Quadrature Method Of Moments Research Articles

Extension of moment projection method to the fragmentation process

The method of moments is a simple but efficient method of solving the population balance equation which describes particle dynamics. Recently, the moment projection method (MPM) was proposed and validated for particle inception, coagulation, growth and, more importantly, shrinkage; here the method is extended to include the fragmentation process. The performance of MPM is tested for 13 different test cases for different fragmentation kernels, fragment distribution functions and initial conditions. Comparisons are made with the quadrature method of moments (QMOM), hybrid method of moments (HMOM) and a high-precision stochastic solution calculated using the established direct simulation algorithm (DSA) and advantages of MPM are drawn.

Simulation of a reacting gas–liquid bubbly flow with CFD and PBM: Validation with experiments

In this work we use computational fluid dynamics (CFD) to simulate a reactive gas–liquid bubbly system in a rectangular bubble column, operating at low superficial velocities (i.e. homogeneous regime). The gas bubbles, injected in the column through a sparger, contain one of the reactants, namely CO2, that via mass transfer moves to the continuous liquid phase, where it reacts with NaOH. A key role is played by the bubble size distribution (BSD) and the specific surface area that define the overall mass transfer rate in the CFD model. In order to correctly predict the BSD and the polydispersity of the bubbly system the population balance equation is solved by the quadrature method of moments (QMOM), within the OpenFOAM (v. 2.2.x) two-fluid solver compressibleTwoPhaseEulerFoam. To reduce the computational time and increase stability, a second-order operator-splitting technique for the solution of the chemically reactive species is also implemented, allowing to solve the different processes involved with their own time-scale. To our knowledge this is the first time that QMOM is employed for the simulation of a real reactive bubbly system and predictions are validated against experiments.

A robust two-node, 13 moment quadrature method of moments for dilute particle flows including wall bouncing

For flows where the particle number density is low and the Stokes number is relatively high, as found when sand or ice is ingested into aircraft gas turbine engines, streams of particles can cross each other's path or bounce from a solid surface without being influenced by inter-particle collisions. The aim of this work is to develop an Eulerian method to simulate these types of flow. To this end, a two-node quadrature-based moment method using 13 moments is proposed. In the proposed algorithm thirteen moments of particle velocity, including cross-moments of second order, are used to determine the weights and abscissas of the two nodes and to set up the association between the velocity components in each node. Previous Quadrature Method of Moments (QMOM) algorithms either use more than two nodes, leading to increased computational expense, or are shown here to give incorrect results under some circumstances. This method gives the computational efficiency advantages of only needing two particle phase velocity fields whilst ensuring that a correct combination of weights and abscissas is returned for any arbitrary combination of particle trajectories without the need for any further assumptions. Particle crossing and wall bouncing with arbitrary combinations of angles are demonstrated using the method in a two-dimensional scheme. The ability of the scheme to include the presence of drag from a carrier phase is also demonstrated, as is bouncing off surfaces with inelastic collisions. The method is also applied to the Taylor–Green vortex flow test case and is found to give results superior to the existing two-node QMOM method and is in good agreement with results from Lagrangian modelling of this case.

Three-Way Coupling Simulation of a Gas-Liquid Stirred Tank using a Multi-Compartment Population Balance Model

Abstract Modelling of gas-liquid stirred tanks is very challenging due to the presence of strong bubble-liquid interactions. Depending upon the needs and desired accuracy, the simulation may be performed by considering one-way, two-way, three-way or four-way coupling between the primary and secondary phase. Accuracy of the prediction on the two-phase flow generally increases as the details of phase interactions increase but at the expense of higher computational cost. This study deals with two-way and three-way coupling of gas-liquid flow in stirred tanks which were then compared with results via four-way coupling. Population balance model (PBM) based on quadrature method of moments (QMOM) was implemented in a multi-compartment model of an aerated stirred tank to predict local bubble size. The multi-compartment model is regarded as three-way coupling because the local turbulent dissipation rates and flow rates were obtained from a two-way computational fluid dynamics (CFD) simulation. The predicted two-phase flows and local bubble size showed good agreement with experimental data.

