Stochastic Particle Algorithm Research Articles

Stochastic weighted particle methods for population balance equations with coagulation, fragmentation and spatial inhomogeneity

This paper introduces stochastic weighted particle algorithms for the solution of multi-compartment population balance equations. In particular, it presents a class of fragmentation weight transfer functions which are constructed such that the number of computational particles stays constant during fragmentation events. The weight transfer functions are constructed based on systems of weighted computational particles and each of it leads to a stochastic particle algorithm for the numerical treatment of population balance equations. Besides fragmentation, the algorithms also consider physical processes such as coagulation and the exchange of mass with the surroundings. The numerical properties of the algorithms are compared to the direct simulation algorithm and an existing method for the fragmentation of weighted particles. It is found that the new algorithms show better numerical performance over the two existing methods especially for systems with significant amount of large particles and high fragmentation rates.

An accurate treatment of diffuse reflection boundary conditions for a stochastic particle Fokker–Planck algorithm with large time steps

In this paper, we present a stochastic particle algorithm for the simulation of flows of wall-confined gases with diffuse reflection boundary conditions. Based on the theoretical observation that the change in location of the particles consists of a deterministic part and a Wiener process if the time scale is much larger than the relaxation time, a new estimate for the first hitting time at the boundary is obtained. This estimate facilitates the construction of an algorithm with large time steps for wall-confined flows. Numerical simulations verify that the proposed algorithm reproduces the correct boundary behaviour.

Stochastic solution of population balance equations for reactor networks

This work presents a sequential modular approach to solve a generic network of reactors with a population balance model using a stochastic numerical method. Full-coupling to the gas-phase is achieved through operator-splitting. The convergence of the stochastic particle algorithm in test networks is evaluated as a function of network size, recycle fraction and numerical parameters. These test cases are used to identify methods through which systematic and statistical error may be reduced, including by use of stochastic weighted algorithms. The optimal algorithm was subsequently used to solve a one-dimensional example of silicon nanoparticle synthesis using a multivariate particle model. This example demonstrated the power of stochastic methods in resolving particle structure by investigating the transient and spatial evolution of primary polydispersity, degree of sintering and TEM-style images.

Probabilistic and deterministic algorithms for space multidimensional irregular porous media equation

The object of this paper is a multi-dimensional generalized porous media equation (PDE) with not smooth and possibly discontinuous coefficient $\beta$, which is well-posed as an evolution problem in $L^1(\mathbb{R}^d)$. This work continues the study related to the one-dimensional case by the same authors. One expects that a solution of the mentioned PDE can be represented through the solution (in law) of a non-linear stochastic differential equation (NLSDE). A classical tool for doing this is a uniqueness argument for some Fokker-Planck type equations with measurable coefficients. When $\beta$ is possibly discontinuous, this is often possible in dimension $d = 1$. If $d > 1$, this problem is more complex than for $d = 1$. However, it is possible to exhibit natural candidates for the probabilistic representation and to use them for approximating the solution of (PDE) through a stochastic particle algorithm. We compare it with some numerical deterministic techniques that we have implemented adapting the method of a paper of Cavalli et al. whose convergence was established when $\beta$ is Lipschitz. Special emphasis is also devoted to the case when the initial condition is radially symmetric. On the other hand assuming that $\beta$ is continuous (even though not smooth), one provides existence results for a mollified version of the (NLSDE) and a related partial integro-differential equation, even if the initial condition is a general probability measure.

On a multivariate population balance model to describe the structure and composition of silica nanoparticles

The aim of this work is to present the mathematical description of a detailed multivariate population balance model to describe the structure and composition of silica nanoparticles. A detailed numerical study of a stochastic particle algorithm for the solution of the multidimensional population balance model is presented. Each particle is described by its constituent primary particles and the connectivity between these primaries. Each primary in turn has internal variables that describe its chemical composition. The algorithms used to solve the population balance equations and to couple the population balance model to gas-phase chemistry are described. Numerical studies are then performed for a number of functionals calculated from the model to establish the convergence with respect to the numerical parameter that determines the number of computational particles in the system. The computational efficiency of the model is found to render it applicable to the simulation of industrial scale systems.

Stochastic weighted particle methods for population balance equations

A class of coagulation weight transfer functions is constructed, each member of which leads to a stochastic particle algorithm for the numerical treatment of population balance equations. These algorithms are based on systems of weighted computational particles and the weight transfer functions are constructed such that the number of computational particles does not change during coagulation events. The algorithms also facilitate the simulation of physical processes that change single particles, such as growth, or other surface reactions. Four members of the algorithm family have been numerically validated by comparison to analytic solutions to simple problems. Numerical experiments have been performed for complex laminar premixed flame systems in which members of the class of stochastic weighted particle methods were compared to each other and to a direct simulation algorithm. Two of the weighted algorithms have been shown to offer performance advantages over the direct simulation algorithm in situations where interest is focused on the larger particles in a system. The extent of this advantage depends on the particular system and on the quantities of interest.

A probabilistic algorithm approximating solutions of a singular PDE of porous media type

The object of this paper is a one-dimensional generalized porous media equation (PDE) with possibly discontinuous coefficient $\beta$, which is well-posed as an evolution problem in $L^1(\mathbb{R})$. In some recent papers of Blanchard et alia and Barbu et alia, the solution was represented by the solution of a non-linear stochastic differential equation in law if the initial condition is a bounded integrable function. We first extend this result, at least when $\beta$ is continuous and the initial condition is only integrable with some supplementary technical assumption. The main purpose of the article consists in introducing and implementing a stochastic particle algorithm to approach the solution to (PDE) which also fits in the case when $\beta$ is possibly irregular, to predict some long-time behavior of the solution and in comparing with some recent numerical deterministic techniques.

