteristic coagulation time120572 Parameters of component effects on coagulation119888 119886-component mean concentration1198880 Overall 119886-component mass concentration

120590 Average particles size119909 Mass of given particles

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This project is supported by National Natural Science Foun-dation of China (no 11272196 and no 11222222) and ScientificResearch Development Fund of Zhejiang Agriculture andForestry University (JYYY1502)

References

[1] F M Gelbard and J H Seinfeld ldquoCoagulation and growth ofa multicomponent aerosolrdquo Journal of Colloid And InterfaceScience vol 63 no 3 pp 472ndash479 1978

[2] S M Iveson ldquoLimitations of one-dimensional population bal-ance models of wet granulation processesrdquo Powder Technologyvol 124 no 3 pp 219ndash229 2002

[3] F Puel G Fevotte and J P Klein ldquoSimulation and analysis ofindustrial crystallization processes through multidimensionalpopulation balance equations Part 1 a resolution algorithmbased on the method of classesrdquo Chemical Engineering Sciencevol 58 no 16 pp 3715ndash3727 2003

[4] H Briesen ldquoSimulation of crystal size and shape by means of areduced two-dimensional population balancemodelrdquoChemicalEngineering Science vol 61 no 1 pp 104ndash112 2006

8 Mathematical Problems in Engineering

[5] B L Cushing V L Kolesnichenko and C J OrsquoConnorldquolsquoRecentadvances in the liquid-phase syntheses of inorganic nanoparti-clesrdquo Chemical Reviews vol 104 no 9 pp 3893ndash3946 2004

[6] P Rajniak C Mancinelli R T Chern F Stepanek L Farberand B T Hill ldquoExperimental study of wet granulation influidized bed impact of the binder properties on the granulemorphologyrdquo International Journal of Pharmaceutics vol 334no 1-2 pp 92ndash102 2007

[7] A A Lushnikov ldquoEvolution of coagulating systems III Coag-ulating mixturesrdquo Journal of Colloid And Interface Science vol54 no 1 pp 94ndash101 1976

[8] P L Krapivsky and E Ben-Naim ldquoAggregation with multipleconservation lawsrdquo Physical Review E vol 53 no 1 pp 291ndash2981996

[9] R D Vigil and R M Ziff ldquoOn the scaling theory of two-com-ponent aggregationrdquo Chemical Engineering Science vol 53 no9 pp 1725ndash1729 1998

[10] T Matsoukas K Lee and T Kim ldquoMixing of components intwo-component aggregationrdquo AIChE Journal vol 52 no 9 pp3088ndash3099 2006

[11] K Lee T Kim P Rajniak and T Matsoukas ldquoCompositionaldistributions in multicomponent aggregationrdquo Chemical Engi-neering Science vol 63 no 5 pp 1293ndash1303 2008

[12] M-L Yang Z-M Lu and Y-L Liu ldquoSelf-similar behaviorfor multicomponent coagulationrdquo Applied Mathematics andMechanics English Edition vol 35 no 11 pp 1353ndash1360 2014

[13] J M Fernandez-Dıaz and G J Gomez-Garcıa ldquoExact solutionof Smoluchowskirsquos continuous multi-component equation withan additive kernelrdquo Europhysics Letters vol 78 no 5 Article ID56002 2007

[14] M Delgado ldquoThe Lagrange-Charpit methodrdquo SIAM Reviewvol 39 no 2 pp 298ndash304 1997

[15] I J Good ldquoGeneralizations to several variables of Lagrangersquosexpansion with applications to stochastic processesrdquo Mathe-matical Proceedings of the Cambridge Philosophical Society vol56 no 4 pp 367ndash380 1960

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

range of 119886-component V119886to V119886+ 119889V119886 and the size range of

119887-component V119887to V119887+ 119889V119887 119870(V119886 V119887 V1015840119886 V1015840119887) is the coagula-

tion rate coefficient Recently Fernandez-Dıaz and Gomez-Garcıa by using Laplace transform obtained an exact ana-lytical solution for (1) with the additive kernel (which isindependent of composition) for any initial particle sizedistribution (PSD) [13] They further analysed the behaviorof the solution for larger sizes and time and found that thescaling solution cannot be used to describe the behavior ofthe number of the particle size distributions In this studywe extend Fernandez-Dıaz and Gomez-Garcıarsquos procedure tosolve (1) with a composition-dependent kernel (see (2) innext section) and analyse the effects of parameter 120572 on theproperties of bicomponent coagulation

2 Exact Solution fora Composition-Dependent Kernel

The kernel considered in this study is given as

119870(V119886 V119887 V1015840119886 V1015840119887) = 120572 (V

119886+ V1015840119886) + (V

119887+ V1015840119887) (2)

where parameter 120572 determines the relative contribution ofcomposition 119886 to the coagulation Evidently the additivekernel studied in [13] is recovered if we are letting 120572 = 1 ofthe kernel given in (2)

To seek the exact solution of (1) with kernel given by (2)we can from (1) obtain the equation for the total number ofparticles

12059711987200(119905)

120597119905= minus [120572119872

10+11987201]11987200(119905) (3)

where 11987200

is the total number of particles 11987210

is the massof 119886-component and 119872

01is the mass of 119887-component The

solution of (3) is easily obtained 11987200(119905) = 119873

0exp(minus120601119905) =

1198730(1 minus 120591) with 120601 = 120572119872

10+ 11987201 1198730= 11987200(0) and the

characteristic coagulation time 120591 = 1 minus exp(minus120601119905)Following [9] the number concentration distribution can

be given as below

119873(V119886 V119887 119905)

= 11987200(119905) exp(minus

120572V119886+ V119887

120572V1198860+ V1198870

120591)119892 (V119886 V119887 120591)

(4)

Substituting (4) into (1) we have

120597119892 (V119886 V119887 120591)

120597120591=

120572V119886+ V119887

2 (120572V1198860+ V1198870)

sdot int

V119886

0

int

V119887

0

119892 (V119886minus V1015840119886 V119887minus V1015840119887 120591) 119892 (V1015840

119886 V1015840119887 120591) 119889V1015840

119886119889V1015840119887

(5)

For (5) we can use two-dimensional Laplace transform

119871 [119892 (1199041 1199042)]

= ∬

infin

0

exp (minus1199041V119886minus 1199042V119887) 119892 (V119886 V119887) 119889V119886119889V119887

= 119866 (1199041 1199042)

(6)

Taking the derivative in (6) follows

120597119866 (1199041 1199042 120591)

120597120591

= minus120572

120572V1198860+ V1198870

119866 (1199041 1199042 120591)

120597119866 (1199041 1199042 120591)

1205971199041

minus1

120572V1198860+ V1198870

119866 (1199041 1199042 120591)

120597119866 (1199041 1199042 120591)

1205971199042

(7)

This is Burgersrsquo equation in multidimension without adiffusive term It can be solved in the transformed space bythe Lagrange-Charpit method [14]

1199081= 1199041minus

120572

1198980

119866 (1199041 1199042) 120591

1199082= 1199042minus

1

1198980

119866 (1199041 1199042) 120591

119866 (1199041 1199042 120591) = 119866 (119908

1 1199082 120591)

(8)

With multidimensional Lagrange inversion [15] we obtain

119866 (1199081 1199082 120591)

=

infin

sum

1198961=0

infin

sum

1198962=0

(minus1)1198961+1198962 (120572120591119898

0)1198961

(1205911198980)1198962

11989611198962

1205971198961+1198962

1205971198961119905112059711989621199052

[119865 (1199051 1199052)] 1199051=1199041

1199052=1199042

(9)

with

119865 (1199051 1199052) = 1198661198961+1198962+1

(1199051 1199052)

minus (1199051minus 1199041) 1198661198961+1198962 (1199051 1199052)120597119866 (1199051 1199052)

1205971199051

minus (1199052minus 1199042) 1198661198961+1198962 (1199051 1199052)120597119866 (1199051 1199052)

1205971199052

(10)

That is

119866 (1199081 1199082 120591) =

infin

sum

1198961=0

infin

sum

1198962=0

(minus1)1198961+1198962 (120572120591119898

0)1198961

(1205911198980)1198962

11989611198962

1

1198961+ 1198962+ 1

1205971198961+1198962

1205971198961119904112059711989621199042

1198661198961+1198962+1

(1199041 1199042 0) (11)

when naming 119866(1199041 1199042 0) = 119866

0(1199041 1199042)

Mathematical Problems in Engineering 3

Applying Laplace inverse transform we obtain

119892 (V119886 V119887 120591)

=

infin

sum

1198961=0

infin

sum

1198962=0

(minus1)1198961+1198962 1205721198961

11989611198962

(120591

120572V1198860+ V1198870

)

1198961+1198962

sdot1

1 + 1198961+ 1198962

119871minus1

[1205971198961+1198962

1205971199041198961

11205971199041198962

2

1198661+1198961+1198962

0(1199041 1199042)]

(12)

By rearranging the multiple series in one we arrive at

119892 (V119886 V119887 120591) =

infin

sum

119896=0

(((120572V119886+ V119887) (120572V

1198860+ V1198870)) 120591)119896

(119896 + 1)

sdot 119871minus1

[1198661+119896

0(1199041 1199042)]

(13)

With (4) we finally obtain the general solutions of (1)