CFD Analysis of Aggregation and Breakage of Li-Ion Precursor in Taylor-Reactor

In this work, aggregation and breakage of Li-ion precursor in Taylor-reactor are studied for various simulation condition, i.e., agitation rate, particle property, and reactor direction. The mean particle size and distribution is important to electrochemical performance of cathode materials. The population balance equation (PBE) is solved for coupling between particle interaction and fluid dynamic. The quadrature method of moments (QMOM) is implemented in a commercial computational fluid dynamics (CFD) code. Sum of three aggregation kernels, such as turbulent, Brownian, and Stokes aggregation, is considered for the calculation. [1] Breakage kernel is a combination of power-law kernel and various daughter distribution functions. Collision efficiency is assumed to be equal and the precursor is considered as spherical particle. Three-dimensional analysis is implemented and the simulation results are compared with experimental data. [1] L. Claudotte, N. Rimbert, P. Gardin, M. Simonnet, J. Lehmann, and B. Oesterle, AIChE J., Vol.56, No.9, pp. 2347-2355 (2010)

Nonrealizability Problem With Quadrature Method of Moments in Wet-Steam Flows and Solution Techniques

The quadrature method of moments (QMOM) has recently attracted much attention in representing the size distribution of liquid droplets in wet-steam flows using the n-point Gaussian quadrature. However, solving transport equations of moments using high-order advection schemes is bound to corrupt the moment set, which is then termed as a nonrealizable moment set. The problem is that the failure and success of the Gaussian quadrature are unconditionally dependent on the realizability of the moment set. First, this article explains the nonrealizability problem with the QMOM. Second, it compares two solutions to preserve realizability of the moment sets. The first solution applies a so-called “quasi-high-order” advection scheme specifically proposed for the QMOM to preserve realizability. However, owing to the fact that wet-steam models are usually built on existing numerical solvers, in many cases modifying the available advection schemes is either impossible or not desired. Therefore, the second solution considers correction techniques directly applied to the nonrealizable moment sets instead of the advection scheme. These solutions are compared in terms of accuracy in representing the droplet size distribution. It is observed that a quasi-high-order scheme can be reliably applied to guarantee realizability. However, as with all the numerical models in an Eulerian reference frame, in general, its results are also sensitive to the grid resolution. In contrast, the corrections applied to moments either fail in identifying and correcting the invalid moment sets or distort the shape of the droplet size distribution after the correction.

QMOM-based population balance model involving a fractal dimension for the flocculation of latex particles

An experimental and computational study of agglomeration and breakage processes for fully destabilized latex particles under turbulent flow conditions in a jar is presented. The particle size distribution (PSD) and the fractal dimension of flocs of latex particles were monitored using an on-line laser diffraction technique. A population balance equation (PBE) was adapted to our problem by including the fractal dimension in its formulation as well as in the aggregation and breakage kernels. The quadrature method of moments was used for the resolution. The adjustment of 4 model parameters was then conducted on the first 6 moments of the PSD for various mean shear rates. The model correctly predicts the evolution of the first 6 moments calculated from the experimental PSD. The experimental results were adequately simulated by a single set of adjusted parameters, proving the relevance of the dependency on the fractal dimension and mean shear rate. A sensitivity analysis was performed on two main adjusted parameters highlighting the major roles of (1) the power to which the mean shear rate is raised in the breakage kernel and (2) the sizes of the colliding aggregates in the collision efficiency model. Finally, analytical relations between the sink and source terms of the breakage or aggregation of the PBE were derived and discussed, highlighting interesting features of the PBE model.

Investigation of gas dispersion characteristics in stirred tank and flotation cell using a corrected CFD-PBM quadrature-based moment method approach

