Simulation of population balance equations using quadrature based moment methods

Upadhyay, Rochan Raj
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Population Balance Equations (PBE) are used for modeling a variety of particulate processes as well as various stochastic phenomena in science and engineering. However PBEs are difficult to solve because they describe the evolution of a probability density function (PDF) in high dimensional spaces. Due to their unique mathematical structure and properties, these equations require special solution techniques. Moment methods are a class of solution techniques that evolve only a few moments of the PDF. While moment methods are simpler, they are known to have closure problems, i.e. a finite set of moment equations do not fully describe the PDF or its evolution. The purpose of this dissertation is to investigate a closure scheme for the moment equations that is based on Gaussian quadrature. This approach, known as the Quadrature Method of Moments (QMOM), is very general as it does not require any a priori assumptions on the form of the PDF. In this study, I first evaluate the accuracy of the moment closure by applying QMOM to solve some well known problems in aerosol science, such as particle nucleation and growth in well stirred reactors and size dependent transport of aerosol particles. I find that results obtained using QMOM compare favorably with results obtained using more expensive techniques. Moment methods are particularly suited for implementation in CFD codes. As an example of a model for smoke detectors, I use QMOM to simulate smoke entry and light scattering in a cylindrical cavity above a uniform flow. As further examples, I describe the use of QMOM in applications such as statistical uncertainty propagation and simulation of turbulent mixing and chemical reaction using the PDF transport equation. While moment methods are widely applicable, they have some limitations. I find that the solutions depend on the choice of moments and that there may not be a globally optimal set of moments. This becomes more problematic for solutions of multivariate PBEs using an extension called the Direct Quadrature Method of Moments (DQMOM). The insights from this work can lead to a greater appreciation of the benefits and limitations of moment methods for solving PBEs.