Treatment of size-dependent aerosol transport processes using quadrature based moment methods

The use of moment methods for simulation of aerosol settling and diffusion phenomena in which the settling velocity and diffusion coefficient are functions of the size of the particles leads to difficult computational problems, especially if the moment equations need to be closed. In this study, a simple one dimensional problem of aerosol diffusion and gravitational settling is carried out using quadrature method of moments (QMOM) and the direct quadrature method of moments (DQMOM). Analytical solutions can be obtained for the number density function, and issues related to the integration of the solutions to get the moments are discussed. Comparison of the solutions of the moment equations to the moments obtained from the analytical solutions reveals that solutions depend on the initial choice of moments. Results also indicate that the proper choice of moments of the initial number density function may be a significant factor in obtaining more accurate solutions from QMOM or DQMOM.     

Evaluation of the use of the Chebyshev algorithm with the quadrature method of moments for simulating aerosol dynamics

The quadrature method of moments (QMOM) has been widely used for the simulation of the evolution of moments of the aerosol general dynamic equations. However, there are several shortcomings in a crucial component of the method, the product-difference (P-D) algorithm. The P-D algorithm is used to compute the quadrature points and weights from the moments of an unknown distribution. The algorithm does not work for all types of distributions or for even reasonably high-order quadrature. In this work, we investigate the use of the Chebyshev algorithm and show that it is more robust than the P-D algorithm and can be used for a wider class of problems. The algorithm can also be used in a number of applications, where accurate computations of weighted integrals are required. We also illustrate the use of QMOM with the Chebyshev algorithm to solve several problems in aerosol science that could not be solved using the P-D algorithm.     

Evaluation of the 1-point quadrature approximation in QMOM for combined aerosol growth laws

This work examines the applicability of various assumptions in implementation of the quadrature method of moments (QMOM) for solving problems in aerosol science involving simultaneous nucleation, surface growth and coagulation. The problem of aerosol growth and coagulation in a box and the problem of vapor condensation in a nozzle are reworked using quadrature method of moments. QMOM uses Gaussian quadrature to evaluate integrals appearing in the moment equations and therefore does not require any assumptions on the form of the size distribution function, the growth laws and coagulation kernels. Results are compared with calculations which assume a lognormal size distribution. The conditions for which one, two and higher quadrature points can be used in the quadrature formula and the issues regarding the accuracy are considered for combined aerosol nucleation, growth and coagulation processes. Results show that for these problems, the simplest 1-point quadrature scheme gives accuracy comparable with the lognormal calculations while using two and higher point quadrature gives highly accurate results. Some difficulties associated with the QMOM are discussed and some insights are provided.     

Solution of population balance equations using the direct quadrature method of moments

The implementation of a population balance equation (PBE) in computational fluid dynamics (CFD) represents a crucial element in the simulation of multiphase flows. Some of the available methods, such as classes methods (CM) and Monte Carlo (MC) methods, are computationally expensive and simulation of real cases of practical interest requires intractable CPU times. On the other hand, other methods such as the method of moments (MOM) are computationally affordable but have proven to be inaccurate for a number of cases. In recent work a new closure, the quadrature method of moments (QMOM), has been introduced, applied and validated. In our earlier work, QMOM was shown to be an efficient and accurate method for tracking the moments of the particle size distribution (PSD) in a CFD simulation. However, QMOM presents two main disadvantages: (i) if applied to multi-variate distributions it loses simplicity and efficiency, and (ii) by tracking only the moments of the PSD, it does not represent realistically polydisperse systems with strong coupling between the internal coordinates and phase velocities. In order to address these issues, in this work the direct quadrature method of moments (DQMOM) is formulated, validated, and tested. DQMOM is based on the idea of tracking directly the variables appearing in the quadrature approximation, rather than tracking the moments of the PSD. Nevertheless, for monovariate cases we show that QMOM and DQMOM yield identical results. In addition, we show how it is possible to extend the DQMOM to multivariate cases and some of relevant theoretical and numerical issues are discussed. These issues are discussed in the present work for homogeneous and one-dimensional flows. References to recent CFD applications of DQMOM to multiphase flows are provided as further proof of the utility of the method.     