In this study, the population balance model (PBM) is coupled with computational fluid dynamics (CFD) to investigate the steady-state bubble size distribution in two types of process equipment namely, a standard Rushton turbine stirred tank reactor and a generic lab-scale flotation cell. The coupling is realized using Fluent 15.07 software, and the numerical model is validated for the stirred tank reactor. The population balance equation (PBE) is solved using the quadrature method of moments (QMOM) technique along with a correction procedure implemented to check and correct invalid moment sets. The breakage and coalescence of bubbles due to turbulence are considered. The breakage rate and daughter size distribution models proposed by Laakkonen et al. (2007) are considered. For modeling coalescence rate, models proposed by Coulaloglou and Tavlarides (1977) are considered. The interaction between the phases is handled by considering the drag model proposed by Lane et al. (2005) while ignoring the other interphase forces. The correction algorithm has been successfully implemented, and improved predictions of gas volume fraction and Sauter mean diameter (SMD, d32) have been observed with a good match between the predictions and experimental measurements. The local SMD predictions are compared against predictions from the past studies and the superiority of the current approach for moderate gassing rates is established. The CFD-PBM approach is then used to study and characterize different flow regimes occurring in a generic mechanical flotation cell at different aeration rates and impeller rotation speeds. Also, power numbers are calculated from torque data and are found to drop considerably with an increase in aeration rate and impeller rotation speed as the flow regime approaches recirculating flow. The predicted SMD for flotation cell indicates that smaller bubbles are concentrated near the high turbulence impeller stream, the lower recirculation region, and close to the tank walls. On the other hand, large bubbles are formed in the upper tank region and are concentrated around the shaft during the flooding, loading, and transition flow regimes. In the future, the corrected QMOM approach will be further extended by implementing kinetic models capable of predicting the flotation rate constant using local bubble size information obtained from CFD-PBM simulations.

Numerical simulations of particle size distributions: Comparison with analytical solutions and kaolinite flocculation experiments

The population balance equation has been used to describe the evolution of particle populations caused by aggregation and breakup processes for several decades. Only recently has it been used to efficiently and directly predict the temporal variation of particle size distribution (PSD) for cohesive sediments by using the quadrature method of moments (QMOM). In this study, an extension of that effort carries out additional validations that compare with analytical solutions for pure coagulation systems, pure breakup systems, and combined coagulation and breakup systems. Besides, a laboratory experiment is conducted in a five liter mixing chamber to provide steady state PSD of suspended kaolinite with a chamber-averaged kaolinite concentration 0.52g/L and shear rate 45s−1. The model results match reasonably well with both of those from analytical solutions and kaolinite flocculation experiment results. Sensitivities regarding the selecting of different model input parameters (such as fractal dimensions, primary particle sizes, aggregation and breakage fitting parameters, and fragmentation distribution functions) are also checked. The results further confirm the capability of QMOM to predict PSD using its quadrature nodes and the corresponding weights.

Multiscale Modeling of Expanding Polyurethane Foams via Computational Fluid Dynamics and Population Balance Equation

SummaryThis study is aimed to formulate a numerical modeling recipe for polyurethane foams. The model is capable of simulating the foam principal characteristics during mold filling. The model is formulated upon coupling of Computational Fluid Dynamics (CFD) and Population Balance Equation (PBE) to predict and simulate the evolution of foam features including apparent density and viscosity, bubble (or cell) size distribution (BSD) during the polymerization, as well as its kinetics. The solution of PBE inside the CFD code is performed with Quadrature Method of Moments (QMOM). The foam, constituted by a liquid polymer and gas bubbles, is simulated as a pseudo‐single‐phase system, while the interface between the foam and the surrounding air is tracked by a Volume‐of‐Fluid (VOF) solver within the open‐source CFD code OpenFOAM. The modeling is applied for a simple foaming experiment and attention is paid to the effect of the rheological model on the predictions.

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.

Comparison of the Method of Classes and the Quadrature of Moment for the Modelling of Neodymium Oxalate Precipitation

Oxalic precipitation is generally used in the nuclear industry to deal with radioactive waste and recover the actinides from a multicomponent solution. To facilitate the development of experimental methods and data acquisitions, actinides are often simulated using lanthanides, gaining experience more easily.The purpose of this article is to compare the results achieved by two methods for solving the population balance during neodymium oxalate precipitation in a continuous MSMPR (Mixed Suspension Mixed Product Removal). The method of classes, also called discretized population balance, used in this study is based on the method of Litster. Whereas, the Quadrature Method of Moment (QMOM) is written in terms of the transport equations of the moments of the number density function. All the integrals are solved through a quadrature approximation thanks to the product-difference algorithm or the Chebyshev algorithm. Primary nucleation, crystal growth and agglomeration are taken into account. Agglomeration phenomena have been found to be represented by a loose agglomerates model. Thermodynamic effects are modeled by activity coefficients which are calculated using the Bromley model. The sizes of particles predicted by the two methods are in good agreement with experimental measurements.

