Taylor-series Expansion Method Of Moments Research Articles

Solution of Smoluchowski coagulation equation for Brownian motion with TEMOM

The particle number density in the Smoluchowski coagulation equation usually cannot be solved as a whole, and it can be decomposed into the following two functions by similarity transformation: one is a function of time (the particle k-th moments), and the other is a function of dimensionless volume (self-preserving size distribution). In this paper, a simple iterative direct numerical simulation (iDNS) is proposed to obtain the similarity solution of the Smoluchowski coagulation equation for Brownian motion from the asymptotic solution of the k-th order moment, which has been solved with the Taylor-series expansion method of moment (TEMOM) in our previous work. The convergence and accuracy of the numerical method are first verified by comparison with previous results about Brownian coagulation in the literature, and then the method is extended to the field of Brownian agglomeration over the entire size range. The results show that the difference between the lognormal function and the self-preserving size distribution is significant. Moreover, the thermodynamic constraint of the algebraic mean volume is also investigated. In short, the asymptotic solution of the TEMOM and the self-preserving size distribution form a one-to-one mapping relationship; thus, a complete method to solve the Smoluchowski coagulation equation asymptotically is established.

Distribution of Nanoparticles in a Turbulent Taylor–Couette Flow Considering Particle Coagulation and Breakage

In this paper, the dynamic evolution of nanoparticles in a turbulent Taylor–Couette flow was studied by means of a numerical simulation. The initial particle size was 200 nm, and the volume concentration was 1 × 10−5. The Reynolds-averaged N–S equation for Taylor–Couette flow was solved numerically using the realizable k-ε model combined with the standard wall function. The numerical result of the velocity distribution is in good agreement with the experimental results. Additionally, the dynamic equation for the particle number distribution function was solved numerically using the Taylor series expansion moment method (TEMOM). The variation characteristics of particle number density, diameter and polydispersity in the flow were analyzed. The results show that particle breakage is obvious in the region with strong vorticity due to the large shear strength, which leads to a significant change in the particle number density, diameter and polydispersity. Furthermore, the effects of the gap width between two cylinders and the Reynolds number on the distribution of the particle number density, size and polydispersity are discussed.

Simulation of Aerosol Evolution within Background Pollution for Nucleated Vehicle Exhaust via TEMOM

This work is intended to study the effect of background particles on vehicle emissions in representative realistic atmospheric environments. The coupling of Reynolds-Averaged Navier–Stokes equation (RANS) and Taylor-series Expansion Method Of Moments (TEMOM) is performed to track the emissions of the vehicle and simulating the evolution of the matters. The transport equation of mass, momentum, heat, and the first three orders of moments are taken into account with the effect of binary homogeneous nucleation, Brownian coagulation, condensation, and thermophoresis. The parameterization model is utilized for nucleation. The measured data for Beijing’s particle size distribution under both polluted and nonpolluted conditions are utilized as background particles. The relationship between the macroscopic measurement results and the microscopic dynamic process is analyzed by comparing the variation trend of several physical quantities in the process of aerosol evolution. It is found with an increase of background particle concentration, the nucleation is inhibited, which is consistent with the existing studies.

Efficient Method of Moments for Simulating Atmospheric Aerosol Growth: Model Description, Verification, and Application

AbstractThe atmospheric aerosol dynamics model (AADM) has been widely used in both comprehensive air quality model systems and chemical transport modeling globally. The AADM consists of Smoluchowski's coagulation equation (SCE), whose solution undergoing Brownian coagulation in the free molecular regime is a challenge because it is inconsistent with aerosols whose size distribution cannot exactly follow the lognormal size distribution. Thus, a new method for solving the SCE without assuming lognormal size distribution is proposed and developed. The underlying principle of this method is that the hybridization of the well‐established method of moments with the assumed lognormal size distribution (log MOM) and Taylor‐series expansion method of moments (TEMOM) is implemented. This method shows excellent agreement with the sectional method (SM) which is used as reference. The accuracy of these two specific models closely approaches that of the TEMOM, but overcomes the limitation of the classical log MOM. The computational time of this scheme is considerably lower than that of the SM. The new method was successfully implemented to reveal the formation and growth of secondary particles emitted from a vehicle exhaust tailpipe. It was found that the formation of new particles only occurs in the interface region of the turbulent exhaust jet (which is very close to the tailpipe exit), whereas no new particles are formed in the mixture of the exhaust jet plume and the surrounding cold air downstream. The new method is verified as an efficient and reliable numerical scheme for studying atmospheric aerosol dynamics.