Numerical study of a stochastic particle algorithm solving a multidimensional population balance model for high shear granulation

This paper is concerned with computational aspects of a multidimensional population balance model of a wet granulation process. Wet granulation is a manufacturing method to form composite particles, granules, from small particles and binders. A detailed numerical study of a stochastic particle algorithm for the solution of a five-dimensional population balance model for wet granulation is presented. Each particle consists of two types of solids (containing pores) and of external and internal liquid (located in the pores). Several transformations of particles are considered, including coalescence, compaction and breakage. A convergence study is performed with respect to the parameter that determines the number of numerical particles. Averaged properties of the system are computed. In addition, the ensemble is subdivided into practically relevant size classes and analysed with respect to the amount of mass and the particle porosity in each class. These results illustrate the importance of the multidimensional approach. Finally, the kinetic equation corresponding to the stochastic model is discussed.

A Detailed Model for the Sintering of Polydispersed Nanoparticle Agglomerates

In this study the coagulation, condensation, and sintering of nanoparticles is investigated using a stochastic particle model. Each stochastic particle consists of interacting polydisperse primary particles that are connected to each other. In the model sintering occurs between each individual pair of neighboring primary particles. This is important for particles in which the range of the size of the primary particles varies significantly. The sintering time is obtained from the viscous flow model. The model is solved using a stochastic particle algorithm. The particles are represented in a binary tree that contains the connectivity as well as the degree of sintering information. Particles are forme, coagulate, sinter, and experience condensation according to known rate laws. The particle binary tree, along with it the degree of sintering, is updated after each time step according to the rates of the different processes. The stochastic particle method uses the technique of fictitious jumps and linear proc...

Calculation of the size distribution function of soot particles in turbulent diffusion flames

The advantages of a combined model approach were exploited to compute piloted natural gas/air turbulent jet diffusion flames with the intention towards a quantitative description of the soot properties (soot volume fraction) as well as the evolution of the size distribution of the soot particles across the flame. The interaction of turbulence and chemistry was described by a probability density approach in combination with the laminar flamelet formulation. Gas-phase chemistry was accounted for by the detailed mechanism of Appel, Bockhorn and Frenklach, which includes the formation of aromatic species up to pyrene. The formation and oxidation of the particulate phase was described by soot particle size population balance equations, which were solved using a modified version of the stochastic particle algorithm developed by Kraft and co-workers. The influence of turbulent fluctuations on particle inception, coagulation and surface reactions is discussed. The predictions of the particulate phase were assessed quantitatively by comparison with measurements of total soot particle concentrations of turbulent diffusion jet flames, which were investigated by Bartenbach et al. The evolution of the size distribution of the primary particles across the turbulent jet flame was visualized. After the region, where particle inception plays the dominating role, shape and extension of the size distribution is governed mainly by coagulation processes until the oxidation reactions are of increasing importance, leading to the consumption and depletion of particles, which significantly alters the distribution.

A new numerical approach for the simulation of the growth of inorganic nanoparticles

In this paper we derive and test an extended mass-flow type stochastic particle algorithm for simulating the growth of nanoparticles that are formed in flames and reactors. The algorithm incorporates the effects of coagulation that dominates such systems, along with a particle source and surface growth. We simulate three different configurations for the creation of nanoparticles. The oxidation of SiH 4 to SiO 2 and Fe(CO) 5 to Fe 2O 3 in premixed H 2/O 2/Ar flames were investigated under different initial concentrations of SiH 4 and Fe(CO) 5, respectively. In addition, the oxidation of TiCl 4 to TiO 2 in a plug-flow reactor was investigated. A simple reaction mechanism for the conversion of Fe(CO) 5 to Fe 2O 3 was suggested, based on prior experimental data along with estimated transport properties for the species considered in this system. The simulation results were compared to experimental data available in the literature.

PROBABILISTIC INTERPRETATION AND PARTICLE METHOD FOR VORTEX EQUATIONS WITH NEUMANN’S BOUNDARY CONDITION

AbstractWe are interested in proving the convergence of Monte Carlo approximations for vortex equations in bounded domains of $\mathbb{R}^2$ with Neumann’s condition on the boundary. This work is the first step towards justifying theoretically some numerical algorithms for Navier–Stokes equations in bounded domains with no-slip conditions.We prove that the vortex equation has a unique solution in an appropriate energy space and can be interpreted from a probabilistic point of view through a nonlinear reflected process with space-time random births on the boundary of the domain.Next, we approximate the solution $w$ of this vortex equation by the weighted empirical measure of interacting diffusive particles with normal reflecting boundary conditions and space-time random births on the boundary. The weights are related to the initial data and to the Neumann condition. We prove a trajectorial propagation-of-chaos result for these systems of interacting particles. We can deduce a simple stochastic particle algorithm to simulate $w$.AMS 2000 Mathematics subject classification: Primary 60K35; 76D05

A stochastic particle method for some one-dimensional nonlinear p.d.e.

We consider the one-dimensional nonlinear P.D.E. in the weak sense: ▪ When the initial condition is a probability on R , the solution U t is the distribution of the random variable X t where ( X t ) is a nonlinear stochastic process in the sense of McKean. Our purpose is to study a stochastic particle algorithm for the computation of the cumulative distribution function of U t . This method is based upon the moving of particles according to the law of a Markov chain approximating ( X t ), and the approximation of ( E b(x, X t) , t ≤ T) by means of empirical distributions. For a bounded Lipschitz function b, we prove the convergence of the method with a rate of convergence of order O(1/ N + A where N is the number of particles and Δ is the time step.