119873(V119886 V119887 120591) = 119873

0(1 minus 120591) exp(minus

120572V119886+ V119887

120572V1198860+ V1198870

120591)

sdot

infin

sum

119896=0

(((120572V119886+ V119887) (120572V

1198860+ V1198870)) 120591)119896

(119896 + 1)

sdot 119871minus1

[1198661+119896

0(1199041 1199042)]

(14)

This solution can be applied for any initial distributionsby determining the multidimensional Laplace transform1198660

1+119896

(1199041 1199042) If exponential initial PSD is assumed

119873(V119886 V119887 0) =

1198730

1198721011987201

exp(minusV119886

11987210

minusV119887

11987201

) (15)

The Laplace transform for this function is

1198660(1199041 1199042) =

1198730

1198721011987201

1

1199041+ 1119872

10

1

1199042+ 1119872

01

(16)

and we can obtain the explicit solution as below

119873(V119886 V119887 120591) =

1198730

1198721011987201

(1 minus 120591)

sdot exp(minus120572V119886+ V119887

120572V1198860+ V1198870

120591 minusV119886

11987210

minusV119887

11987201

)

sdot

infin

sum

119896=0

((V119886V1198871198721011987201) ((120572V

119886+ V119887) (120572V

1198860+ V1198870)) 120591)119896

119896 (119896 + 1)

(17)

It is easily verified that the solution in [13] is recovered if weare letting 120572 = 1 We will further analyse some interestingproperties of solution (17) in the next section and attentionis paid particularly to the effects of parameter 120572 on thecoagulation properties of such systems

3 The Effects of 120572 on the Coagulation

31 The Total Number and Mass of Particles with the Concen-tration 119888 Mixing degree of the mixtures is one of the keyissues in bicomponent coagulation problems To this end wedefine two important magnitudes introduced in [7 13] Oneis the total number of particles having the concentration 119888 ofthe first component

119873(119888 119905) = ∬

infin

0

119873(V119886 V119887 120591) 120575 (119888 minus

V119886

V119886+ V119887

)119889V119886119889V119887

(18)

and the mass of these particles

119872(119888 119905) = ∬

infin

0

(V119886+ V119887)119873 (V

119886 V119887 120591)

sdot 120575 (119888 minusV119886

V119886+ V119887

)119889V119886119889V119887

(19)

With the compositional-dependent additive kernel (1) andthe initial condition (15) we obtain

119873(119888 120591) =1 minus 120591

11987210119872011198602

infin

sum

119896=0

(3119896 + 1)

(119896 + 1) (119896)2(119863

1198603)

119896

119872 (119888 120591) =1 minus 120591

11987210119872011198603

infin

sum

119896=0

(3119896 + 2)

(119896 + 1) (119896)2(119863

1198603)

119896

(20)

with

119860 =119888

11987210

+1 minus 119888

11987201

+120572119888 + (1 minus 119888)

12057211987201+ 1119872

10

120591

1198721011987201

119863 =120572119888 + (1 minus 119888)

12057211987201+ 1119872

10

120591119888 (1 minus 119888)

(1198721011987201)2

(21)

As suggested in [13] we observe the process at differentaverage particles size 120590 with 120591 = 1 minus 120590minus12

Figures 1ndash3 show the evolution of total number and totalmass with concentration 119888 for 120572 = 01 1 and 20 respectivelyIt is shown that the overall behavior of the evolution oftotal number and mass is similar for different 120572 while theevolution for total number ((a) in Figures 1ndash3) and totalmass ((b) in Figures 1ndash3) is obviously different The curvesof total number do not tend to a Dirac-120575 function whereasthe curves of total mass do and this result is consistent withthe result in [13] The effect of 120572 can be clearly seen fromcomparison between Figures 2(b) and 3(b) and it is shownthat the maxima for the total mass approach the overallfraction from the left end for 120572 = 1 but from the right endfor 120572 = 20This is due to the fact that the larger the parameter120572 the bigger the contribution of the 119886-component to thecoagulation which leads to the quicker growth of particleswith higher concentration

32 Conditional Distribution for Mass-Given Particles Con-ditional distribution is another important issue in bicom-ponent coagulation problems Vigil and Ziff [9] postulatedthe general scaling law for two-component aggregation with

4 Mathematical Problems in Engineering

15

10

05

00

00 02 04 06 08 10

120590 = 2

120590 = 20

120590 = 100

120590 = 500

N(c)

c

(a)

0

10

20

30

40

50

60

120590 = 2

120590 = 50

120590 = 100

120590 = 500

00 02 04 06 08 10

M(c)

c

(b)

Figure 1 Number (a) and mass (b) for initial exponential distribution with 120572 = 01 and1198730= 1 and119872

10= 13 and119872

01= 1

00 02 04 06 08 10

120590 = 2

120590 = 20

120590 = 100

120590 = 500

15

10

05

00

N(c)

c

(a)

00 02 04 06 08 10

120590 = 2

120590 = 20

120590 = 100

120590 = 500

0

10

20

30

40

50M(c)

c

(b)

Figure 2 Number (a) and mass (b) for initial exponential distribution with 120572 = 1 and1198730= 1 and119872

10= 13 and119872

01= 1

00 02 04 06 08 10

120590 = 2

120590 = 20

120590 = 100

120590 = 500

15

20

10

05

00

N(c)

c

(a)

00 02 04 06 08 10

120590 = 2

120590 = 20

120590 = 100

120590 = 500

0

10

20

30

40

50

M(c)

c

(b)

Figure 3 Number (a) and mass (b) for initial exponential distribution with 120572 = 20 and1198730= 1 and119872

10= 13 and119872

01= 1

Mathematical Problems in Engineering 5

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

035

030

025

020

015

010

005

000

N(c|

a+b=x)

c

(a)

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

035

030

025

020

015

010

005

000

N(c|

a+b=x)

c

(b)

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

035

030

025

020

015

010

005

000

N(c|

a+b=x)

c

(c)

Figure 4 The compositional distributions for 119909 = 1 with1198730= 1119872

10= 13119872

01= 1 and (a) 120572 = 01 (b) 120572 = 1 and (c) 120572 = 20

nongelling kernels that bivariate PSD function can be theproduct of a normal distribution in particle composition andthe one-parameter scaling function for the correspondinghomogeneous coagulation problem And for large time thecompositional distribution in large particles is a Gaussianfunction We explore the conditional distribution for acomposition-dependent kernel given in (2) and again theeffects of 120572 are analysed in this section To this aim weintroduce the conditional 119886-component number distribution119873(119888 | 119909 = V

119886+ V119887) where 119888 = V

119886(V119886+ V119887) is 119886-component

concentration for particles given mass 119909Figure 4 shows the conditional distribution for particles

given mass 119909 = 1 with three different 120572 values It is shownthat the general behavior is similar for all three 120572 values forexample the total number decreases with the coagulationgoing on while the curves become flat The effects ofparameter 120572 in these small particles are nearly negligibleFigure 5 shows the conditional distribution for particles givenmass 119909 = 4 with three different 120572 values The differenceof 120572 being smaller than 1 and bigger than 1 is clearly seenby comparison of Figures 5(a) and 5(b) with Figure 5(c)

The obvious difference is that there exist two extreme pointsof the curve 119873(119888 | 119909 = V

119886+ V119887) for the case 120572 = 20

while only one extreme point exists for 120572 le 1 which meansmixing degree is worse for 120572 gt 1 than 120572 le 1 Comparisonof Figures 4 and 5 shows that 119886-component is mixed better inlarger particles than inminute particlesThis tendency can befurther illustrated in Figure 6 which shows the distributionfor 119909 = 100 It is clearly shown in Figure 6 that the mixture iswell mixed when 120590 gt 10 irrespective of the value 120572 It is alsonoticed that the distribution curves for 3 different 120572 valuescollapse each other to a Gaussian function which can beverified analytically in fact since from the exact solutions (17)we can obtain for large particles at large time the conditionaldistribution as below119873(119888 | 119909)

sim1

radic4120587 (1198882

0(1 minus 1198880)2

119909)

exp(minus119909 (119888 minus 119888

0)2

41198882

0(1 minus 1198880)2)

(22)

Equation (22) is a Gaussian function and is independent ofthe value 120572

6 Mathematical Problems in Engineering

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

0000

0005

0010

0015

N(c|

a+b=x)

c

(a)

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

0000

0005

0010

0015

N(c|

a+b=x)

c

(b)

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

0000

0005

0010

0015

N(c|

a+b=x)

c

(c)

Figure 5 The compositional distributions for 119909 = 4 with1198730= 1119872

10= 13119872

01= 1 and (a) 120572 = 01 (b) 120572 = 1 and (c) 120572 = 20

From Figure 6 we can find an obvious difference betweenthe results for 120572 = 20 and the results for 120572 le 1 Thedistribution approaches to the asymptotic steady solution(22) from right end when 120572 = 20 while the curves approachthis steady state from left end In other words the equilib-rium concentration 119888 of the compositional distribution isbigger than overall 119886-component mass concentration 119888

0=

11987210(11987210+ 11987201) when 120572 = 20 and smaller than overall

concentration when 120572 le 1 The mechanism is the same asexplained in Section 31 In fact it can be proved that when120572 gt 43 the equilibrium concentration is larger than theoverall fraction 119888

0at a finite time

4 Conclusion

In this paper we have obtained the exact solution ofSmoluchowskirsquos continuous two-component equation witha composition-dependent additive kernel which is given as119870(V119886 V119887 V1015840119886 V1015840119887) = 120572(V

