Direct Quadrature Method Of Moments Research Articles

Reactive mixing performance for a nanoparticle precipitation in a swirling vortex flow reactor

Mixing performance for a consecutive competing reaction system has been investigated in a swirling vortex flow reactor (SVFR). The direct quadrature method of moments combined with the interaction by exchange with the mean (DQMOM-IEM) method was employed to model such reacting flows. This type of reactors is able to generate a strong swirling flow with a great shear gradient in the radial direction. Firstly, mixing at both macroscale and microscale was assessed by mean mixture fraction and its variance, respectively. It is found that macromixing can be rapidly achieved throughout the whole reactor chamber due to its swirling feature. However, micromixing estimated by Bachelor length scale is sensitive to turbulence. Moreover, the additional introduction of ultrasound irradiation can significantly improve the mixing uniformity, namely, free of any stagnant zone presented in the reactor chamber on a macroscale, and little variance deviating from the mean environment value can be observed on a microscale. Secondly, reaction progress variable and the reactant conversion serve as indicators for the occurrence of side reaction. It is found that strong turbulence and a relatively fast micromixing process compared to chemical reaction can greatly reduce the presence of by-product, which will then provide homogenous environment for particle precipitation. Moreover, due to the generation of cavitation bubbles and their subsequent collapse, ultrasound irradiation can further intensify turbulence, creating rather even environment for chemical reactions. Low conversion rate was observed and little by-products were generated consequently. Therefore, it is suggested that the SVFR especially intensified by ultrasound irradiation has the ability to provide efficient mixing performance for the fine-particle synthesis process.

Dynamics of rising bubble population undergoing mass transfer and coalescence in highly viscous liquid

The two-phase flow dynamics involving mass transfer and coalescence is investigated. The model is specifically developed to describe the dynamics of bubble population dispersed in glass forming liquids. The amounts of gas dissolved in the liquid are determined using the chemical equilibrium involving oxidation–reduction reactions. The gravitational bubble rising is used to write the coalescence kernel for which a collision efficiency is also introduced.Based on a Direct Quadrature Method of Moments (DQMOM), a numerical method is developed. This numerical tool is applied to melting of borosilicate glass beads for which temperature and residence time of the sample in a crucible are investigated. The bubble density decreases sharply at short times. This early stage decrease is well explained and quantified when the coalescence is taken into account in numerical computations. The bubble size density is very well described with a log-normal distribution. Using the first three moments, the bubble size distribution obtained numerically is in good agreement with experimental data.Numerical computations are also applied to soda-lime-silica glass in which the bubble release is driven by the mass transfer between the two phases. The faster decrease of bubble density than would be expected by temperature is reproduced by the numerical computation. The enhancement of the bubble release rate is mainly due to the increase of dissolved gas species with temperature.

Modeling the reactive flow of semi-continuous mixtures by the adaptive characterization method

This work proposes and analyzes the direct quadrature method of moments (DQMoM) applied to the adaptive characterization of pseudo-components for simulating semi-continuous reactive mixtures’ flow. We also extended the fixed point quadrature method (FPQM) by expanding its scope to solve field problems and use discrete distributions as initial or inlet data. We implemented both methods in OpenFOAM® and used them to simulate the thermal cracking of hydrocarbon mixtures, guaranteeing mesh convergence. The FPQM results showed poor convergence behavior, while the DQMoM results converged regarding the number of quadrature points. DQMOM was up to 4.9 faster than FPQM for our reactive flow test case.

A comprehensive modeling study of particulate coalescence and breakup mechanisms in solid propellant motor nozzles

Gas-particle flows are modeled to account for the coalescence and breakup of liquid aluminum oxide particles dispersed within the gas phase in solid propellant motors. The one-way coupled population balance equation, which addresses a change in number concentration of particles by the particle to particle and surrounding to particle interactions, is solved using the direct quadrature method of moments (DQMOM). Simulations are performed on the converging-diverging nozzles attached to solid propellant motors and the mass mean diameter at the nozzle exit is compared to previous measurements and a correlation derived from numerous existing measurements data. The results from a series of simulations are presented to assess the effects of chamber pressure, initial particle concentration and size, and nozzle scale on resulting particle size growth.

