Error analysis in stochastic solutions of population balance equations

Zhou, Kun; Jiang, Xiao; Chan, Tat Leung

doi:10.1016/j.apm.2019.11.045

Cited by 8 publications

(10 citation statements)

References 64 publications

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“…system time) to restrict the stochastic error (e.g. variance and standard deviation) (Zhou et al , 2020). However, the tails of the particle size spectrum always contain only a small number of numerical particles, which are poorly represented in the numerical simulation.…”

Section: Introductionmentioning

confidence: 99%

A new weighted fraction Monte Carlo method for particle coagulation

Jiang

Chan

2021

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Purpose The purpose of this study is to investigate the aerosol dynamics of the particle coagulation process using a newly developed weighted fraction Monte Carlo (WFMC) method. Design/methodology/approach The weighted numerical particles are adopted in a similar manner to the multi-Monte Carlo (MMC) method, with the addition of a new fraction function (α). Probabilistic removal is also introduced to maintain a constant number scheme. Findings Three typical cases with constant kernel, free-molecular coagulation kernel and different initial distributions for particle coagulation are simulated and validated. The results show an excellent agreement between the Monte Carlo (MC) method and the corresponding analytical solutions or sectional method results. Further numerical results show that the critical stochastic error in the newly proposed WFMC method is significantly reduced when compared with the traditional MMC method for higher-order moments with only a slight increase in computational cost. The particle size distribution is also found to extend for the larger size regime with the WFMC method, which is traditionally insufficient in the classical direct simulation MC and MMC methods. The effects of different fraction functions on the weight function are also investigated. Originality Value Stochastic error is inevitable in MC simulations of aerosol dynamics. To minimize this critical stochastic error, many algorithms, such as MMC method, have been proposed. However, the weight of the numerical particles is not adjustable. This newly developed algorithm with an adjustable weight of the numerical particles can provide improved stochastic error reduction.

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Section: Introductionmentioning

confidence: 99%

A new weighted fraction Monte Carlo method for particle coagulation

Jiang

Chan

2021

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“…The WFMC method (Jiang and Chan, 2021) is developed based on the basic concept of MMC method (Zhao et al , 2009), but the implementation is similar to the weighted flow algorithm (WFA) (DeVille et al , 2011) which also removes the coagulated particle followed the probability to keep the conservation of total particle mass. It has already been proved that the methods with differential weight distribution can reduce the stochastic error for higher-order moments and PSDs in larger particle size regime (Zhou et al , 2020).…”

Section: Methodsmentioning

confidence: 99%

“…In MC simulation, resampling (Smith and Matsoukas, 1998) or resizing (Liffman, 1992) are always applied to maintain constant number of numerical particles. However, it has been proved that these methods will introduce stochastic error in numerical simulation (Zhou et al , 2020). In the present study, the number of weighted numerical particles is controlled by the stochastic merging method Kotalczyk and Kruis, 2017) to reduce such resampling error.…”

Section: Methodsmentioning

confidence: 99%

Lagrangian particle tracking with new weighted fraction Monte Carlo method for studying the soot particle size distributions in premixed flames

Jiang

Chan

2021

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Purpose The purpose of this paper is to study the soot formation and evolution by using this newly developed Lagrangian particle tracking with weighted fraction Monte Carlo (LPT-WFMC) method. Design/methodology/approach The weighted soot particles are used in this MC framework and is tracked using Lagrangian approach. A detailed soot model based on the LPT-WFMC method is used to study the soot formation and evolution in ethylene laminar premixed flames. Findings The LPT-WFMC method is validated by both experimental and numerical results of the direct simulation Monte Carlo (DSMC) and Multi-Monte Carlo (MMC) methods. Compared with DSMC and MMC methods, the stochastic error analysis shows this new LPT-WFMC method could further extend the particle size distributions (PSDs) and improve the accuracy for predicting soot PSDs at larger particle size regime. Originality/value Compared with conventional weighted particle schemes, the weight distributions in LPT-WFMC method are adjustable by adopting different fraction functions. As a result, the number of numerical soot particles in each size interval could be also adjustable. The stochastic error of PSDs in larger particle size regime can also be minimized by increasing the number of numerical soot particles at larger size interval.

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“…In Case III, the particles initially have an exponential size distribution which satisfies Equation (26). The coagulation kernel in the free molecular regime is used which is expressed as Equation (27).…”

Section: Coagulation and Condensation/evaporation Processes In Single...mentioning

confidence: 99%

“…The disadvantage of the MC method is that it is difficult to balance computational accuracy and computational cost. To solve this problem, researchers put forward the concept of "weighted fictitious particles", meaning that each fictitious particle can represent a certain number of real particles of the same physical properties, which dramatically reduces the computational consumption of the MC methods [25,26]. In order to reduce use of computing memory and simulation time of the MC algorithm, Zhao et al [27][28][29] put forward a differentially weighted Monte Carlo (DWMC) method to solve the particle balance equation.…”

Section: Introductionmentioning

confidence: 99%

Numerical Modeling of Droplet Aerosol Coagulation, Condensation/Evaporation and Deposition Processes

et al. 2022

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The differentially weighted operator-splitting Monte Carlo (DWOSMC) method is further developed to describe the droplet aerosol dynamic behaviors, including coagulation, deposition, condensation, and evaporation processes. It is first proposed that the droplet aerosols will experience firstly condensation and then evaporation, and this phenomenon is first implemented into the Monte Carlo method and sectional method with considering coagulation, deposition, and condensation/evaporation processes in both single-component and two-component aerosol particle systems. It is found that the calculated results of the DWOSMC method agree well with both the analytical solutions and the sectional method. The further developed DWOSMC method can predict the variation of particle number density, total particle volume, mean particle diameter, particle size distributions, and the component-related particle volume densities in both single component and two-component droplet aerosol systems considering coagulation, deposition, and condensation/evaporation processes.

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Error analysis in stochastic solutions of population balance equations

Cited by 8 publications

References 64 publications

A new weighted fraction Monte Carlo method for particle coagulation

A new weighted fraction Monte Carlo method for particle coagulation

Lagrangian particle tracking with new weighted fraction Monte Carlo method for studying the soot particle size distributions in premixed flames

Numerical Modeling of Droplet Aerosol Coagulation, Condensation/Evaporation and Deposition Processes

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