119886+ V1015840119886) + (V119887+ V1015840119887) for any initial particle

size distributions The main characteristics of the solution

and the effects of parameter 120572 on the bicomponent coagu-lation have been analysed in detail for an exponential initialdistribution

The effects of 120572 on the total number and mass of particleswith concentration 119888 were first investigated and the resultsshow that the curves for the total number do not reduceto a Dirac-120575 function for any given 120572rsquos while the curves forthe total mass tend to a Dirac-120575 function at large time Theapproaching process is slightly different which depends onthe parameter 120572

Then the effects of 120572 on the conditional distribution ofthe first component in mass-given particles were discussedand it is found that the conditional distribution is a Gaussianfunction at large time which is independent of the valueof 120572 Besides the conditional distribution in particles withmoderate mass is quite different which strongly depends onthe parameter 120572

The exact solutions obtained in this paper are of potentialhelp to understanding the coagulation in a bicomponentsystem The solutions can also be used to test the validity of

Mathematical Problems in Engineering 7

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

5 times 10minus6

4 times 10minus6

3 times 10minus6

2 times 10minus6

1 times 10minus6

0

N(c|

a+b=x)

c

(a)

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

5 times 10minus6

4 times 10minus6

3 times 10minus6

2 times 10minus6

1 times 10minus6

0

N(c|

a+b=x)

c

(b)

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

5 times 10minus6

4 times 10minus6

3 times 10minus6

2 times 10minus6

1 times 10minus6

0

N(c|

a+b=x)

c

(c)

Figure 6The compositional distributions for 119909 = 100with1198730= 1119872

10= 13119872

01= 1 and (a) 120572 = 01 (b) 120572 = 1 and (c) 120572 = 20 Compared

with overall 119886-component mass concentration 1198880= 11987210(11987210+11987201) (here 119888

0= 025)

the numerical methods developed to solve multicomponentSmoluchowskirsquos equation

Symbols

V119886 V119887 The size of particles 119886 119887

11987200(119905) Particle number at time 119905

11987210 Mass of 119886-component

11987201 Mass of 119887-component

1198730 Initial number of particles

120601 Equivalent total mass of particles120591 Characteristic coagulation time120572 Parameters of component effects on coagulation119888 119886-component mean concentration1198880 Overall 119886-component mass concentration

120590 Average particles size119909 Mass of given particles

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This project is supported by National Natural Science Foun-dation of China (no 11272196 and no 11222222) and ScientificResearch Development Fund of Zhejiang Agriculture andForestry University (JYYY1502)

References

[1] F M Gelbard and J H Seinfeld ldquoCoagulation and growth ofa multicomponent aerosolrdquo Journal of Colloid And InterfaceScience vol 63 no 3 pp 472ndash479 1978

[2] S M Iveson ldquoLimitations of one-dimensional population bal-ance models of wet granulation processesrdquo Powder Technologyvol 124 no 3 pp 219ndash229 2002

[3] F Puel G Fevotte and J P Klein ldquoSimulation and analysis ofindustrial crystallization processes through multidimensionalpopulation balance equations Part 1 a resolution algorithmbased on the method of classesrdquo Chemical Engineering Sciencevol 58 no 16 pp 3715ndash3727 2003

[4] H Briesen ldquoSimulation of crystal size and shape by means of areduced two-dimensional population balancemodelrdquoChemicalEngineering Science vol 61 no 1 pp 104ndash112 2006

8 Mathematical Problems in Engineering

[5] B L Cushing V L Kolesnichenko and C J OrsquoConnorldquolsquoRecentadvances in the liquid-phase syntheses of inorganic nanoparti-clesrdquo Chemical Reviews vol 104 no 9 pp 3893ndash3946 2004

[6] P Rajniak C Mancinelli R T Chern F Stepanek L Farberand B T Hill ldquoExperimental study of wet granulation influidized bed impact of the binder properties on the granulemorphologyrdquo International Journal of Pharmaceutics vol 334no 1-2 pp 92ndash102 2007

[7] A A Lushnikov ldquoEvolution of coagulating systems III Coag-ulating mixturesrdquo Journal of Colloid And Interface Science vol54 no 1 pp 94ndash101 1976

[8] P L Krapivsky and E Ben-Naim ldquoAggregation with multipleconservation lawsrdquo Physical Review E vol 53 no 1 pp 291ndash2981996

[9] R D Vigil and R M Ziff ldquoOn the scaling theory of two-com-ponent aggregationrdquo Chemical Engineering Science vol 53 no9 pp 1725ndash1729 1998

[10] T Matsoukas K Lee and T Kim ldquoMixing of components intwo-component aggregationrdquo AIChE Journal vol 52 no 9 pp3088ndash3099 2006

[11] K Lee T Kim P Rajniak and T Matsoukas ldquoCompositionaldistributions in multicomponent aggregationrdquo Chemical Engi-neering Science vol 63 no 5 pp 1293ndash1303 2008

[12] M-L Yang Z-M Lu and Y-L Liu ldquoSelf-similar behaviorfor multicomponent coagulationrdquo Applied Mathematics andMechanics English Edition vol 35 no 11 pp 1353ndash1360 2014

[13] J M Fernandez-Dıaz and G J Gomez-Garcıa ldquoExact solutionof Smoluchowskirsquos continuous multi-component equation withan additive kernelrdquo Europhysics Letters vol 78 no 5 Article ID56002 2007

[14] M Delgado ldquoThe Lagrange-Charpit methodrdquo SIAM Reviewvol 39 no 2 pp 298ndash304 1997

[15] I J Good ldquoGeneralizations to several variables of Lagrangersquosexpansion with applications to stochastic processesrdquo Mathe-matical Proceedings of the Cambridge Philosophical Society vol56 no 4 pp 367ndash380 1960

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

Applying Laplace inverse transform we obtain

119892 (V119886 V119887 120591)

=

infin

sum

1198961=0

infin

sum

1198962=0

(minus1)1198961+1198962 1205721198961

11989611198962

(120591

120572V1198860+ V1198870

)

1198961+1198962

sdot1

1 + 1198961+ 1198962

119871minus1

[1205971198961+1198962

1205971199041198961

11205971199041198962

2

1198661+1198961+1198962

0(1199041 1199042)]

(12)

By rearranging the multiple series in one we arrive at

119892 (V119886 V119887 120591) =

infin

sum

119896=0

(((120572V119886+ V119887) (120572V

1198860+ V1198870)) 120591)119896

(119896 + 1)

sdot 119871minus1

[1198661+119896

0(1199041 1199042)]

(13)

With (4) we finally obtain the general solutions of (1)

119873(V119886 V119887 120591) = 119873

0(1 minus 120591) exp(minus

120572V119886+ V119887

120572V1198860+ V1198870

120591)

sdot

infin

sum

119896=0

(((120572V119886+ V119887) (120572V

1198860+ V1198870)) 120591)119896

(119896 + 1)

sdot 119871minus1

[1198661+119896

0(1199041 1199042)]

(14)

This solution can be applied for any initial distributionsby determining the multidimensional Laplace transform1198660

1+119896

(1199041 1199042) If exponential initial PSD is assumed

119873(V119886 V119887 0) =

1198730

1198721011987201

exp(minusV119886

11987210

minusV119887

11987201

) (15)

The Laplace transform for this function is

1198660(1199041 1199042) =

1198730

1198721011987201

1

1199041+ 1119872

10

1

1199042+ 1119872

01

(16)

and we can obtain the explicit solution as below

119873(V119886 V119887 120591) =

1198730

1198721011987201

(1 minus 120591)

sdot exp(minus120572V119886+ V119887

120572V1198860+ V1198870

120591 minusV119886

11987210

minusV119887

11987201

)

sdot

infin

sum

119896=0

((V119886V1198871198721011987201) ((120572V

119886+ V119887) (120572V

1198860+ V1198870)) 120591)119896

119896 (119896 + 1)

(17)

It is easily verified that the solution in [13] is recovered if weare letting 120572 = 1 We will further analyse some interestingproperties of solution (17) in the next section and attentionis paid particularly to the effects of parameter 120572 on thecoagulation properties of such systems

3 The Effects of 120572 on the Coagulation

31 The Total Number and Mass of Particles with the Concen-tration 119888 Mixing degree of the mixtures is one of the keyissues in bicomponent coagulation problems To this end wedefine two important magnitudes introduced in [7 13] Oneis the total number of particles having the concentration 119888 ofthe first component

119873(119888 119905) = ∬

infin

0

119873(V119886 V119887 120591) 120575 (119888 minus

V119886

V119886+ V119887

)119889V119886119889V119887

(18)

and the mass of these particles

119872(119888 119905) = ∬

infin

0

(V119886+ V119887)119873 (V

119886 V119887 120591)

sdot 120575 (119888 minusV119886

V119886+ V119887

)119889V119886119889V119887

(19)

With the compositional-dependent additive kernel (1) andthe initial condition (15) we obtain

119873(119888 120591) =1 minus 120591

11987210119872011198602

infin

sum

119896=0

(3119896 + 1)

(119896 + 1) (119896)2(119863

1198603)

119896

119872 (119888 120591) =1 minus 120591

11987210119872011198603

infin

sum

119896=0

(3119896 + 2)

(119896 + 1) (119896)2(119863

1198603)

119896

(20)

with

119860 =119888

11987210

+1 minus 119888

11987201

+120572119888 + (1 minus 119888)

12057211987201+ 1119872

10

120591

1198721011987201

119863 =120572119888 + (1 minus 119888)