An open-source computational framework for the solution of the bivariate population balance equation

The bivariate population balance equation (PBE) is a mathematical framework to explain the evolution of polydisperse multiphase systems. In this work, the direct quadrature method of moments (DQMOM) is implemented in an open-source CFD code, Fluidity, for the numerical solution of bivariate PBE. This efficient numerical framework is a highly-parallelised finite element (FE) CFD code that allows for the use of mesh adaptivity on fully-unstructured meshes. Various test cases to solve spatially homogeneous bivariate PBEs with aggregation, breakage, growth and dispersion were simulated and verified against analytical solutions, resulting in excellent agreement. Benchmarking, by comparison with the Monte Carlo method solutions from the literature, with realistic kernels in a gas–liquid system for simultaneous bivariate aggregation and breakage was also performed to show the feasibility of this implementation for realistic applications. This open-source framework demonstrates its impressive potential in the case of bivariate PBE and can be exploited for the simulation of complex polydisperse multiphase systems.

CFD-PBE coupled model for size-driven segregation in polydisperse granular flows

Mixtures of granular materials made of different-sized particles may segregate when subjected to vibration or shear rate and in the presence of a gravitational field. This leads to highly inhomogeneous mixtures, which are undesirable in many industrial processes. This work focuses on size-driven segregation in polydisperse mixtures. We described the evolution of the particle size distribution through a Population Balance Equation (PBE), which we solved numerically with the Direct Quadrature Method of Moments. To allow segregation and micromixing to occur, we closed the size-conditioned velocity of the particles with a segregation-remixing model. The PBE was then included in an Eulerian-Eulerian model and solved in a commercial Computational Fluid Dynamics (CFD) code. We used the model to study granular flows down inclined planes. The numerical results were then compared with those obtained from Discrete Element Method simulations. The CFD-PBE model predicts well segregation and micromixing in packed beds of polydisperse powders.

Eulerian formulation for the triboelectric charging of polydisperse powder flows

We propose an Eulerian formulation for the triboelectric charging of powder flows. Recently, continuum descriptions of the charging of monodisperse and bidisperse powder appeared. Now, we propose the first Eulerian formulation for this type of flows that can fully account for polydisperse particle size distributions. To this end, the joint particle size, velocity, and charge distribution function are solved through the direct quadrature method of moments. The newly established approach includes the uptake of electrostatic charge by the particles when contacting a solid wall and the exchange of charge in-between particles during collisions. The electrostatic field induced by the charge of the particles and the drag forces of the surrounding gas affects the particle dynamics. Test cases of two-dimensional steady channel flows demonstrate the charging of the flow from the walls toward the center and in the downstream direction. Further, the method predicts the effect of variations of the particle size and initial velocity distribution and the charge diffusion coefficient. The ability to handle polydispersity is a step toward the simulation of the electric charge buildup of real powder flows in full-scale technical applications.

Evaluation of direct quadrature method of moment for the internally circulating fluidized bed simulation with ultrafine particles

In the framework of the Euler-Euler gas–solid two-fluid model, the particle population balance equation is solved by the direct quadrature method of moment. The dynamic process of ultrafine particle movement and aggregation in an internally circulating fluidized bed is simulated. The distribution of the concentration and velocity of the agglomerates in the flow process is given, and the changes of the moments in the bed are shown. The effects of different breakage coefficients and inlet gas rates on the concentration distribution of agglomerates are compared. The results show that the particle size decreases with the increase of breakage coefficient, and the time required to reach steady fluidization state increases; the higher the inlet velocity, the better the effect of circulating particles in the bed. When there is a certain gas velocity difference between the two sides, the effect of circulating particles in the bed is better.