Smoke buildup and light scattering in a cylindrical cavity above a uniform flow

In this study, we use computational fluid dynamics (CFD) and aerosol dynamics modeling to investigate the buildup of smoke and light scattering in a cylindrical cavity geometry, considered to be an idealized representation of a photoelectric smoke detector. CFD coupled with the quadrature method of moments (QMOM) is used for simulation of aerosol dynamics. The Rayleigh–Debye–Gans/polydisperse fractal aggregate (RDGPFA) theory is used for calculation of smoke extinction and angular light scattering. It is seen that the flow external to the cavity sets up a recirculating flow pattern within the cavity and that the flow processes determine the spatial distribution of smoke. Aerosol extinction and scattering calculations are performed to examine the time varying profiles of the intensity along a simulated LED light beam and the scattered intensity at different angles. The variation of the detector activation time with inlet velocity and smoke volume fraction is obtained from a calculation of the angular light scattering. The results are compared with calculations using an empirically determined detector response function and with a simpler model that assumes a uniform distribution of smoke inside the cavity. Results indicate that although the distribution of smoke inside the cavity is not uniformly mixed, the simple first-order mixing model with appropriately chosen parameters is valid for predicting detector activation time.     

Description of Aerosol Dynamics by the Quadrature Method of Moments

ABSTRACT

The method of moments (MOM) may be used to determine the evolution of the lower-order moments of an unknown aerosol distribution. Previous applications of the method have been limited by the requirement that the equations governing the evolution of the lower-order moments be in closed form. Here a new approach, the quadrature method of moments (QMOM), is described. The dynamical equations for moment evolution are replaced by a quadrature-based approximate set that satisfies closure under a much broader range of conditions without requiring that the size distribution or growth law maintain any special mathematical form. The conventional MOM is recovered as a special case of the QMOM under those conditions, e.g., free-molecular growth, for which conventional closure is satisfied. The QMOM is illustrated for the growth of sulfuric acid-water aerosols and simulations of diffusion-controlled cloud droplet growth are presented.     

Automatic differentiation‐based quadrature method of moments for solving population balance equations

The quadrature method of moments (QMOM) is a promising tool for the solution of population balance equations. QMOM requires solving differential algebraic equations (DAEs) consisting of ordinary differential equations related to the evolution of moments and nonlinear algebraic equations resulting from the quadrature approximation of moments. The available techniques for QMOM are computationally expensive and are able to solve for only a few moments due to numerical robustness deficiencies. In this article, the use of automatic differentiation (AD) is proposed for solution of DAEs arising in QMOM. In the proposed method, the variables of interest are approximated using high‐order Taylor series. The use of AD and Taylor series gives rise to algebraic equations, which can be solved sequentially to obtain high‐fidelity solution of the DAEs. Benchmark examples involving different mechanisms are used to demonstrate the superior accuracy, computational advantage, and robustness of AD‐QMOM over the existing state‐of‐the‐art technique, that is, DAE‐QMOM. © 2011 American Institute of Chemical Engineers AIChE J, 2012     

流化催化裂化提升管反应器的CFD-PBM耦合模型:矩方法适用性比较

建立了描述FCC提升管中气-固流动行为的CFD-PBM耦合模型,模型同时考虑了颗粒动力学和颗粒聚并破碎内核。讨论了求解耦合模型中众体平衡方程（PBE）的3种典型矩方法[即：正交矩方法（quadrature method of moments,QMOM）,直接正交矩方法（direct quadrature method of moments,DQMOM）和固定轴点正交矩方法（fixed pivot quadrature method of moments,FPQMOM）]对模拟结果的影响。研究结果表明3种矩方法均能合理预测提升管内径向和轴向颗粒体积分数和颗粒速度分布。通过将模拟结果与实验结果进行比较,表明QMOM在反应器结构简单情况下,计算结果更接近于实际情况。     