A study of the evolution of nanoparticle dynamics in a homogeneous isotropic turbulence flow via a DNS-TEMOM method

In this article, a coupling of the direct numerical simulation (DNS) and the population balance modeling (PBM) is implemented to study the effect of turbulence on nanoparticle dynamics in homogenous isotropic turbulence (HIT). The DNS is implemented based on a pseudo-spectral method and the PBM is implemented using the Taylor-series expansion method of moments. The result verifies that coagulation due to turbulent shear force has a bigger impact on the evolution of number concentration, polydispersity, and average diameter of nanoparticles than Brownian coagulation in the HIT. The Reynolds number plays an important role in determining the number concentration, polydispersity, and average diameter of nanoparticles, and these quantities change more rapidly with an increase of Reynolds number. It is also found that the initial geometric standard deviation slows down the evolution of particle dynamics, but almost has no influence on the polydispersity of nanoparticles.

Effect of Fluctuating Aerosol Concentration on the Aerosol Distributions in a Turbulent Jet

ABSTRACT We apply the Reynolds averaging method to derive the averaged particle general dynamic equation (APGDE), which models the effect of fluctuations in aerosol concentrations on coagulation, referred to as the fluctuating coagulation term (FCT), based on the turbulent kinetic energy. The APGDE is numerically solved using the Taylor-series expansion method of moments for a turbulent jet; the equation is then validated by comparing some of the results with experimental values. Obtaining the distribution and evolution of the aerosol particle number concentration, mass concentration, mean diameter and geometric standard deviation both when the FCT is incorporated and when it is excluded, we find that the contribution of fluctuating concentrations to the coagulation becomes a factor when x/D = 5 and reaches a stable state when x/D = 20. When the FCT is considered, the number concentration, mean diameter and geometric standard deviation are clearly lower, larger and greater, respectively. The fluctuating concentrations intensify the coagulation, reduce the number concentration and increase the mean diameter. Therefore, such fluctuations in a turbulent jet cannot be neglected.

Numerical study on fractal-like soot aggregate dynamics of turbulent ethylene-oxygen flame

The soot aggregate dynamics of turbulent ethylene-oxygen flame is numerically studied for different equivalence ratios and jet Reynolds numbers (Rej). Our developed Taylor-series expansion method of moments (TEMOM) model in our previous research studies is further extended to solve the bivariate population balance equation (PBE) and formulate the novel Bivariate TEMOM model scheme. Full numerical validations are performed with the stochastically weighted operator splitting Monte Carlo method and moving sectional method, the Bivariate TEMOM model scheme coupled with large eddy simulation (LES) method and soot formation model is used to simulate fractal-like soot aggregate dynamics in turbulent ethylene-oxygen flame. The results show that soot nucleation and surface growth processes are enhanced with increasing equivalence ratio while the coagulation rate is hardly varied as the total soot volume fraction is quite low. The increasingly uniform distributions of fractal dimension and particle size can also be observed. As Rej increases from 14,400 to 36,000, both the mean diameter and mean fractal dimension of soot aggregates gradually decrease as the surface growth rates of soot aggregates decrease significantly. However, the effect of increasing Rej on coagulation and nucleation rates are slight due to the decreasing residence time of soot particles in the combustor.