A two-way coupled CFD-DQMOM approach for long-term dynamic simulation of a fluidized bed reactor

For the long-term dynamic simulation of a fluidized bed reactor (FBR), a two-way coupled computational fluid dynamics (CFD)-direct quadrature method of moments (DQMOM) approach is proposed. In this approach, CFD is first used only for hydrodynamic information without simulating any other chemical reactions or physical phenomena. Subsequently, the derived information is applied to the DQMOM calculation in MATLAB. From the calculation, a particle size distribution is obtained and subsequently adopted in a new CFD model to reflect the flow change caused by a change in the particle size distribution. Through several iterative calculations, long-term dynamic simulations are performed. To evaluate the efficacy of the proposed approach, the results from the suggested approach are compared for 60 s with those of the CFD-quadrature method of moments (QMOM) approach, which calculates hydrodynamics and physical phenomena simultaneously in CFD. The proposed approach successfully simulated the FBR for 6 h. The results confirmed that the proposed method can simulate complex flow patterns, which cannot be obtained in conventional CFD models. Another advantage of the approach is that it can be applied to various industrial multiphase reactors without any tuning parameters.

Modelling polydisperse nanoparticle size distributions as produced via flame spray pyrolysis

The correct description of the particulates produced via the flame spray pyrolysis (FSP) process is essential for scale-up studies and new reactor design. With the combination of computational fluid dynamics (CFD) and population balance models (PBM) it is possible to predict the formation and evolution of nanoparticles in such reactors, and estimate final product characteristics. The solution of such PBM, however, can be very costly, given the fact that the phenomena involved in the process require for a bivariate approach. In this work, the direct quadrature method of moments (DQMoM) is employed in order to obtain a polydisperse solution of the PBM, due to its relatively straight forward adaptation from uni- to bivariate cases. Furthermore, the reacting turbulent multiphase flow of the burning spray is described by an Eulerian-Lagrangian framework and solved with the use of CFD. The production of zirconium dioxide nanoparticles from a solution of zirconium n-propoxide in ethanol and propanol is investigated, and experimental data obtained through transmission electron microscopy (TEM) is used for model validation. Accuracy of about 90% is obtained for the first two moments of the distribution and the number of primary particles per agglomerate show good agreement with experiments.

Numerical Methods for the Solution of Population Balance Equations Coupled with Computational Fluid Dynamics.

This review article discusses the solution of population balance equations, for the simulation of disperse multiphase systems, tightly coupled with computational fluid dynamics. Although several methods are discussed, the focus is on quadrature-based moment methods (QBMMs) with particular attention to the quadrature method of moments, the conditional quadrature method of moments, and the direct quadrature method of moments. The relationship between the population balance equation, in its generalized form, and the Euler-Euler multiphase flow models, notably the two-fluid model, is thoroughly discussed. Then the closure problem and the use of Gaussian quadratures to overcome it are analyzed. The review concludes with the presentation of numerical issues and guidelines for users of these modeling approaches.

Examining the physical and chemical contributions to size spectrum evolution during the development of hazes

China has experienced severe hazes with high concentrations of particulate matter in recent years. The understanding of the size spectrum evolution of submicron particulate matter is critical to making efficient remediation policies to minimize the regional and global environmental impacts from hazes. During a time period of about one month, we monitored five severe haze episodes in Xi’an and four severe haze episodes in Beijing, which were characterized by two distinct kinds of aerosol mass concentration growth processes: accumulative-rise and abrupt-rise. A new method was developed to quantitatively evaluate the physical and chemical contributions to growth processes by analysing the size spectrum evolution data. The results showed that the accumulative-rise processes are governed by primary emissions and the abrupt-rise processes are governed by secondary chemical reactions. The population balance equations (PBE) were used to describe the variation of size spectrum of fine particulate matter, and the respective contributions of the physical aggregation rate and the chemical growth rate. The PBE model is solved using the adjustable direct quadrature method of moments (ADQMOM) to simulate the abrupt-rise process of haze development and to calibrate the contribution of the physical and chemical effects on the size spectrum of aerosol particles.

