Method Of Moments With Interpolative Closure Research Articles

Effect of O2 concentration in ambient mixture and multiphase radiation on pollutant formation in ECN spray-A

This study investigates the formation and evolution of soot and NO X in a high-pressure constant-volume combustion chamber. This work focuses on the effect of multiphase thermal radiation and O2 dilution in ambient/exhaust gases, sometimes also referred to as exhaust gas recirculation (EGR), qualitatively and quantitatively. The spray-A case (n-dodecane as fuel) from Engine Combustion Network (ECN) is used as the target condition. Two different soot modelling approaches have been considered: a semi-empirical two-equation model and a detailed method of moments with interpolative closure (MOMIC) model. A multiphase photon Monte Carlo (PMC) solver with line-by-line (LBL) spectral data is used to resolve radiative heat transfer. Results show that the effect of radiation on soot is minimal in spray-A. Inclusion of radiation modelling, on the other hand, marginally reduces NO prediction. Both peak soot and NO formation increase with O2 content in the ambient gas. Oxygen content in ambient gas is also found to have significant effect on soot sizes as the mean soot diameter increases along with considerable widening of the diameter distribution with the increase of O2 percentage in the ambient gas.

Systematic assessment of the Method of Moments with Interpolative Closure and guidelines for its application to soot particle dynamics in laminar and turbulent flames

The Method of Moments with Interpolative Closure (MOMIC) is a widely used approach for numerical modeling of aerosol dynamics, where soot formation is an important area of application. Two basic variants of the algorithm were proposed in the past: In the original method, transport equations are solved for several non-negative order integer moments of the soot Number Density Function and one for the moment of order minus infinity. Based on these moments, two interpolation functions are constructed and used for the evaluation of positive and negative fractional order moments required for closure of the moment equation source terms. In the other variant, which is a simplification of the first, equations are solved only for non-negative order integer moments. Negative order moments then need to be evaluated by extrapolation. In addition to the choice between these two algorithms, it is not clear a-priori what is the optimal choice regarding the number of transported moments and hence the order of the interpolation polynomial. In this work, the effects of different choices in the setup of the MOMIC algorithm on the evaluation of soot source terms and the prediction of the soot number density and volume fraction are assessed systematically. To this end, MOMIC is combined with a state-of-the-art physico-chemical soot model and applied to the simulation of soot formation in a laminar premixed flame and along Lagrangian trajectories extracted from a Direct Numerical Simulation of a turbulent sooting flame. Monte Carlo simulations serve as reference solutions for a combined a-priori and a-posteriori study. Including the moment of order minus infinity is shown to be beneficial for an accurate calculation of the coagulation source term and hence the soot number density, because moments of negative order can be evaluated more accurately. Furthermore, a high interpolation order yields a more accurate prediction of the surface growth source term. This source term is systematically overpredicted, but errors are substantially reduced to a few percent with increasing interpolation order.

Joint-scalar transported PDF modelling of soot in a turbulent non-premixed natural gas flame

The focus of the present work is on the prediction of soot in the turbulent Delft III/Adelaide natural gas flame at a Reynolds number of 9700. A parabolic flow solver with the SSG (Speziale, Sarkar and Gatski) Reynolds stress transport model for turbulence is coupled to a joint-scalar transported PDF (probability density function) approach allowing exact treatment of the interactions of turbulence with the solid and gas phase chemistries. Scalar mixing is treated via the modified Curl's coalescence/dispersion model and two different closures for the scalar dissipation rate are explored. The gas phase chemistry is represented by a systematically reduced mechanism featuring 144 reactions, 15 solved and 14 steady-state species. The dynamics of soot particles, including coagulation and aggregation in the coalescent and fractal aggregate limits, is treated either via a simplified two-equation model or via the MOMIC (method of moments with interpolative closure). The inclusion of soot surface reactions based on a second ring PAH (polycyclic aromatic hydrocarbon) analogy is also investigated. Soot oxidation via O, OH and O is taken into account and the sensitivity to the applied rates is investigated. An updated acetylene-based soot nucleation rate is formulated based on consistency with detailed chemistry up to pyrene and combined with a sectional model to compute soot particle size distributions in the NIST well-stirred/plug flow reactor configuration. The derived rate is subsequently used in turbulent flame calculations and subjected to a sensitivity analysis. Radiative emission from soot and gas phase species is accounted for using the RADCAL method and by the inclusion of enthalpy as a solved scalar. Computed soot levels reproduce experimental data comparatively well and approximately match absolute values. The axial location of peak soot in the Delft III/Adelaide flame is consistent with previous large eddy simulations and possible causes for the discrepancy with experimental data are analysed.