Development and application of moment method on nanoparticles evolution due to coagulation and deposition

In this study, the Taylor-series expansion method of moments is developed to describe the dynamic behaviour of nanoparticles considering both the effects of coagulation and deposition in closed environments, where the two effects have always been studied separately before. Compared with traditional methods of moments, the new method could give more accurate results for solving the general dynamics equation. In addition, moment equations with respect to coagulation and deposition considering the fractal dimension (Df) of particles are acquired, and the results are compared with experimental data and are more accurate than the simulation results under the assumption of spherical mode. The new dynamic model could be applied in more real conditions with different microstructures of nanoparticles. With decreasing Df, the particle number concentration decreases more rapidly, additionally, Df has a relatively greater effect on the agglomeration of smaller particles under the same initial concentration. Moreover, the effect of the initial number concentration on coagulation is also studied. The results show that a higher initial number concentration or a smaller particle size could have a greater effect of coagulation on the evolution of particles.

Cercignani's conjecture is almost true for Smoluchowski equation

The asymptotic approach to thermodynamic equilibrium for the Smoluchowski equation describing Brownian coagulation, is examined using the Taylor-series expansion method of moments and the principle of entropy production. The equality of the relationship between the deviation of the entropy from its equilibrium value and the entropy production rate, similar to that in Cercignani's conjecture, allows one to infer that Cercignani's conjecture is almost true for the Smoluchowski equation with soft sphere collision. In this paper, we provide an explicit expression for the evolution of growth rate of the entropy to equilibrium for the spatially homogeneous Smoluchowski equation of Brownian coagulation.

The asymptotic stability of the Taylor-series expansion method of moment model for Brownian coagulation

In the present study, the linear stability of population balance equation due to Brownian motion is analyzed with the Taylor-series expansion method of moment. Under certain conditions, the stability of the Taylor-series expansion method of moment model is reduced to a well-studied problem involving eigenvalues of matrices. Based on the principle of dimensional analysis, the perturbation equation is solved asymptotically. The results show that the Taylor-series expansion method of moment model is asymptotic stable, which implies that the asymptotic solution is uniqueness, and supports the self-preserving size distribution hypothesis theoretically.

A New Mathematical Scheme for Approximating Overall Aerosol Extinction Coefficient during Brownian Coagulation

The approximation of the overall aerosol extinction coefficient is conventionally achieved through integrating the single particle extinction efficiency over the whole size distribution, which requires much computational time. In this work, a new approximation scheme with higher efficiency than the conventional scheme is proposed, in which the combination of polynomials for fitting Mie’s solution and the method of moments produces the overall extinction coefficient of evolving nanoparticles. The closure of arbitrary moments is achieved by implementing the Taylor-series expansion method of moments. The new approximation scheme was verified by comparing it to a more exact referenced scheme for two different typical aerosol modes, namely the nucleation mode and the accumulation mode. This study verifies that the new scheme is a reliable method for approximating the overall extinction coefficient during aerosol evolution with acceptable efficiency and accuracy; thus, it is suitable for use in some atmospheric aerosol models.

The Asymptotic Behavior of Particle Size Distribution Undergoing Brownian Coagulation Based on the Spline-Based Method and TEMOM Model

In this paper, the particle size distribution is reconstructed using finite moments based on a converted spline-based method, in which the number of linear system of equations to be solved reduced from 4m × 4m to (m + 3) × (m + 3) for (m + 1) nodes by using cubic spline compared to the original method. The results are verified by comparing with the reference firstly. Then coupling with the Taylor-series expansion moment method, the evolution of particle size distribution undergoing Brownian coagulation and its asymptotic behavior are investigated.

