Multi-Monte Carlo method for particle coagulation: description and validation

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Abstract

In the interest of decreasing computation cost and increasing computation precision of Monte Carlo method for general dynamics equation (GDE), a new multi-Monte Carlo (MMC) method for particle coagulation is prompted, which has characteristic of time-driven, constant-number and constant-volume Monte Carlo technique. The paper has described detailedly the scheme of MMC method, including the setting of time step, the choice of coagulation partner, the judgment the occurrence of coagulation event, and the consequential treatment of particle coagulation event. MMC method is validated by five special coagulation cases: (1) constant coagulation kernel of monodisperse particles; (2) constant coagulation kernel of exponential polydisperse particle distribution; (3) linear coagulation kernel of exponential polydisperse particle distribution; (4) quadratic coagulation kernel of exponential polydisperse particle distribution; (5) Brownian coagulation kernel of log-normal polydisperse particles in the continuum regime. The simulation results of MMC method for GDE agree with analytical solution well, and its computation cost is low enough to apply engineering computation and general scientific quantitative analysis.

Introduction

Solid particles (or droplets) coagulation is an important mechanism in both nature and engineering, including formation and evolvement of air aerosols and emulsion droplets, manufacture of nanoparticle agglomerates. Because many important properties such as light scattering, electrostatic charging, toxicity, radioactivity of suspended particles, sediment and capturing strategy depend on their size distribution, the time evolution of size distribution due to the particle coagulation is of fundamental interest and a key issue (see [1]). Particle size distribution (PSD) along with time is described by general dynamics equation (GDE), which takes account of the physical processes such as coagulation, condensation/evaporation, nucleation, breakage and deposition (see [2]). So GDE is a key point describing those physical events of particulate matter. The paper focuses on algorithm solving GDE for particle coagulation, where GDE for particle coagulation is as follows:dnp(v,t)dt=120vβ(v-u,u)np(v-u,t)np(u,t)du-np(v,t)0β(v,u)np(u,t)du,where np(v, t) is the particle size distribution function at time t, so that np(v, t)dv is the number concentration of particles whose size range between v and v + dv per volume unit at time t; the dimension of np(v, t) is particles/m3/m3, where “particles” denotes the number of particles; β(v, u) is the coagulation kernel for two particle of volume v and u, describing the probability of a binary coagulation event in unit time; the dimension of β(v, u) is m3/particles/s. The terms on the left-hand side of Eq. (1) describes the change in the number concentration of particles of volume v with time, and the two terms on the right-hand side describe respectively the gain and loss in number concentration due to coagulation.

PSD is usually polydisperse and spans widely, for example, pulverized coal fly ash particle formation is accurately described as a tri-modal PSD that includes a submicron fume region centered at approximately 0.08 μm diameter, a fine fragmentation region centered at approximately 2.0 μm diameter, and a bulk or supermicron fragmentation region for particles of approximately 5 μm diameter and greater (see [3]). In addition, some kinds of mechanisms (such as coagulation and condensation/evaporation) have different and complicated nonlinear influence on PSD. The above complication leads to a condition which normal numerical methods (such as finite volume method and finite difference method) are difficult to solve GDE. Nowadays the most popular numerical methods of GDE are moments of method (see [4]), Monte Carlo method (see [5], [6]), sectional method (see [7]) and so on. Those methods have both advantages and disadvantages. The merit of moments method is less computation time; however its model is complicated and it assumes special particle initial size distribution. In addition there is no information about the history of each particle which collides to form a bigger particle. Sectional method has a receivable computation cost and computation precision, however, sectional representation results in very complex algorithms and it is difficult to handle multi-component, more-dimensional, chemical reaction and coating, etc. Monte Carlo method are time-consuming comparatively, whereas it can gain information about history, trajectory crossing and internal structure of particles; the Monte Carlo algorithms for solving polydisperse and multi-component particle GDE are easily programmed even considering restructuring, coating, chemical reaction and fractal aggregation. With more and more strong computer power, simulation with some 104–107 particles is possible on fast PCs, which relieves greatly the contradiction of expensive computation. So Monte Carlo methods are adapted more and more in solving GDE.