12057211987201+ 1119872

10

120591119888 (1 minus 119888)

(1198721011987201)2

(21)

As suggested in [13] we observe the process at differentaverage particles size 120590 with 120591 = 1 minus 120590minus12

Figures 1ndash3 show the evolution of total number and totalmass with concentration 119888 for 120572 = 01 1 and 20 respectivelyIt is shown that the overall behavior of the evolution oftotal number and mass is similar for different 120572 while theevolution for total number ((a) in Figures 1ndash3) and totalmass ((b) in Figures 1ndash3) is obviously different The curvesof total number do not tend to a Dirac-120575 function whereasthe curves of total mass do and this result is consistent withthe result in [13] The effect of 120572 can be clearly seen fromcomparison between Figures 2(b) and 3(b) and it is shownthat the maxima for the total mass approach the overallfraction from the left end for 120572 = 1 but from the right endfor 120572 = 20This is due to the fact that the larger the parameter120572 the bigger the contribution of the 119886-component to thecoagulation which leads to the quicker growth of particleswith higher concentration

32 Conditional Distribution for Mass-Given Particles Con-ditional distribution is another important issue in bicom-ponent coagulation problems Vigil and Ziff [9] postulatedthe general scaling law for two-component aggregation with

4 Mathematical Problems in Engineering

15

10

05

00

00 02 04 06 08 10

120590 = 2

120590 = 20

120590 = 100

120590 = 500

N(c)

c

(a)

0

10

20

30

40

50

60

120590 = 2

120590 = 50

120590 = 100

120590 = 500

00 02 04 06 08 10

M(c)

c

(b)

Figure 1 Number (a) and mass (b) for initial exponential distribution with 120572 = 01 and1198730= 1 and119872

10= 13 and119872

01= 1

00 02 04 06 08 10

120590 = 2

120590 = 20

120590 = 100

120590 = 500

15

10

05

00

N(c)

c

(a)

00 02 04 06 08 10

120590 = 2

120590 = 20

120590 = 100

120590 = 500

0

10

20

30

40

50M(c)

c

(b)

Figure 2 Number (a) and mass (b) for initial exponential distribution with 120572 = 1 and1198730= 1 and119872

10= 13 and119872

01= 1

00 02 04 06 08 10

120590 = 2

120590 = 20

120590 = 100

120590 = 500

15

20

10

05

00

N(c)

c

(a)

00 02 04 06 08 10

120590 = 2

120590 = 20

120590 = 100

120590 = 500

0

10

20

30

40

50

M(c)

c

(b)

Figure 3 Number (a) and mass (b) for initial exponential distribution with 120572 = 20 and1198730= 1 and119872

10= 13 and119872

01= 1

Mathematical Problems in Engineering 5

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

035

030

025

020

015

010

005

000

N(c|

a+b=x)

c

(a)

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

035

030

025

020

015

010

005

000

N(c|

a+b=x)

c

(b)

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

035

030

025

020

015

010

005

000

N(c|

a+b=x)

c

(c)

Figure 4 The compositional distributions for 119909 = 1 with1198730= 1119872

10= 13119872

01= 1 and (a) 120572 = 01 (b) 120572 = 1 and (c) 120572 = 20

nongelling kernels that bivariate PSD function can be theproduct of a normal distribution in particle composition andthe one-parameter scaling function for the correspondinghomogeneous coagulation problem And for large time thecompositional distribution in large particles is a Gaussianfunction We explore the conditional distribution for acomposition-dependent kernel given in (2) and again theeffects of 120572 are analysed in this section To this aim weintroduce the conditional 119886-component number distribution119873(119888 | 119909 = V

119886+ V119887) where 119888 = V

119886(V119886+ V119887) is 119886-component

concentration for particles given mass 119909Figure 4 shows the conditional distribution for particles

given mass 119909 = 1 with three different 120572 values It is shownthat the general behavior is similar for all three 120572 values forexample the total number decreases with the coagulationgoing on while the curves become flat The effects ofparameter 120572 in these small particles are nearly negligibleFigure 5 shows the conditional distribution for particles givenmass 119909 = 4 with three different 120572 values The differenceof 120572 being smaller than 1 and bigger than 1 is clearly seenby comparison of Figures 5(a) and 5(b) with Figure 5(c)

The obvious difference is that there exist two extreme pointsof the curve 119873(119888 | 119909 = V

119886+ V119887) for the case 120572 = 20

while only one extreme point exists for 120572 le 1 which meansmixing degree is worse for 120572 gt 1 than 120572 le 1 Comparisonof Figures 4 and 5 shows that 119886-component is mixed better inlarger particles than inminute particlesThis tendency can befurther illustrated in Figure 6 which shows the distributionfor 119909 = 100 It is clearly shown in Figure 6 that the mixture iswell mixed when 120590 gt 10 irrespective of the value 120572 It is alsonoticed that the distribution curves for 3 different 120572 valuescollapse each other to a Gaussian function which can beverified analytically in fact since from the exact solutions (17)we can obtain for large particles at large time the conditionaldistribution as below119873(119888 | 119909)

sim1

radic4120587 (1198882

0(1 minus 1198880)2

119909)

exp(minus119909 (119888 minus 119888

0)2

41198882

0(1 minus 1198880)2)

(22)

Equation (22) is a Gaussian function and is independent ofthe value 120572

6 Mathematical Problems in Engineering

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

0000

0005

0010

0015

N(c|

a+b=x)

c

(a)

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

0000

0005

0010

0015

N(c|

a+b=x)

c

(b)

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

0000

0005

0010

0015

N(c|

a+b=x)

c

(c)

Figure 5 The compositional distributions for 119909 = 4 with1198730= 1119872

10= 13119872

01= 1 and (a) 120572 = 01 (b) 120572 = 1 and (c) 120572 = 20

From Figure 6 we can find an obvious difference betweenthe results for 120572 = 20 and the results for 120572 le 1 Thedistribution approaches to the asymptotic steady solution(22) from right end when 120572 = 20 while the curves approachthis steady state from left end In other words the equilib-rium concentration 119888 of the compositional distribution isbigger than overall 119886-component mass concentration 119888

0=

11987210(11987210+ 11987201) when 120572 = 20 and smaller than overall

concentration when 120572 le 1 The mechanism is the same asexplained in Section 31 In fact it can be proved that when120572 gt 43 the equilibrium concentration is larger than theoverall fraction 119888

0at a finite time

4 Conclusion

In this paper we have obtained the exact solution ofSmoluchowskirsquos continuous two-component equation witha composition-dependent additive kernel which is given as119870(V119886 V119887 V1015840119886 V1015840119887) = 120572(V

119886+ V1015840119886) + (V119887+ V1015840119887) for any initial particle

size distributions The main characteristics of the solution

and the effects of parameter 120572 on the bicomponent coagu-lation have been analysed in detail for an exponential initialdistribution

The effects of 120572 on the total number and mass of particleswith concentration 119888 were first investigated and the resultsshow that the curves for the total number do not reduceto a Dirac-120575 function for any given 120572rsquos while the curves forthe total mass tend to a Dirac-120575 function at large time Theapproaching process is slightly different which depends onthe parameter 120572

Then the effects of 120572 on the conditional distribution ofthe first component in mass-given particles were discussedand it is found that the conditional distribution is a Gaussianfunction at large time which is independent of the valueof 120572 Besides the conditional distribution in particles withmoderate mass is quite different which strongly depends onthe parameter 120572

The exact solutions obtained in this paper are of potentialhelp to understanding the coagulation in a bicomponentsystem The solutions can also be used to test the validity of

Mathematical Problems in Engineering 7

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

5 times 10minus6

4 times 10minus6

3 times 10minus6

2 times 10minus6

1 times 10minus6

0

N(c|

a+b=x)

c

(a)

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

5 times 10minus6

4 times 10minus6

3 times 10minus6

2 times 10minus6

1 times 10minus6

0

N(c|

a+b=x)

c

(b)

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

5 times 10minus6

4 times 10minus6

3 times 10minus6

2 times 10minus6

1 times 10minus6

0

N(c|

a+b=x)

c

(c)

Figure 6The compositional distributions for 119909 = 100with1198730= 1119872

10= 13119872

01= 1 and (a) 120572 = 01 (b) 120572 = 1 and (c) 120572 = 20 Compared

with overall 119886-component mass concentration 1198880= 11987210(11987210+11987201) (here 119888

0= 025)

the numerical methods developed to solve multicomponentSmoluchowskirsquos equation

Symbols

V119886 V119887 The size of particles 119886 119887

11987200(119905) Particle number at time 119905

11987210 Mass of 119886-component

11987201 Mass of 119887-component

1198730 Initial number of particles

120601 Equivalent total mass of particles120591 Characteristic coagulation time120572 Parameters of component effects on coagulation119888 119886-component mean concentration1198880 Overall 119886-component mass concentration

120590 Average particles size119909 Mass of given particles

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This project is supported by National Natural Science Foun-dation of China (no 11272196 and no 11222222) and ScientificResearch Development Fund of Zhejiang Agriculture andForestry University (JYYY1502)

References

[1] F M Gelbard and J H Seinfeld ldquoCoagulation and growth ofa multicomponent aerosolrdquo Journal of Colloid And InterfaceScience vol 63 no 3 pp 472ndash479 1978

[2] S M Iveson ldquoLimitations of one-dimensional population bal-ance models of wet granulation processesrdquo Powder Technologyvol 124 no 3 pp 219ndash229 2002