Modeling of the Coal Particle Behavior in an Ultra-Supercritical Boiler With Large Eddy Simulation

Abstract Large eddy simulation (LES) is becoming a promising tool for the design and retrofit of utility boilers. It explicitly calculates the large-scale eddies which play an important role in the particle behavior inside the boilers. An ultra-supercritical tangentially fired boiler was simulated under the boiler maximum continuous rate (BMCR) condition by LES. The particle phase was tracked by the simplified direct quadrature method of moments in the Eulerian framework. Five particle sizes were adopted to represent the wide particle size distribution. The predicted gas velocities were in good agreement with the thermal calculation. The LES results showed that the particles were more likely to be concentrated in the main burner zone while quickly dispersed in the over fire air (OFA) zone. A theoretical analysis found that the particle Stokes number based on the sub-grid scale was much smaller than one. The particles would behave as tracers for the eddies resolved by LES. However, some differences between the small and large particles were observed in the particle number density distributed along the vertical and horizontal directions. It meant that the inertial effects on the particle motion cannot be neglected.

Mass transfer in continuous mixtures with Maxwell–Stefan diffusion using the adaptive characterization method

AbstractThe adaptive pseudocomponent characterization method for continuous mixtures was extended to mass transfer problems using the Maxwell–Stefan diffusion model. It is based on the direct quadrature method of moments (DQMoM), using a quadrature rule to discretize the molar fraction distribution of the continuous mixture. The solution method was applied to two one‐dimensional mass transfer problems: the transient diffusion in a Loschmidt tube and the steady‐state diffusion in a thin film. In the latter, it was showed that the DQMoM equations reduce to an equivalent problem with a fixed characterization and solution methods for linearized theory problems can be employed. For these two problems, the proposed method was verified against the discrete component model (DCM), whose implementation was also verified against existing solutions. Results showed that the adaptive method with five pseudocomponents predicts the mixture properties with maximum relative deviation smaller than 1% when compared to the DCM with 57 components.

A framework for polydisperse pulp phase modelling in flotation

Froth flotation is one of the most widely-used mineral processing operations. The pulp zone in flotation tanks is polydisperse in general and serves as a medium for the interaction between the solid particles and the gas bubbles in a liquid continuum, leading to particle–bubble attachment/detachment and bubble coalescence/breakage phenomena. To better predict the hydrodynamics and inform the design of efficient flotation equipment, it is therefore important to accurately model and simulate the evolution of the size distribution of the dispersed phases. This has created an urgent need for a framework that can model the pulp phase in an efficient manner, which is not currently available in the literature. The available software products are not efficient enough to allow for a tractable modelling of industrial-scale flotation cells and in some cases they cannot model the polydispersity of the dispersed phase at all. This work presents an efficient numerical framework for the macroscale simulation of the polydisperse pulp phase in froth flotation in an open-source finite element computational fluid dynamics (CFD) code that provides an efficient solution method using mesh adaptivity and code parallelisation. A (hybrid finite element–control volume) finite element framework for modelling the pulp phase has been presented for the first time in this work. An Eulerian–Eulerian turbulent flow model was implemented in this work including a transport equation for attached and free solid particles. Special care was taken to model the settling velocity of the free solids and the modification of the liquid viscosity due to the presence of these particles. Bubble polydispersity was modelled using the population balance equation (PBE), which was solved using the direct quadrature method of moments (DQMOM). Appropriate functions for bubble coalescence and breakage were chosen in the PBE. Mesh adaptivity was applied to the current problem to produce fully-unstructured anisotropic meshes, which improved the solution efficiency, while all simulations were executed on a multicore architecture. The model was validated for 2D simulations of a bubble column against experimental results available in the literature. After successful validation, the model was applied to the simulation of the pulp phase in a flotation column for monodisperse and polydisperse solids. Polydispersity of the solids was modelled for the first time in this work using three separate solid size classes. A clear dependence of the flotation rate on the particle size was noticed and the monodisperse solids simulations were shown to over-predict the flotation rate. Other than flotation, this open-source framework can be used for the simulation of a variety of polydisperse multiphase flow problems in the process industry.