Many researchers have investigated Monte Carlo method for solving GDE. To sum up, MC method can be divided into two classes according to approach of time-step setting: one is referred to as “time-driven” Monte Carlo, which takes into account all of possible event such as coagulation and breakage within a special adjustable time step, and time step must be less than or equal to minimum time in which every possible event takes place once for every simulation particle; the other more common Monte Carlo is called “event-driven” Monte Carlo. In general especial events are implemented stochastically with probabilities derived from the mean-field rates of the corresponding process. In simulation of event-driven MC, a single event is selected to occur, and the time is advanced by an appropriate increment. In contrast to time-driven MC, this MC does not need explicit time discretization and its time step, which is calculated during the simulation, adjusts itself to the rates of the various event processes. On the other hand, MC method can also be classified into two general classes according to whether the number of simulation particles and simulation volume are changed along with the evolvement of time. The first approach is to track a constant volume and thus grow or shrink the number of simulation particle in direct proportion to the number concentration of the physical system, while conserving the mass, this method is sometimes referred to as “constant-volume MC”, which cannot maintain constant statistical accuracy. The second class is “constant-number MC” prompted by Matsoukas etc. [8], [9], [10], in which the number of simulation particles is kept constant and the simulation volume is continuously adjusted so as to contain the same number of particles. The constant-number method maintains constant statistical accuracy and can simulate growth over arbitrarily long times with a finite number of simulation particles. Nevertheless, it is difficult for constant-number MC to take account of space dispersion of size function because of the expansion and contraction of the simulated subsystem volume. Furthermore, “event-driven” Monte Carlo method can hardly consider particle Lagrangian tracking, which is important in coupling with two-phase Euler/Lagrange model to investigate particle-flow interaction and particle motion.

In order to decrease computation cost and increase computation accuracy of Monte Carlo method for solving general dynamics equation (GDE) for coagulation, a new multi-Monte Carlo (MMC) method for coagulation is prompted. Firstly the MMC method is described detailedly; and then five special cases for which complete or partial analytical solutions exist are chosen to validate the MMC method; lastly some conclusions is drawn.

Section snippets

Description for multi-Monte Carlo method

In the first place, fictitious particles, of which the number is far less than real particles, are created by handling real particles. Those real particles which have same or similar volume can be considered to have the same properties and hence the same behaviors. Those real particles can be represented by one or several simulation particles (naming “weighted fictitious particle”) according to local particle size distribution. One fictitious particle, whose serial number is i, is endowed with

Computational cases

Because GDE describes the evolution of particle size distribution with time, few experiment result or even numerical simulation can be referred. In general the best usual and effective measure of validating algorithm for GDE is comparison with analytical solution in some special cases.

Simulation

Computational conditions of five cases are listed in Table 1, where Nf is the total number of fictitious particle. For collecting statistics properties such as particle size distribution, polydisperse particles must explicit particle bin discretization. For Cases 2–5, polydisperse particles are divided into 200 classes between the largest and smallest particle volume in the simulation. (Noting: no information about bin discretization needs during simulating, which will avoid numerical diffusion

Conclusion

Multi-Monte Carlo (MMC) method for general dynamics equation (GDE) is performed in the paper. Its characteristics are as follows: handling fictitious particles of which the number is far less than that of real particles to decrease computation cost, coupling “time-driven” Monte Carlo technique and constant-number Monte Carlo technique, and conserving computational domain. MMC method had been used to simulation five special coagulation cases. The agreement between MMC solution and analytical

Acknowledgments

We wish to thank “National Key Basic Research and Development Program 2002CB211602” for funds and the National Natural Science Foundation of China under grant number 50325621.

References (16)

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