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Dimension reduction of bivariate population balances using the quadrature method of moments

Wolfram Heineken, Dietrich Flockerzi, Andreas Voigt, Kai Sundmacher
2011 Computers and Chemical Engineering  
We propose a model reduction based on the quadrature method of moments (QMOM). Using this method a two-dimensional population balance is reduced to a system of one-dimensional advection equations.  ...  Despite the dimension reduction the method keeps important volume dependent information of the crystal size distribution (CSD).  ...  The population balance The growth process of crystals showing direction-dependent growth can be modelled using a multi-dimensional population balance.  ... 
doi:10.1016/j.compchemeng.2010.06.012 fatcat:7btbaqj2qveavlrfl7bpqj42yi

Comminution process modeling based on the monovariate and bivariate direct quadrature method of moments

Christine Frances, Alain Liné
2014 AIChE Journal  
However, other properties, such as the particle strength, also influence the grinding kinetics. A bivariate PBE was developed and resolved by the direct quadrature method of moments.  ...  Population balance equations (PBE) applied to comminution processes are commonly based on selection and breakage functions, which allow the description of changes of particle-size distributions vs. time  ...  Introduction Population balance equations (PBE) are commonly used to model particulate processes.  ... 
doi:10.1002/aic.14358 fatcat:ilkc44ebr5dgjkwzg7el4hrxmm

Batch Cooling Crystallization of Plate-like Crystals: A Simulation Study

Botond Szilágyi, Béla Lakatos
2015 Periodica Polytechnica: Chemical Engineering  
The population balance equation is reduced to a system of equations for the bivariate mixed moments of internal variables, which are solved by means of the quadrature method of moments.  ...  The population balance approach is used to describe a seeded crystallization process, developing a bivariate morphological population balance model including nucleation and size dependent crystal growth  ...  Acknowledgements The financial support of the Hungarian Scientific Research Fund under Grant K 77955 and the Collegium Talentum is gratefully acknowledged.  ... 
doi:10.3311/ppch.7581 fatcat:7eowlijnmveaboe5hy773gc7mm

Numerical solution of the bivariate population balance equation for the interacting hydrodynamics and mass transfer in liquid–liquid extraction columns

Menwer M. Attarakih, Hans-Jörg Bart, Naim M. Faqir
2006 Chemical Engineering Science  
These methods are the generalized fixed-pivot technique (GFP) and the quadrature method of moments (QMOM).  ...  A comprehensive model for predicting the interacting hydrodynamics and mass transfer is formulated on the basis of a spatially distributed population balance equation in terms of the bivariate number density  ...  Acknowledgements The authors wish to thank the DFG and DAAD for supporting this work.  ... 
doi:10.1016/j.ces.2004.12.055 fatcat:rhlnw6d7enhk5m2n2hs6rxpwuq

Approximate Moment Methods for Population Balance Equations in Particulate and Bioengineering Processes

Robert Dürr, Andreas Bück
2020 Processes  
A specific class of solution algorithms, so-called moment methods, is based on the reduction of complex models to a set of ordinary differential equations characterizing dynamics of integral quantities  ...  In this contribution, a concise overview of the most prominent approximate moment methods is given.  ...  The method uses a Gauss-Christoffel quadrature rule and shows good results in comparison to experimental data. • The bivariate moment projection algorithm is presented in [138] .  ... 
doi:10.3390/pr8040414 fatcat:lszkh5hdbffljb3aurq45mojwq

Comparison of methods for multivariate moment inversion—Introducing the independent component analysis

Jovani L. Favero, Luiz Fernando L.R. Silva, Paulo L.C. Lage
2014 Computers and Chemical Engineering  
Despite the advantages of the moment based methods in solving multivariate population balances models, these methods still suffer with the so-called multivariate moment inversion problem.  ...  This work presents a comparative analysis of the existing methods regarding their accuracy and robustness.  ...  The multivariate moment inversion using optimization was first reported by Wright et al. (2001) for a bivariate population balance problem.  ... 
doi:10.1016/j.compchemeng.2013.08.006 fatcat:ioxtaueakjcmtjik44e4xushbu

Modelling of Particulate Processes

Murkus Kraft
2005 Kona : Powder and Particle  
We review the method of moments, sectional, finite element, and Monte Carlo methods and give examples of their applications in f lame synthesis.  ...  We review models and numerical methods used in f lame synthesis of organic and inorganic nanoparticles.  ...  Acknowledgements The author would like to thank the members of the Computational Modelling Group in Cambridge, in par ticular my students Neal Morgan, Rober t Patterson, Jasdeep Singh, and Sebastian Mosbach  ... 
doi:10.14356/kona.2005007 fatcat:alsno7jdubcgpiqe6amtlne3yq

Simulation and control of nanoparticle size distribution in a high temperature reactor

Oluranti Sadiku-Agboola, Andrei Kolesnikov, Emmanuel Sadiku
2012 Polish Journal of Chemical Technology  
The proposed method "which is the application of reduced population balance model" shows that there is some dependence of the average particle diameter on the wall temperature and the model can thus be  ...  The simulation results from the above model demonstrated that the reduced population balance can be effectively used to control the PSD.  ...  Wright et al 26 used the bivariate extension of the quadrature method of moments (QMOM) to effi ciently model the dynamic of a population of inorganic nanoparticles undergoing simultaneous coagulation  ... 
doi:10.2478/v10026-012-0052-y fatcat:wibhlpuli5e55i7imiamokby4q