[3] F Puel G Fevotte and J P Klein ldquoSimulation and analysis ofindustrial crystallization processes through multidimensionalpopulation balance equations Part 1 a resolution algorithmbased on the method of classesrdquo Chemical Engineering Sciencevol 58 no 16 pp 3715ndash3727 2003

[4] H Briesen ldquoSimulation of crystal size and shape by means of areduced two-dimensional population balancemodelrdquoChemicalEngineering Science vol 61 no 1 pp 104ndash112 2006

8 Mathematical Problems in Engineering

[5] B L Cushing V L Kolesnichenko and C J OrsquoConnorldquolsquoRecentadvances in the liquid-phase syntheses of inorganic nanoparti-clesrdquo Chemical Reviews vol 104 no 9 pp 3893ndash3946 2004

[6] P Rajniak C Mancinelli R T Chern F Stepanek L Farberand B T Hill ldquoExperimental study of wet granulation influidized bed impact of the binder properties on the granulemorphologyrdquo International Journal of Pharmaceutics vol 334no 1-2 pp 92ndash102 2007

[7] A A Lushnikov ldquoEvolution of coagulating systems III Coag-ulating mixturesrdquo Journal of Colloid And Interface Science vol54 no 1 pp 94ndash101 1976

[8] P L Krapivsky and E Ben-Naim ldquoAggregation with multipleconservation lawsrdquo Physical Review E vol 53 no 1 pp 291ndash2981996

[9] R D Vigil and R M Ziff ldquoOn the scaling theory of two-com-ponent aggregationrdquo Chemical Engineering Science vol 53 no9 pp 1725ndash1729 1998

[10] T Matsoukas K Lee and T Kim ldquoMixing of components intwo-component aggregationrdquo AIChE Journal vol 52 no 9 pp3088ndash3099 2006

[11] K Lee T Kim P Rajniak and T Matsoukas ldquoCompositionaldistributions in multicomponent aggregationrdquo Chemical Engi-neering Science vol 63 no 5 pp 1293ndash1303 2008

[12] M-L Yang Z-M Lu and Y-L Liu ldquoSelf-similar behaviorfor multicomponent coagulationrdquo Applied Mathematics andMechanics English Edition vol 35 no 11 pp 1353ndash1360 2014

[13] J M Fernandez-Dıaz and G J Gomez-Garcıa ldquoExact solutionof Smoluchowskirsquos continuous multi-component equation withan additive kernelrdquo Europhysics Letters vol 78 no 5 Article ID56002 2007

[14] M Delgado ldquoThe Lagrange-Charpit methodrdquo SIAM Reviewvol 39 no 2 pp 298ndash304 1997

[15] I J Good ldquoGeneralizations to several variables of Lagrangersquosexpansion with applications to stochastic processesrdquo Mathe-matical Proceedings of the Cambridge Philosophical Society vol56 no 4 pp 367ndash380 1960

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

15

10

05

00

00 02 04 06 08 10

120590 = 2

120590 = 20

120590 = 100

120590 = 500

N(c)

c

(a)

0

10

20

30

40

50

60

120590 = 2

120590 = 50

120590 = 100

120590 = 500

00 02 04 06 08 10

M(c)

c

(b)

Figure 1 Number (a) and mass (b) for initial exponential distribution with 120572 = 01 and1198730= 1 and119872

10= 13 and119872

01= 1

00 02 04 06 08 10

120590 = 2

120590 = 20

120590 = 100

120590 = 500

15

10

05

00

N(c)

c

(a)

00 02 04 06 08 10

120590 = 2

120590 = 20

120590 = 100

120590 = 500

0

10

20

30

40

50M(c)

c

(b)

Figure 2 Number (a) and mass (b) for initial exponential distribution with 120572 = 1 and1198730= 1 and119872

10= 13 and119872

01= 1

00 02 04 06 08 10

120590 = 2

120590 = 20

120590 = 100

120590 = 500

15

20

10

05

00

N(c)

c

(a)

00 02 04 06 08 10

120590 = 2

120590 = 20

120590 = 100

120590 = 500

0

10

20

30

40

50

M(c)

c

(b)

Figure 3 Number (a) and mass (b) for initial exponential distribution with 120572 = 20 and1198730= 1 and119872

10= 13 and119872

01= 1

Mathematical Problems in Engineering 5

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

035

030

025

020

015

010

005

000

N(c|

a+b=x)

c

(a)

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

035

030

025

020

015

010

005

000

N(c|

a+b=x)

c

(b)

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

035

030

025

020

015

010

005

000

N(c|

a+b=x)

c

(c)

Figure 4 The compositional distributions for 119909 = 1 with1198730= 1119872

10= 13119872

01= 1 and (a) 120572 = 01 (b) 120572 = 1 and (c) 120572 = 20

nongelling kernels that bivariate PSD function can be theproduct of a normal distribution in particle composition andthe one-parameter scaling function for the correspondinghomogeneous coagulation problem And for large time thecompositional distribution in large particles is a Gaussianfunction We explore the conditional distribution for acomposition-dependent kernel given in (2) and again theeffects of 120572 are analysed in this section To this aim weintroduce the conditional 119886-component number distribution119873(119888 | 119909 = V

119886+ V119887) where 119888 = V

119886(V119886+ V119887) is 119886-component

concentration for particles given mass 119909Figure 4 shows the conditional distribution for particles

given mass 119909 = 1 with three different 120572 values It is shownthat the general behavior is similar for all three 120572 values forexample the total number decreases with the coagulationgoing on while the curves become flat The effects ofparameter 120572 in these small particles are nearly negligibleFigure 5 shows the conditional distribution for particles givenmass 119909 = 4 with three different 120572 values The differenceof 120572 being smaller than 1 and bigger than 1 is clearly seenby comparison of Figures 5(a) and 5(b) with Figure 5(c)

The obvious difference is that there exist two extreme pointsof the curve 119873(119888 | 119909 = V

119886+ V119887) for the case 120572 = 20

while only one extreme point exists for 120572 le 1 which meansmixing degree is worse for 120572 gt 1 than 120572 le 1 Comparisonof Figures 4 and 5 shows that 119886-component is mixed better inlarger particles than inminute particlesThis tendency can befurther illustrated in Figure 6 which shows the distributionfor 119909 = 100 It is clearly shown in Figure 6 that the mixture iswell mixed when 120590 gt 10 irrespective of the value 120572 It is alsonoticed that the distribution curves for 3 different 120572 valuescollapse each other to a Gaussian function which can beverified analytically in fact since from the exact solutions (17)we can obtain for large particles at large time the conditionaldistribution as below119873(119888 | 119909)

sim1

radic4120587 (1198882

0(1 minus 1198880)2

119909)

exp(minus119909 (119888 minus 119888

0)2

41198882

0(1 minus 1198880)2)

(22)

Equation (22) is a Gaussian function and is independent ofthe value 120572

6 Mathematical Problems in Engineering

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

0000

0005

0010

0015

N(c|

a+b=x)

c

(a)

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

0000

0005

0010

0015

N(c|

a+b=x)

c

(b)

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

0000

0005

0010

0015

N(c|

a+b=x)

c

(c)

Figure 5 The compositional distributions for 119909 = 4 with1198730= 1119872

10= 13119872

01= 1 and (a) 120572 = 01 (b) 120572 = 1 and (c) 120572 = 20

From Figure 6 we can find an obvious difference betweenthe results for 120572 = 20 and the results for 120572 le 1 Thedistribution approaches to the asymptotic steady solution(22) from right end when 120572 = 20 while the curves approachthis steady state from left end In other words the equilib-rium concentration 119888 of the compositional distribution isbigger than overall 119886-component mass concentration 119888

0=

11987210(11987210+ 11987201) when 120572 = 20 and smaller than overall

concentration when 120572 le 1 The mechanism is the same asexplained in Section 31 In fact it can be proved that when120572 gt 43 the equilibrium concentration is larger than theoverall fraction 119888

0at a finite time

4 Conclusion

In this paper we have obtained the exact solution ofSmoluchowskirsquos continuous two-component equation witha composition-dependent additive kernel which is given as119870(V119886 V119887 V1015840119886 V1015840119887) = 120572(V

119886+ V1015840119886) + (V119887+ V1015840119887) for any initial particle

size distributions The main characteristics of the solution

and the effects of parameter 120572 on the bicomponent coagu-lation have been analysed in detail for an exponential initialdistribution

The effects of 120572 on the total number and mass of particleswith concentration 119888 were first investigated and the resultsshow that the curves for the total number do not reduceto a Dirac-120575 function for any given 120572rsquos while the curves forthe total mass tend to a Dirac-120575 function at large time Theapproaching process is slightly different which depends onthe parameter 120572

Then the effects of 120572 on the conditional distribution ofthe first component in mass-given particles were discussedand it is found that the conditional distribution is a Gaussianfunction at large time which is independent of the valueof 120572 Besides the conditional distribution in particles withmoderate mass is quite different which strongly depends onthe parameter 120572

The exact solutions obtained in this paper are of potentialhelp to understanding the coagulation in a bicomponentsystem The solutions can also be used to test the validity of

Mathematical Problems in Engineering 7

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

5 times 10minus6

4 times 10minus6

3 times 10minus6

2 times 10minus6

1 times 10minus6

0

N(c|

a+b=x)

c

(a)