New quadrature‐based moment method for the mixing of inert polydisperse fluidized powders in commercial CFD codes

To describe the behavior of polydisperse multiphase systems in an Eulerian framework, we solved the population balance equation (PBE), letting it account only for particle size dependencies. To integrate the PBE within a commercial computational fluid dynamics code, we formulated and implemented a novel version of the quadrature method of moments (QMOM). This no longer assumes that the particles move with the same velocity, allowing the latter to be size‐dependent. To verify and test the model, we simulated the mixing of inert polydisperse fluidized suspensions initially segregated, validating the results experimentally. Because the accuracy of QMOM increases with the number of moments tracked, we ran three classes of simulations, preserving the first four, six, and eight integer moments of the particle density function. We found that in some cases the numerics corrupts the higher‐order moments and a corrective algorithm, designed to restore the validity of the moment set, has to be implemented. © 2012 American Institute of Chemical Engineers AIChE J, 58: 3054–3069, 2012     

A novel algorithm for solving population balance equations: the parallel parent and daughter classes. Derivation, analysis and testing

A novel numerical method, the parallel parent and daughter classes (PPDC) technique, for solving population balance equations (PBEs) is presented in this paper. In many practical applications, the PBE of particles under investigation is coupled with the thermo-fluid dynamics of the surrounding fluid. Hence, the PBE needs to be implemented in a computational fluid dynamics (CFD) code, which leads to an additional computational load. The computational cost becomes intractable when techniques such as methods of classes (CM) or Monte Carlo method are used. Quadrature method of moments (QMOM) and direct quadrature method of moments (DQMOM) are accurate and require a relatively low additional computational cost when applied to CFD. The PPDC is shown to be as accurate as QMOM and DQMOM, and even more accurate in some cases, when the same number of classes is used. In the present work, the PPDC technique has been derived and tested. This technique can be used for solving a wide class of problems involving PBE such as polymerization, aerosol dynamics, bubble columns, etc. Numerical simulations have been carried out on aggregation processes with different kernels and on simultaneous aggregation and breakage processes. The numerical predictions are compared either with analytical solutions, when available, or with the numerical solutions obtained by methods of classes.     

A multi‐QMOM framework to describe multi‐component agglomerates in liquid steel

A variant of the quadrature method of moments (QMOM) for solving multiple population balance equations (PBE) is developed with the objective of application to steel industry processing. During the process of oxygen removal in a steel ladle, a large panel of oxide inclusions may be observed depending on the type of oxygen removal and addition elements. The final quality of the steel can be improved by accurate numerical simulation of the multi‐component precipitation. The model proposed in this article takes into account the interactions between three major aspects of steelmaking modeling, namely fluid dynamics, thermo‐kinetics and population balance. A commercial CFD code is used to predict the liquid steel hydrodynamics, whereas a home‐made thermo‐kinetic code adjusts chemical composition with nucleation and diffusion growth, and finally a set of PBE tracks the evolution of inclusion size with emphasis on particle aggregation. Each PBE is solved by QMOM, the first PBE/QMOM system describing the clusters and each remaining PBE/QMOM system being dedicated to the elementary particles of each inclusion species. It is shown how this coupled model can be used to investigate the cluster size and composition of a particular grade of steel (i.e., Fe‐Al‐Ti‐O). © 2010 American Institute of Chemical Engineers AIChE J, 2010     

Limitations of quadrature-based moment methods for modeling inhomogeneous polydisperse fluidized powders