Nanoparticle formation and growth in turbulent flows using the bimodal TEMOM

A developed bimodal Taylor-series expansion method of moments (B-TEMOM) is first coupled with a large eddy simulation (LES) model to investigate nanoparticle formation and subsequent growth due to nucleation, coagulation and condensation processes in turbulent flows. An incompressible gas mixture containing sulfuric acid and water vapor is injected into a stationary flow field with background aerosols. The spatial and temporal particle size distribution (PSD), particle number and mass concentrations, and competition between the formation of primary and secondary particles in turbulent flows are studied. The instantaneous results demonstrate that the large coherent structures strongly affect the particle number and mass concentration distributions as well as particle polydispersity. This finding also verifies that the coherent structures enhance diffusion in the flows and finally increases particle transfer between the two modes of particles. Furthermore, the numerical simulation results obtained by the B-TEMOM are validated with those obtained by the sectional method with excellent agreement.

Bivariate Taylor-series expansion method of moment for particle population balance equation in Brownian coagulation

In this study, we extend the Taylor-series expansion method of moment to two-component aggregation problem undergoing Brownian coagulation with kernels that are independent of composition. A set of closed particle population balance equation for lower-order moments is then derived. Numerical results and its asymptotic solutions are validated by comparing with Monte Carlo simulation method both in free molecular regime and continuum regime. It is shown that three dimensionless particle moments MC1,MC2,MC3 almost approach to a same value over large evolution time. The normalized variance of excess component A decreases as 1/v¯ and it tends to zero over large evolution time.

An Efficient Algorithm Scheme for Implementing the TEMOM for Resolving Aerosol Dynamics

This paper presents a new algorithm scheme for implementing the Taylor-series expansion method of moments (TEMOM), which is a general method for solving the population balance equation. Instead of deriving moment ordinary differential equations (ODEs) for particular aerosol dynamics, the new numerical algorithm develops a subroutine that corresponds to types of kernels of aerosol dynamics rather than a particular kernel. Consequently, the TEMOM can be conveniently applied to types of dynamics rather than a particular dynamics, and the derivation from the PBE to moment ODEs is avoided. A key aspect to the new algorithm scheme is that the particle kernels are written in a general form, allowing for a universal expression of aerosol dynamics. The closure of ODEs for moments was accomplished by implementing a new Taylor-series expansion closure function with two varying parameters H and ϕ, which enables the TEMOM applicable to any type of moment sequence. The feasibility of the new algorithm scheme was verified by comparing it with other recognized methods for six classic aerosol dynamics. The new algorithm scheme makes the TEMOM much easier to be used for users as compared to its original version (Aerosol Sci Tech 49(2015):1021–1036). The idea of the algorithm scheme can be applied to other non-quadrature-based methods of moments.

Hybrid method of moments with interpolation closure–Taylor-series expansion method of moments scheme for solving the Smoluchowski coagulation equation

This paper presents a hybrid method of moments with interpolation closure–Taylor-series expansion method of moments (MoMIC–TEMoM) scheme for solving the Smoluchowski coagulation equation. In the proposed scheme, the exponential function, which arises in the conversion from a particle size distribution space to a space of moments, is expressed in an additive form using the third-order Taylor-series expansion; the implicit moments are approximated using two Lagrange interpolation functions, namely the newly defined normalized moment function and the normalized moment function defined by Frenklach and Harris (1987). The new hybrid scheme allows implementation of the method of moments with an arbitrary type of moment sequence, and it overcomes the shortcomings of the Taylor-series expansion moment method proposed by Frenklach and Harris. The proposed scheme is verified with three aerosol dynamics, namely Brownian coagulation in the free molecular regime, Brownian coagulation in the continuum-slip regime, and turbulence coagulation. The results reveal that the hybrid MoMIC–TEMoM scheme has similar accuracy to currently recognized methods including the quadrature method of moments, MoMIC, and TEMoM, and its accuracy can be further enhanced as the fractional moment sequence type is used for Brownian coagulation in the free molecular regime. Thus, the proposed scheme is a reliable for solving the Smoluchowski coagulation equation.

