CHINESE JOURNAL OF COMPUTATIONAL PHYSICS

30 1 2013 1 CHINESE JOURNAL OF COMPUTATIONAL PHYSICS Vol 30 No 1 Jan 2013 1001-246X201301-0089-09 Monte Carlo 430074 Monte CarloMC - MC O Monte Carlo O211 62O359 A 0 PSD PBE PBE 1 nv t = 1 t 2 v v max βv - u u tnv - u tnu tdu - nv t v min βv u tnu tdu v min u v m 3 nv t m - 3 m - 3 nv tdv t v ~ v + dv βv u t t v u 1 u u < v v - u v v PBE PBM Monte CarloMC MC 2 MC PBE PBE 3 MC 1 2012-04 - 13 2012-08 - 06 50876037 NCET-09-0395 MPCS-2011-D-02 1986 - E-mailxu_zuwei@ 163 com 1977 - E-maillinsmannzhb@ 163 com

90 30 MC 4 MC 5 4 6 7 8-9 7 10 MC ON 2 1 s N /2 s MC 1 MPI MPI-Cluster OpenMP OpenCL / CUDA GPU 11 2 MC Kruis 12 Smart booeeping technology Eibec Wagner 13-14 Maorant ernel Marov MC Irizarry 15-16 Marov Lécot 17 18 ICF MC MC Differentially weighted Monte Carlo DWMC 19 MC Fast-DWMC - - DWMC Normal-DWMC Fast-DWMC Fast-DWMC Normal-DWMC Fast-DWMC Fast-DWMC O 1 1 1 Normal-DWMC MC Monte Carlo DWMC DWMC i w w

1 Monte Carlo 91 i min w min w v i + v 1 2 i old > w old new = old - w old v i new = v i old 2 { w new = w old v new = v i old + v old old = w old new = old /2 v i new = v i old + v old 3 { w new = w old /2 v new = v i old + v old new old V s R' coag 20 R' coag = 1 Σ β' 2V sσ 2 i i = 1 = 1 i β' i 4 2max w = β i w 5 + w Marov - 1 Δt ED R' coag Δt ED = 1 V s R' coag = 2V s / Σ Σ i = 1 = 1 i i P i = β' i /Σ Σ β' i i = 1 = 1 i ( β' i ) 6 i - 4-0 1 r r β' i / β' max β' max = max i i = 1 = 1 i 7 8 β' i - - Normal-DWMC β' max Δt β' max Normal-DWMC ON 2 s Fast-DWMC 1 2 Fast-DWMC 1 2 1 Eibec Wagner 13-14 MC β i ^β i ^β i β i ^β i ^β i = Σ h i g h i g i β max ^β max = Σ maxh i maxg maxh i maxg ^β max

92 30 MC - MC - β' i 1 a = v i / v ^β i = 槡 2K fm v 1 /6 a 2 /3 + a 1 /6 + 1 + a -1 /2 9 2w max w 2w + w v min a = v i v v β i 2w max w ^β i + w v max v 2w max w 2 ^β i w + w ^β i = 槡 2K fm v 1 /6 a 2 /3 + a 1 /6 + 1 + a -1 /2 槡 2K fm v 1 /6 v max / v 2 /3 + v max / v 1 /6 + 1 + v min / v -1 /2 ^β' i ^β' i = 2 ^β i w = 2 槡 2K fm v 1 /6 w 1 + v max / v 1 /6 + v max / v 2 /3 + v min / v -1 /2 10 ^β' max β' max ^β' max O 1 Table 1 Normal and maorant coagulation ernels β' max β i = A ^β i = A ^β'i = 2Aw β i = K fm v i 1 /3 + v 1 /3 2 v -1 i + v -1 1 /2 ^β i = 槡 2K fm v 1 /6 i + v 1 /6 + v 2 /3-1 /2 i v + v 2 /3-1 /2 vi ^β' i = 2 槡 2K fm v 1 /6 w v max / v 2 /3 + v max / v 1 /6 + 1 + v min / v -1 /2 1 /6 3 6 A K fm = B T 4π ρ p 1 /2 B Boltzmann T ρ p 1 2 2 Fast-DWMC Fast-DWMC ^β' max - r r β' i / ^β'max i N r N r 0 - N r + 1 - - - 1/2! = Σ i = 1 2V s = Σ β' i = 1 i 2V s 2V sn r + 1-1β - N r +1 ' i - 1Σ = 1 11 12 β' i β' i 1 ^β' max ^β' max 12