Solution of population balance equations in applications with fine particles: Mathematical modeling and numerical schemes

T.T. Nguyen, F. Laurent, R.O. Fox, M. Massot
2016 Journal of Computational Physics  
quadrature method of moments (EQMOM) based on the work of Yuan et al.  ...  For this purpose, two types of methods are investigated for solving the population balance equation with aggregation, continuous particle size change (growth and size reduction), and nucleation: the extended  ...  Moreover, the moment space is described in each Population balance equation (PBE) Let us consider the volume v as the only internal variable.  ... 
doi:10.1016/j.jcp.2016.08.017 fatcat:x6qetqir4be7lkbvx5rte6ipca

Quadrature-Based Methods for Obtaining Approximate Solutions to Nonlinear Asset Pricing Models

George Tauchen, Robert Hussey
1991 Econometrica  
The method works by using numerical quadrature rules to approximate the integral operators that arise in stochastic intertemporal models.  ...  An empirical application uses the method to study the relationship between the risk premium and the conditional variability of the equity return under an ARCH endowment process.  ...  Burnside (1989) uses versions of the method to approximate predicted population moments for method of moments estimation of financial models.  ... 
doi:10.2307/2938261 fatcat:23wqkgfcpzcdzpva6odrczmseu

Strongly coupled fluid-particle flows in vertical channels. II. Turbulence modeling

Jesse Capecelatro, Olivier Desjardins, Rodney O. Fox
2016 Physics of Fluids  
Here, data from the EL simulations are used to validate a multiphase Reynolds-stress model (RSM) that predicts the wall-normal distribution of the two-phase, one-point turbulence statistics up to second  ...  Finally, an algebraic stress model for the PA particle-phase pressure tensor and the Reynolds stresses is derived from the RSM using the weak-equilibrium assumption.  ...  Population balance equation (PBE) Let us consider the volume v as the only internal variable.  ... 
doi:10.1063/1.4943234 fatcat:jmohocqth5bypbdkpvbs4us4em

Experimental and numerical study on the effect of oxymethylene ether-3 (OME3) on soot particle formation

Federica Ferraro, Carmela Russo, Robert Schmitz, Christian Hasse, Mariano Sirignano
2021 Fuel  
Particle size distribution (PSD) and Laser Induced Fluorescence (LIF) and Incandescence (LII) along with numerical (Conditional Quadrature Method of Moments - CQMOM, based on D'Anna physico-chemical soot  ...  The soot volume fraction and PSD indicate a reduction in the total number and the size of the soot particles at all equivalence ratios, while the number of small nanoparticles remains almost unchanged.  ...  Acknowledgments The  ... 
doi:10.1016/j.fuel.2020.119353 fatcat:oh3qsaq5ojadncim5jemfvtnke

Particle formation in oxymethylene ethers (OMEn, n = 2-4) / ethylene premixed flames [article]

Robert Schmitz, Federica Ferraro, Mariano Sirignano, Christian Hasse
2022 arXiv   pre-print
In the numerical simulations, the kinetic mechanism is combined with a detailed quadvariate soot model which uses the Conditional Quadrature Method of Moments and validated with Laser-Induced Fluorescence  ...  Furthermore, the extent of soot reduction is comparable among the three OMEn. The trends and overall reduction are very well captured by the model.  ...  Acknowledgments The authors gratefully acknowledge the funding by the German Federal Ministry of Education and Research (BMBF) as part of the NAMOSYN Project (project number 03SF0566R0).  ... 
arXiv:2204.10733v1 fatcat:ilkjwhttj5egfn5l5vpiggytqa

Modelling of Soot Aerosol Dynamics in Turbulent Flow

Stelios Rigopoulos
2019 Flow Turbulence and Combustion  
Comprehensive modelling of soot formation in combustion processes requires coupling of the population balance equation, which is the fundamental equation governing aerosol dynamics, with the equations  ...  emissions require control of the number of smaller particles.  ...  Acknowledgments The author wishes to thank Prof. W. P. Jones for feedback on an early version of this paper.  ... 
doi:10.1007/s10494-019-00054-8 fatcat:s63b4g65irhshc4srsyzeas3du

Chapter 40 Classical estimation methods for LDV models using simulation [chapter]

Vassilis A. Hajivassiliou, Paul A. Ruud
1994 Handbook of Econometrics  
We are grateful to the National Science Foundation for partial financial support, under grants SES-929411913 (Hajivassiliou) and SES-9122283 (Ruud).  ...  It is crucial to stress that such numerical quadrature methods offer only poor approximations to integrals of such dimension.  ...  Let s be the time in seconds a given computer requires to approximate a 16-dimensional integral by numerical quadrature methods.  ... 
doi:10.1016/s1573-4412(05)80009-1 fatcat:2r67s6yl3rferkwo6slim5gf5i
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