Systematic assessment of the Method of Moments with Interpolative Closure and guidelines for its application to soot particle dynamics in laminar and turbulent flames
Introduction
The appearance of polydisperse multiphase flows is multifarious [1] including applications in the chemical process industry, the automotive and aerospace sectors, pharmaceutical engineering, and environmental flows. Particulate emissions originating from incomplete combustion of hydrocarbon fuels is an important example of an aerosol appearing in a number of engineering devices such as diesel and aircraft engines and industrial burners. The evolution of the Number Density Function (NDF) of these nanoparticles is governed by a Population Balance Equation (PBE), which needs to be appropriately solved in numerical simulations of particle-laden reactive flows.
A plethora of solution strategies for PBEs has emerged over the last decades [2], [3]. Statistical methods that have been applied to soot evolution include Monte Carlo (MC) methods [4], [5], [6], sectional methods [7], [8], and moment methods. Monte Carlo methods can be considered as the most accurate methods for the solution of PBEs as no closure assumption is required. A population of so-called stochastic particles evolves according to the various aerosol dynamics processes described by the underlying physico-chemical soot model. For a sufficiently large number of stochastic particles, the evolution of the NDF of this representative particle sample converges to the NDF that a direct solution of the PBE would produce in combination with the same physico-chemical model. Hence, compared to a direct solution of the PBE, which is generally computationally infeasible, soot statistics as well as the NDF itself can be obtained with arbitrarily high accuracy. Results from MC simulations are therefore well suited as reference solution to validate other statistical approaches. However, due to high computational costs, the applicability of MC methods is restricted to simplified configurations such as laminar one-dimensional flames.
Regarding simulations of system-scale configurations as well as most laboratory-scale cases, a widely used class of statistical methods is the method of moments. In this approach, transport equations are solved for a few low order moments of the soot NDF. Generally, the source terms appearing in these equations cannot be expressed as a function of the transported moments only, and therefore require closure. The extensive amount of literature on approaches to close the moment equations includes quadrature-based moment methods such as the (Direct) Quadrature Method of Moments (QMOM/DQMOM) [9], [10], the Conditional Quadrature Method of Moments (CQMOM) [11], the Extended Quadrature Method of Moments (EQMOM) [12], [13], [14], and other related methods. Conceptually, they have in common that equations for a set of low order integer moments are solved, and that a multi-dimensional Dirac-delta function as a representation of the NDF or a superposition of continuous kernel functions as an approximation of the NDF is reconstructed. Source terms are then computed using this reconstruction.
Another type of moment methods, which was introduced even before the first quadrature-based algorithms, is based on an interpolation procedure. It was proposed by Frenklach and Harris [15] as the first approach to solve PBEs through a numerical closure of moments of the NDF without imposing a presumed form of the NDF and was later referred to as the Method of Moments with Interpolative Closure (MOMIC) [16]. In MOMIC, transport equations are solved for a given set of moments of the NDF. The source terms appearing in these equations depend on other moments including fractional moments of positive and negative order, and are therefore unclosed. These moments are determined using an interpolation function, which is constructed from the transported moments. MOMIC has been shown to combine several desirable characteristics of statistical methods for soot dynamics, such as computational efficiency, numerical robustness and straightforward implementation into 3D-CFD (Computational Fluid Dynamics) codes.
All of the moment methods described above share a certain ambiguity regarding the optimal choice of moments for which transport equations are solved, especially for bivariate or multivariate model formulations. Fox [17] has defined guidelines for optimal moment sets in the context of a multivariate DQMOM algorithm. However, optimality here refers primarily to numerical stability of the algorithm rather than accuracy of the solution. When MOMIC was introduced, Frenklach and Harris [15] recongnized that the choice of moments used to construct the interpolation function for the fractional moments affects the accuracy of the solution. They proposed to solve equations for a certain number of non-negative integer order moments up to an order p and one for the moment of order minus infinity. Two interpolation functions were then constructed, one based on and one based on . These functions are used to evaluate unknown fractional moments of positive and negative order, respectively. This algorithm is hereafter referred to as MOMIC-infty.
Later, Frenklach and Wang [18] and Kazakov and Frenklach [19] introduced a simplification, where only M0 and positive order moments are used. This variant is therefore referred to as MOMIC-pos in the following. Realizing the difficulty with high order extrapolation that would be required for negative order moments, Frenklach and Wang [18] proposed to use a reduced moment set (M0, M1, M2) to construct a polynomial for the extrapolation of negative order moments, even if more moments are available.
While the benefits of using the minus infinity moment were pointed out by Frenklach [16], the MOMIC-pos variant of the method with up to six moments has more frequently been used in soot or nanoparticle dynamics simulations [20], [21], [22], [23], [24], [25]. The Hybrid Method of Moments (HMOM) [26] is a combination of MOMIC and DQMOM. Also in HMOM, a number of non-negative integer order moments is solved for and used for interpolation and extrapolation of fractional order moments similar to the MOMIC-pos algorithm. In addition, an equation is solved for the number density at the nucleation size. This number density is equivalent to DQMOM with a single delta function located at the smallest particle size. The purpose of the delta peak at the minimum size is similar to solving for the moment of order minus infinity.