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

5 times 10minus6

4 times 10minus6

3 times 10minus6

2 times 10minus6

1 times 10minus6

0

N(c|

a+b=x)

c

(b)

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

5 times 10minus6

4 times 10minus6

3 times 10minus6

2 times 10minus6

1 times 10minus6

0

N(c|

a+b=x)

c

(c)

Figure 6The compositional distributions for 119909 = 100with1198730= 1119872

10= 13119872

01= 1 and (a) 120572 = 01 (b) 120572 = 1 and (c) 120572 = 20 Compared

with overall 119886-component mass concentration 1198880= 11987210(11987210+11987201) (here 119888

0= 025)

the numerical methods developed to solve multicomponentSmoluchowskirsquos equation

Symbols

V119886 V119887 The size of particles 119886 119887

11987200(119905) Particle number at time 119905

11987210 Mass of 119886-component

11987201 Mass of 119887-component

1198730 Initial number of particles

120601 Equivalent total mass of particles120591 Characteristic coagulation time120572 Parameters of component effects on coagulation119888 119886-component mean concentration1198880 Overall 119886-component mass concentration

120590 Average particles size119909 Mass of given particles

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This project is supported by National Natural Science Foun-dation of China (no 11272196 and no 11222222) and ScientificResearch Development Fund of Zhejiang Agriculture andForestry University (JYYY1502)

References

[1] F M Gelbard and J H Seinfeld ldquoCoagulation and growth ofa multicomponent aerosolrdquo Journal of Colloid And InterfaceScience vol 63 no 3 pp 472ndash479 1978

[2] S M Iveson ldquoLimitations of one-dimensional population bal-ance models of wet granulation processesrdquo Powder Technologyvol 124 no 3 pp 219ndash229 2002

[3] F Puel G Fevotte and J P Klein ldquoSimulation and analysis ofindustrial crystallization processes through multidimensionalpopulation balance equations Part 1 a resolution algorithmbased on the method of classesrdquo Chemical Engineering Sciencevol 58 no 16 pp 3715ndash3727 2003

[4] H Briesen ldquoSimulation of crystal size and shape by means of areduced two-dimensional population balancemodelrdquoChemicalEngineering Science vol 61 no 1 pp 104ndash112 2006

8 Mathematical Problems in Engineering

[5] B L Cushing V L Kolesnichenko and C J OrsquoConnorldquolsquoRecentadvances in the liquid-phase syntheses of inorganic nanoparti-clesrdquo Chemical Reviews vol 104 no 9 pp 3893ndash3946 2004

[6] P Rajniak C Mancinelli R T Chern F Stepanek L Farberand B T Hill ldquoExperimental study of wet granulation influidized bed impact of the binder properties on the granulemorphologyrdquo International Journal of Pharmaceutics vol 334no 1-2 pp 92ndash102 2007

[7] A A Lushnikov ldquoEvolution of coagulating systems III Coag-ulating mixturesrdquo Journal of Colloid And Interface Science vol54 no 1 pp 94ndash101 1976

[8] P L Krapivsky and E Ben-Naim ldquoAggregation with multipleconservation lawsrdquo Physical Review E vol 53 no 1 pp 291ndash2981996

[9] R D Vigil and R M Ziff ldquoOn the scaling theory of two-com-ponent aggregationrdquo Chemical Engineering Science vol 53 no9 pp 1725ndash1729 1998

[10] T Matsoukas K Lee and T Kim ldquoMixing of components intwo-component aggregationrdquo AIChE Journal vol 52 no 9 pp3088ndash3099 2006

[11] K Lee T Kim P Rajniak and T Matsoukas ldquoCompositionaldistributions in multicomponent aggregationrdquo Chemical Engi-neering Science vol 63 no 5 pp 1293ndash1303 2008

[12] M-L Yang Z-M Lu and Y-L Liu ldquoSelf-similar behaviorfor multicomponent coagulationrdquo Applied Mathematics andMechanics English Edition vol 35 no 11 pp 1353ndash1360 2014

[13] J M Fernandez-Dıaz and G J Gomez-Garcıa ldquoExact solutionof Smoluchowskirsquos continuous multi-component equation withan additive kernelrdquo Europhysics Letters vol 78 no 5 Article ID56002 2007

[14] M Delgado ldquoThe Lagrange-Charpit methodrdquo SIAM Reviewvol 39 no 2 pp 298ndash304 1997

[15] I J Good ldquoGeneralizations to several variables of Lagrangersquosexpansion with applications to stochastic processesrdquo Mathe-matical Proceedings of the Cambridge Philosophical Society vol56 no 4 pp 367ndash380 1960

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

035

030

025

020

015

010

005

000

N(c|

a+b=x)

c

(a)

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

035

030

025

020

015

010

005

000

N(c|

a+b=x)

c

(b)

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

035

030

025

020

015

010

005

000

N(c|

a+b=x)

c

(c)

Figure 4 The compositional distributions for 119909 = 1 with1198730= 1119872

10= 13119872

01= 1 and (a) 120572 = 01 (b) 120572 = 1 and (c) 120572 = 20

nongelling kernels that bivariate PSD function can be theproduct of a normal distribution in particle composition andthe one-parameter scaling function for the correspondinghomogeneous coagulation problem And for large time thecompositional distribution in large particles is a Gaussianfunction We explore the conditional distribution for acomposition-dependent kernel given in (2) and again theeffects of 120572 are analysed in this section To this aim weintroduce the conditional 119886-component number distribution119873(119888 | 119909 = V

119886+ V119887) where 119888 = V

119886(V119886+ V119887) is 119886-component

concentration for particles given mass 119909Figure 4 shows the conditional distribution for particles

given mass 119909 = 1 with three different 120572 values It is shownthat the general behavior is similar for all three 120572 values forexample the total number decreases with the coagulationgoing on while the curves become flat The effects ofparameter 120572 in these small particles are nearly negligibleFigure 5 shows the conditional distribution for particles givenmass 119909 = 4 with three different 120572 values The differenceof 120572 being smaller than 1 and bigger than 1 is clearly seenby comparison of Figures 5(a) and 5(b) with Figure 5(c)

The obvious difference is that there exist two extreme pointsof the curve 119873(119888 | 119909 = V

119886+ V119887) for the case 120572 = 20

while only one extreme point exists for 120572 le 1 which meansmixing degree is worse for 120572 gt 1 than 120572 le 1 Comparisonof Figures 4 and 5 shows that 119886-component is mixed better inlarger particles than inminute particlesThis tendency can befurther illustrated in Figure 6 which shows the distributionfor 119909 = 100 It is clearly shown in Figure 6 that the mixture iswell mixed when 120590 gt 10 irrespective of the value 120572 It is alsonoticed that the distribution curves for 3 different 120572 valuescollapse each other to a Gaussian function which can beverified analytically in fact since from the exact solutions (17)we can obtain for large particles at large time the conditionaldistribution as below119873(119888 | 119909)

sim1

radic4120587 (1198882

0(1 minus 1198880)2

119909)

exp(minus119909 (119888 minus 119888

0)2

41198882

0(1 minus 1198880)2)

(22)

Equation (22) is a Gaussian function and is independent ofthe value 120572

6 Mathematical Problems in Engineering

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

0000

0005

0010

0015

N(c|

a+b=x)

c

(a)

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

0000

0005

0010

0015

N(c|

a+b=x)

c

(b)

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

0000

0005

0010

0015

N(c|

a+b=x)

c

(c)

Figure 5 The compositional distributions for 119909 = 4 with1198730= 1119872

10= 13119872

01= 1 and (a) 120572 = 01 (b) 120572 = 1 and (c) 120572 = 20

From Figure 6 we can find an obvious difference betweenthe results for 120572 = 20 and the results for 120572 le 1 Thedistribution approaches to the asymptotic steady solution(22) from right end when 120572 = 20 while the curves approachthis steady state from left end In other words the equilib-rium concentration 119888 of the compositional distribution isbigger than overall 119886-component mass concentration 119888

0=

11987210(11987210+ 11987201) when 120572 = 20 and smaller than overall

concentration when 120572 le 1 The mechanism is the same asexplained in Section 31 In fact it can be proved that when120572 gt 43 the equilibrium concentration is larger than theoverall fraction 119888

0at a finite time

4 Conclusion

In this paper we have obtained the exact solution ofSmoluchowskirsquos continuous two-component equation witha composition-dependent additive kernel which is given as119870(V119886 V119887 V1015840119886 V1015840119887) = 120572(V

119886+ V1015840119886) + (V119887+ V1015840119887) for any initial particle

size distributions The main characteristics of the solution

and the effects of parameter 120572 on the bicomponent coagu-lation have been analysed in detail for an exponential initialdistribution

The effects of 120572 on the total number and mass of particleswith concentration 119888 were first investigated and the resultsshow that the curves for the total number do not reduceto a Dirac-120575 function for any given 120572rsquos while the curves forthe total mass tend to a Dirac-120575 function at large time Theapproaching process is slightly different which depends onthe parameter 120572

Then the effects of 120572 on the conditional distribution ofthe first component in mass-given particles were discussedand it is found that the conditional distribution is a Gaussianfunction at large time which is independent of the valueof 120572 Besides the conditional distribution in particles withmoderate mass is quite different which strongly depends onthe parameter 120572

The exact solutions obtained in this paper are of potentialhelp to understanding the coagulation in a bicomponentsystem The solutions can also be used to test the validity of