CFD modeling of particle dispersion and deposition coupled with particle dynamical models in a ventilated room

Two dynamical models, the traditional method of moments coupled model (MCM) and Taylor-series expansion method of moments coupled model (TECM) for particle dispersion distribution and gravitation deposition are developed in three-dimensional ventilated environments. The turbulent airflow field is modeled with the renormalization group (RNG) k–ε turbulence model. The particle number concentration distribution in a ventilated room is obtained by solving the population balance equation coupled with the airflow field. The coupled dynamical models are validated using experimental data. A good agreement between the numerical and experimental results can be achieved. Both models have a similar characteristic for the spatial distribution of particle concentration. Relative to the MCM model, the TECM model presents a more close result to the experimental data. The vortex structure existed in the air flow makes a relative large concentration difference at the center region and results in a spatial non-uniformity of concentration field. With larger inlet velocity, the mixing level of particles in the room is more uniform. In general, the new dynamical models coupled with computational fluid dynamics (CFD) in the current study provide a reasonable and accurate method for the temporal and spatial evolution of particles effected by the deposition and dispersion behaviors. In addition, two ventilation modes with different inlet velocities are proceeded to study the effect on the particle evolution. The results show that with the ceiling ventilation mode (CVM), the particles can be better mixed and the concentration level is also higher. On the contrast, with the side ceiling ventilation mode (SVM), the particle concentration has an obvious stratified distribution with a relative lower level and it makes a much better environment condition to the human exposure.

Solution of general dynamic equation for nanoparticles in turbulent flow considering fluctuating coagulation

A new averaged general dynamic equation (GDE) for nanoparticles in the turbulent flow is derived by considering the combined effect of convection, Brownian diffusion, turbulent diffusion, turbulent coagulation, and fluctuating coagulation. The equation is solved with the Taylor-series expansion moment method in a turbulent pipe flow. The experiments are performed. The numerical results of particle size distribution correlate well with the experimental data. The results show that, for a turbulent nanoparticulate flow, a fluctuating coagulation term should be included in the averaged particle GDE. The larger the Schmidt number is and the lower the Reynolds number is, the smaller the value of ratio of particle diameter at the outlet to that at the inlet is. At the outlet, the particle number concentration increases from the near-wall region to the near-center region. The larger the Schmidt number is and the higher the Reynolds number is, the larger the difference in particle number concentration between the near-wall region and near-center region is. Particle polydispersity increases from the near-center region to the near-wall region. The particles with a smaller Schmidt number and the flow with a higher Reynolds number show a higher polydispersity. The degree of particle polydispersity is higher considering fluctuating coagulation than that without considering fluctuating coagulation.

Error estimation of TEMOM for Brownian coagulation

In the present study, the error estimation of Taylor series expansion method of moment (TEMOM) for particle population balance equation due to Brownian coagulation is proposed. The app-roximation introduced by Taylor series polynomials has two parts: one is the approximation of nonlinear collision kernel, and the other is the approximation of higher and fractional particle moment. The results show that the two kinds of errors have the same order in the original TEMOM model, which is related to the third derivative of collision kernel and the first four integer order particle moments, i.e., M0, M1, M2, and M3. Based on the lognormal size distribution assumption, the system errors are obtained for different TEMOM models due to Brownian coagulation; and all the errors are small and acceptable. The results have confirmed the validity of TEMOM model.Copyright © 2016 American Association for Aerosol Research

A new analytical solution for agglomerate growth undergoing Brownian coagulation

We proposed a new analytical solution for a population balance equation for fractal-like agglomerates. The new analytical solution applies to agglomerates with any mass fractal dimensions. Two well-known numerical methods, including the Taylor-series expansion method of moments and the quadrature method of moments, were selected as references. The reliability of the analytical solution with three mass fractal dimensions of 1.0, 2.0, and 3.0 in the continuum-slip regime was verified. The accuracy of the new analytical solution is affected by both the Knudsen number and the mass fractal dimension. The new analytical solution can be further improved in accuracy by introducing a correction factor to the originally derived mathematical formula. The new analytical solution was finally confirmed to possess potential for replacing the numerical solution in the continuum-slip regime.