While it is an obvious choice to solve equations for those moments directly linked to the quantities of interest - typically, these are the number density, which is M0, and the soot volume fraction, which is proportional to M1 - additional equations are solved for other moments in order to more accurately compute the source terms for the quantities of interest. However, it is not clear a priori which and how many additional moments should be transported to get the most accurate results. Therefore, the goal of the present paper is to provide recommendations for combinations of moment sets and interpolation order. This analysis is done via comparisons of MOMIC computations with MC simulations as reference solutions in a combination of a-priori and a-posteriori analyses.
Comparisons between MOMIC and MC in an a-posteriori sense provide a quantification of statistical errors for quantities of interest such as soot volume fraction and number density. To reveal the origin of potential errors, the moment source terms as well as individual fractional moments are analyzed a-priori using the soot NDF provided by the MC simulation for laminar and turbulent cases.
In this work, MOMIC is combined with a state-of-the-art physico-chemical soot model, which includes soot inception based on polycyclic aromatic hydrocarbons (PAH), condensation of PAH dimers onto soot particles, heterogeneous surface chemistry, and coagulation [27]. The same physico-chemical model is implemented in the MC code.
The paper is structured as follows: In the next section, the physico-chemical soot model and the mathematical background of MOMIC in the context of this model are reviewed. Then, in Section 3, the investigated test cases for model analysis and validation are discussed. To maximize the generality of the conclusions of this work, these include both laminar and turbulent flame conditions. Sections 4–6 provide a thorough analysis of the performance of MOMIC in the laminar and turbulent flame, respectively. A summarizing discussion of these analyses and concluding remarks regarding an optimal and efficient use of the MOMIC algorithm follow in Section 7.
Section snippets
Interpolation and extrapolation in MOMIC
In this work, the soot model considers the following physico-chemical processes: nucleation based on PAH dimers and condensation of PAH dimers onto soot particles using the model by Blanquart and Pitsch [27], coagulation in the free molecular regime [28], [29], and surface growth according to the hydrogen-abstraction-carbon-addition (HACA) mechanism [30], [31] with reaction rates given in [27] and references therein. The specific model formulation of these processes within the PBE, e.g., the
Validation cases
To test the method’s applicability and accuracy for a broad range of flame conditions, MOMIC is used to compute soot evolution in both laminar and turbulent environments, in which different soot formation and growth mechanisms are relevant. From a chemical point of view, the soot mass growth can be categorized into PAH-based growth, i.e., nucleation and condensation, and growth due to surface reactions modeled with the HACA mechanism. First, a laminar premixed burner-stabilized ethylene flame
Model analysis for a laminar premixed flame
The performance of MOMIC is first assessed in computations of the laminar premixed BSS ethylene flame described in Section 3.1. A combined a-posteriori and a-priori study for the MOMIC-pos interpolation algorithm is followed by a discussion on how the MOMIC-infty algorithm, using a second interpolation function based on the moment of order minus infinity, affects the results.
Model analysis for Lagrangian trajectories in the DNS of a turbulent flame
The laminar premixed BSS ethylene flame discussed in the previous section is a typical validation case for both physico-chemical and statistical soot models [13], [44], [45], [46], as it is characterized by well defined boundary conditions. While laminar premixed flames are widely used for model assessment and validation due to the simplicity of the configuration and the availability of detailed data, the variety of flame conditions in such a configuration is somewhat limited. In particular,
Concepts for the evaluation of fractional moments of orders between zero and one
Two alternative strategies for the evaluation of fractional moments in the most challenging region of moment orders between zero and one are discussed in the following: First, transport equations could be solved for fractional moments in this region, e.g., M2/3 or M1/2. Second, the two interpolation functions constructed in the MOMIC-infty algorithm overlap in the region x ∈ [0, 1]. Hence, instead of the polynomial based on the second function based on could be used for
Implications for users of MOMIC
The results for the laminar premixed BSS ethylene flame discussed in Section 4 can be summarized as follows: Excellent results for the soot volume fraction are obtained with both interpolation schemes for p ≥ 2, and errors are less than 15 % for . Furthermore, the best results for the number density are obtained using the MOMIC-pos interpolation scheme combined with the lowest interpolation order and with the MOMIC-infty interpolation scheme combined with higher interpolation orders.
Conclusions
The Method of Moments with Interpolative Closure was combined with a state-of-the-art physico-chemical soot model considering nucleation, condensation, surface growth, and coagulation. Two variants of the MOMIC closure algorithm, each using a varying number of transported moments for the interpolation of fractional moments, were analyzed in a combination of a-priori and a-posteriori studies with Monte Carlo simulations as reference solution. In both algorithms, referred to as MOMIC-pos and
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
Funding from the German Research Foundation (DFG) under grant no. PI 368/6-2 and from the German Research Association for Combustion Engines e.V. (FVV) under grant no. 1239 is gratefully acknowledged. The authors wish to thank Antonio Attili and Fabrizio Bisetti for sharing the DNS data and for valuable discussions throughout the course of this work.
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