Computational fluid dynamics (CFD) is extensively used to investigate the behavior of dense fluidized suspensions. Often modelers assume that these are formed by few solid phases of particles with constant size. But real powders are continuously distributed over the particle size, and their distribution functions change continuously in time and space reflecting the physical and chemical phenomena occurring within the system. To account for this key feature, models have to include a population balance equation (PBE), which needs to be solved in place of or along with the customary fluid dynamic transport equations. The recently developed quadrature method of moments (QMOM) and direct quadrature method of moments (DQMOM) permit to solve PBEs in commercial CFD codes at relatively low computational cost. These methods, however, still need testing in the context of multiphase flows. Investigating a simple problem, namely the dynamics of two inert polydisperse fluidized suspensions initially segregated, we highlight an important limitation of these methods, which fail to properly model diffusion in real space. We explain where the problem originates and comment on a possible way to overcome it. To conclude the work, we discuss some simulations based on the original and revised formulations of the methods, describing how the code numerics affects the results.     

基于积分近似矩量法的声波团聚数值模拟

引言 目前我国在控制粉尘排放方面取得了较好的成效,燃煤锅炉等工业排放的烟气粉尘总浓度一般均能够满足国家排放标准.但是,现广泛采用的静电除尘设备、旋风分离器等均难以除去烟气中的超细颗粒[1].     

Quadrature-based moment methods for the population balance equation: An algorithm review

The dispersed phase in multiphase flows can be modeled by the population balance model (PBM). A typical population balance equation (PBE) contains terms for spatial transport, loss/growth and breakage/coalescence source terms. The equation is therefore quite complex and difficult to solve analytically or numerically. The quadrature-based moment methods (QBMMs) are a class of methods that solve the PBE by converting the transport equation of the number density function (NDF) into moment transport equations. The unknown source terms are closed by numerical quadrature. Over the years, many QBMMs have been developed for different problems, such as the quadrature method of moments (QMOM), direct quadrature method of moments (DQMOM), extended quadrature method of moments (EQMOM), conditional quadrature method of moments (CQMOM), extended conditional quadrature method of moments (ECQMOM) and hyperbolic quadrature method of moments (HyQMOM). In this paper, we present a comprehensive algorithm review of these QBMMs. The mathematical equations for spatially homogeneous systems with first-order point processes and second-order point processes are derived in detail. The algorithms are further extended to the inhomogeneous system for multiphase flows, in which the computational fluid dynamics (CFD) can be coupled with the PBE. The physical limitations and the challenging numerical problems of these QBMMs are discussed. Possible solutions are also summarized.     

Implementation of the quadrature method of moments in CFD codes for aggregation-breakage problems

In this work the quadrature method of moments (QMOM) is implemented in a commercial computational fluid dynamics (CFD) code (FLUENT) for modeling simultaneous aggregation and breakage. Turbulent and Brownian aggregation kernels are considered in combination with different breakage kernels (power law and exponential) and various daughter distribution functions (symmetric, erosion, uniform). CFD predictions are compared with experimental data taken from other work in the literature and conclusions about CPU time required for the simulations and the advantages of this approach are drawn.     

Modeling spatial distribution of floc size in turbulent processes using the quadrature method of moment and computational fluid dynamics

A study was performed that utilizes the quadrature method of moments (QMOM) to model the transient spatial evolution of the floc size in a heterogeneous turbulent stirred reactor. The QMOM approach was combined with a commercial computational fluid dynamics (CFD) code (PHOENICS), which was used to simulate the turbulent flow and transport of these aggregates in the reactor. The CFD/QMOM model was applied to a 28 l square reactor containing an axial flow impeller and 100 mg/l concentration of 1 μm nominal clay particles. Simulations were performed for different average characteristic velocity gradients (40,70,90, and 150 s^-1). The average floc size and growth rate were compared with experimental measurements performed in the bulk region and the impeller discharge region. The CFD/QMOM results confirmed the experimentally measured spatial heterogeneity in the floc size and growth rate. In addition, the model predicts spatial variations in the aggregation and breakup rates. Finally, the model also predicts that the transport of flocs into the high shear impeller discharge zone was responsible for the transient evolution of the average floc size curve displaying a maximum before decreasing to a steady-state floc size.     

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.

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