In situ adaptive tabulation (ISAT) for combustion chemistry in a network of perfectly stirred reactors (PSRs) for the investigation of soot formation and growth

This paper presents an efficient computational implementation of the in situ adaptive tabulation (ISAT) approach (Pope, 1997) for combustion chemistry in a network of perfectly stirred reactors (PSRs) for the investigation of soot formation and growth. This study, for the first time, extends the thermochemical composition vector to contain the soot moments, using the method of moments with interpolative closure (MOMIC) as a soot model with six concentration moments. A series of PSR calculations is carried out using the direct integration (DI) and ISAT approaches. Assessment of the accuracy of ISAT approach is conducted through direct comparisons with DI calculations. Moreover, complimentary cumulative distribution function (CCDF), sensitivity of ISAT calculations with respect to the absolute error tolerance values and speedup are analyzed for two different test cases of ethylene–air using two different chemical kinetic mechanisms. A maximum speedup of 50 × was achieved for an error tolerance of 10−4.

New scheme for implementing the method of moments with interpolative closure

ABSTRACTThis article presents a new scheme for implementing the method of moments with interpolative closure (MOMIC), which was proposed by Frenklach and Harris in 1987. The new scheme can be implemented with arbitrary moment sequences. A new interpolation scheme, which is implemented among the square root of logarithms of newly defined normalized moments, was verified to be suitable for interpolation with only three known data points. The new scheme was applied to studies on Brownian coagulation in both free-molecular and continuum-slip regimes. The new MOMIC with integer moment sequence was verified to exhibit nearly identical accuracy to its original version as well as other state-of-the-art numerical methods, including QMOM, TEMOM, and log MM, and higher accuracy than these referenced numerical methods as the fractional moment sequence was employed. The study enhances the capability of the MOMIC for solving population balance equations.© 2017 American Association for Aerosol Research

Modelling TiO2 formation in a stagnation flame using method of moments with interpolative closure

The stagnation flame synthesis of titanium dioxide nanoparticles from titanium tetraisopropoxide (TTIP) is modelled based on a simple one-step decomposition mechanism and one-dimensional stagnation flow. The particle model, which accounts for nucleation, surface growth, and coagulation, is fully-coupled to the flow and the gas phase chemistry and solved using the method of moments with interpolative closure (MoMIC). The model assumes no formation of aggregates considering the high temperature of the flame. In order to account for the free-jet region in the flow, the computational distance, H=1.27 cm, is chosen based on the observed flame location in the experiment (for nozzle-stagnation distance, L=3.4 cm). The model shows a good agreement with experimentally measured mobility particle size for stationary stagnation surface with varying TTIP loading, although the particle geometric standard deviation, GSD, is underpredicted for high TTIP loading. The particle size is predicted to be sensitive to the sampling location near the stagnation surface in the modelled flame. The sensitivity to the sampling location is found to increase with increasing precursor loading and stagnation temperature. Lastly, the effect of surface growth is evaluated by comparing the result with an alternative reaction model. It is found that surface growth plays an important role in the initial stage of particle growth which, if neglected, results in severe underprediction of particle size and overprediction of particle GSD.

A Hybrid Newton/Time Integration Approach Coupled to Soot Moment Methods for Modeling Soot Formation and Growth in Perfectly-Stirred Reactors

ABSTRACTA perfectly-stirred reactor (PSR) model based on the hybrid Newton/time integration methodology (Grcar et al., 1988) is developed and coupled to two state-of-the-art soot moment techniques, namely the method of moments with interpolative closure (MOMIC) and the hybrid method of moments (HMOM), with the latter coupling being implemented for the first time for investigating soot formation and growth. Five different PSR calculations based on initial data from previous simulations and experiments are carried out in the present study at various equivalence ratios, temperatures, and residence times, in order to test this coupling. These calculations consist of combustion of mixtures of ethylene, ethylene-benzene blends, methane, and acetylene, with air. The PSR algorithm employs a procedure of switching between steady-state and pseudo-transient calculations of the nonlinear algebraic steady PSR equations in order to achieve a more conditioned estimate of the initial guess. Soot moment equations are coupled with species conservation equations to obtain soot quantities, such as soot volume fraction, particle number density, particle diameter, and soot surface area density. As expected, soot volume fraction, particle number density, and soot surface area density all increased with an increase in fuel-air equivalence ratio, (), and residence time, where HMOM mostly predicted lower values of these quantities than MOMIC. The particle diameter variations with were case-specific, but HMOM always predicted larger-sized soot particles than MOMIC. Comparisons with experimental data, where available, showed that both HMOM and MOMIC overpredicted the soot volume fraction.