Mathematical Problems in Engineering 7

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

5 times 10minus6

4 times 10minus6

3 times 10minus6

2 times 10minus6

1 times 10minus6

0

N(c|

a+b=x)

c

(a)

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

5 times 10minus6

4 times 10minus6

3 times 10minus6

2 times 10minus6

1 times 10minus6

0

N(c|

a+b=x)

c

(b)

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

5 times 10minus6

4 times 10minus6

3 times 10minus6

2 times 10minus6

1 times 10minus6

0

N(c|

a+b=x)

c

(c)

Figure 6The compositional distributions for 119909 = 100with1198730= 1119872

10= 13119872

01= 1 and (a) 120572 = 01 (b) 120572 = 1 and (c) 120572 = 20 Compared

with overall 119886-component mass concentration 1198880= 11987210(11987210+11987201) (here 119888

0= 025)

the numerical methods developed to solve multicomponentSmoluchowskirsquos equation

Symbols

V119886 V119887 The size of particles 119886 119887

11987200(119905) Particle number at time 119905

11987210 Mass of 119886-component

11987201 Mass of 119887-component

1198730 Initial number of particles

120601 Equivalent total mass of particles120591 Characteristic coagulation time120572 Parameters of component effects on coagulation119888 119886-component mean concentration1198880 Overall 119886-component mass concentration

120590 Average particles size119909 Mass of given particles

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This project is supported by National Natural Science Foun-dation of China (no 11272196 and no 11222222) and ScientificResearch Development Fund of Zhejiang Agriculture andForestry University (JYYY1502)

References

[1] F M Gelbard and J H Seinfeld ldquoCoagulation and growth ofa multicomponent aerosolrdquo Journal of Colloid And InterfaceScience vol 63 no 3 pp 472ndash479 1978

[2] S M Iveson ldquoLimitations of one-dimensional population bal-ance models of wet granulation processesrdquo Powder Technologyvol 124 no 3 pp 219ndash229 2002

[3] F Puel G Fevotte and J P Klein ldquoSimulation and analysis ofindustrial crystallization processes through multidimensionalpopulation balance equations Part 1 a resolution algorithmbased on the method of classesrdquo Chemical Engineering Sciencevol 58 no 16 pp 3715ndash3727 2003

[4] H Briesen ldquoSimulation of crystal size and shape by means of areduced two-dimensional population balancemodelrdquoChemicalEngineering Science vol 61 no 1 pp 104ndash112 2006

8 Mathematical Problems in Engineering

[5] B L Cushing V L Kolesnichenko and C J OrsquoConnorldquolsquoRecentadvances in the liquid-phase syntheses of inorganic nanoparti-clesrdquo Chemical Reviews vol 104 no 9 pp 3893ndash3946 2004

[6] P Rajniak C Mancinelli R T Chern F Stepanek L Farberand B T Hill ldquoExperimental study of wet granulation influidized bed impact of the binder properties on the granulemorphologyrdquo International Journal of Pharmaceutics vol 334no 1-2 pp 92ndash102 2007

[7] A A Lushnikov ldquoEvolution of coagulating systems III Coag-ulating mixturesrdquo Journal of Colloid And Interface Science vol54 no 1 pp 94ndash101 1976

[8] P L Krapivsky and E Ben-Naim ldquoAggregation with multipleconservation lawsrdquo Physical Review E vol 53 no 1 pp 291ndash2981996

[9] R D Vigil and R M Ziff ldquoOn the scaling theory of two-com-ponent aggregationrdquo Chemical Engineering Science vol 53 no9 pp 1725ndash1729 1998

[10] T Matsoukas K Lee and T Kim ldquoMixing of components intwo-component aggregationrdquo AIChE Journal vol 52 no 9 pp3088ndash3099 2006

[11] K Lee T Kim P Rajniak and T Matsoukas ldquoCompositionaldistributions in multicomponent aggregationrdquo Chemical Engi-neering Science vol 63 no 5 pp 1293ndash1303 2008

[12] M-L Yang Z-M Lu and Y-L Liu ldquoSelf-similar behaviorfor multicomponent coagulationrdquo Applied Mathematics andMechanics English Edition vol 35 no 11 pp 1353ndash1360 2014

[13] J M Fernandez-Dıaz and G J Gomez-Garcıa ldquoExact solutionof Smoluchowskirsquos continuous multi-component equation withan additive kernelrdquo Europhysics Letters vol 78 no 5 Article ID56002 2007

[14] M Delgado ldquoThe Lagrange-Charpit methodrdquo SIAM Reviewvol 39 no 2 pp 298ndash304 1997

[15] I J Good ldquoGeneralizations to several variables of Lagrangersquosexpansion with applications to stochastic processesrdquo Mathe-matical Proceedings of the Cambridge Philosophical Society vol56 no 4 pp 367ndash380 1960

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

0000

0005

0010

0015

N(c|

a+b=x)

c

(a)

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

0000

0005

0010

0015

N(c|

a+b=x)

c

(b)

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

0000

0005

0010

0015

N(c|

a+b=x)

c

(c)

Figure 5 The compositional distributions for 119909 = 4 with1198730= 1119872

10= 13119872

01= 1 and (a) 120572 = 01 (b) 120572 = 1 and (c) 120572 = 20

From Figure 6 we can find an obvious difference betweenthe results for 120572 = 20 and the results for 120572 le 1 Thedistribution approaches to the asymptotic steady solution(22) from right end when 120572 = 20 while the curves approachthis steady state from left end In other words the equilib-rium concentration 119888 of the compositional distribution isbigger than overall 119886-component mass concentration 119888

0=

11987210(11987210+ 11987201) when 120572 = 20 and smaller than overall

concentration when 120572 le 1 The mechanism is the same asexplained in Section 31 In fact it can be proved that when120572 gt 43 the equilibrium concentration is larger than theoverall fraction 119888

0at a finite time

4 Conclusion

In this paper we have obtained the exact solution ofSmoluchowskirsquos continuous two-component equation witha composition-dependent additive kernel which is given as119870(V119886 V119887 V1015840119886 V1015840119887) = 120572(V

119886+ V1015840119886) + (V119887+ V1015840119887) for any initial particle

size distributions The main characteristics of the solution

and the effects of parameter 120572 on the bicomponent coagu-lation have been analysed in detail for an exponential initialdistribution

The effects of 120572 on the total number and mass of particleswith concentration 119888 were first investigated and the resultsshow that the curves for the total number do not reduceto a Dirac-120575 function for any given 120572rsquos while the curves forthe total mass tend to a Dirac-120575 function at large time Theapproaching process is slightly different which depends onthe parameter 120572

Then the effects of 120572 on the conditional distribution ofthe first component in mass-given particles were discussedand it is found that the conditional distribution is a Gaussianfunction at large time which is independent of the valueof 120572 Besides the conditional distribution in particles withmoderate mass is quite different which strongly depends onthe parameter 120572

The exact solutions obtained in this paper are of potentialhelp to understanding the coagulation in a bicomponentsystem The solutions can also be used to test the validity of

Mathematical Problems in Engineering 7

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

5 times 10minus6

4 times 10minus6

3 times 10minus6

2 times 10minus6

1 times 10minus6

0

N(c|

a+b=x)

c

(a)

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

5 times 10minus6

4 times 10minus6

3 times 10minus6

2 times 10minus6

1 times 10minus6

0

N(c|

a+b=x)

c

(b)

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

5 times 10minus6

4 times 10minus6

3 times 10minus6

2 times 10minus6

1 times 10minus6

0

N(c|

a+b=x)

c

(c)

Figure 6The compositional distributions for 119909 = 100with1198730= 1119872

10= 13119872

01= 1 and (a) 120572 = 01 (b) 120572 = 1 and (c) 120572 = 20 Compared

with overall 119886-component mass concentration 1198880= 11987210(11987210+11987201) (here 119888

0= 025)

the numerical methods developed to solve multicomponentSmoluchowskirsquos equation

Symbols

V119886 V119887 The size of particles 119886 119887

11987200(119905) Particle number at time 119905

11987210 Mass of 119886-component

11987201 Mass of 119887-component

1198730 Initial number of particles

120601 Equivalent total mass of particles120591 Characteristic coagulation time120572 Parameters of component effects on coagulation119888 119886-component mean concentration1198880 Overall 119886-component mass concentration

120590 Average particles size119909 Mass of given particles

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This project is supported by National Natural Science Foun-dation of China (no 11272196 and no 11222222) and ScientificResearch Development Fund of Zhejiang Agriculture andForestry University (JYYY1502)

References

[1] F M Gelbard and J H Seinfeld ldquoCoagulation and growth ofa multicomponent aerosolrdquo Journal of Colloid And InterfaceScience vol 63 no 3 pp 472ndash479 1978

[2] S M Iveson ldquoLimitations of one-dimensional population bal-ance models of wet granulation processesrdquo Powder Technologyvol 124 no 3 pp 219ndash229 2002

[3] F Puel G Fevotte and J P Klein ldquoSimulation and analysis ofindustrial crystallization processes through multidimensionalpopulation balance equations Part 1 a resolution algorithmbased on the method of classesrdquo Chemical Engineering Sciencevol 58 no 16 pp 3715ndash3727 2003

[4] H Briesen ldquoSimulation of crystal size and shape by means of areduced two-dimensional population balancemodelrdquoChemicalEngineering Science vol 61 no 1 pp 104ndash112 2006