1 Monte Carlo 93 Fig 1 1 Fast-DWMC Normal-DWMC Composition computing process of Fast-DWMC and Normal-DWMC ^β' max Normal-DWMC ^β' max 2 12 Fast-DWMC Normal-DWMC = 10 000 3 Monte Carlo 2 Table 2 2 Calculating parameters of conditions Case v 0 N 0! coag β 1 1 1 0 10 10 1 / N 0 A A 2 1 414 10-26 m 3 1 0 10 17 1 / N 0 K fm v 0 1 /6 K fm u 1 /3 + v 1 /3 2 u - 1 + v - 1 1 /2 A = 1 0 10-10 m 3 s - 1 K fm = 3 1 /6 6 B T 4π ρ ρ = 1 000 g m - 3 μ = 1 832 10-5 Pa s p 1 /2 T = 300 K Boltzmann B = 1 38 10-23 J K - 1

94 30 2 1 Fast-DWMC CPU Smith 21 MC Kruis 12 CPU IntelRCoreTM2 Quad Q8300 @ 2 5GHz 4GB MC CPU t CPU 2 Fig 2 2 Comparison of computing speeds 2 MC ON 2 s Fast-DWMC O N s Fast-DWMC 10 000 9 000 Fast-DWMC 2 2 MC 10 DWMC MC N σ N t = 1 Q ΣQ i = 1 槡 Σ L Δt i = 1 - N th N i N th 2 Σ L = 1 i Δt σ d t = 1 Q ΣQ i = 1 Σ L 槡 = 1 Δt i max - min + 1 Σ max P i t- P th t 2 = min Σ L = 1 i Δt Q MC L Δt i i MC N i i MC N th max min P i t i MC t MC P th t PSDF 22 2 2 1 Case 1 Nt P σ N t σ d t 3 2% Normal-DWMC Fast-DWMC 3 a Fast- DWMC = Aw 2max w / + β' i 13 14

1 Monte Carlo 95 w Normal-DWMC β' max Fast-DWMC w max ^β' max = 2Aw max β' max ^β' max 2β' max - Normal- DWMC Fast-DWMC Fast-DWMC 2a Fig 3 3 Case 1 Case 1constant coagulation ernel 2 2 2 Case 2 PSDF 23 4 η = v / v 珋 = Nv / Mψ = Mnv t/ N 2 N M v max v min 1 ^β' 4 Case 2 max β' max Fig 4 Case 2self-preserving particle size distribution -

96 30 v max v min v max v max v min v max v min v min v min v max v min 2 1 000! coag - 20 1 08 100 4 Fast-DWMC Fast-DWMC Normal-DWMC 2 b ^β' max - β' max 5 ^β'max / β' max Case 2 ^β' max β' max Fig 5 ^β'max / β' max trends over time θ = ^β'max / β' max 5 3 N ( s ) = - 1/2 2 - / / MC Fast-DWMC Normal- DWMC 1 Ramrishna D Population balancetheory and applications to particulate systems in engineering M San DiegoAcademic Press200047-116 2 Deng LLi G A summarization on Monte Carlo simulation in particle transportj Chinese Journal of Computational Physics 2010276 791-798 3 M 2008 4 Garcia A LVan Den Broec CAertsens Met al A Monte Carlo simulation of coagulationj Physica AStatistical Mechanics and its Applications19871433 535-546 5 Liffman K A direct simulation Monte-Carlo method for cluster coagulationj Journal of Computational Physics1992100 1 116-127