A Comparison of Modeling Techniques for Polydispersed Droplet Spectra in Steam Turbines

Within steam turbine flows, condensation phenomena give rise to complex droplet spectra that can span more than two orders of magnitude in size. To predict the behavior of the two-phase flow and the resulting losses, the interactions between the vapor phase and droplets of all sizes must be accurately calculated. The estimation of thermodynamic losses and droplet deposition rates, in particular, depends on the size range and shape of the droplet spectrum. These calculations become computationally burdensome when a large number of droplet groups are present, and it is therefore advantageous to capture the complete droplet spectrum in a compressed form. This paper compares several methods for reducing the complexity of the droplet spectrum: a single representative droplet size (equivalent monodispersion), the moment method (including various growth rate approximations), the quadrature method of moments (QMOM), and spectrum pruning. In spectrum pruning, droplet groups are individually nucleated, but their number is subsequently reduced by combining groups together in a manner that preserves droplet number, wetness fraction, and the shape of the initial spectrum. The various techniques are compared within a Lagrangian framework by tracking the two-phase behavior along predefined pressure–time trajectories. Primary and secondary nucleation, droplet evaporation, and a representative turbomachinery case are modeled. The calculations are compared in terms of speed, accuracy, and robustness. It is shown that both the moment methods and spectrum pruning provide an appreciable improvement in accuracy over the use of an “equivalent” monodispersion without compromising calculation speed. Although all the examined methods are adequate for primary nucleation and droplet growth calculations, spectrum pruning and the QMOM are most accurate over the range of conditions considered.

A pseudo-3D model with 3D accuracy and 2D cost for the CFD–PBM simulation of a pilot-scale rotating disc contactor

In this study, the computational fluid dynamics–population balance model (CFD–PBM) was developed for a five-compartment pilot-scale rotating disc contactor (RDC) using the solving methods of finite volume method (FVM) and quadrature method of moments (QMOM) as implemented in Fluent 14.0. The differences between a simplified two-dimensional (2D) axisymmetric swirl model and a full three-dimensional (3D) model were investigated, and a pseudo-three-dimensional (pseudo-3D) model was developed to combine the benefits of both the 2D and 3D frameworks. The comparison results of the velocity distributions, droplet mean diameter and turbulent dissipation rate indicated that the 3D approach was suitable for quantitative investigations, whereas the 2D axisymmetric swirl model had only qualitative predictive capabilities. The differences are probably attributed to the 2D framework ignoring the cylindrical geometry characteristics of the problem. Based on this knowledge, a pseudo-3D model was proposed by constructing a fan-shaped geometry to overcome this disadvantage, and a parametric study was conducted to determine the angle of the sector. The results showed that the pseudo-3D model had nearly the same computational accuracy as the full 3D simulation, which verified the approach. Moreover, the sector angle had almost no effect on the simulated results, and only two layers of grids in the tangential direction were sufficient. Thus, the pseudo-3D model was found to be a powerful tool with computational costs approximate to those of the 2D framework, but with a calculation accuracy near that of the 3D model.

Reconstruction of a distribution from a finite number of its moments: A comparative study in the case of depolymerization process

The resolution of the population balance equation (PBE) using moment-based methods offers a high computational efficiency however, information on the time evolution of the probability density function (PDF) is out of reach. For this, several PDF reconstruction methods using a finite number of moments are proposed in the literature. In this contribution, three different methods (i.e. beta kernel density function based method, spline based technique and the maximum entropy based approach) are tested and compared to the analytical solution of a depolymerization process. The maximum entropy method gives the most accurate approximations using only a set of six moments. This method is combined with the quadrature method of moments (QMOM) for a simultaneous reconstruction during the PBE resolution. A three nodes and a four nodes quadrature are tested. The results show that the quality of the reconstruction is highly dependent on the accuracy of the computed moments.

