Quadrature-based Moment Methods Research Articles

A quadrature-based method of moments for nonlinear filtering

According to the nonlinear filtering theory, optimal estimates of a general continuous-discrete nonlinear filtering problem can be obtained by solving the Fokker–Planck equation, coupled with a Bayesian update rule. This procedure does not rely on linearizations of the dynamical and/or measurement models. However, the lack of fast and efficient methods for solving the Fokker–Planck equation presents challenges in real time nonlinear filtering problems. In this paper, a direct quadrature method of moments is introduced to solve the Fokker–Planck equation efficiently and accurately. This approach involves representation of the state conditional probability density function in terms of a finite collection of Dirac delta functions. The weights and locations (abscissas) in this representation are determined by moment constraints and modified using the Bayes’ rule according to measurement updates. As demonstrated by numerical examples, this approach appears to be promising in the field of nonlinear filtering.

BASF works on new gas-to-olefin process

Microemulsion systems may enable innovative and highly efficient synthesis paths for the chemical industry. When expensive catalysts are involved in the process, an efficient separation of products together with an extremely low catalyst leaching are of utmost importance. In order to optimize the separation process and improve the settler design, a thorough understanding of droplet-droplet interactions is required. In this work, computational fluid dynamics (CFD) has been used to study the multiphase flow in a horizontal settler using the Eulerian two-fluid framework. Droplet coalescence, which is the most important physical phenomenon in gravity-driven separation, has been modeled using a spatially inhomogeneous buoyancy-based kernel, and the resulting population balance equation has been solved using the quadrature-based method of moments (QMOM). After calibrating the model parameters using selected experiments, a very good description of the observed separation is obtained, opening the door for process understanding and optimization.

A quadrature-based third-order moment method for dilute gas-particle flows

Dilute gas-particle flows can be described by a kinetic equation containing terms for spatial transport, gravity, fluid drag, and particle–particle collisions. However, the direct numerical solution of the kinetic equation is intractable for most applications due to the large number of independent variables. A useful alternative is to reformulate the problem in terms of the moments of the velocity distribution function. Closure of the moment equations is challenging for flows away from the equilibrium (Maxwellian) limit. In this work, a quadrature-based third-order moment closure is derived that can be applied to gas-particle flows at any Knudsen number. A key component of quadrature-based closures is the moment-inversion algorithm used to find the weights and abscissas. A robust inversion procedure is proposed for moments up to third order, and tested for three example applications (Riemann shock problem, impinging jets, and vertical channel flow). Extension of the moment-inversion algorithm to fifth (or higher) order is possible, but left to future work. The spatial fluxes in the moment equations are treated using a kinetic description and hence a gradient-diffusion model is not used to close the fluxes. Because the quadrature-based moment method employs the moment transport equations directly instead of a discretized form of the Boltzmann equation, the mass, momentum and energy are conserved for arbitrary Knudsen number (including the Euler limit). While developed here for dilute gas-particle flows, quadrature-based moment methods can, in principle, be applied to any application that can be modeled by a kinetic equation (e.g., thermal and non-isothermal flows currently treated using lattice Boltzmann methods), and examples are given from the literature.

A quadrature-based moment method for dilute fluid-particle flows

Gas-particle and other dispersed-phase flows can be described by a kinetic equation containing terms for spatial transport, acceleration, and particle processes (such as evaporation or collisions). In principle, the kinetic description is valid from the dilute (non-collisional) to the dense limit. However, its numerical solution in multi-dimensional systems is intractable due to the large number of independent variables. As an alternative, Lagrangian methods “discretize” the density function into “parcels” that are simulated using Monte-Carlo methods. While quite accurate, as in any statistical approach, Lagrangian methods require a relatively large number of parcels to control statistical noise, and thus are computationally expensive. A less costly alternative is to solve Eulerian transport equations for selected moments of the kinetic equation. However, it is well known that in the dilute limit Eulerian methods have great difficulty to describe correctly the moments as predicted by a Lagrangian method. Here a two-node quadrature-based Eulerian moment closure is developed and tested for the kinetic equation. It is shown that the method can successfully handle highly non-equilibrium flows (e.g. impinging particle jets, jet crossing, particle rebound off walls, finite Stokes number flows) that heretofore could not be treated accurately with the Eulerian approach.

Improved Rate Laws and Population Balance Simulation Methods; CRE Applications, Including the Combustion Synthesis of Valuable Nano-Particles

Using ‘flame-synthesized’ nanoparticles (nps) as one prototypical application, we illustrate our recent progress in two broad areas of current CRE-interest, viz., the development of: 1. Improved rate laws/transport coefficients for next-generation Eulerian, multi-(state) variable population-balance formulations, and 2. Quadrature-based multi-variate moment methods (hereafter QMOM) suitable for articulation with evolving Eulerian CFD simulation methods Admittedly, in previous work much insight was obtained by introducing deliberately (over-) simplified rate laws (for nucleation, Brownian coagulation, vapor growth/evaporation, sintering, thermophoresis, ) into the generally nonlinear integro-partial differential equation called the ‘population balance’ equation (PBE). However, despite the complexity of this equation, and the need to satisfy it along with many other local PDE-balance principles in multi-dimensional CRE environments, in our view current requirements for reactor design, as well as the frequent need to infer meaningful physico-chemical parameters based on laboratory measurements on populations rather than individual ‘particles’, make the introduction of more accurate rate/transport laws essential for next-generation particle synthesis reactor models. Our present examples are motivated both by measurements/calculations of the structure of laminar counterflow flames synthesizing Al2O3 nps and/or the predicted performance of well-mixed steady-flow devices in which sintering or sublimation occurs. Corresponding illustrative results, which focus on the rate laws for sphere dissolution or aggregate Brownian coagulation support our contentions that: i) systematic introduction of more accurate rate laws (including nucleation, sintering, growth, )/transport coefficients will be essential to meet the quantitative demands of next-generation PBE-based CRE-simulation models for high-value particulate synthesis equipment, and, ii) QMOM is able to incorporate realistic rate laws and faithfully generate their effects on important ‘moments’ characterizing the product joint distribution functions.

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.