A Systematic Comparison of Detailed Soot Models and Gas-Phase Chemical Mechanisms in Laminar Premixed Flames

ABSTRACTThis research is part of an ongoing assessment to identify the most promising soot models for applications to laboratory turbulent flames, and ultimately to practical combustion systems. To that end, it is of interest to compare soot model simulation results with experiments over a broad range of conditions. Here five steady, one-dimensional, laminar, premixed ethylene flames have been targeted that cover a range of C/O ratios (0.68 to 0.88), pressures (1 atm to 10 bar), and peak soot levels (0.03 to 3.2 ppm). Seven different chemical mechanisms have been considered. For each mechanism, results from several soot model variants are compared. Two different approaches for soot aerosol dynamics are used: a discrete sectional method (DSM), and a method of moments with interpolative closure (MOMIC). DSM and MOMIC results are also compared with those from a widely used semi-empirical two-equation soot model. The main figure-of-merit is the ratio of computed-to-measured peak soot volume fraction. The computed soot volume fraction is most sensitive to variations in the surface chemistry scheme. Sensitivities to variations in model configurations are reduced at high pressure. The DSM-based models yield slightly lower soot volume fraction and slightly larger particle size compared to the corresponding MOMIC-based models. A modified semi-empirical two-equation model produced a good match to experiment for all five flames. Particle sizes and number densities from the models are discussed for a low-sooting flame where such data are available. The DSM-based models produce bimodal particle size distributions that are qualitatively consistent with those observed experimentally. The best results overall are obtained using an underlying chemical mechanism that incorporates recent understanding of polycyclic aromatic hydrocarbons kinetics. There is no clear advantage of DSM over MOMIC in predicting global quantities, such as soot volume fraction, particle number density, and average particle size, while the computational cost of DSM is significantly higher than that of MOMIC.

A computational study of ethylene–air sooting flames: Effects of large polycyclic aromatic hydrocarbons

An updated reduced gas-phase kinetic mechanism was developed and integrated with aerosol models to predict soot formation characteristics in ethylene nonpremixed and premixed flames. A primary objective is to investigate the sensitivity of the soot formation to various chemical pathways for large polycyclic aromatic hydrocarbons (PAH). The gas-phase chemical mechanism adopted the KAUST-Aramco PAH Mech 1.0, which utilized the AramcoMech 1.3 for gas-phase reactions validated for up to C2 fuels. In addition, PAH species up to coronene (C24H12 or A7) were included to describe the detailed formation pathways of soot precursors. In this study, the detailed chemical mechanism was reduced from 397 to 99 species using directed relation graph with expert knowledge (DRG-X) and sensitivity analysis. The method of moments with interpolative closure (MOMIC) was employed for the soot aerosol model. Counterflow nonpremixed flames at low strain rate sooting conditions were considered, for which the sensitivity of soot formation characteristics to different nucleation pathways were investigated. Premixed flame experiment data at different equivalence ratios were also used for validation. The findings show that higher PAH concentrations result in a higher soot nucleation rate, and that the total soot volume and average size of the particles are predicted in good agreement with experimental results. Subsequently, the effects of different pathways, with respect to pyrene- or coronene-based nucleation models, on the net soot formation rate were analyzed. It was found that the nucleation processes (i.e., soot inception) are sensitive to the choice of PAH precursors, and consideration of higher PAH species beyond pyrene is critical for accurate prediction of the overall soot formation.