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.Copyright 2015 American Association for Aerosol Research

Inter-Particle Aggregation and Breakage in Taylor-Reactor Using CFD

Taylor reactor is composed of two co-axial cylinders where the outer one is stationary and the inner one is rotating. Recently, Taylor reactor is applied on the lithium ion battery to produce cathode material. In the reactor, complex chemical reactions and particle growth occur. The particle size and distribution have significant impact on the performance of cathode material. This study simulates inter-particle aggregation and breakage in Taylor reactor and predicts particle size and distribution. The simulated results are compared with experiment. The quadrature method of moments (QMOM) is implemented to solve a population balance equation (PBE). For the aggregation kernel, the sum of turbulence kernel and Brownian aggregation kernel is considered. The power-law kernel is taken as a breakage kernel. The mixture model is used to describe the interaction between continuous phase and dispersed phase. The effect of internal flow on the dispersed phase (i.e. particle) is investigated by changing rpm.

Computational fluid dynamic modeling of the flame spray pyrolysis process for silica nanopowder synthesis

A computational fluid dynamic model that couples the fluid dynamics with various processes involving precursor droplets and product particles during the flame spray pyrolysis (FSP) synthesis of silica nanopowder from volatile precursors is presented. The synthesis of silica nanopowder from tetraethylorthosilicate and tetramethylorthosilicate in bench- and pilot-scale FSP reactors, with the ultimate purpose of industrial-scale production, was simulated. The transport and evaporation of liquid droplets are simulated from the Lagrangian viewpoint. The quadrature method of moments is used to solve the population balance equation for particles undergoing homogeneous nucleation and Brownian collision. The nucleation rate is computed based on the rates of thermal decomposition and oxidation of the precursor with no adjustable parameters. The computed results show that the model is capable of reproducing the magnitude as well as the variations of the average particle diameter with different experimental conditions using a single value of the collision efficiency factor α for a given reactor size.

Direct expansion method of moments with n/3th moments for nanoparticle Brownian coagulation in the free molecule regime

Purpose– The purpose of this paper is to develop new types of direct expansion method of moments (DEMM) by using the n/3th moments for simulating nanoparticle Brownian coagulation in the free molecule regime. The feasibilities of new proposed DEMMs with n/3th moments are investigated to describe the evolution of aerosol size distribution, and some of the models will be applied to further simulation of physical processes.Design/methodology/approach– The accuracy and efficiency of some kinds of methods of moments are mainly compared including the quadrature method of moments (QMOM), Taylor-expansion method of moments (TEMOM), the log-normal preserving method of moments proposed by Lee (LMM) and the derived DEMM in this paper. QMOM with 12 quadrature approximation points is taken as a reference to evaluate other methods.Findings– The newly derived models, namely DEMM(4/3,4) and DEMM(2,6), as well as the previous DEMM(2,4), are considered to be qualified models due to their high accuracy and efficiency. They are confirmed to be valid and alternative models to describe the evolution of aerosol size distribution for particle dynamical process involving the n/3th moments.Originality/value– The n/3th moments, which have clear physical interpretations when n stands for first several integers, are first introduced in the DEMM method for simulating nanoparticle Brownian coagulation in the free molecule regime.

The Use of Quadrature Method of Moments (QMOM) in Studying the Mechanisms of Aerosol Agglomeration and its Practical Use

The efficiency of dust collector is relatively low when dealing with PM2.5. Agglomerating the aerosols under acoustic field is a preliminary method to make the efficiency improve. Acoustic agglomeration of aerosol mainly includes four mechanisms: orthokinetic interaction, hydrodynamic interaction, acoustic wake effect and Brown agglomeration. All mechanisms have their own characteristics with the change of acoustic field. Among witch orthokinetic interaction plays the leading role, while Brown agglomeration is the weakest. Besides, Brown agglomeration is only related to the change of temperature and other air characteristics, thus it changes little under the effect of acoustic field. Compared with experimental methods, simulating the agglomerate process on computer is much faster and more economical. To overcome the flaws of traditional ways in simulation of long-time, hard to make exact hypotheses and difficult to make the formulas close, quadrature method of moments (QMOM) is employed when doing simulations, and taking it into practical use. The results of QMOM show a good match with group method, while it takes less time and does not need to make unnecessary hypotheses. Besides studying the characteristics of acoustic agglomeration, QMOM will also be easy to be used in other fields of studies with similar dynamic equations.