Characterisation of soot particle size distribution through population balance approach and soot diagnostic techniques for a buoyant non-premixed flame

Combustion-generated soot particles is a subject of great interest due to its existing and future use in research and industrial sectors. An in-house Direct Quadrature Method of Moments (DQMOM) based population balance model is constructed to investigate the evolution of soot particulate. Most numerical models assume the particle size to be monodispersed, while the proposed model enables the evaluation of real-time soot particle size distribution which further enhances prediction accuracies. The proposed model is fully coupled with all essential fire sub-modelling components and is specifically constructed for low-speed buoyant flames. Additionally, to better describe the combustion process of parental fuel, ethylene, the strained laminar flamelet model considering detailed chemical reaction mechanisms is adopted. Numerical simulation with the implementation of the proposed population balance soot model has been validated against an in-house co-flow burner experimental study and been compared with other numerical studies. The results demonstrated that the improved DQMOM soot model has significant improvement to the accuracy of simulation when compared to Moss-Brookes soot model. It was also discovered that by employing Moss nucleation law, modified NSC oxidation law and adopting fractal dimension value of 2.0, the DQMOM model produces the results with the best agreement against experimental data.

Sensitivity Analysis of Key Parameters for Population Balance Based Soot Model for Low-Speed Diffusion Flames

In this article, the evolution of in-flame soot species in a slow speed, buoyancy-driven diffusion flame is thoroughly studied with the implementation of the population balance approach in association with computational fluid dynamics (CFD) techniques. This model incorporates interactive fire phenomena, including combustion, radiation, turbulent mixing, and all key chemical and physical formation and destruction processes, such as particle inception, surface growth, oxidation, and aggregation. The in-house length-based Direct Quadrature Method of Moments (DQMOM) soot model is fully coupled with all essential fire sub-modelling components and it is specifically constructed for low-speed flames. Additionally, to better describe the combustion process of the parental fuel, ethylene, the strained laminar flamelet model, which considers detailed chemical reaction mechanisms, is adopted. Numerical simulation is validated against a self-conducted co-flow slot burner experimental measurement. A comprehensive assessment of the effect of adopting different nucleation laws, oxidation laws, and various fractal dimension and diffusivity values is performed. The results suggest the model employing Moss law of nucleation, modified NSC law of oxidation, and adopting a fractal dimension value of 2.0 and Schmidt number of 0.9 yields the simulation result that best agreed with experimental data.

CFD-PBM Coupled Simulation of Liquid-Liquid Dispersions in Spray Fluidized Bed Extractor: Comparison of Three Numerical Methods

A coupled numerical code of the Euler-Euler model and the population balance model (PBM) of the liquid-liquid dispersions in a spray fluidized bed extractor (SFBE) has been performed to investigate the hydrodynamic behavior. A classes method (CM) and two representatively numerical moment-based methods, namely, a quadrature method of moments (QMOM) and a direct quadrature method of moments (DQMOM), are used to solve the PBE for evaluating the effect of the numerical method. The purpose of this article is to compare the results achieved by three methods for solving population balance during liquid-liquid two-phase mixing in a SFBE. The predicted results reveal that the CM has the advantage of computing the droplet size distribution (DSD) directly, but it is computationally expensive if a large number of intervals are needed. The MOMs (QMOM and DQMOM) are preferable to coupling the PBE solution with CFD codes for liquid-liquid dispersions simulations due to their easy application, reasonable accuracy, and high reliability. Comparative results demonstrated the suitability of the DQMOM for modeling the spray fluidized bed extractor with simultaneous droplet breakage and aggregation. This work increases the understanding of the chemical engineering characteristics of multiphase systems and provides a theoretical basis for the quantitative design, scale-up, and optimization of multiphase devices.

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.

A New Model Approach for Convective Wall Heat Losses in DQMOM-IEM Simulations for Turbulent Reactive Flows

A new model approach is presented in this work for including convective wall heat losses in the direct quadrature method of moments (DQMoM) approach, which is used here to solve the transport equation of the one-point, one-time joint thermochemical probability density function (PDF). This is of particular interest in the context of designing industrial combustors, where wall heat losses play a crucial role. In the present work, the novel method is derived for the first time and validated against experimental data for the thermal entrance region of a pipe. The impact of varying model-specific boundary conditions is analyzed. It is then used to simulate the turbulent reacting flow of a confined methane jet flame. The simulations are carried out using the DLR in-house computational fluid dynamics code THETA. It is found that the DQMoM approach presented here agrees well with the experimental data and ratifies the use of the new convective wall heat losses model.