1 Monte Carlo 97 6 Lin YLee KMatsouas T Solution of the population balance equation using constant-number Monte CarloJ Chemical Engineering Science20025712 2241-2252 7 Zhao HZheng C A new event-driven constant-volume method for solution of the time evolution of particle size distribution J Journal of Computational Physics20092285 1412-1428 8 Zhao HZheng CXu M Multi-Monte Carlo method for particle coagulationdescription and validationj Applied Mathematics and Computation20051672 1383-1399 9 Zhao HZheng CXu M Multi-Monte Carlo method for coagulation and condensation / evaporation in dispersed systemsj Journal of Colloid and Interface Science20052861 195-208 10Zhao HKruis F EZheng C A differentially weighted Monte Carlo method for two-component coagulationj Journal of Computational Physics201022919 6931-6945 11 GPU M 2009 12Kruis F EMaisels AFissan H Direct simulation Monte Carlo method for particle coagulation and aggregationj AIChE Journal2000469 1735-1742 13Eibec AWagner W An efficient stochastic algorithm for studing coagulation dynamics and gelation phenomenaj SIAM Journal on Scientific Computing2000223 802-821 14Eibec AWagner W Stochastic particle approximations for Smoluchosi's coagulation equationj The Annals of Applied Probability2001114 1137-1165 15Irizarry R Fast Monte Carlo methodology for multivariate particulate systems IPoint ensemble Monte CarloJ Chemical Engineering Science2008631 95-110 16Irizarry R Fast Monte Carlo methodology for multivariate particulate systems-ii - PEMCJ Chemical Engineering Science 2008631 111-121 17Lécot CTarhini A A quasi-stochastic simulation of the general dynamics equation for aerosolsj Monte Carlo Methods and Applications2008135-6 369-388 18Li STian DDeng L High efficiency Monte Carlo sample method for super-high energy neutronsj Chinese Journal of Computational Physics2011283 323-328 19Zhao HKruis F EZheng C Reducing statistical noise and extending the size spectrum by applying weighted simulation particles in Monte Carlo simulation of coagulationj Aerosol Science and Technology2009438 781-793 20Zhao HZheng C The event-driven constant volume method for particle coagulation dynamicsj Science in China Series E- Technological Sciences2008518 1255-1271 21Smith MMatsouas T Constant-number Monte Carlo simulation of population balancesj Chemical Engineering Science 1998539 1777-1786 22Seinfeld J HPandis S N Atmospheric chemistry and physics from air pollution to climate change M New YorWiley1998 23Vemury SPratsinis S E Self-preserving size distributions of agglomerates J Journal of Aerosol Science1995262 175-185 Fast Monte Carlo Method for Particle Coagulation Dynamics XU ZuweiZHAO HaiboLIU XinZHENG Chuguang State Key Laboratory of Coal CombustionHuazhong University of Science and TechnologyWuhan 430074China Abstract We propose a fast random simulation strategy based on differentially weighted MC The strategy improves computation efficiency significantlyand guarantees enough calculation accuracythus coordinates contradiction between computation cost and computation accuracy The main idea is based on maorant ernel It is possible to transfer a traditional coagulation ernel to a maorant ernel through splitting and amplifying slightly The maximum of maornant ernel is obtained by single looping over all simulation particles The maximum maornant ernel is used to approximate the maximum coagulation ernel in particle populationand is further used to search coagulation particle pairs randomly with acceptance-reection method The waiting time time-stepfor a coagulation event is calculated by summing coagulation erenls of particle pairs involved in acceptance / reection processes Double looping in normal Monte Carlo simulation is avoided and computation efficiency is improved greatly Key words population balance modelingcoagulationmonte Carlo methoddifferentially weighted simulation particlesparticle size distribution Received date 2012-04 - 13Revised date 2012-08 - 06