Development of High Fidelity Soot Aerosol Dynamics Models using Method of Moments with Interpolative Closure

The method of moments with interpolative closure (MOMIC) for soot formation and growth provides a detailed modeling framework maintaining a good balance in generality, accuracy, robustness, and computational efficiency. This study presents several computational issues in the development and implementation of the MOMIC-based soot modeling for direct numerical simulations (DNS). The issues of concern include a wide dynamic range of numbers, choice of normalization, high effective Schmidt number of soot particles, and realizability of the soot particle size distribution function (PSDF). These problems are not unique to DNS, but they are often exacerbated by the high-order numerical schemes used in DNS. Four specific issues are discussed in this article: the treatment of soot diffusion, choice of interpolation scheme for MOMIC, an approach to deal with strongly oxidizing environments, and realizability of the PSDF. General, robust, and stable approaches are sought to address these issues, minimizing the use of ad hoc treatments such as clipping. The solutions proposed and demonstrated here are being applied to generate new physical insight into complex turbulence-chemistry-soot-radiation interactions in turbulent reacting flows using DNS. Copyright 2014 American Association for Aerosol Research

A coupled CFD-population balance approach for nanoparticle synthesis in turbulent reacting flows

This paper investigates the first part of a two-stage methodology for the detailed fully coupled modelling of nanoparticle formation in turbulent reacting flows. We use a projected fields (PF) method to approximate the joint composition probability density function (PDF) transport equation that describes the evolution of the nanoparticles. The method combines detailed chemistry and the method of moments with interpolative closure (MoMIC) population balance model in a commercial computational fluid dynamics (CFD) code. We show details of the implementation and present an extensive set of numerical experiments and validation. We consider the example of the chloride process for the industrial synthesis of titania. We show good agreement with experimental data and present fully coupled detailed chemistry CFD simulations of nanoparticle formation in a representative ‘slot’ reactor geometry. The simulations show that inception occurs in a mixing zone near the reactor inlets. Most of the nanoparticle mass is due to surface growth downstream of the mixing zone with a narrower size distribution occurring in the regions of higher surface growth. The predicted temperature and particle properties are compared to a perfect mixing case. The implications for the second part of the methodology, where it is proposed to post-process the data using a more detailed particle model, are discussed critically.

Hybrid Method of Moments for modeling soot formation and growth

In this work, a new statistical model for soot formation and growth is developed and presented. The Hybrid Method of Moments (HMOM) seeks to combine the advantages of two moment methods, the Method of Moments with Interpolative Closure (MOMIC) and the Direct Quadrature Method of Moments (DQMOM), in an accurate and consistent formulation. MOMIC is numerically simple and easy to implement but is unable to account for bimodal soot Number Density Functions (NDF). DQMOM is accurate but is numerically ill-posed and difficult to implement. HMOM combines the best of both two methods to capture bimodal NDF while retaining ease of implementation and numerical robustness. The new hybrid method is shown to predict mean quantities nearly as accurately as DQMOM and high-fidelity Monte Carlo simulations. In addition, a model for combining particle coalescence with particle aggregation is presented and shown to accurately reproduce experimental measurements in a variety of sooting flames.

A joint volume-surface model of soot aggregation with the method of moments

In this work, a bivariate model of soot aggregation is formulated within the framework of the Method of Moments with Interpolative Closure (MOMIC). In the bivariate model, soot particles are represented by two independent variables: their volume and surface area. This joint formulation also allows for the blending of aggregation and coalescence with the two as limits. The new formulation is compared to the old formulation with the univariate model as well as both the Direct Quadrature Method of Moments (DQMOM) and Direct Simulation Monte Carlo (DSMC) for a laminar premixed ethylene flame. With the bivariate model, MOMIC is shown to predict volume fraction and number density very accurately and gives some insight into the properties of the aggregates.

Modeling ferrocene reactions and iron nanoparticle formation: Application to CVD synthesis of carbon nanotubes

This paper presents a practically useful model that can predict the formation process and the growth rate of iron nanoparticles from ferrocene. There is a strong need to create such a model that can help improve the process of carbon nanotube (CNT) production by chemical vapor deposition (CVD) and flame synthesis methods. This study serves this purpose. A simple reaction mechanism between ferrocene and hydrogen was proposed based on experimental and theoretical studies conducted by other researchers; then the method of moments with interpolative closure (MOMIC) was applied to compute the size distribution of iron particles. MOMIC was found to reproduce the result of a sectional method with better accuracy than a lognormal method. We simulated the isothermal decomposition of ferrocene and the consequent formation of iron particles with MOMIC; the computed diameter of iron particles agreed reasonably well with experimental data. Analytical solutions for particle diameter were obtained as a function of residence time, assuming monodisperse particles and a constant collision frequency function. They compared very well to the numerical prediction by a monodisperse model and reasonably well to that by MOMIC, suggesting the analytical solutions to be a simple first approximation in assessing the performance of CVD and flame synthesis methods of producing CNTs.