Taylor Expansion Method Of Moments Research Articles

Numerical study on evolution of ultrafine particles emitted from vehicle exhaust with multi-dynamical behaviors

In this study, a numerical investigation is performed to simulate the evolution process of ultrafine particles emitted from diesel vehicle exhausts. The Taylor expansion method of moments (TEMOM)-CFD coupled model is primarily applied to the non-spherical particles for simultaneously simulating four kinds of particle dynamical behaviors including coagulation, deposition, condensation and nucleation. Compared with previous simulation studies, the developed model shows closer results to the experimental data. The effects of initial particle number, initial particle size, environmental temperature, relative humidity, fuel sulfur content, and exhaust velocity on the evolution process are systematically analyzed. The results show that the increase rate of the geometric mean diameter (GMD) and the increase margin of GMD are proportional to the initial particle number concentration (PNC) and the environmental temperature. When the relative humidity or the fuel sulfur content is higher, the larger increase margin of the GMD appears, but the increase rate of GMD is not affected obviously. Moreover, the initial PNC gives nearly no contributions to the diffusion range. The initial GMD also shows no markedly effects on the decrease rate of the volume concentration. The fuel sulfur content could affect the PNC distribution in a larger region of the flow field, while the environmental relative humidity and temperature have no evident effects on the distribution range of the PNC.

High-pressure dispersion of nanoparticle agglomerates through a continuous aerosol disperser

By discharging aerosols from a high-pressure vessel through a nozzle, an improved continuous aerosol dispersion system can disperse nanoparticle agglomerates to almost primary nanoparticle size. The system’s high-pressure dispersion was modelled and numerically simulated using the Taylor expansion method of moments. The strong turbulent shear near the nozzle could achieve the same level as the van der Waals force, which is the main mechanism for deagglomerating nanoparticle agglomerates. The change in nanoparticle size was determined to be proportional to the pressure drop between the nozzle inlet and outlet. Nanoparticle breakup was also observed to be affected by the fractional dimension and primary size of particles.

Numerical simulation of particle formation and evolution in a vehicle exhaust plume using the bimodal Taylor expansion method of moments

The bimodal Taylor expansion method of moments (B-TEMOM) model scheme was developed to simulate the formation and evolution of vehicle exhaust particles. Two independent types of log-normal particle size distributions were selected in the B-TEMOM model scheme, comprising large and small particles to represent background (i.e., the surrounding environment) and vehicle exhaust particles, respectively. Concentration distributions of exhaust and background particles derived using this model scheme were verified against results from a moving sectional method and the bimodal quadrature method of moments, showing excellent agreement. The effects of vehicle tailpipe exit conditions (e.g., exhaust particle concentrations and velocity), sulfur content, and relative humidity on the evolution of particles were investigated numerically. Both two-dimensional and three-dimensional numerical simulations showed that tailpipe exit velocity and relative humidity did not greatly affect the steady-state concentrations or the diameters of particles in urban atmospheres. Although an increase in sulfur content had little effect on the particle concentration, it led to background particles with larger geometric average diameter entering the environment. This coupled CFD-B-TEMOM numerical model provides a simple but accurate and efficient method for studying bimodal aerosol dynamics.

Generalized TEMOM Scheme for Solving the Population Balance Equation

This article proposes a novel generalized Taylor expansion method of moments (TEMOM) scheme for solving the population balance equation. The proposed scheme can completely overcome the shortcoming of the existing TEMOM and substantially improve the accuracy for both integer and fractional moments. In the generalized TEMOM, the optimal number of equations is 2+1, where is an integer greater than zero. The existing TEMOM is a special case of the generalized TEMOM when is 1. The generalized TEMOM was tested for aerosols undergoing Brownian coagulation in the continuum regime, and it was verified to achieve nearly the same accuracy as the quadrature method of moments (QMOM) with a fractional moment sequence, and higher accuracy than the QMOM with an integer moment sequence. Regarding accuracy and efficiency, the generalized TEMOM scheme was verified to be a competitive method for solving the population balance equation.Copyright 2015 American Association for Aerosol Research

A bimodal moment method model for submicron fractal-like agglomerates undergoing Brownian coagulation

A new bimodal moment method for submicron fractal-like agglomerates undergoing Brownian coagulation is developed. The entire bimodal agglomerate size distribution is firstly separated into two separated distributions, where each distribution is represented by a population balance equation. The two joint population balance equations are then resolved by introducing the Taylor expansion method of moments (TEMOM). This newly developed bimodal model (B-TEMOM) is compared with the commonly used bimodal log-normal method of moments (B-log MM), and bimodal quadrature method of moments (B-QMOM) for the first three moments as well as geometric standard deviation of number distribution. Compared to the B-QMOM, the solution suggests that the B-TEMOM and the B-log MM produce much closer results. With the increase in the difference of the total particle number concentration, the geometric standard deviation, and the geometric mean volume of two modes, the difference among the investigated bimodal models is also slightly increased. The difference between B-log MM and B-TEMOM is found to increase with a decrease in mass fractal dimension, especially at a later stage in the evolution. The solution fully verifies that this newly developed B-TEMOM is reliable for predicting agglomerate dynamics undergoing Brownian coagulation in the continuum-slip regime.

