CHINESE JOURNAL OF COMPUTATIONAL PHYSICS

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1 CHINESE JOURNAL OF COMPUTATIONAL PHYSICS Vol 30 No 1 Jan X Monte Carlo 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 NCET MPCS-2011-D xu_zuwei@ 163 com linsmannzhb@ 163 com

2 90 30 MC 4 MC 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 Maorant ernel Marov MC Irizarry Marov Lécot 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 Normal-DWMC MC Monte Carlo DWMC DWMC i w w

3 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 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 Eibec Wagner 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

4 92 30 MC - MC - β' i 1 a = v i / v ^β i = 槡 2K fm v 1 /6 a 2 /3 + a 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 / a -1 /2 槡 2K fm v 1 /6 v max / v 2 /3 + v max / v 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 / v min / v -1 /2 1 /6 3 6 A K fm = B T 4π ρ p 1 /2 B Boltzmann T ρ p Fast-DWMC Fast-DWMC ^β' max - r r β' i / ^β'max i N r N r 0 - N r /2! = Σ i = 1 2V s = Σ β' i = 1 i 2V s 2V sn r + 1-1β - N r +1 ' i - 1Σ = β' i β' i 1 ^β' max ^β' max 12

5 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 = Monte Carlo 2 Table 2 2 Calculating parameters of conditions Case v 0 N 0! coag β / N 0 A A m / N 0 K fm v 0 1 /6 K fm u 1 /3 + v 1 /3 2 u v /2 A = m 3 s - 1 K fm = 3 1 /6 6 B T 4π ρ ρ = g m - 3 μ = Pa s p 1 /2 T = 300 K Boltzmann B = J K - 1

6 Fast-DWMC CPU Smith 21 MC Kruis 12 CPU IntelRCoreTM2 Quad 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 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 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

7 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 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 -

8 96 30 v max v min v max v max v min v max v min v min v min v max v min ! coag 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 Press Deng LLi G A summarization on Monte Carlo simulation in particle transportj Chinese Journal of Computational Physics M Garcia A LVan Den Broec CAertsens Met al A Monte Carlo simulation of coagulationj Physica AStatistical Mechanics and its Applications Liffman K A direct simulation Monte-Carlo method for cluster coagulationj Journal of Computational Physics

9 1 Monte Carlo 97 6 Lin YLee KMatsouas T Solution of the population balance equation using constant-number Monte CarloJ Chemical Engineering Science Zhao HZheng C A new event-driven constant-volume method for solution of the time evolution of particle size distribution J Journal of Computational Physics Zhao HZheng CXu M Multi-Monte Carlo method for particle coagulationdescription and validationj Applied Mathematics and Computation Zhao HZheng CXu M Multi-Monte Carlo method for coagulation and condensation / evaporation in dispersed systemsj Journal of Colloid and Interface Science Zhao HKruis F EZheng C A differentially weighted Monte Carlo method for two-component coagulationj Journal of Computational Physics GPU M Kruis F EMaisels AFissan H Direct simulation Monte Carlo method for particle coagulation and aggregationj AIChE Journal Eibec AWagner W An efficient stochastic algorithm for studing coagulation dynamics and gelation phenomenaj SIAM Journal on Scientific Computing Eibec AWagner W Stochastic particle approximations for Smoluchosi's coagulation equationj The Annals of Applied Probability Irizarry R Fast Monte Carlo methodology for multivariate particulate systems IPoint ensemble Monte CarloJ Chemical Engineering Science Irizarry R Fast Monte Carlo methodology for multivariate particulate systems-ii - PEMCJ Chemical Engineering Science Lécot CTarhini A A quasi-stochastic simulation of the general dynamics equation for aerosolsj Monte Carlo Methods and Applications Li STian DDeng L High efficiency Monte Carlo sample method for super-high energy neutronsj Chinese Journal of Computational Physics Zhao 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 Technology Zhao HZheng C The event-driven constant volume method for particle coagulation dynamicsj Science in China Series E- Technological Sciences Smith MMatsouas T Constant-number Monte Carlo simulation of population balancesj Chemical Engineering Science Seinfeld J HPandis S N Atmospheric chemistry and physics from air pollution to climate change M New YorWiley Vemury SPratsinis S E Self-preserving size distributions of agglomerates J Journal of Aerosol Science Fast Monte Carlo Method for Particle Coagulation Dynamics XU ZuweiZHAO HaiboLIU XinZHENG Chuguang State Key Laboratory of Coal CombustionHuazhong University of Science and TechnologyWuhan China 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 Revised date