8 Mathematical Problems in Engineering

[5] B L Cushing V L Kolesnichenko and C J OrsquoConnorldquolsquoRecentadvances in the liquid-phase syntheses of inorganic nanoparti-clesrdquo Chemical Reviews vol 104 no 9 pp 3893ndash3946 2004

[6] P Rajniak C Mancinelli R T Chern F Stepanek L Farberand B T Hill ldquoExperimental study of wet granulation influidized bed impact of the binder properties on the granulemorphologyrdquo International Journal of Pharmaceutics vol 334no 1-2 pp 92ndash102 2007

[7] A A Lushnikov ldquoEvolution of coagulating systems III Coag-ulating mixturesrdquo Journal of Colloid And Interface Science vol54 no 1 pp 94ndash101 1976

[8] P L Krapivsky and E Ben-Naim ldquoAggregation with multipleconservation lawsrdquo Physical Review E vol 53 no 1 pp 291ndash2981996

[9] R D Vigil and R M Ziff ldquoOn the scaling theory of two-com-ponent aggregationrdquo Chemical Engineering Science vol 53 no9 pp 1725ndash1729 1998

[10] T Matsoukas K Lee and T Kim ldquoMixing of components intwo-component aggregationrdquo AIChE Journal vol 52 no 9 pp3088ndash3099 2006

[11] K Lee T Kim P Rajniak and T Matsoukas ldquoCompositionaldistributions in multicomponent aggregationrdquo Chemical Engi-neering Science vol 63 no 5 pp 1293ndash1303 2008

[12] M-L Yang Z-M Lu and Y-L Liu ldquoSelf-similar behaviorfor multicomponent coagulationrdquo Applied Mathematics andMechanics English Edition vol 35 no 11 pp 1353ndash1360 2014

[13] J M Fernandez-Dıaz and G J Gomez-Garcıa ldquoExact solutionof Smoluchowskirsquos continuous multi-component equation withan additive kernelrdquo Europhysics Letters vol 78 no 5 Article ID56002 2007

[14] M Delgado ldquoThe Lagrange-Charpit methodrdquo SIAM Reviewvol 39 no 2 pp 298ndash304 1997

[15] I J Good ldquoGeneralizations to several variables of Lagrangersquosexpansion with applications to stochastic processesrdquo Mathe-matical Proceedings of the Cambridge Philosophical Society vol56 no 4 pp 367ndash380 1960

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

5 times 10minus6

4 times 10minus6

3 times 10minus6

2 times 10minus6

1 times 10minus6

0

N(c|

a+b=x)

c

(a)

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

5 times 10minus6

4 times 10minus6

3 times 10minus6

2 times 10minus6

1 times 10minus6

0

N(c|

a+b=x)

c

(b)

00 02 04 06 08 10

120590 = 10

120590 = 50

120590 = 200

Initial

5 times 10minus6

4 times 10minus6

3 times 10minus6

2 times 10minus6

1 times 10minus6

0

N(c|

a+b=x)

c

(c)

Figure 6The compositional distributions for 119909 = 100with1198730= 1119872

10= 13119872

01= 1 and (a) 120572 = 01 (b) 120572 = 1 and (c) 120572 = 20 Compared

with overall 119886-component mass concentration 1198880= 11987210(11987210+11987201) (here 119888

0= 025)

the numerical methods developed to solve multicomponentSmoluchowskirsquos equation

Symbols

V119886 V119887 The size of particles 119886 119887

11987200(119905) Particle number at time 119905

11987210 Mass of 119886-component

11987201 Mass of 119887-component

1198730 Initial number of particles

120601 Equivalent total mass of particles120591 Characteristic coagulation time120572 Parameters of component effects on coagulation119888 119886-component mean concentration1198880 Overall 119886-component mass concentration

120590 Average particles size119909 Mass of given particles

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This project is supported by National Natural Science Foun-dation of China (no 11272196 and no 11222222) and ScientificResearch Development Fund of Zhejiang Agriculture andForestry University (JYYY1502)

References

[1] F M Gelbard and J H Seinfeld ldquoCoagulation and growth ofa multicomponent aerosolrdquo Journal of Colloid And InterfaceScience vol 63 no 3 pp 472ndash479 1978

[2] S M Iveson ldquoLimitations of one-dimensional population bal-ance models of wet granulation processesrdquo Powder Technologyvol 124 no 3 pp 219ndash229 2002

[3] F Puel G Fevotte and J P Klein ldquoSimulation and analysis ofindustrial crystallization processes through multidimensionalpopulation balance equations Part 1 a resolution algorithmbased on the method of classesrdquo Chemical Engineering Sciencevol 58 no 16 pp 3715ndash3727 2003

[4] H Briesen ldquoSimulation of crystal size and shape by means of areduced two-dimensional population balancemodelrdquoChemicalEngineering Science vol 61 no 1 pp 104ndash112 2006

8 Mathematical Problems in Engineering

[5] B L Cushing V L Kolesnichenko and C J OrsquoConnorldquolsquoRecentadvances in the liquid-phase syntheses of inorganic nanoparti-clesrdquo Chemical Reviews vol 104 no 9 pp 3893ndash3946 2004

[6] P Rajniak C Mancinelli R T Chern F Stepanek L Farberand B T Hill ldquoExperimental study of wet granulation influidized bed impact of the binder properties on the granulemorphologyrdquo International Journal of Pharmaceutics vol 334no 1-2 pp 92ndash102 2007

[7] A A Lushnikov ldquoEvolution of coagulating systems III Coag-ulating mixturesrdquo Journal of Colloid And Interface Science vol54 no 1 pp 94ndash101 1976

[8] P L Krapivsky and E Ben-Naim ldquoAggregation with multipleconservation lawsrdquo Physical Review E vol 53 no 1 pp 291ndash2981996

[9] R D Vigil and R M Ziff ldquoOn the scaling theory of two-com-ponent aggregationrdquo Chemical Engineering Science vol 53 no9 pp 1725ndash1729 1998

[10] T Matsoukas K Lee and T Kim ldquoMixing of components intwo-component aggregationrdquo AIChE Journal vol 52 no 9 pp3088ndash3099 2006

[11] K Lee T Kim P Rajniak and T Matsoukas ldquoCompositionaldistributions in multicomponent aggregationrdquo Chemical Engi-neering Science vol 63 no 5 pp 1293ndash1303 2008

[12] M-L Yang Z-M Lu and Y-L Liu ldquoSelf-similar behaviorfor multicomponent coagulationrdquo Applied Mathematics andMechanics English Edition vol 35 no 11 pp 1353ndash1360 2014

[13] J M Fernandez-Dıaz and G J Gomez-Garcıa ldquoExact solutionof Smoluchowskirsquos continuous multi-component equation withan additive kernelrdquo Europhysics Letters vol 78 no 5 Article ID56002 2007

[14] M Delgado ldquoThe Lagrange-Charpit methodrdquo SIAM Reviewvol 39 no 2 pp 298ndash304 1997

[15] I J Good ldquoGeneralizations to several variables of Lagrangersquosexpansion with applications to stochastic processesrdquo Mathe-matical Proceedings of the Cambridge Philosophical Society vol56 no 4 pp 367ndash380 1960

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

[5] B L Cushing V L Kolesnichenko and C J OrsquoConnorldquolsquoRecentadvances in the liquid-phase syntheses of inorganic nanoparti-clesrdquo Chemical Reviews vol 104 no 9 pp 3893ndash3946 2004

[6] P Rajniak C Mancinelli R T Chern F Stepanek L Farberand B T Hill ldquoExperimental study of wet granulation influidized bed impact of the binder properties on the granulemorphologyrdquo International Journal of Pharmaceutics vol 334no 1-2 pp 92ndash102 2007

[7] A A Lushnikov ldquoEvolution of coagulating systems III Coag-ulating mixturesrdquo Journal of Colloid And Interface Science vol54 no 1 pp 94ndash101 1976

[8] P L Krapivsky and E Ben-Naim ldquoAggregation with multipleconservation lawsrdquo Physical Review E vol 53 no 1 pp 291ndash2981996

[9] R D Vigil and R M Ziff ldquoOn the scaling theory of two-com-ponent aggregationrdquo Chemical Engineering Science vol 53 no9 pp 1725ndash1729 1998

[10] T Matsoukas K Lee and T Kim ldquoMixing of components intwo-component aggregationrdquo AIChE Journal vol 52 no 9 pp3088ndash3099 2006

[11] K Lee T Kim P Rajniak and T Matsoukas ldquoCompositionaldistributions in multicomponent aggregationrdquo Chemical Engi-neering Science vol 63 no 5 pp 1293ndash1303 2008

[12] M-L Yang Z-M Lu and Y-L Liu ldquoSelf-similar behaviorfor multicomponent coagulationrdquo Applied Mathematics andMechanics English Edition vol 35 no 11 pp 1353ndash1360 2014

[13] J M Fernandez-Dıaz and G J Gomez-Garcıa ldquoExact solutionof Smoluchowskirsquos continuous multi-component equation withan additive kernelrdquo Europhysics Letters vol 78 no 5 Article ID56002 2007

[14] M Delgado ldquoThe Lagrange-Charpit methodrdquo SIAM Reviewvol 39 no 2 pp 298ndash304 1997

[15] I J Good ldquoGeneralizations to several variables of Lagrangersquosexpansion with applications to stochastic processesrdquo Mathe-matical Proceedings of the Cambridge Philosophical Society vol56 no 4 pp 367ndash380 1960

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of