Direct expansion method of moments with n/3th moments for nanoparticle Brownian coagulation in the free molecule regime

Purpose– The purpose of this paper is to develop new types of direct expansion method of moments (DEMM) by using the n/3th moments for simulating nanoparticle Brownian coagulation in the free molecule regime. The feasibilities of new proposed DEMMs with n/3th moments are investigated to describe the evolution of aerosol size distribution, and some of the models will be applied to further simulation of physical processes.Design/methodology/approach– The accuracy and efficiency of some kinds of methods of moments are mainly compared including the quadrature method of moments (QMOM), Taylor-expansion method of moments (TEMOM), the log-normal preserving method of moments proposed by Lee (LMM) and the derived DEMM in this paper. QMOM with 12 quadrature approximation points is taken as a reference to evaluate other methods.Findings– The newly derived models, namely DEMM(4/3,4) and DEMM(2,6), as well as the previous DEMM(2,4), are considered to be qualified models due to their high accuracy and efficiency. They are confirmed to be valid and alternative models to describe the evolution of aerosol size distribution for particle dynamical process involving the n/3th moments.Originality/value– The n/3th moments, which have clear physical interpretations when n stands for first several integers, are first introduced in the DEMM method for simulating nanoparticle Brownian coagulation in the free molecule regime.

A direct expansion method of moments for Brownian coagulation

We propose a new method of moments for solving a coagulation equation based on the Taylor-expansion method of moments (TEMOM). In this model, the closure of the moment equations is achieved by using the Taylor expansion technique, and the fractional-order moments can be used to improve the accuracy of the results. The newly proposed method has no prior requirement for the shape of the particle size distribution. By comparing it with other known methods both in free molecular and continuum regimes, the new approach is found to be the most accurate method. The results show that the predictions of the lower-order moments are not significantly influenced by the higher-order ones, and the higher-order moments are not necessary if one intends to increase the accuracy of the numerical predictions.

A new analytical solution for solving the population balance equation in the continuum-slip regime

A new analytical solution is first proposed to solve the population balance equation due to Brownian coagulation in the continuum-slip regime. An assumption for a novel variable g (g=m0m2/m12, where m0, m1 and m2 are the first three moments, respectively) is successfully applied in executing a separate variable method for ordinary differential equations of the Taylor expansion method of moments. The sectional method is selected as a reference to verify the new solution. The accuracy between the new solution and Lee et al. analytical solution (Lee et al., 1997, Journal of Colloid and Interface Science, 188, 486–492) is mainly compared. The geometric standard deviation of number distribution for the new analytical solution is revealed to be limited to 1.6583. Within the valid range of the geometric standard deviation, the new analytical solution is confirmed to solve the population balance equation undergoing Brownian coagulation with the very nearly same accuracy as Lee et al. analytical solution. For the total particle number concentration, the new solution usually yields higher accuracy. The new solution and Lee et al. analytical solution approximately become one solution as the Knudsen number is smaller than 0.1000. The new solution has the potential to become a competitive analytical solution for solving population balance equation regarding its accuracy and very straightforward derivation.

An analytical solution for the population balance equation using a moment method

Brownian coagulation is the most important inter-particle mechanism affecting the size distribution of aerosols. Analytical solutions to the governing population balance equation (PBE) remain a challenging issue. In this work, we develop an analytical model to solve the PBE under Brownian coagulation based on the Taylor-expansion method of moments. The proposed model has a clear advantage over conventional asymptotic models in both precision and efficiency. We first analyze the geometric standard deviation (GSD) of aerosol size distribution. The new model is then implemented to determine two analytic solutions, one with a varying GSD and the other with a constant GSD. The varying solution traces the evolution of the size distribution, whereas the constant case admits a decoupled solution for the zero and second moments. Both solutions are confirmed to have the same precision as the highly reliable numerical model, implemented by the fourth-order Runge–Kutta algorithm, and the analytic model requires significantly less computational time than the numerical approach. Our results suggest that the proposed model has great potential to replace the existing numerical model, and is thus recommended for the study of physical aerosol characteristics, especially for rapid predictions of haze formation and evolution.

A New Analytical Solution for Solving the Smoluchowski Equation Due to Nanoparticle Brownian Coagulation for Non-Self-Preserving System

The Smoluchowski equation has become a fundamental equation in nanoparticle processes since it was proposed in 1917, whereas the achievement of its analytical solution remains a challenging issue. In this work, a new analytical solution, which is absolutely different from the conventional asymptotic solutions, is first proposed and verified for nonself-preserving nanoparticle systems in the free molecular regime. The Smoluchowski equation is first converted to the form of moment ordinary differential equations by the performance of Taylor expansion method of moments and subsequently resolved by the separate variable technique. In the derivative, a novel variable, g = m0m2/m1 2 , where m0, m1 and m2 are the first three moments, is first revealed which can be treated as constant. Three specific models are proposed, two with a constant g (an Analytical Model with Constant g (AMC), and a Modified Analytical Model with Constant g (MAMC)), and another with varying g (a finite Analytical Model with Varying g (AMV)). The AMC model yields significant errors, while its modified version, i.e., the MAMC model, is able to produce highly reliable results. The AMV is verified to have the capability to solve the Smoluchowski equation with the same precision as the numerical method, but an iterative procedure has to be employed in the calculation.